Model Theory of Operator Algebras 9783110768282, 9783110768213

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Isaac Goldbring (Ed.) Model Theory of Operator Algebras

De Gruyter Series in Logic and Its Applications

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Edited by Denis R. Hirschfeldt, University of Chicago, USA Amador Martin-Pizarro, University of Freiburg, Germany Ieke Moerdijk, Utrecht University, The Netherlands Itay Neeman, UCLA, USA Anand Pillay, University of Notre Dame, USA

Volume 11

Model Theory of Operator Algebras �

Edited by Isaac Goldbring

Editor Prof. Isaac Goldbring University of California Department of Mathematics 340 Rowland Hall Bldg. 400 Irvine, CA 92697-3875 USA [email protected]

ISBN 978-3-11-076821-3 e-ISBN (PDF) 978-3-11-076828-2 e-ISBN (EPUB) 978-3-11-076833-6 ISSN 1438-1893 Library of Congress Control Number: 2023935142 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

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To my math girls: Karina, Kaylee, and Daniella

Preface While applications of model theory had some interaction with analysis (the successes of o-minimality and nonstandard analysis, for example), the introduction of the current incarnation of continuous logic by Ben Yaacov and Usvyatsov (and later expanded upon by Ben Yaacov, Berenstein, Henson, and Usvyatsov) allowed for the systematic application of model theory to classes of analytic objects such as C∗ -algebras, which are out of reach of discrete first-order logic. That being said, we also certainly need to recognize predecessors of this continuous logic, such as Henson’s positive bounded logic, and its applications to areas of functional analysis such as the geometry of Banach spaces. A particularly exciting avenue of research emerged when the model theory associated to this logic was applied to studying operator algebras, which are certain collections of bounded operators on Hilbert spaces closed under the relevant algebraic operations, as well as closed in various topologies on the space of all bounded operators. It is this area of applied model theory that is the focus of this volume. The model-theoretic study of operator algebras began in earnest with the series of three papers, aptly titled Model theory of operator algebras: I, II, and III, by Ilijas Farah, Bradd Hart, and David Sherman. The first paper discussed how model-theoretic ideas could be used to shed light on a recurring problem in operator algebras asked by many researchers, namely whether or not various ultraproduct and ultrapower constructions (or related constructions such as relative commutants in ultrapowers) depend on the choice of ultrafilter; for definitiveness, we restrict attention to this question for separable objects and their ultraproducts with respect to nonprincipal ultrafilters on the natural numbers. While it is fairly straightforward to show that the ultrapower in question is independent of the choice of ultrafilter if one assumes the continuum hypothesis (CH), the situation is far more murky if one assumes the negation of CH. Adapting known arguments due to Shelah, the authors show that indeed one can obtain nonisomorphic ultraproducts using different ultrafilters. The driving force behind these facts should not come as a surprise to the model theorist: operator algebras are highly unstable (in the model-theoretic sense of the word). In general, operator algebras tend to be very complicated from the model-theoretic perspective. Having avoided precise model-theoretic terminology in the first paper, the second paper clearly describes the languages for studying both C∗ -algebras and tracial von Neumann algebras and shows that both of these classes of structures are elementary in their respective languages. We stress that, while often routine in classical model theory, treating these objects as structures in the appropriate language is far from obvious and uses nontrivial results in their respective subjects. The remainder of the second paper enumerates various familiar model-theoretic ideas and results in the continuous context and spells out the connection between stability and isomorphic ultrapowers in full generality. The final paper begins a true model-theoretic study of II1 factors (a certain class of tracial von Neumann algebras, which are in fact generic model-theoretically speakhttps://doi.org/10.1515/9783110768282-201

VIII � Preface ing). In particular, they show that two pivotal properties that a II1 factor may or may not possess, namely property Gamma and the McDuff property, are both elementary properties. They also show that every theory of II1 factors has continuum many nonisomorphic separable models. The latter part of this paper gained notoriety in operator-algebraic circles for providing a “Poor man’s solution” to the (in)famous Connes Embedding Problem. After these three papers appeared, the model-theoretic study of operator algebras advanced rapidly and quickly became a vibrant area of research. New results were proven both by model theorists and operator algebraists, often in collaboration. The synthesis of basic model-theoretic techniques and highly nontrivial operator algebraic results clearly demonstrated that these two subjects were meant to intermingle. That being said, a serious issue plagued the subject: how should someone not working in the field prepare themselves to contribute to this ever-expanding area? For the model-theorists, the problems are two-fold. First, while growing in popularity, continuous model theory is not quite at the point where it is essentially understood by the greater model-theoretic community (much to this author’s chagrin). Moreover, even if a model-theorist has a working knowledge of continuous model theory, an even larger hurdle remains, namely learning a sufficient amount of operator algebras in order to understand what the issues at hand are and how their model-theoretic expertise might be put to good use. For the operator-algebraist, the issue is learning the basic modeltheoretic techniques and, perhaps even more challenging, learning how to “think like a model-theorist.” (I recall a certain operator algebraist – to remain nameless – acting shocked when first learning of the Downward Löwenheim–Skolem theorem, acting as if it were some form of voodoo.) The first portion of this volume is meant to remedy these issues. The first two chapters, written by Szabó and Ioana, respectively, are intended to be concise (but thorough) introductions to C∗ -algebras and tracial von Neumann algebras. These articles include a plethora of exercises to help the reader gain a better understanding of the material being presented. The third chapter, written by Hart, is an introduction to continuous model theory (without assuming any previous knowledge of classical, discrete model theory) with an eye towards examples from operator algebras. The remaining articles present results and research directions belonging to the model theory of operator algebras proper, some of which are new and have not been presented elsewhere in the literature thus far. The article of Goldbring and Hart surveys a variety of known results on tracial von Neumann algebras; in particular, they discuss the nonisomorphic ultrapower results mentioned above in more detail, as well as results about existentially closed II1 factors and recent applications to the study of embeddings into ultraproducts with factorial relative commutant. The Berenstein and Henson article details various aspects of the model theory of classical probability spaces, perfectly complementing the next article of Jekel on model theory and free probability. This latter article in particular contains a new result concerning free entropy and irreducible embeddings into ultraproducts of matrix algebras

Preface � IX

which are proven by extending Voiculescu’s notion of free entropy from quantifier-free types to full types. The next article of Chifan, Drimbe, and Ioana first details Popa’s influential deformation-rigidity programme and then uses it to prove a variety of tensor product indecomposability results for the class of existentially closed factors, further developing our understanding of this important class of factors. While the articles above are entirely focused on the class of tracial von Neumann algebras, where the model theory is better understood, Ando’s article introduces the reader to the more general class of σ-finite von Neumann algebras and details the relevant ultraproduct construction for such algebras. This article also explains some of the first model-theoretic results on the class of σ-finite von Neumann algebras stemming from these ultraproducts. The book then switches gears to C∗ -algebras. Sinclair’s article treats the model theory of C∗ -algebras within the more general context of operator systems, thus explaining the model-theoretic framework for these as well. This article is followed by Lupini’s article on G-C∗ -algebras, namely C∗ -algebras equipped with an action of some fixed locally compact group G, which shows how the model-theoretic approach leads to a unified treatment of a variety of dimensions considered on such algebras. Goldbring’s article details how many natural ultrapower embedding problems in operator algebras, such as the aforementioned Connes Embedding Problem, can be viewed as model theory problems and shows how this viewpoint can be used to shed light on these problems. In particular, the model-theoretic approach to the recent refutation of the Connes Embedding Problem from the recent landmark quantum complexity result known as MIP∗ = RE is presented. The final article in the volume is Vignati’s introduction to Fraïssé theory in the continuous context and discusses a variety of examples of Fraïssé limits in operator algebras and functional analysis more generally. While exposing the reader to the work that has been done in the model theory of operator algebras thus far, an emphasis has been placed on highlighting major open questions left unanswered and future lines of research to pursue. In other words, this volume should be viewed as an invitation to the reader to join us in further developing this burgeoning area of model theory. I would like to thank Amador Martin-Pizarro and Anand Pillay for inviting me to edit this volume and to Steven Eliott and Nadja Schedensack at DeGruyter for all of their assistance in preparing this book. Finally, I would like to thank all of the authors for their hard work in preparing a truly exceptional collection of articles. Isaac Goldbring Irvine, CA

Contents Preface � VII Gábor Szabó Introduction to C∗ -algebras � 1 Adrian Ioana An introduction to von Neumann algebras � 43 Bradd Hart An introduction to continuous model theory � 83 Isaac Goldbring and Bradd Hart A survey on the model theory of tracial von Neumann algebras � 133 Alexander Berenstein and C. Ward Henson Model theory of probability spaces � 159 David Jekel Free probability and model theory of tracial W∗ -algebras � 215 Ionuţ Chifan, Daniel Drimbe, and Adrian Ioana Tensor product indecomposability results for existentially closed factors � 269 Hiroshi Ando Introduction to nontracial ultraproducts of von Neumann algebras � 303 Thomas Sinclair Model theory of operator systems and C∗ -algebras � 343 Martino Lupini Model theory of G-C*-algebras and order zero dimension � 387 Isaac Goldbring Model theory and ultrapower embedding problems in operator algebras � 425 Alessandro Vignati Fraïssé theory in operator algebras � 453 Index � 479

Gábor Szabó

Introduction to C∗-algebras Abstract: We give a concise introduction to the general theory of C∗ -algebras, focusing mostly on the fundamental results and some selected further topics that are relevant in subsequent chapters of this volume. Keywords: Banach algebra, spectrum, spectral theory, C∗ -algebra, Gelfand–Naimark theorem, functional calculus, positivity, universal C∗ -algebras, states, GNS representation MSC 2020: 46H05, 46L05, 46L06

1 Spectral theory in Banach algebras In this section we give an introduction to the basics of the spectral theory in Banach algebras, at least to the extent that is necessary for subsequent applications in the theory of C∗ -algebras. Definition 1.1. A normed algebra is an associative algebra A over the complex numbers ℂ with a norm ‖ ⋅ ‖ such that for all x, y ∈ A, one has the inequality ‖xy‖ ≤ ‖x‖ ⋅ ‖y‖. If (A, +, ‖ ⋅ ‖) is also a Banach space, then we call A a Banach algebra. We furthermore say that A is – unital, if there is a unit element 1 ∈ A with 1 ⋅ x = x = x ⋅ 1 for all x ∈ A, and ‖1‖ = 1; – commutative (or abelian), if we have xy = yx for all x, y ∈ A. Initially we will almost always assume that our Banach algebras are unital. In this case we identify ℂ ≅ ℂ ⋅ 1 ⊆ A via λ 󳨃→ λ ⋅ 1. Remark 1.2. If a Banach algebra A has no unit, it is always possible to add a unit via the following procedure: Set A† = A × ℂ. This becomes a vector space with pointwise operations. We define a multiplication via (x, λ) ⋅ (y, μ) = (xy + λy + μx, λμ) and a norm by ‖(x, λ)‖ = ‖x‖ + |λ|. It is a simple exercise to show that A† becomes a Banach algebra with unit 1 = (0, 1) such that the assignment x 󳨃→ (x, 0) defines an isometric inclusion A ⊆ A† . Definition 1.3. Let A be a complex algebra over ℂ. An involution on A is an antilinear, self-inverse, and multiplication-reversing map ∗ : A → A, i. e., a map satisfying the ̄ ∗ + b∗ , (a∗ )∗ = a and (ab)∗ = b∗ a∗ for all a, b ∈ A and λ ∈ ℂ. identities (λa + b)∗ = λa If A comes equipped with an involution, we call it a ∗-algebra. If A also happens to be a Gábor Szabó, KU Leuven, Department of Mathematics, Celestijnenlaan 200B, box 2400, 3001 Leuven, Belgium, e-mail: [email protected]; URL: https://wis.kuleuven.be/analyse/gabor.szabo https://doi.org/10.1515/9783110768282-001

2 � G. Szabó Banach algebra, then we say that A is a Banach-∗-algebra, if one additionally has ‖x ∗ ‖ = ‖x‖ for all x ∈ A. Example 1.4. Let X be a locally compact topological space.1 Then A = 𝒞b (X) := {bounded continuous functions X → ℂ} is a unital complex algebra with pointwise addition and multiplication. If we consider the sup-norm ‖f ‖∞ = supx∈X |f (x)|, then A becomes a Banach algebra, where the completeness is by virtue of the uniform convergence theorem. If we furthermore consider the involution f ∗ (x) = f (x), then A becomes a Banach-∗algebra. If X happens to be compact, then continuous functions on X are automatically bounded, in which case we write 𝒞 (X) instead. Example 1.5. If E is a Banach space, then A = ℒ(E) := {bounded linear operators E → E} becomes a Banach algebra with the operator norm and (composition) multiplication T ⋅ S = T ∘ S. This will always be noncommutative when E is not one-dimensional. Exercise 1.6. (1) Let J ⊆ A be a closed two-sided ideal in a Banach algebra. Show that on the quotient Banach space A/J, the multiplication (x + J)(y + J) = xy + J is well defined, and that A/J is a Banach algebra with the usual quotient norm given by ‖x + J‖ = dist(x, J) = inf ‖x − b‖. b∈J

(2) Let I be a nonempty index set. Suppose that we are given a Banach algebra Ai for each i ∈ I. Consider the direct product ∏ Ai = {(ai )i∈I | ai ∈ Ai for all i ∈ I and sup ‖ai ‖ < ∞} i∈I

i∈I

Show that with pointwise addition, multiplication, and equipped with the supremum norm, this becomes a Banach algebra. (3) Let X be a locally compact space. A continuous function f : X → ℂ is said to vanish at infinity, if for every ε > 0, there exists a compact set K ⊆ X such that ‖f |X\K ‖∞ ≤ ε. The set of all such functions is denoted 𝒞0 (X). Show that 𝒞0 (X) is a Banach subalgebra of 𝒞b (X). Notation 1.7. Given a unital complex algebra A, we set GL(A) = {invertible elements in A}, which forms a group under multiplication. 1 We shall follow the French convention popularized by Bourbaki that “locally compact” or “compact” entails being Hausdorff.

Introduction to C∗ -algebras

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The following simple trick has many far-reaching consequences and underpins a striking amount of the spectral theory we are about to develop. Proposition 1.8 (Neumann series). Let A be a unital Banach algebra. If x ∈ A satisfies ‖1 − x‖ < 1, then x is invertible and the inverse is ∞

x −1 = ∑ (1 − x)n . n=0

Proof. Since ‖(1 − x)n ‖ ≤ ‖1 − x‖n for all n ≥ 1, the series is absolutely convergent. As A is complete, the limit is a well-defined element in A. We compute via a telescoping sum argument that ∞

∞

x ⋅ ∑ (1 − x)n = [1 − (1 − x)] ⋅ ∑ (1 − x)n n=0

n=0

∞

n

= ∑ [(1 − x) − (1 − x)n+1 ] n=0

= lim 1 − (1 − x)k+1 = 1 k→∞

n Analogously, one sees [∑∞ n=0 (1 − x) ] ⋅ x = 1.

Remark 1.9. Let A be a unital Banach algebra with two elements x, z ∈ A. If z is invertible and ‖z − x‖ < ‖z−1 ‖−1 , then it follows that 󵄩󵄩 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩1 − z x 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩z 󵄩󵄩󵄩 ⋅ ‖z − x‖ < 1. As a consequence of Proposition 1.8, it follows that z−1 x is invertible, hence x is invertible. An immediate consequence of this observation is that the invertible elements GL(A) ⊆ A form an open set in A. Exercise 1.10. Let A be a unital Banach algebra. Prove that the inversion map on GL(A) is continuous. Corollary 1.11. Let A be a unital Banach algebra. Suppose x ∈ A and λ ∈ ℂ satisfy ‖x‖ < |λ|. Then λ ⋅ 1 − x = λ − x ∈ A is invertible. Proof. We may assume λ = 1 by replacing x by xλ . Then ‖1 − (1 − x)‖ = ‖x‖ < 1, so 1 − x is invertible by Proposition 1.8. Definition 1.12. Let A be a unital Banach algebra. For an element x ∈ A, its spectrum is defined as σ(x) = σA (x) = {λ ∈ ℂ | λ − x ∈ A is not invertible}.

4 � G. Szabó For those readers who are totally unfamiliar with this notion, the following exercise may be somewhat illuminating. In particular for its last part, it may be useful to read ahead and use some general facts that are treated after the exercise. Exercise 1.13. Let ℋ be a Hilbert space and x ∈ ℒ(ℋ). (1) Suppose dim ℋ < ∞. Show that σ(x) is the set of eigenvalues of x. (2) Let λ ∈ ℂ. Suppose there exists a sequence of unit vectors ξn ∈ ℋ with ‖(λ − x)ξn ‖ → 0. Show λ ∈ σ(x). (3) Consider ℋ = ℓ2 (ℕ) and let S by the one-sided shift operator given by S(γ1 , γ2 , γ3 , . . . ) = (0, γ1 , γ2 , . . . ). Use this example to show that the previous criterion is not necessary for λ ∈ σ(x). (4) Compute the spectrum of S. Proposition 1.14. For every element x ∈ A in a unital Banach algebra, the spectrum σ(x) ⊆ ℂ is a compact subset of {λ ∈ ℂ | |λ| ≤ ‖x‖}. Proof. It follows directly from Corollary 1.11 that |λ| ≤ ‖x‖ when λ ∈ σ(x). So it suffices to show that σ(x) is closed. Observe that by Remark 1.9 the noninvertibles in A form a closed set. Therefore σ(x) is defined as the preimage of a closed set under the continuous mapping λ 󳨃→ λ − x, so it is closed. Definition 1.15. For an element x ∈ A in a unital Banach algebra, we define its resolvent set ρ(x) = ℂ \ σ(x) and its resolvent map Rx : ρ(x) → A via Rx (z) = (z − x)−1 . Lemma 1.16. The resolvent map is holomorphic2 and its derivative is given as −(z − x)−2 .

d R (z) dz x

=

Proof. Let x ∈ A be given. Let z ≠ z0 ∈ ρ(x) be two distinct points in the resolvent set. We compute [(z − x)−1 − (z0 − x)−1 ](z − x)(z0 − x)(z0 − z)−1 = [1 − (z0 − x)−1 (z − x)](z0 − x)(z0 − z)−1 = [(z0 − x) − (z − x)](z0 − z)−1 = 1. With the analogous calculation in the reverse order, we conclude that (z − x)−1 − (z0 − x)−1 = (z0 − z)(z0 − x)−1 (z − x)−1 Using continuity of inversion, we conclude 2 The meaning of “holomorphic”, as is made precise by the proof, means that derivatives exist as limits of differential quotients within A.

Introduction to C∗ -algebras

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(z − x)−1 − (z0 − x)−1 d 󵄨󵄨󵄨󵄨 lim = −(z0 − x)−2 . 󵄨󵄨 Rx (z) = z→z dz 󵄨󵄨z=z0 z − z0 0 Theorem 1.17. Let A be a unital Banach algebra. For every x ∈ A, the spectrum σ(x) is nonempty. Proof. We argue by contradiction. Fix x ∈ A and assume σ(x) = 0, which equivalently means ρ(x) = ℂ. The resolvent map z 󳨃→ (z − x)−1 is then an entire function by Lemma 1.16. As x −1 ≠ 0, we may use the Hahn–Banach theorem and choose a bounded linear functional f ∈ A∗ with f (x −1 ) ≠ 0. As f is continuous and linear, we see that the map ℂ → ℂ,

z 󳨃→ f [(z − x)−1 ]

is also a nonzero entire function. By Liouville’s theorem, it is therefore either constant (≠ 0) or unbounded. However, if |z| ≥ 2‖x‖, then it follows with Proposition 1.8 that −1 󵄩 ∞󵄩 󵄩 󵄩󵄩 󵄩󵄩 x 󵄩󵄩󵄩n |z|→∞ x 󵄩󵄩 󵄩 −1 󵄩 −1 󵄩 −1 −1 󵄩󵄩(z − x) 󵄩󵄩󵄩 = |z| ⋅ 󵄩󵄩󵄩(1 − ) 󵄩󵄩󵄩 ≤ |z| ⋅ ∑ 󵄩󵄩󵄩 󵄩󵄩󵄩 ≤ |z| ⋅ 2 󳨀→ 0. 󵄩󵄩 󵄩󵄩 z 󵄩󵄩 z 󵄩󵄩 n=0

But this leads to a contradiction, hence indeed σ(x) ≠ 0. Theorem 1.18 (Gelfand–Mazur). Let A be a unital Banach algebra in which every nonzero element is invertible. Then A = ℂ ⋅ 1. Proof. Suppose we have x ∈ A \ ℂ ⋅ 1. Then the assumption implies σ(x) = 0, a contradiction to Theorem 1.17. To summarize, we have so far succeeded in proving that the spectrum of any element x in a unital Banach algebra is a nonempty compact subset. We will now be concerned with other useful characterizations of the spectrum, which, among other things, are convenient in computations. Definition 1.19. Let A be a Banach algebra. A character on A is a nonzero multiplicative linear functional A → ℂ. Proposition 1.20. Let A be a unital Banach algebra. Then every character φ : A → ℂ is continuous, in fact ‖φ‖ = φ(1) = 1. Proof. As φ(x) = φ(x ⋅ 1) = φ(x)φ(1) for all x ∈ A and φ ≠ 0, it follows that φ(1) = 1. Suppose that some x ∈ A were to satisfy |φ(x)| > ‖x‖. Then Corollary 1.11 implies that φ(x) − x is invertible. However, this means 1 = φ((φ(x) − x)−1 )φ(φ(x) − x) and 0 = φ(x) − φ(x) = φ(φ(x) − x), which would be a contradiction. Definition 1.21. Let A be a unital commutative Banach algebra. We define its Gelfand spectrum as

6 � G. Szabó  = {characters A → ℂ} ⊆ A∗ . We equip  with the weak-∗ topology, i. e., the topology of pointwise convergence. Due to the Banach–Alaoglu theorem, one easily sees that this topology turns  into a compact Hausdorff space. Proposition 1.22. Let A be a unital Banach algebra. Suppose that 𝒥 ⊆ A is a maximal ideal. Then (a) 𝒥 is closed. (b) If A is commutative, then A/𝒥 ≅ ℂ. Proof. If 𝒥 is not closed, then 𝒥 = A follows by maximality. However, if 𝒥 is dense, then it has to contain some invertible element by Remark 1.9. Hence 𝒥 = A, a contradiction. Next assume that A is commutative. It follows from elementary algebra that the quotient A/𝒥 is a field. At the same time it is a unital Banach algebra by Exercise 1.6. So A/𝒥 ≅ ℂ follows from Theorem 1.18. Proposition 1.23. Let A be a unital commutative Banach algebra. Then the assignment φ 󳨃→ ker(φ) becomes a bijection  → {maximal ideals in A} Proof. We first need to see that this map is well defined. As we have φ(1) = 1 for all φ ∈  and thus A = ℂ ⋅ 1 + ker(φ) via x = φ(x)1 + (φ(x)1 − x), it follows that ker(φ) ⊆ A has codimension 1. Hence it is even maximal as a proper linear subspace. The fact that it is an ideal is trivial. Furthermore, the above equality A = ℂ ⋅ 1 + ker(φ) implies that ker(φ) = ker(ψ) implies φ = ψ for all φ, ψ ∈ A.̂ This shows injectivity. For surjectivity, let a maximal ideal 𝒥 ⊆ A be given. By Proposition 1.22, we have A/𝒥 ≅ ℂ, so the quotient map A → A/𝒥 ≅ ℂ can be viewed as a character, which has kernel equal to 𝒥 . Example 1.24. Let X be a compact Hausdorff space. Given x ∈ X, denote by evx : 𝒞 (X) → ℂ the evaluation map given by evx (f ) = f (x). Then x 󳨃→ evx defines a homeomorphism X ≅ 𝒞̂ (X). Proof. It is clear that evx is a character for all x ∈ X. Furthermore, since the Gelfand spectrum is equipped with the topology of pointwise convergence, this assignment is clearly continuous. Injectivity is an easy consequence of the Urysohn–Tietze extension theorem. The hard part is to show surjectivity. For this purpose, we appeal to Proposition 1.23, which tells us that it is enough to know that every maximal ideal 𝒥 ⊂ 𝒞 (X) is of

Introduction to C∗ -algebras �

7

the form ker(evx ) for some x ∈ X. By maximality, this is the same as finding some x ∈ X with 𝒥 ⊆ ker(evx ). Suppose that this were false. Given any x ∈ X, we have 𝒥 ⊈ ker(evx ), so we can find gx ∈ 𝒥 with gx (x) ≠ 0. Then fx = ḡx gx = |gx |2 ∈ 𝒥 is nonnegative with fx (x) > 0. By continuity, fx is strictly positive on an open neighborhood of x. We can then use the compactness of X to find a finite number of points x1 , . . . , xk ∈ X such that the function ∑kj=1 fxj is strictly positive. This implies that 𝒥 has an invertible element, which yields a contradiction. Theorem 1.25. Let A be a unital commutative Banach algebra. Let x ∈ A. Then σ(x) = ̂ {φ(x) | φ ∈ A}. Proof. Let λ ∈ ℂ. For “⊇”, suppose λ = φ(x) for some φ ∈ A.̂ Then λ − x ∈ ker φ, which is a proper ideal. So λ − x ∉ GL(A) and λ ∈ σ(x) by definition. For “⊆”, suppose λ ∈ σ(x), i. e., λ−x is not invertible. By the theory of commutative rings,3 the element λ−x is contained in some maximal ideal 𝒥 ⊆ A. By Proposition 1.23, this means λ − x ∈ ker φ for some φ ∈ A.̂ Hence 0 = φ(λ − x) = λ − φ(x), which implies λ = φ(x). Exercise 1.26. Let A be a unital Banach algebra and x ∈ A. (1) Show that there is a unital commutative Banach subalgebra C ⊆ A with x ∈ C such that σC (x) = σA (x). If A is a Banach-∗-algebra and x ∗ x = xx ∗ , then C can be chosen to be ∗-closed. (2) Show that for every λ ∈ σ(x), there exists a continuous linear functional φ ∈ A∗ with 1 = φ(1) = ‖φ‖ such that φ(x) = λ. Proposition 1.27. Let A be a unital Banach algebra. For x, y ∈ A, the element 1 − xy is invertible if and only if 1 − yx is invertible. Proof. Evidently it suffices to show “⇒”, as the converse then follows by exchanging x and y. Assuming 1 − xy is invertible, we claim that y(1 − xy)−1 x + 1 ∈ A is the inverse of 1 − yx. Indeed, we have (1 − yx)[y(1 − xy)−1 x + 1] = y(1 − xy)−1 x + 1 − yxy(1 − xy)−1 x − yx = y[(1 − xy)−1 − xy(1 − xy)−1 ]x + 1 − yx = y(1 − xy)(1 − xy)−1 x + 1 − yx = 1. Similarly, one gets 1 = [y(1 − xy)−1 x + 1](1 − yx). 3 Remember that this is an application of Zorn’s Lemma! Commutativity is crucial here. This statement fails for a rank-one matrix in the algebra of 2 × 2 matrices.

8 � G. Szabó Corollary 1.28. Let A be a unital Banach algebra. For x, y ∈ A, we have σ(xy) ∪ {0} = σ(yx) ∪ {0}. Proof. For λ ∈ ℂ \ {0}, we have def

λ ∈ σ(xy) ⇔ λ − xy is not invertible xy ⇔ 1− is not invertible λ 1.27 yx is not invertible ⇔ 1− λ ⇔ λ ∈ σ(yx). As λ was arbitrary, this proves the claimed equality of sets. Definition 1.29. Let A be a unital Banach algebra. For x ∈ A, we define its spectral radius as r(x) = max{|λ| | λ ∈ σ(x)}. The proof of the following theorem is perhaps the most nontrivial in this section, and takes a fair amount of complex analysis for granted. Theorem 1.30 (Spectral radius formula). Let A be a unital Banach algebra. Let x ∈ A. Then 󵄩 󵄩1/n r(x) = lim 󵄩󵄩󵄩x n 󵄩󵄩󵄩 . n→∞ Proof. We first observe that by Exercise 1.26, we may assume without loss of generality that A is commutative. As a consequence of Theorem 1.25, we can observe for λ ∈ σ(x) that λn ∈ σ(x n ), and hence |λ|n ≤ ‖x n ‖. As λ ∈ σ(x) and n ≥ 1 are arbitrary, we see r(x) ≤ infn∈ℕ ‖x n ‖1/n . We want to show lim supn→∞ ‖x n ‖1/n ≤ r(x). We proceed by proving that 󵄩 󵄩1/n |z| lim sup󵄩󵄩󵄩x n 󵄩󵄩󵄩 ≤ 1 n→∞

for all |z|
r, then either r ≤ |λ| or r > |λ|. If also λ ≥ 0, then thus either |λ − r| = λ − r ≤

λ ≤r 2

or |λ − r| = r − λ ≤ r. But this would be a contradiction. So we conclude σ(a) ⊈ ℝ≥0 if one has ‖a − r ⋅ 1‖ > r. On the other hand, if λ ∈ σ(a) satisfies λ < 0, then clearly |λ − r| = r − λ > r. This proves the equivalence. Lemma 3.3. Let A be a unital C∗ -algebra. Let a, b ∈ A be self-adjoint elements with σ(a), σ(b) ⊆ ℝ≥0 . Then σ(a + b) ⊆ ℝ≥0 . Proof. Set r = ‖a‖ + ‖b‖ ≥

‖a+b‖ . 2

Using Lemma 3.2 and the triangle inequality, we see

‖a + b − r ⋅ 1‖ ≤ ‖a − ‖a‖ ⋅ 1‖ + ‖b − ‖b‖ ⋅ 1‖ ≤ ‖a‖ + ‖b‖ = r. So indeed σ(a + b) ⊆ ℝ≥0 . Lemma 3.4. Let A be a unital C∗ -algebra and v ∈ A an element. If σ(v∗ v) ⊆ ℝ≤0 , then v = 0. Proof. Write v = v1 + iv2 for self-adjoints v1 , v2 ∈ A. Then v∗ v + vv∗ = v21 − iv2 v1 + iv1 v2 + v22 + v21 + iv2 v1 − iv1 v2 + v22 = 2(v21 + v22 ). We have σ(v21 ), σ(v22 ) ⊆ ℝ≥0 . Using Lemma 3.3, it follows that σ(vv∗ ) = σ(2(v21 + v22 ) + (−v∗ v)) ⊆ ℝ≥0 . By Proposition 1.27, we have σ(v∗ v) ∪ {0} = σ(vv∗ ) ∪ {0}. But this leaves only σ(v∗ v) = {0}. As v∗ v is clearly self-adjoint (and hence normal), we get v∗ v = 0 with Theorem 2.7 and so v = 0.

18 � G. Szabó Lemma 3.5. Let A be a unital C∗ -algebra, and a ∈ A a self-adjoint element. Then there exist self-adjoint elements a+ , a− ∈ A with a = a+ −a− , σ(a+ ), σ(a− ) ⊆ ℝ≥0 , and a+ ⋅a− = 0. Proof. We consider the two continuous functions f + , f − : ℝ → ℝ≥0 given by t,

f + (t) = {

0,

t ≥ 0, t < 0,

0,

and f − (t) = {

−t,

t ≥ 0, t < 0.

Then, clearly, t = f + (t) − f − (t) for all t ∈ ℝ, and f + ⋅ f − = 0. Since σ(a) ⊆ ℝ, it follows by functional calculus that we have the desired properties for a+ = f + (a) and a− = f − (a). Theorem 3.6. Let A be a unital C∗ -algebra. An element x ∈ A is positive if and only if it is normal and σ(x) ⊆ ℝ≥0 . Proof. We already know the “if” part. So let us assume that x ∈ A is positive. Write x = y∗ y. By Lemma 3.5, we may write x = x + − x − as stated there. Set v = yx − . Then we observe 3

σ(v∗ v) = σ(x − xx − ) = σ(x − (x + − x − )x − ) = σ(−(x − ) ) Since σ(x − ) ⊆ ℝ≥0 , it follows that σ(v∗ v) ⊆ ℝ≤0 . By Lemma 3.4, we conclude v = 0, but then also x − = 0. Hence x = x + is indeed self-adjoint with positive spectrum. Corollary 3.7. Every element in a (unital) C∗ -algebra x is a linear combination of at most four positive elements with norm at most ‖x‖. Exercise 3.8. Let A be a unital C∗ -algebra. Show that the set of all positive elements in A is closed. Next we prove some basics about positivity-preserving functionals on C∗ -algebras, which (among other things) play a seminal role in a later section of this chapter, where we develop GNS representation theory. Definition 3.9. Let A be a C∗ -algebra. We say that a linear functional φ : A → ℂ is positive if φ(a) ≥ 0 whenever a ≥ 0. If the operator norm of φ is additionally assumed to be one, we call φ a state on A. Lemma 3.10. Let A be a (unital) C∗ -algebra.5 Then every positive linear functional φ : A → ℂ is continuous. Proof. Suppose φ is unbounded. By Corollary 3.7, we may conclude that φ must be unbounded on the positive part of the unit ball of A. Given n ≥ 1, we can choose an ≥ 0 with 5 Once we cover the next section, the proof presented here is actually the correct one for not necessarily unital C∗ -algebras. It can be a nice exercise to find an even simpler proof for only unital C∗ -algebras.

Introduction to C∗ -algebras �

19

−n ‖an ‖ = 1 and φ(an ) ≥ n ⋅ 2n . We consider the element a = ∑∞ n=1 2 an , which is positive by Exercise 3.8, as well as all partial sums of this series. Hence ∞

φ(a) = φ( ∑ 2−n an ) + 2−k φ(ak ) ≥ 2−k φ(ak ) ≥ k n=1 n=k ̸

holds for all k ∈ ℕ, which is a contradiction. Proposition 3.11. Let φ : A → ℂ be a positive linear functional on a unital C∗ -algebra. Then φ(x ∗ ) = φ(x) for all x ∈ A. Proof. Since we can write x = x1 + ix2 for self-adjoint elements x1 , x2 ∈ A, it suffices to prove φ(x1 ), φ(x2 ) ∈ ℝ. But this follows directly from Lemma 3.5. Lemma 3.12. Let φ : A → ℂ be a positive linear functional on a unital C∗ -algebra. For all x, y ∈ A, we have 󵄨󵄨 ∗ 󵄨󵄨2 ∗ ∗ 󵄨󵄨φ(y x)󵄨󵄨 ≤ φ(x x)φ(y y). Proof. This is a special case of the Cauchy–Schwarz inequality. Theorem 3.13. Let φ : A → ℂ be a linear functional on a unital C∗ -algebra. Then φ is positive if and only if ‖φ‖ = φ(1). Proof. For the “only if” part, let x ∈ A with ‖x‖ ≤ 1. Using Cauchy–Schwarz inequality, we see that 󵄨󵄨 󵄨2 󵄨 󵄨2 ∗ 󵄨󵄨φ(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨φ(x ⋅ 1)󵄨󵄨󵄨 ≤ φ(x x)φ(1) ≤ φ(1)‖φ‖. Now taking the supremum over all contractions x ∈ A yields ‖φ‖ ≤ φ(1). The reverse inequality φ(1) ≤ ‖φ‖ is trivial. For the “if” part, we may already conclude from ‖φ‖ = φ(1) and Proposition 2.13 that φ sends self-adjoint elements to real numbers. Let a ∈ A be a self-adjoint element with ‖a‖ = 1 and suppose φ(a) < 0. Then ‖φ‖ = φ(1) = φ(a + (1 − a)) = φ(a) + φ(1 − a) < φ(1 − a). If we additionally assume that a is positive, then 1−a has norm at most one, so we would arrive at a contradiction. Hence we must have φ(a) ≥ 0 if a ≥ 0. Corollary 3.14. Let B ⊆ A be an inclusion of unital C∗ -algebras. Then every state on B extends to a state on A. Proof. Let φ : B → ℂ be a state. Then ‖φ‖ = φ(1) = 1. By Hahn–Banach theorem, we can ̃ extend φ to a linear functional φ̃ : A → ℂ with the same norm. Then ‖φ‖̃ = φ(1) = 1 still holds. Hence φ̃ is a state by Theorem 3.13.

20 � G. Szabó Corollary 3.15. Let A be a unital C∗ -algebra. For every normal x ∈ A, there is a state φ with ‖x‖ = |φ(x)|. Proof. By Theorem 2.7, we can choose λ ∈ σ(x) with |λ| = ‖x‖. Applying Exercise 1.26 along with Theorem 3.13 yields a state with the required property. Definition 3.16. Let A be a C∗ -algebra. We define the state space 𝒮 (A) of A to be the subset of all states in the unit ball of A∗ . We equip it with the weak-∗ topology. Proposition 3.17. Let A be a unital C∗ -algebra. Then 𝒮 (A) is Hausdorff, convex, and compact. Proof. Being Hausdorff is trivial since the weak-∗ topology is Hausdorff to begin with. Given φ1 , φ2 ∈ 𝒮 (A) and t ∈ (0, 1), we observe ‖tφ1 + (1 − t)φ2 ‖ ≤ 1 and (tφ1 + (1 − t)φ2 )(1) = tφ1 (1) + (1 − t)φ2 (1) = t + (1 − t) = 1. It follows from Theorem 3.13 that tφ1 + (1 − t)φ2 ∈ 𝒮 (A). In other words, 𝒮 (A) is indeed convex. For compactness, we only need to show that 𝒮 (A) is closed in the unit ball of A∗ , but this is also clear from Theorem 3.13. We note that if a C∗ -algebra A is not unital, then its state space 𝒮 (A) is only locally compact, but not compact. Definition 3.18. Let A be a C∗ -algebra. For two self-adjoint elements a, b ∈ A, we write a ≤ b if b − a ≥ 0. Proposition 3.19. Let A be a unital C∗ -algebra. (i) If a, b, c, d ∈ A are self-adjoint and a ≤ b and c ≤ d, then a + c ≤ b + d. (ii) For a ∈ A self-adjoint, we have a ≤ ‖a‖ ⋅ 1. (iii) If a ≤ b and x is any element, then x ∗ ax ≤ x ∗ bx. (iv) If 0 ≤ a ≤ b and a is invertible, then so is b and b−1 ≤ a−1 . Proof. (i) follows from the fact that positive elements are additively closed (Lemma 3.3), while (ii) is a consequence of functional calculus. For (iii), write b − a = c∗ c. Then x ∗ bx − x ∗ ax = x ∗ (b − a)x = x ∗ c∗ cx = (cx)∗ cx ≥ 0. For (iv), we note that by functional calculus, if a is invertible, there exists a positive scalar ε > 0 with ε ≤ a, so the same is true for b and it is also invertible. Furthermore, we conclude from (iii) that a ≤ b implies 1 ≤ a−1/2 ba−1/2 . Set x = b1/2 a−1/2 . By Corollary 1.28, we have σ(x ∗ x) ∪ {0} = σ(xx ∗ ) ∪ {0}. Given that x itself is invertible, one gets equality of the spectra, and therefore the spectrum of xx ∗ = b1/2 a−1 b1/2 equals that of x ∗ x = a−1/2 ba−1/2 ≥ 1. Hence 1 ≤ b1/2 a−1 b1/2 , so applying (iii) again finally yields b−1 ≤ a−1 .

Introduction to C∗ -algebras

� 21

4 Nonunital C∗ -algebras For the most part, we have so far stuck to studying Banach and C∗ -algebras with a unit. However, C∗ -algebras without a unit are also interesting in their own right, and in fact have a structure theory that runs parallel to the unital case. This is in part due to the construction discussed in the first part of this section, where we see that it is always possible to enlarge a given C∗ -algebra by “adding” a unit to it. Definition 4.1. Let A be a C∗ -algebra. We define an involutive unital ℂ-algebra A† = A × ℂ, which carries the same ℂ-algebra structure as in Remark 1.2 and the involution (a, λ)∗ = (a∗ , λ). We will always view A ⊆ A† by identifying a ∈ A with (a, 0) ∈ A† . One usually denotes A, A is unital, Ã = { † A , A is nonunital. If φ : A → B is a ∗-homomorphism between C∗ -algebras, we write φ† : A† → B† for the unital ∗-homomorphism given by φ† (a, λ) = (φ(a), λ). Theorem 4.2. Let A be a C∗ -algebra. Then there exists a unique norm on A† that turns it into a C∗ -algebra so that the inclusion A ⊆ A† is isometric. Proof. We already know from Corollary 2.8 that such a norm is necessarily unique. So we need to show existence. Suppose first that A was already unital. In this case, we claim that A† ≅ A ⊕ ℂ via a ⊕ λ 󳨃→ (a − λ1A , λ) as unital ∗-algebras. It is easy to see that this map is linear, ∗-preserving, and bijective. It is multiplicative because a direct computation shows (a − λ ⋅ 1A , λ)(b − η ⋅ 1A , η)

= ((ab − λb − ηa + λη ⋅ 1A ) + (λb − λη ⋅ 1A ) + (ηa − ηλ ⋅ 1A ), λη) = (ab − λη ⋅ 1A , λη),

which is the image of ab ⊕ λη. Since A ⊕ ℂ is a C∗ -algebra with the max-norm given by (a, λ) 󳨃→ max{‖a‖, |λ|}, the claim follows. Suppose next that A is nonunital. We consider the following semi-norm on A† : 󵄩󵄩 󵄩 󵄩󵄩(a, λ)󵄩󵄩󵄩 = sup{‖ay + λy‖ | y ∈ A, ‖y‖ ≤ 1}. Notice that this is the operator norm of the linear operator on A that is given by leftmultiplication with (a, λ). So by the property of the operator norm, we get ‖(a, λ)(b, η)‖ ≤ ‖(a, λ)‖ ⋅ ‖(b, η)‖. We claim that this is actually a norm. Suppose ‖(x, λ)‖ = 0. If λ = 0, then either x = 0 or ‖(x, 0)‖ ≥ ‖x ⋅ (x ∗ ⋅ ‖x‖−1 )‖ = ‖x‖ ≠ 0, which is a contradiction. If λ ≠ 0, then clearly

22 � G. Szabó also x ≠ 0. Thus we have for all y ∈ A that 0 = (λ + x)y, which is equivalent to y = (− xλ )y. But this means that − xλ ∈ A is a unit, which is a contradiction. In summary, we can only have λ = 0 and x = 0. From the above we can also see ‖(x, 0)‖ ≥ ‖x‖, but “≤” is clear, so the inclusion A ⊆ A† is indeed isometric. Since the norm on A† is therefore complete on a linear subspace of codimension one, it is necessarily complete on A† . The only thing left to show is the C∗ -identity. If we keep in mind Remark 2.2, then the computation 󵄩󵄩 󵄩 󵄩 󵄩 ∗ ∗ 󵄩󵄩(a, λ) (a, λ)󵄩󵄩󵄩 = sup 󵄩󵄩󵄩(a, λ) (a, λ)y󵄩󵄩󵄩 ‖y‖=1

󵄩 󵄩 ≥ sup 󵄩󵄩󵄩y∗ (a, λ)∗ (a, λ)y󵄩󵄩󵄩 ‖y‖=1

󵄩 󵄩2 = sup 󵄩󵄩󵄩(a, λ)y󵄩󵄩󵄩 ‖y‖=1

󵄩 󵄩2 = 󵄩󵄩󵄩(a, λ)󵄩󵄩󵄩 proves that the C∗ -identity holds. Definition 4.3. Let A be a C∗ -algebra. We define the canonical character ε = εA : A† → ℂ given by εA (a, λ) = λ. In this way A = ker(εA ) ⊆ A† is an inclusion as a maximal ideal. Exercise 4.4. Let A be a nonunital C∗ -algebra and B a C∗ -algebra. Let φ : A → B be a ̃ λ) = φ(a) + λ ⋅ 1B̃ is the unique ℂ-algebra homomorphism. Then φ̃ : à → B̃ given by φ(a, extension of φ to a unital ℂ-algebra homomorphism. If φ is a ∗-homomorphism, then so is φ.̃ Definition 4.5. Let A be a (possibly nonunital) commutative C∗ -algebra. Like in Definition 1.21, we define its Gelfand spectrum  as the set of all characters on A, equipped with the weak-∗ topology. By Exercise 4.4, every character φ ∈  extends uniquely to ̂† if A is nonunital. As we see below, this gives an inclusion  ⊆ A ̂† as an open set. φ̃ ∈ A ̂ In particular A is locally compact. ̂† =  ∪{ε ̇ A }. In Proposition 4.6. For a nonunital commutative C∗ -algebra A, we have A ̂ particular, A is locally compact as an open subset of a compact space. Proof. We know from Proposition 1.23 that every character on A† is uniquely deter̂† is not equal to ε , then ker(φ) ≠ ker(ε ) = A, mined by its kernel. If a character φ ∈ A A A ∼ which means φ|A ∈  and φ = (φ|A ) by Exercise 4.4. Corollary 4.7. Every character on a commutative C∗ -algebras is ∗-preserving. ̇ Definition 4.8. Let X be a locally compact space. Set X † = X ∪{∞} and equip it with the topology τ = {U ⊆ X | U open} ∪ {{∞} ∪ (X \ K) | K ⊆ X compact}. Then (X † , τ) is a compact space and is called the one-point compactification of X.

Introduction to C∗ -algebras �

23

Exercise 4.9. Let X be a locally compact space. (i) Verify the claim that X † is compact. (ii) Show that ∞ ∈ X † is an isolated point if and only if X is compact. (iii) Suppose X is compact and nonempty. Let x0 ∈ X be any point. Show that (X \{x0 })† ≅ X via ∞ 󳨃→ x0 and x 󳨃→ x for all x ≠ x0 . (iv) Show {f ∈ 𝒞 (X † ) | f (∞) = 0} ≅ 𝒞0 (X) via the restriction map. (v) Show that 𝒞 (X † ) ≅ 𝒞0 (X)† via f 󳨃→ ((f − f (∞)1)|X , f (∞)) is an isomorphism of C∗ -algebras. Conclude that the spectrum of 𝒞0 (X) is canonically homeomorphic to X. The following extends our earlier Gelfand–Naimark theorem and establishes a more general one-to-one correspondence between commutative C∗ -algebras and locally compact spaces. In analogy to the unital case, this allows us to subsequently derive a nonunital version of functional calculus. Theorem 4.10 (Gelfand–Naimark; nonunital version). Let A be a nonunital commutative ≅ ̂† ) restricts to an isometric C∗ -algebra. Then the (unital) Gelfand transform ∧ : A† 󳨀→ 𝒞 (A ̂ ∗-isomorphism ∧ : A → 𝒞0 (A). Proof. We know from Theorem 2.21 that the unital Gelfand transform is an isometric ̂† ) ≅ ∗-isomorphism. By combining Proposition 4.6 and Exercise 4.9, we see that 𝒞 (A † ̂ 𝒞0 (A) via f 󳨃→ (f − f (εA ), f (εA )). Under this isomorphism, we see for all (a, λ) ∈ A† that ε𝒞

0 (A)

̂

def

[(a, λ)∧ ] = (a, λ)∧ (εA ) = εA (a, λ) = λ.

Hence, under the unital Gelfand transform, one has that A = ker(εA ) ⊆ A† is mapped ̂† ). onto 𝒞0 (A)̂ = ker(ε𝒞 (A)̂ ) ⊆ 𝒞0 (A)̂ † ≅ 𝒞 (A 0

Definition 4.11. Let A be a C∗ -algebra and x ∈ A an element. We define its spectrum as σ(x) = σà (x). Proposition 4.12 (Functional calculus; nonunital version). Let A be a C∗ -algebra and x ∈ A a normal element. Then there is a (unique) isometric ∗-homomorphism, {f ∈ 𝒞 (σ(x)) | f (0) = 0} → A

with

idσ(x) 󳨃→ x.

Proof. If A is unital, then the statement follows from restriction of the functional calculus map that we already know to exist. Hence we may assume that A is nonunital. We know from Theorem 2.23 that there is an isometric unital ∗-embedding 𝒞 (σ(x)) → A† with idσ(x) 󳨃→ x. The function f ∈ 𝒞 (σ(x)) satisfies f (x) ∈ A if and only if f (x) ∈ ker(εA ), or, in other words, 2.27 0 = εA (f (x)) = f (εA (x)) = f (0). This proves the claim.

24 � G. Szabó Remark 4.13. Without carrying out the details, we now have the tools ready to generalize almost all results so far from unital C∗ -algebras to arbitrary ones. For example, this includes: – All ∗-homomorphisms between C∗ -algebras are norm-contractive. – All injective ∗-homomorphisms between C∗ -algebras are isometric. – All aspects of spectral theory from the second chapter that do not directly involve a unit. – The characterization of positive elements and functionals. From now on, we will use much of this without further mention. Definition 4.14. Let A be a Banach algebra. A net eα ∈ A is called an approximate unit, if ‖eα ‖ ≤ 1 for all α and a = limα→∞ eα ⋅ a = limα→∞ a ⋅ eα holds for all a ∈ A. If A is a C∗ -algebra, we call (eα ) moreover an increasing approximate unit if also 0 ≤ eα0 ≤ eα1 for all α0 ≤ α1 . We will usually assume that approximate units in C∗ -algebras are increasing unless specified otherwise. If A is unital, we see that in fact an approximate unit is simply a net eα ∈ A with eα → 1, hence the concept is only really interesting when A is not unital. Theorem 4.15. Let A be a unital C∗ -algebra and I ⊆ A a (not necessarily closed) left-ideal. Then there exists an increasing net eα ∈ I of positive elements with norm at most one such that x = lim xeα ,

x ∈ I.

α→∞

Remark 4.16. Before we proceed with the proof, we record for all t ≥ 0 and n ≥ 1 the estimate 0≤t−

t + n−1 t tn−1 1 t2 = t( − ) = ≤ . −1 −1 −1 −1 n t+n t+n t+n t+n

Proof of Theorem 4.15. Let Λ be the set of all finite subsets of I, which becomes a directed set via the inclusion order. For α ∈ Λ, write α = {x1 , . . . , xn } and set vα = x1∗ x1 + . . . + xn∗ xn ∈ I. This is a net of positive elements. Using functional calculus, we furthermore define eα = (

1 + vα ) vα ∈ I. n −1

This is a net of positive elements with norm at most one. Using functional calculus again with Remark 4.16, we estimate:

Introduction to C∗ -algebras �

25

−1 󵄩 󵄩󵄩 󵄩󵄩 1 1 󵄩 ≥ 󵄩󵄩󵄩vα − v2α ( + vα ) 󵄩󵄩󵄩 󵄩 󵄩󵄩 n n 󵄩 󵄩󵄩 󵄩󵄩 = 󵄩󵄩vα (1 − eα )󵄩󵄩 󵄩 󵄩 ≥ 󵄩󵄩󵄩(1 − eα )vα (1 − eα )󵄩󵄩󵄩 3.19 󵄩 󵄩 ≥ 󵄩󵄩󵄩(1 − eα )xj∗ xj (1 − eα )󵄩󵄩󵄩

= ‖xj − xj eα ‖2 ,

where xj ∈ α. But this ensures that x = limα→∞ xeα for all x ∈ I. The only thing left to show is that eα is indeed increasing. For this, we observe 1 1 1 1 1 + vα )[1 − ( + vα ) ] = + vα − = vα , n n n n n −1

(

which implies 1 − n1 ( n1 + vα )−1 = eα for all α ∈ Λ. Suppose α, β ∈ Λ satisfy α ≤ β. Then let us write α = {x1 , . . . , xn } and β = {x1 , . . . , xn , . . . , xp } for p ≥ n. Clearly, vα ≤ vβ . By Proposition 3.19(iv), we have eα

1 1 ( + vα ) n n

−1

=

1−

=

1 − (1 + nvα )−1

≤

1 − (1 + pvβ )−1 = eβ .

nvα ≤pvβ

Since this holds whenever α ≤ β, we have proved the claim. Corollary 4.17. Every C∗ -algebra has an increasing approximate unit. Proof. We may assume that A is nonunital. Then A ⊆ A† becomes an ideal in a unital C∗ -algebra. By Theorem 4.15, there is an increasing net eα ∈ A of positive elements such that a = limα→∞ aeα for all a ∈ A. By applying the adjoint to this equation for a∗ in place of a, we also have a = limα→∞ eα a. Corollary 4.18. Let A be a C∗ -algebra and I ⊆ A a closed two-sided ideal. Then I is selfadjoint, i. e., I = I ∗ . In particular, I is a C∗ -subalgebra. Proof. If A is nonunital, then it is easy to see that I ⊆ A† is also a two-sided ideal. So let us assume A is unital. By Theorem 4.15, there is a net eα ∈ I of positive elements with x = limα→∞ xeα , x ∈ I. But then also x ∗ = limα→∞ eα x ∗ . Since I is a closed two-sided ideal, we conclude x ∗ ∈ I. Corollary 4.19. Let A be a C∗ -algebra. Suppose I ⊆ A is a closed two-sided ideal, and J ⊆ I is a closed two-sided ideal. Then J ⊆ A is a closed two-sided ideal. Proof. We only need to show that J is an ideal in A. Let c ∈ J, a ∈ A. Let eα be an increasing approximate unit in I. Then

26 � G. Szabó c ⋅ a = lim c ⋅ ⏟⏟⏟⏟⏟⏟⏟⏟⏟ eα ⋅ a ∈ J. α→∞

∈I

Similarly, a ⋅ c ∈ J. Lemma 4.20. Let A be a C∗ -algebra and J ⊆ A a closed two-sided ideal. Let eα ∈ J be an increasing approximate unit. Then the quotient norm on A/J is given by def

‖x + J‖ = inf ‖x − y‖ = lim ‖x − eα x‖, α→∞

y∈J

x ∈ A.

Proof. On the one hand, one clearly has ‖x + J‖ ≤ lim inf ‖x − eα x‖. α→∞

On the other hand, if y ∈ J is arbitrary, then lim sup ‖x − eα x‖ = lim sup ‖x − eα x − y + eα y‖, α→∞

α→∞

which equals 󵄩 󵄩 lim sup󵄩󵄩󵄩(1 − eα )(x − y)󵄩󵄩󵄩 ≤ ‖x − y‖. α→∞

This implies lim supα→∞ ‖x − eα x‖ ≤ ‖x + J‖ and yields the desired equality. Theorem 4.21. Let A be a C∗ -algebra and J ⊆ A a closed two-sided ideal. Then the quotient A/J is a C∗ -algebra. Proof. By Corollary 4.18, J is automatically ∗-closed, and hence (x +J)∗ := x ∗ +J is a welldefined involution on A/J. We already know that A/J is a Banach algebra. It suffices to show ‖x + J‖2 ≤ ‖x ∗ x + J‖ for all x ∈ A. Let y ∈ J be arbitrary and let eα ∈ J be an increasing approximate unit. Then 󵄩 󵄩2 ‖x + J‖2 = lim 󵄩󵄩󵄩x(1 − eα )󵄩󵄩󵄩 α→∞

󵄩 󵄩 = lim 󵄩󵄩󵄩(1 − eα )x ∗ x(1 − eα )󵄩󵄩󵄩 α→∞ 󵄩 󵄩 = lim 󵄩󵄩󵄩(1 − eα )(x ∗ x + y)(1 − eα )󵄩󵄩󵄩 α→∞ 󵄩 󵄩 ≤ 󵄩󵄩󵄩x ∗ x + y󵄩󵄩󵄩.

Taking the infimum over all y ∈ J shows the claim. Corollary 4.22. Let Ψ : A → B be a ∗-homomorphism between C∗ -algebras. Then the induced map Ψ : A/(ker Ψ) → B is an isometric ∗-isomorphism onto the image of Ψ. In particular, Ψ(A) ⊆ B is a C∗ -subalgebra. Proof. This follows from Theorem 4.21 and Theorem 2.28.

Introduction to C∗ -algebras

� 27

5 Universal C∗ -algebras In this section we develop a framework to construct C∗ -algebras that are universal for a specifically given and well-behaved ∗-algebraic relation. Since numerous interesting C∗ -algebras arise in this way, this construction provides a big source of examples, some of which shall be discussed near the end of the section. The content of this section is inspired by similar parts in the book [5], which gives a much more elaborate treatment of the subject of relations and their C∗ -algebras. Definition 5.1. Let 𝒳 be a nonempty set. We set 𝒳 ∗ = {x ∗ | x ∈ 𝒳 } and view it as a set which is disjoint from 𝒳 . A noncommutative ∗-polynomial with (free) variables in 𝒳 is a formal expression of the form m

∑ λk ⋅ xk,1 ⋅ xk,2 . . . xk,nk ,

k=1

where m ∈ ℕ, nk ∈ ℕ≥0 , xn,k ∈ 𝒳 ∪ 𝒳 ∗ , and λk ∈ ℂ. Note that for the case nk = 0, we formally define the resulting empty product to be the unit and the corresponding summand is just the scalar λk . Definition 5.2. Let 𝒳 be a nonempty set. A polynomial relation ℛ on 𝒳 is a collection of formal statements of the form 󵄩󵄩 󵄩 󵄩󵄩pj (𝒳 )󵄩󵄩󵄩 ≤ rj ,

j ∈ J,

where J is some index set and for every j ∈ J, rj ≥ 0 is a real constant and pj is a noncommutative ∗-polynomial with variables in 𝒳 . Definition 5.3. Let 𝒳 be a nonempty set and ℛ a polynomial relation on 𝒳 . A representation of (𝒳 | ℛ) is a map π : 𝒳 → A into a C∗ -algebra such that the relation ℛ becomes true in the image of π. For clarification, let us note here that even if the target algebra A is not necessarily unital, the relation given by some polynomial with a scalar part can still be understood and checked within A† . It is a matter of convention how to interpret this condition for the special case A = 0. In these notes we adopt the convenient convention that the trivial map π : 𝒳 → 0 into the zero C∗ -algebra is always declared to be a representation for any polynomial relation ℛ on 𝒳 , which ensures that one never quantifies over the empty set in the subsequent parts of this section. Example. If 𝒳 = {x} and ℛ = {‖x ∗ x − xx ∗ ‖ = 0}, then a map of the form π : 𝒳 → A, x 󳨃→ a is a representation of (𝒳 | ℛ) if and only if a ∈ A is a normal element. One can likewise easily describe in this language when elements are self-adjoint, unitary, or projections.

28 � G. Szabó Definition 5.4. Let 𝒳 be a nonempty set and ℛ a polynomial relation on 𝒳 . Let B be a C∗ -algebra and πu : 𝒳 → B a representation of (𝒳 | ℛ). We call πu universal if the following holds: Whenever π : 𝒳 → A is a representation of (𝒳 | ℛ) into a C∗ -algebra A, there exists a unique ∗-homomorphism φ : B → A such that φ ∘ πu = π. In this case, we write B = C∗ (𝒳 | ℛ) and call it the universal C∗ -algebra generated by (𝒳 | ℛ). Exercise 5.5. Show that if Bi = C∗ (πu(i) (𝒳 )), i = 1, 2, are C∗ -algebras that are each generated by a universal representation of (𝒳 | ℛ) as above, then there is an isomorphism Φ : B1 → B2 with Φ ∘ πu(1) = πu(2) . For clarification, let it be noted that it often makes sense to speak of a unital universal C∗ -algebra as well. Even if the universal C∗ -algebra B generated by (𝒳 | ℛ) (in the above sense) exists and is nonunital, we may still consider B† as the universal unital C∗ -algebra generated by the relation (𝒳 | ℛ). Indeed, one then has B† = C∗ (𝒳 1 | ℛ1 ), ̇ where 𝒳 1 = 𝒳 ∪{1} and one enlarges ℛ to ℛ1 by adding the polynomial relations 1 = 12 = 1∗ and x ⋅ 1 = 1 ⋅ x = x for every x ∈ 𝒳 . Definition 5.6. A polynomial relation ℛ on 𝒳 is said to be bounded, if for every x ∈ 𝒳 , we have 󵄩 󵄩 sup{󵄩󵄩󵄩π(x)󵄩󵄩󵄩 | π : 𝒳 → A representation of (𝒳 | ℛ)} < ∞. Example. The example 𝒳 = {a}, ℛ = {a − a∗ = 0} is not bounded, since self-adjoint elements in C∗ -algebras can have arbitrarily large norms. On the other hand, 𝒳 = {x, y} with ℛ = {1 − x ∗ x − y∗ y = 0} is bounded. Definition 5.7. Let X be a topological space. The density character of X is the smallest cardinality of a dense subset of X. Exercise 5.8. Let κ be a cardinal. Then the isomorphism classes of C∗ -algebras with density character at most κ form a set. Theorem 5.9. Let 𝒳 be a nonempty set and ℛ a polynomial relation on 𝒳 . Then (𝒳 | ℛ) is bounded if and only if C∗ (𝒳 | ℛ) exists. Proof. Suppose that C∗ (𝒳 | ℛ) exists, and let πu : 𝒳 → C∗ (𝒳 | ℛ) be the universal representation. Then for all x ∈ 𝒳 , we get from the universal property that 󵄩 󵄩 sup{󵄩󵄩󵄩π(x)󵄩󵄩󵄩 | π : 𝒳 → A representation of (𝒳 | ℛ)} 󵄩 󵄩 = sup{󵄩󵄩󵄩φ(πu (x))󵄩󵄩󵄩 | φ : C∗ (𝒳 | ℛ) → A ∗-homomorphism} 󵄩 󵄩 ≤ 󵄩󵄩󵄩πu (x)󵄩󵄩󵄩. Hence (𝒳 | ℛ) is bounded. Now suppose conversely that (𝒳 | ℛ) is bounded. We will construct C∗ (𝒳 | ℛ) explicitly. Let κ = max(|𝒳 |, ℵ0 ). This ensures that if π : 𝒳 → B is any map into

Introduction to C∗ -algebras �

29

a C∗ -algebra, then C∗ (π(𝒳 )) has density character at most κ. Since the isomorphism classes of C∗ -algebras with density character at most κ form a set, there exists a set I and a family of representations πi : 𝒳 → Ai of (𝒳 | ℛ) for i ∈ I with the following property. Whenever π : 𝒳 → A is a representation of (𝒳 | ℛ) into any C∗ -algebra, there exists i ∈ I and an embedding ι : Ai → A such that π = ι ∘ πi . More specifically, if we let A′ be a representative of an isomorphism class of a C∗ -algebra with density character at most κ and π ′ : 𝒳 → A′ is any representation of (𝒳 | ℛ), then we declare that (A′ , π ′ ) ∈ I and its associated representation is π ′ . We set6 𝒜 = ∏ Ai i∈I

and πu = ∏ πi : 𝒳 → 𝒜. i∈I

This is well defined as a map because we assumed (𝒳 | ℛ) to be bounded. Note that πu is a representation of (𝒳 | ℛ) because it is a product of representations. Set B = C∗ (πu (x) | x ∈ 𝒳 ) ⊆ 𝒜. We claim that πu : 𝒳 → B is a universal representation for (𝒳 | ℛ). Indeed, if π : 𝒳 → A is any representation, then (by construction) there is i ∈ I and an embedding ι : Ai → A such that π = ι ∘ πi . We may consider pi : B → Ai to be the restriction of the projection map onto the ith component. Then π = ι ∘ πi = ι ∘ pi ∘ πu and ι ∘ pi : B → A is the unique ∗-homomorphism that fits into this equation. In summary, B is the universal C∗ -algebra of (𝒳 | ℛ), and πu is the universal representation. Exercise 5.10. Let I be some index set. Show that the universal C∗ -algebra ∗ A = C∗ (ei,j | i, j ∈ I, ei,j = ej,i , ei,j ek,l = δj,k ei,l )

exists and is isomorphic to the C∗ -algebra of all compact operators on ℓ2 (I). In particular, we obtain such a description for the algebra of n × n matrices, for any n ≥ 2. Here we would associate to ei,j the n × n matric whose components vanish except for a single entry in the ith row and jth column, where the value is 1. Elements obeying the above relation are typically referred to as matrix units. Example 5.11 (Universal group C∗ -algebras). Let Γ be a discrete group. Then C∗u (Γ) = C∗ (ug | g ∈ Γ, ug∗ ug = ug ug∗ = 1, ug uh = ugh ) exists and is called the (full or universal) group C∗ -algebra of Γ. 6 Here we appeal to the direct product construction for C∗ -algebras. We saw in Exercise 1.6 that we can form direct products of Banach algebras. It is easy to check that the direct product of a family of C∗ -algebras is a C∗ -algebra.

30 � G. Szabó Example 5.12 (Toeplitz algebra). The universal C∗ -algebra generated by an isometry 𝒯 = C (s | s s = 1) ∗

∗

exists and is called the Toeplitz algebra. Example 5.13 (Cuntz algebras). Let n ≥ 2. The Cuntz algebra 𝒪n is defined as the universal C∗ -algebra n

𝒪n = C (s1 , . . . , sn | sk sk = 1 for all k = 1, . . . , n, and ∑ sj sj = 1). ∗

∗

∗

j=1

We note that the above relation forces sj∗ sk = 0 for 1 ≤ j ≠ k ≤ n. The Cuntz algebra 𝒪∞ is defined as 𝒪∞ = C (s1 , s2 , s3 , . . . | sj sk = δj,k 1 for all j, k ≥ 1). ∗

∗

Example 5.14 (General C∗ -algebras). Let A be any C∗ -algebra. We consider the collection of free variables 𝒳 A = {Xa | a ∈ A} that comes with a given bijection to A. Then we can encode the ∗-algebraic structure in the form of the polynomial relation ℛA that consists of all statements of the form Xλa+b = λXa + Xb ,

Xa∗ = Xa∗ ,

Xab = Xa Xb ,

λ ∈ ℂ,

a, b ∈ A.

Now let B be any C∗ -algebra. If one simply compares definitions, it is clear that a representation (𝒳 A | ℛA ) → B is nothing but a ∗-homomorphism A → B. The structure result Theorem 2.9 tells us that ℛA is bounded, and from this discussion it becomes clear that we can identify C∗ (𝒳 A | ℛA ) = A. In conclusion, we see that every C∗ -algebra is the universal algebra of some relation, although the previous examples show that the relation is by no means uniquely determined in any way. This also allows us to endow a commonly used shortcut in the literature with a rigorous meaning, whereby one often considers a C∗ -algebra that is universally generated by “a copy of A” along with some extra relations. We shall see a few examples of this form below. Example 5.15 (Amalgamated free products). Let A, B, C be a triple of C∗ -algebras with a pair of ∗-homomorphisms φ : C → A and ψ : C → B. The amalgamated free product associated to this input data is the universal C∗ -algebra A ∗C B generated by a copy of A, a copy of B, and the relation φ(c) = ψ(c) for all c ∈ C. The reader can check by themselves how to elaborate this into a bounded polynomial relation. Example 5.16 (Crossed products). Let Γ be a discrete group and A a unital C∗ -algebra. Consider a group action α : Γ ↷ A by ∗-automorphisms. Then the (full or universal) crossed product C∗ -algebra A ⋊α Γ is the universal C∗ -algebra generated by a unital copy of A and a collection of unitaries {ug | g ∈ Γ} subject to the relation

Introduction to C∗ -algebras �

ugh = ug uh

and

ug aug∗ = αg (a),

a ∈ A,

31

g, h ∈ Γ.

One notices that the full group C∗ -algebra construction becomes a special case of this because C∗u (Γ) = ℂ ⋊ Γ. Given the identical situation to the above with A nonunital, we consider the canonical unitized action α† : Γ ↷ A† and can define the full crossed product A ⋊α Γ as the kernel of the canonical quotient map A† ⋊α† Γ → C∗u (Γ). Exercise 5.17 (Nontrivial!). In the situation of Example 5.14, let κ be the density character of A. Show that there exists a ℚ[i]-∗-subalgebra A0 ⊂ A of cardinality at most κ with the following property. If one considers the free variables X0A = {Xa | a ∈ A0 } and the relation ℛA0 consisting of all statements of the form Xλa+b = λXa + Xb ,

Xa∗ = Xa∗ ,

Xab = Xa Xb ,

λ ∈ ℚ[i],

a, b ∈ A0 ,

then one has C∗ (𝒳0A | ℛA0 ) ≅ A. In particular, every separable C∗ -algebra can be expressed as the universal C∗ -algebra of a countable polynomial relation over countably many variables. Next we will consider the inductive limit construction of C∗ -algebras. Definition 5.18. A (countable) inductive system of C∗ -algebras is a sequence of C∗ -algebras An for n ≥ 1 and a sequence of ∗-homomorphisms φn : An → An+1 . In this context one says that the An are the building blocks and that φn are the connecting maps of the inductive system. For ease of notation one writes φm,n = φn−1 ∘ . . . ∘ φm+1 ∘ φm : Am → An whenever m < n. This ensures φk,n ∘ φm,k = φm,n whenever m < k < n. Theorem 5.19 (Inductive limits). Let φn : An → An+1 be an inductive system of C∗ -algebras. Then there exists a C∗ -algebra A along with ∗-homomorphisms φn,∞ : An → A with φn+1,∞ ∘ φn = φn,∞ satisfying the following universal property: If B is any C∗ -algebra and ψn : An → B are ∗-homomorphisms satisfying ψn+1 ∘φn = ψn for all n ≥ 1, then there exists a unique ∗-homomorphism ψ : A → B satisfying ψ ∘ φn,∞ = ψn for all n ≥ 1. The C∗ -algebra A is unique up to isomorphism and is called the inductive limit of the system {An , φn }. One writes A = lim󳨀→ {An , φn }. The detailed proof of this theorem is omitted here, but can be checked in various textbooks. Although this is by no means the standard way of justification, we leave it as an optional exercise for the readers to convince themselves that Theorem 5.19 can be obtained as a special case of Theorem 5.9, using the basic idea in Example 5.14.

6 The GNS construction In this section we treat another highlight of fundamental C∗ -algebra theory, namely the seminal construction of representations by Gelfand–Naimark–Segal, which associates

32 � G. Szabó to every state on a C∗ -algebra a cyclic representation on a Hilbert space under which the given state becomes a vector state. We note that this section only covers the bare basics of the theory and omits further topics such as those related to irreducible representations. In any case, it is an outcome of this section that every C∗ -algebra in the abstract sense is in fact expressible as a concrete C∗ -algebra of operators on a Hilbert space. Proposition 6.1. Let φ be a state on a C∗ -algebra A. Then Nφ := {x ∈ A | φ(x ∗ x) = 0} is a closed left ideal in A. Moreover, we have φ(y∗ x) = 0 whenever x, y ∈ Nφ . Proof. The last part of the statement follows directly from the Cauchy–Schwarz inequality 󵄨󵄨 ∗ 󵄨󵄨2 ∗ ∗ 󵄨󵄨φ(y x)󵄨󵄨 ≤ φ(x x)φ(y y). Hence, if x, y ∈ Nφ , then φ((x + y)∗ (x + y)) = φ(x ∗ x) + φ(y∗ x) + φ(x ∗ y) + φ(y∗ y) = 0, so we conclude x + y ∈ Nφ . Clearly, Nφ becomes a closed linear subspace. If a ∈ A and x ∈ Nφ , then φ((ax)∗ ax) = φ(x ∗ a∗ ax) ≤ ‖a‖2 φ(x ∗ x) = 0, so a ⋅ x ∈ Nφ . So Nφ is indeed a left ideal. Definition 6.2. Let A be a C∗ -algebra. A (∗-)representation of A is a ∗-homomorphism π : A → ℒ(ℋ), where ℋ is a Hilbert space. We say that π is – faithful, if π is injective; – nondegenerate, if π(A)ℋ = ℋ; – cyclic with respect to a unit vector ξ ∈ ℋ, if π(A)ξ = ℋ. Definition 6.3. Let φ be a state on a C∗ -algebra A. We let Hφ0 = A/Nφ as a vector space

and denote the cosets as [x] = x+Nφ for x ∈ A. By construction, Hφ0 becomes a pre-Hilbert space with inner product ⟨[x] | [y]⟩φ = φ(y∗ x), We let ℋφ = Hφ0

⟨−|−⟩φ

x, y ∈ A.

be the Hilbert space completion of Hφ0 .

Introduction to C∗ -algebras

� 33

Theorem 6.4 (Gelfand–Naimark–Segal). Let φ be a state on a C∗ -algebra A. For every x ∈ A, the map πφ (x) : Hφ0 → Hφ0 ,

πφ (x)[y] = [xy]

is well defined and extends to a bounded linear operator πφ (x) ∈ ℒ(ℋφ ). The resulting assignment πφ : A → ℒ(ℋφ ),

x 󳨃→ πφ (x)

defines a ∗-representation. Moreover, πφ is cyclic with respect to a unit vector ξφ ∈ ℋφ such that ⟨πφ (x)ξφ | ξφ ⟩φ = φ(x),

x ∈ A.

Proof. For convenience, we shall assume that A is unital. Let x ∈ A. If [y1 ] = [y2 ] in Hφ0 , then y1 − y2 ∈ Nφ , and hence xy1 − xy2 ∈ Nφ by Proposition 6.1. So [xy1 ] = [xy2 ] and we get that πφ (x) is a well-defined map on Hφ0 . It is bounded because 󵄩󵄩 󵄩2 󵄩2 ∗ ∗ 2 ∗ 2󵄩 󵄩󵄩[xy]󵄩󵄩󵄩φ = φ(y x xy) ≤ ‖x‖ ⋅ φ(y y) = ‖x‖ 󵄩󵄩󵄩[y]󵄩󵄩󵄩φ . We may conclude ‖πφ (x)‖ ≤ ‖x‖ and thus πφ (x) extends to a bounded operator on ℋφ . The resulting map πφ : A → ℒ(ℋφ ) is clearly linear. It is multiplicative because πφ (ab)[y] = [aby] = πφ (a)[by] = πφ (a)πφ (b)[y] for all a, b, y ∈ A. It is moreover ∗-preserving. We have for all a, x, y ∈ A that ⟨[ax] | [y]⟩φ = φ(y∗ ax) = φ((a∗ y) x) = ⟨[x] | [a∗ y]⟩φ , ∗

which forces πφ (a∗ ) = πφ (a)∗ in ℒ(ℋφ ). In summary, we get that πφ is indeed a ∗-representation. The rest of the claim is proved once we realize that the unit vector ξφ = [1A ] does the job. Definition 6.5. Let A be a C∗ -algebra. Let I be a nonempty index set, and suppose that {πi : A → ℒ(ℋi )}i∈I is a family of ∗-representations of A. Then their direct sum is the ∗-representation π = ⨁ πi : A → ℒ(⨁ ℋi ) i∈I

i∈I

defined by π(a)(ξi )i∈I = (πi (a)ξi )i∈I .

34 � G. Szabó Definition 6.6. Let A be a C∗ -algebra. We define the universal representation of A as π = ⨁ πφ : A → ℒ( ⨁ ℋφ ). φ∈𝒮(A)

φ∈𝒮(A)

Lemma 6.7. Let A be a C∗ -algebra and X ⊂ 𝒮 (A) a set of states. Suppose that for all positive elements a ∈ A \ {0}, there is φ ∈ X with φ(a) > 0. Then the representation ρ = ⨁ πφ : A → ℒ(⨁ ℋφ ) φ∈X

φ∈X

is faithful. Proof. Let x ∈ A \ {0}. Then we may choose φx ∈ X with φx (x ∗ x) > 0. If we let ξφx ∈ ℋφx be the canonical unit vector and consider η = (ηφ )φ∈X ∈ ⨁φ∈X ℋφ via ξφ , φ = φx , ηφ = { x 0, else, then we obtain 󵄩󵄩 󵄩2 󵄩󵄩ρ(x)η󵄩󵄩󵄩 = ⟨ρ(x)η | ρ(x)η⟩ def

= ⟨πφx (x)ξφx | πφx (x)ξφx ⟩

= ⟨πφx (x ∗ x)ξφx | ξφx ⟩

6.4 = φx (x ∗ x) > 0. We conclude ρ(x) ≠ 0.

Corollary 6.8. For every C∗ -algebra A, its universal representation is faithful. In particular, every (abstract) C∗ -algebra can be expressed as a concrete C∗ -algebra. Corollary 6.9. Let A be a separable C∗ -algebra. Then there exists a faithful inclusion A ⊆ ℒ(ℓ2 (ℕ)). Proof. Since A is separable, there exists a sequence {an }n∈ℕ ⊆ A+ with {an } = A+ . For every n ∈ ℕ, we apply Corollary 3.15 and choose φ(n) ∈ 𝒮 (A) with |φ(n) (an )| = φ(n) (an ) = ‖an ‖. We claim that {φ(n) }n∈ℕ = X ⊆ 𝒮 (A) satisfies the assumption in Lemma 6.7. Indeed, k→∞

for any a ∈ A+ \ {0}, there are nk ∈ ℕ such that ank 󳨀→ a, so

‖a‖ = lim ‖ank ‖ = lim φ(nk ) (ank ) = lim φ(nk ) (a) > 0. k→∞

k→∞

k→∞

We conclude that ρ = ⨁ πφ(n) : A → ℒ(⨁ ℋφ(n) ) n∈ℕ

n∈ℕ

Introduction to C∗ -algebras

� 35

is faithful. Furthermore, we know that ℋφ(n) has a dense continuous image of A inside it,

namely Hφ0(n) = A/Nφ(n) . So ℋφ(n) is separable, and so ⨁n∈ℕ ℋφ(n) is separable and infinitedimensional. Hence we get ⨁n∈ℕ ℋφ(n) ≅ ℓ2 (ℕ) by basic Hilbert space theory.

Exercise 6.10. Let A be a C∗ -algebra and n ≥ 2. We consider the complex algebra ∗ Mn (A) of A-valued n × n matrices, which carries the involution (ak,ℓ )∗k,ℓ = (aℓ,k )k,ℓ . Use Corollary 6.8 to show that Mn (A) can be equipped with a norm that turns it into a C∗ -algebra.

7 Selected further topics 7.1 Stably finite C∗ -algebras and traces Definition 7.1. A C∗ -algebra A is called finite if à is a directly finite ring, i. e., if the equation xy = 1 implies yx = 1 for all x, y ∈ A.̃ We call A stably finite if Mn (A) is finite for all n ≥ 1. As we can see, (stable) finiteness of a C∗ -algebra is really a property of its smallest unitization, hence it makes sense to restrict to unital C∗ -algebras from now on. We shall first characterize finiteness of C∗ -algebras in terms of an a priori weaker property which is more commonly used. Proposition 7.2. A unital C∗ -algebra A is finite if and only if for every element x ∈ A, one has that x ∗ x = 1 implies xx ∗ = 1. Proof. The “only if” part is tautological, so assume for the “if” part that every isometry in A is a unitary. Suppose that x, y ∈ A satisfy xy = 1. Then 1 = xyy∗ x ∗ ≤ ‖y‖2 xx ∗ . Since y ≠ 0, it follows that xx ∗ is an invertible positive element. Using functional calculus, we see that x ∗ (xx ∗ )−1/2 is an isometry, so by assumption, we can conclude that it is invertible, which amounts to x being invertible. The equation xy = 1 then necessarily implies yx = 1 by basic ring theory. Definition 7.3. Let A be a C∗ -algebra. A tracial state on A is a state τ : A → ℂ satisfying the trace identity τ(xy) = τ(yx) for all x, y ∈ A. We call τ faithful if τ(a) = 0 implies a = 0 whenever a ≥ 0. Remark 7.4. The trace identity above holds if and only if τ(x ∗ x) = τ(xx ∗ ) for all x ∈ A. For the “if” part, we can observe that this weaker property implies τ(uau∗ ) = τ(a) for all a ≥ 0 and unitary elements u ∈ A. Since positive elements generate the whole C∗ -algebra linearly, we may insert in place of a an element of the form yu and obtain the identity τ(uy) = τ(yu) for all y ∈ A. Since unitary elements also generate the whole C∗ -algebra linearly, the equivalence follows.

36 � G. Szabó Corollary 7.5. Let A be a C∗ -algebra. If there exists a faithful tracial state τ : A → ℂ, then A is stably finite. Proof. Given an element x ∈ A with x ∗ x = 1, we obtain via the tracial identity that τ(1 − xx ∗ ) = 1 − τ(x ∗ x) = 0, which forces xx ∗ = 1 because τ is faithful. This proves that A is finite. We leave it as an exercise to prove that τ extends to a faithful tracial state τ (n) on any matrix algebra Mn (A) for n ≥ 1 via the assignment τ (n) ((ak,ℓ )k,ℓ≤n ) =

1 n ∑ τ(ak,k ). n k=1

The same argument as above yields that A is in fact stably finite. Remark 7.6. Let Γ be a discrete group. Consider ℋ = ℓ2 (Γ) with the standard orthonormal basis {δg }g∈Γ . Then one has two unitary representations of G on ℋ λ, ρ : Γ → 𝒰 (ℓ2 (G)) given by λg (ξ)(h) = ξ(g −1 h) and ρg (ξ)(h) = ξ(hg) for all g, h ∈ Γ and ξ ∈ ℋ. These are called the left-regular representation and the right-regular representation of Γ, respectively. On the standard orthonormal basis vectors one can observe λg (δh ) = δgh and ρg (δh ) = δhg −1 . The (self-inverse) unitary U ∈ 𝒰 (ℋ) given by U(ξ)(h) = ξ(h−1 ) induces a unitary equivalence between λ and ρ. Definition 7.7. Let Γ be a discrete group and keep in mind the unitary representations in the above example. Then C∗λ (Γ) = C∗r (Γ) = C∗ (λg | g ∈ Γ) is called the reduced group C∗ -algebra of Γ. As λ and ρ are unitarily equivalent via U, one has C∗λ (Γ) ≅ C∗ (ρg | g ∈ Γ) via x 󳨃→ UxU. Proposition 7.8. Let Γ be a discrete group. Then τ : C∗λ (Γ) → ℂ given by τ(x) = ⟨xδ1 | δ1 ⟩ is a faithful tracial state. Proof. Since τ arises as the restriction of a vector state, it is clear that τ is a state. In particular it is continuous. Let us first prove that τ is tracial. Since C∗λ (Γ) = span{λg | g ∈ Γ}, it suffices to prove the equality τ(xy) = τ(yx) for x = λg and y = λh for some g, h ∈ Γ. This we can directly compute as 1, gh = 1, τ(λg λh ) = τ(λgh ) = ⟨λgh δ1 | δ1 ⟩ = ⟨δgh | δ1 ⟩ = { 0, gh ≠ 1. Evidently, gh = 1 holds if and only if hg = 1, hence the same computation in reverse gives us τ(λg λh ) = τ(λh λg ).

Introduction to C∗ -algebras �

37

Now let us show that τ is faithful. For this, we observe that by definition of the leftand right-regular representations, their images pairwise commute, i. e., one has λg ρh = ρh λg for all g, h ∈ Γ. For this reason, we may conclude xρh = ρh x for all x ∈ C∗λ (Γ). If we are given x ∈ C∗λ (Γ) and assume τ(x ∗ x) = 0, then we observe for all h ∈ Γ that 0 = τ(x ∗ x) = ⟨x ∗ xδ1 | δ1 ⟩ = ‖xδ1 ‖2

= ‖ρh−1 xδ1 ‖2 = ‖xρh−1 δ1 ‖2 = ‖xδh ‖2 .

In other words, x vanishes on all vectors of an orthonormal basis, which means x = 0. φn

Exercise 7.9. Let {An → An+1 }n≥1 be an inductive system of C∗ -algebras. Show that if An is (stably) finite for all n ≥ 1, then so is the limit A = lim󳨀→ An .

7.2 AF algebras and UHF algebras In this subsection we shall deal with a well-understood but important class of C∗ algebras, which will reappear throughout this book. Definition 7.10. A C∗ -algebra A is said to be approximately finite-dimensional (abbreviated AF), if it can be expressed as some inductive limit A = lim󳨀→ An with a sequence of finite-dimensional C∗ -algebras. If we can demand a limit decomposition where each An is a matrix algebra, then we call A uniformly hyperfinite (abbreviated UHF). Remark 7.11. Clearly, every UHF algebra is an AF algebra. What makes the class of AF algebras rather concrete is the fact that there exists an easy description of what finitedimensional C∗ -algebras look like. In fact, if a C∗ -algebra B is finite-dimensional, then one can write it up to isomorphism as a direct sum of matrix algebras B ≅ Mn1 ⊕ Mn2 ⊕ . . . ⊕ Mnk . Unfortunately, we cannot give a clean proof of this fact here since it arises as a consequence of the theory of irreducible representations, but the interested reader is referred to [2, Theorem III.1.1]. Exercise 7.12. Let X be a compact metric space. Show that 𝒞 (X) is AF if and only if X is totally disconnected. Definition 7.13. Let ℕ denote the union of the natural numbers with {0, ∞}. Let ℙ be the set of prime numbers. A supernatural number is a map n : ℙ → ℕ. We identify n with the formal product ∏p∈ℙ pn(p) , so we see that a supernatural number can be identified with a natural number if and only if n has finite values and evaluates to zero almost everywhere. The product of two supernatural numbers is correspondingly defined as the sum of the maps. We say that n divides another supernatural number m, written n|m, if n(p) ≤ m(p) for all p ∈ ℙ.

38 � G. Szabó We leave the proof of the following statement, which involves playing around with the universal property of inductive limits, as an exercise. Proposition 7.14. Let n be a supernatural number. Let (nk )k≥1 be a sequence of natural numbers with the property that nk |nk+1 for all k ≥ 1, and when identified with a sequence of supernatural numbers, one has nk (p) → n(p) for all p ∈ ℙ. Consider the unital ∗-homomorphism ιk : Mnk → Mnk+1 , which assigns to an nk × nk matrix x the blockn diagonal matrix ιk (x), which repeats x exactly nk+1 times on the diagonal. Then the UHF k algebra Mn := lim󳨀→ {Mnk , ιk } does not depend on the choice of the sequence (nk )k≥1 up to isomorphism. The following historical theorem due to Glimm [4] is a bit deeper and shall only be stated here for context and without proof. Theorem 7.15. The assignment n 󳨃→ Mn from above yields a bijection between the set of supernatural numbers and the isomorphism classes of unital UHF algebras. Example 7.16. We point out two specific examples of UHF algebras with special significance. Firstly, we can consider the universal UHF algebra 𝒬, which is the one belonging to the supernatural number that attains the value ∞ over every prime number. It has the property that it unitally contains every other UHF algebra, which is why it naturally occurs in the context of C∗ -algebras that are matricially finite (MF) or quasidiagonal, which are more advanced concepts subsequently studied in this book. Another historically important UHF algebra is the CAR algebra M2∞ . The abbreviation stands for “canonical anticommutation relation” and stems from an alternative universal description via generators and relations that occurs in the quantum mechanical study of fermionic particles.

7.3 Tensor products In this small section we will discuss how to form tensor products of C∗ -algebras. We assume that the reader is familiar with tensor products of vector spaces (denoted by the ̂ We will always use ⊗ to denote symbol ⊙) and Hilbert spaces (denoted by the symbol ⊗). tensor products of elements. When we talk of forming a tensor product of C∗ -algebras A and B, this means that we aim to equip the algebraic tensor product A ⊙ B with a norm that allows us to complete it to a C∗ -algebra. What makes this theory so intricate is the fact that there is in general no unique norm accomplishing this task, i. e., there is no unambiguous way to construct a tensor product associated to a pair of C∗ -algebras. Our purpose will be to give a short introduction to the two most natural constructions, along with some structural properties of C∗ -algebras that relate to tensor products. For a far more rigorous treatment of this subject matter, the reader is referred to [1].

Introduction to C∗ -algebras

� 39

Remark 7.17. Let A and B be C∗ -algebras. Then their algebraic tensor product A ⊙ B can be endowed with the structure of a ∗-algebra via (a ⊗ b)∗ = a∗ ⊗ b∗ and (a ⊗ b)(c ⊗ d) = ac ⊗ bd. Here is an important toy example where forming the tensor product of C∗ -algebras is straightforward: Exercise 7.18. Let A be a C∗ -algebra and n ≥ 2. Prove that Mn ⊙ A is ∗-isomorphic to Mn (A). Conclude that Mn ⊙ A carries a unique C∗ -norm. We shall use the following basic fact about representations of the algebraic tensor product. Its proof, which is not discussed here, is rather nontrivial in full generality, but the readers can convince themselves as an exercise that the case of unital C∗ -algebras is rather straightforward. Theorem 7.19. Let A and B be C∗ -algebras and let ℋ be a Hilbert space. Let πA : A → ℒ(ℋ) and πB : B → ℒ(ℋ) be two representations with commuting ranges, i. e., πA (a)πB (b) = πB (b)πA (a) for all a ∈ A and b ∈ B. Then π = πA × πB : A ⊙ B → ℒ(ℋ) given by π(a ⊗ b) = πA (a)πB (b) is a ∗-representation. Moreover, every ∗-representation A ⊙ B → ℒ(ℋ) is of this form. Proposition 7.20. Let π : A → ℒ(ℋ1 ) and ρ : B → ℒ(ℋ2 ) be two representations on some Hilbert spaces. Then π ⊗ ρ : A ⊙ B → ℒ(ℋ1 ⊗̂ ℋ2 ) given by (π ⊗ ρ)(a ⊗ b)(ξ1 ⊗ ξ2 ) = (π(a)ξ1 ) ⊗ (ρ(b)ξ2 ) is a ∗-representation. If π and ρ are faithful, then π ⊗ ρ is injective. Proof. We leave it as an exercise to the readers to convince themselves that this construction can be viewed as a special case of Theorem 7.19, so π ⊗ ρ is indeed a ∗-representation. For injectivity, let ℬ be an orthonormal basis of ℋ1 . We consider the collection of bounded linear operators Tμ : ℋ1 ⊗̂ ℋ2 → ℋ2 for μ ∈ ℬ given by Tμ (ξ1 ⊗ ξ2 ) = ⟨ξ1 | μ⟩ξ2 for all ξ1 ∈ ℋ1 and ξ2 ∈ ℋ2 . The adjoint is then given by Tμ∗ (ξ2 ) = μ ⊗ ξ2 . For a given element x ∈ A ⊙ B, we may write x = ∑m k=1 ak ⊗ bk ∈ A ⊙ B such that {bk | k ≤ m} is linearly independent. We observe for all μ ∈ ℬ that we have the following equality of operators in ℒ(ℋ2 ):7 m

Tμ (π ⊗ ρ)(x)Tμ∗ = ∑ Tμ (π ⊗ ρ)(ak ⊗ bk )Tμ∗ k=1 m

= ∑ Tμ (π(ak )μ) ⊗ ρ(bk ) k=1 m

= ∑ ⟨π(ak )μ | μ⟩ ⋅ ρ(bk ). k=1

7 The reader should be careful with the interpretation of the involved expressions. For instance, the tensor product of a vector with an operator gives rise to an operator, so is not to be confused with a vector in a Hilbert space.

40 � G. Szabó Suppose that (π ⊗ ρ)(x) = 0. Then these operators vanish for all μ ∈ ℬ. Since ρ was faithful, the operators ρ(bk ) are also linearly independent, which leads to the equation ⟨π(ak )μ | μ⟩ = 0 for all k ≤ m and μ ∈ ℬ. Hence π(ak ) = 0, and finally ak = 0 due to the faithfulness of π, which leads to x = 0. The observations so far allow us to justify the validity of the following definition: Definition 7.21. Let A and B be two C∗ -algebras. (i) The maximal norm on A ⊙ B is given by ‖x‖max = sup{π(x) | π : A ⊙ B → ℒ(ℋ) is a ∗-representation}. One defines the maximal (or full) tensor product of A and B as the completion A⊗max

B = A ⊙ B max . (ii) The minimal norm on A ⊙ B is given by ‖⋅‖

󵄩 󵄩 ‖x‖min = 󵄩󵄩󵄩(π ⊗ ρ)(x)󵄩󵄩󵄩, where π : A → ℒ(ℋ1 ) and ρ : B → ℒ(ℋ2 ) are faithful representations on Hilbert spaces. One defines the minimal (or spatial) tensor product of A and B as the completion A ⊗min B = A ⊙ B

‖⋅‖min

.

It is unfortunately outside of the scope of these notes to prove that the minimal tensor product norm does not depend on the chosen representations. We notice that Theorem 7.19 implies that the supremum used to define the maximal norm is finite. In fact, m given x = ∑m k=1 ak ⊗bk , we see ‖x‖max ≤ ∑k=1 ‖ak ‖‖bk ‖. Clearly, both maps are seminorms with ‖ ⋅ ‖min ≤ ‖ ⋅ ‖max , but they are in fact norms as a consequence of Proposition 7.20. Due to the fact that every C∗ -algebra can be faithfully represented on a Hilbert space, it is clear that, like the name suggests, ‖ ⋅ ‖max is the largest possible norm on A ⊙ B whose completion is a C∗ -algebra. The analogous statement happens to be true for ‖ ⋅ ‖min , but we will not discuss its proof here. Theorem 7.22 (Takesaki). For any pair of C∗ -algebras A and B, the norm ‖ ⋅ ‖min is the smallest norm on A ⊙ B whose completion is a C∗ -algebra. We are now ready to define nuclearity for C∗ -algebras. Definition 7.23. A C∗ -algebra A is called nuclear, if for every C∗ -algebra B, the maximal and minimal norms agree on A ⊙ B. In other words, there is a unique way to complete A ⊙ B to a C∗ -algebra. In this context we will simply denote it by A ⊗ B. Exercise 7.24. Show that finite-dimensional C∗ -algebras are nuclear. The class of nuclear C∗ -algebras is rather large; see [1, Sections 2.3, 2.4]. In addition to all finite-dimensional ones, it contains all commutative C∗ -algebras. It is closed un-

Introduction to C∗ -algebras �

41

der many constructions such as inductive limits, which allows us to observe that the previously introduced AF algebras are always nuclear. In order to round off the concepts studied in this section so far, let us state the following famous result without proof; see [1, Section 2.6]. Theorem 7.25. Let Γ be a discrete group. The following are equivalent: (i) Γ is amenable. (ii) C∗ (Γ) is nuclear. (iii) C∗r (Γ) is nuclear. (iv) The canonical map C∗ (Γ) → C∗r (Γ) is an isomorphism. In particular, it follows that nonamenable groups, such as the free groups 𝔽n for n ≥ 2, give rise to nonnuclear C∗ -algebras.

Bibliography [1] N. P. Brown and N. Ozawa, C∗ -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2008. xvi+509 pp. [2] K. R. Davidson, C∗ -algebras by example, Fields Institute Monographs, American Mathematical Society, Providence, RI, 1996. xiv+309 pp. [3] J. Dixmier, C∗ -algebras, North-Holland Mathematical Library, vol. 15, Amsterdam–New York–Oxford, 1977. xiii+492 pp. [4] J. Glimm, On a certain class of operator algebras, Trans. Am. Math. Soc. 95 (1960), 318–340. [5] T. A. Loring, Lifting solutions to perturbing problems in C∗ -algebras, Fields Institute Monographs, AMS, Providence, RI, 1997. x+165 pp. [6] G. J. Murphy, C∗ -algebras and operator theory, Academic Press, Inc., Boston, MA, 1990. x+286 pp. [7] G. K. Pedersen, C∗ -algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, London–New York, 1979. ix+416 pp. [8] S. Sakai, C∗ -algebras and W∗ -algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 60, Springer, New York–Heidelberg, 1971. xii+253 pp.

Adrian Ioana

An introduction to von Neumann algebras Abstract: These notes provide a brief introduction to von Neumann algebras. Keywords: von Neumann algebras, spectral theory, types of von Neumann algebras, II1 factor, hyperfinite II1 factor, group von Neumann algebra, group measure space construction, amenability, property (T), property Gamma MSC 2020: 46L10, 46L36

Von Neumann algebras were introduced by von Neumann who developed their theory in a series of joint works with Murray in the 1930–1940s. A von Neumann algebra is a self-adjoint algebra of bounded operators on a Hilbert space which is closed in the weak operator topology. As shown by von Neumann, unital von Neumann algebras admit an entirely algebraic interpretation: they are precisely the commutants of self-adjoint sets of operators. Von Neumann algebras were originally considered in order to formalize quantum mechanics and understand group representations. Unitary representations π : Γ → 𝒰 (H) naturally give rise to von Neumann algebras: the span of π(Γ) is a selfadjoint operator algebra and so its weak operator closure is a von Neumann algebra. When Γ is a countable group and π is its left regular representation, this construction retrieves Murray and von Neumann’s group von Neumann algebra L(Γ). More generally, one can associate a von Neumann algebra to any nonsingular measurable action Γ ↷ (X, μ). These constructions, going back to the 1940s, provide connections between von Neumann algebras, group theory, and ergodic theory which continue to stimulate research in the area. Over the years, the theory of von Neumann algebras has broadened and diversified in remarkable fashion. It is now organized into three main areas (subfactor theory, free probability, deformation/rigidity theory) and has deep connections to many fields of mathematics and physics, including ergodic theory, geometric group theory, logic (model theory, descriptive set theory), random matrices, tensor categories, quantum field theory, and quantum information theory.

Acknowledgement: This text is partially based on lecture notes prepared by the author for a graduate course at UCSD. I am grateful for support from a Simons Fellowship, the NSF (through Grant DMS 2153805, FRG Grant 1854074, Career Grant DMS 1253402, and Grant DMS 1161074) and the Sloan Foundation. I would like to thank Isaac Goldbring for putting this volume together and inviting me to write the present text, Isaac Goldbring and my PhD students Greg Patchell and Hui Tan for providing a number of corrections, and the referee for several comments that helped improve the text. Adrian Ioana, Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA, e-mail: [email protected]; URL: https://mathweb.ucsd.edu/~aioana/ https://doi.org/10.1515/9783110768282-002

44 � A. Ioana

1 Basics of von Neumann algebras This section is devoted to basic notions concerning von Neumann algebras. We introduce the weak and strong operator topologies, define the notion of von Neumann algebras, prove von Neumann’s double commutant theorem and present the realization of L∞ -algebras as von Neumann algebras. Throughout these notes, H will denote a complex Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ξ‖ = √⟨ξ, ξ⟩. We denote by 𝔹(H) the algebra of all bounded linear operators T : H → H. The operator norm of T ∈ 𝔹(H) is given by ‖T‖ = sup ‖Tξ‖ ‖ξ‖≤1

and its adjoint is the unique T ∗ ∈ 𝔹(H) such that ⟨Tξ, η⟩ = ⟨ξ, T ∗ η⟩, ∀ξ, η ∈ H. Exercise 1.1. Let T, S ∈ 𝔹(H). Prove that ‖TS‖ ≤ ‖T‖ ‖S‖, ‖T ∗ ‖ = ‖T‖, and ‖T ∗ T‖ = ‖T‖2 . Definition 1.2. An operator T ∈ 𝔹(H) is called: – self-adjoint (or, hermitian) if T ∗ = T; – a projection if T = T ∗ = T 2 ; – a unitary if T ∗ T = TT ∗ = 1; – an isometry if T ∗ T = 1; – normal if T ∗ T = TT ∗ ; – positive (in symbols, T ≥ 0) if ⟨Tξ, ξ⟩ ≥ 0, ∀ξ ∈ H. Remark 1.3. The set of unitary operators T ∈ 𝔹(H) is a group which is denoted by 𝒰 (H). Remark 1.4. An operator T ∈ 𝔹(H) is positive if and only if T = S ∗ S, for some S ∈ 𝔹(H). Given T1 , T2 ∈ 𝔹(H), we write T1 ≤ T2 to mean that T2 − T1 ≥ 0. Definition 1.5. We endow 𝔹(H) with the following three topologies: – the norm topology: Ti → T if ‖Ti − T‖ → 0; – the strong operator topology (SOT): Ti → T if ‖Ti ξ − Tξ‖ → 0, ∀ξ ∈ H; – the weak operator topology (WOT): Ti → T if |⟨Ti ξ, η⟩ − ⟨Tξ, η⟩| → 0, ∀ξ, η ∈ H. The norm topology is stronger than the SOT, which in turn is stronger than the WOT. Exercise 1.6. Let (Ti )i∈I ⊂ 𝔹(H) be a net such that Ti → T (WOT), for some T ∈ 𝔹(H). For parts (3), (4), and (5) below assume that H is infinite dimensional. (1) Assume that (Ti )i∈I and T are projections. Prove that Ti → T (SOT). (2) Assume that (Ti )i∈I and T are unitaries. Prove that Ti → T (SOT). (3) Give an example of a net of projections (Ti )i∈I converging in the WOT but not the SOT. (4) Give an example of a net of unitaries (Ti )i∈I converging in the WOT but not the SOT. (5) Prove that the closed unit ball {T ∈ 𝔹(H) | ‖T‖ ≤ 1} is WOT but not SOT compact.

An introduction to von Neumann algebras

� 45

While the SOT is strictly stronger than the WOT, we have the following: Proposition 1.7. If C ⊂ 𝔹(H) is a convex set, then C

SOT

=C

WOT

.

WOT

, ξ1 , . . . , ξn ∈ H, and ε > 0. Then D = {(xξ1 , . . . , xξn ) | x ∈ C} is a convex Proof. Let y ∈ C n n subset of H = ⨁i=1 H. By the Hahn–Banach theorem, the weak and norm closures of D coincide (see, e. g., [16, Theorem 1.3.4]). Since (yξ1 , . . . , yξn ) is in the weak closure of D, it is also in its norm closure. Therefore, we can find x ∈ C such that (∑ni=1 ‖xξi −yξi ‖2 )1/2 < ε. This implies that y ∈ C

SOT

. Since the inclusion C

SOT

⊂C

WOT

also holds, we are done.

Definition 1.8. Let H be a complex Hilbert space. – A subalgebra A ⊂ 𝔹(H) is called a ∗-algebra if T ∗ ∈ A, ∀T ∈ A. – A ∗-subalgebra A ⊂ 𝔹(H) is called a (concrete) C∗ -algebra if it closed in the norm topology. – A ∗-subalgebra A ⊂ 𝔹(H) is called a von Neumann algebra if it is WOT-closed. Definition 1.9. A map π : A → B between two C∗ -algebras A and B is called a ∗-homomorphism if it is linear, multiplicative, and ∗-preserving (i. e., π(a∗ ) = π(a)∗ , ∀a ∈ A). A bijective ∗-homomorphism is called a ∗-isomorphism. A ∗-homomorphism π : A → 𝔹(H), for some complex Hilbert space H, is called a representation of A. Exercise 1.10. Let 𝒮 ⊂ 𝔹(H) be a set which is ∗-closed (i. e., T ∗ ∈ 𝒮 , ∀T ∈ 𝒮 ). Prove that the commutant of 𝒮 , defined as 𝒮 ′ = {T ∈ 𝔹(H) | TS = ST, ∀S ∈ 𝒮 }, is a von Neumann algebra. Conversely, the next fundamental result shows that every von Neumann algebra arises this way: Theorem 1.11 (von Neumann’s double commutant theorem, [31]). If M ⊂ 𝔹(H) is a unital ∗-subalgebra, then the following three conditions are equivalent: (1) M is WOT-closed. (2) M is SOT-closed. (3) M = M ′′ := (M ′ )′ . tor.

Here, we say that a subalgebra M ⊂ 𝔹(H) is unital if it contains the identity opera-

This beautiful result asserts that, for unital ∗-algebras, the analytic condition of being closed in the WOT is equivalent to the algebraic condition of being equal to their double commutant. Proof. It is clear that (3) ⇒ (1) by Exercise 1.10 and that (1) ⇔ (2) by Proposition 1.7. To prove that (2) ⇒ (3), it suffices to show that if x ∈ M ′′ , ε > 0, and ξ1 , . . . , ξn ∈ H, then there exists y ∈ M such that ‖xξi − yξi ‖ < ε, for all i = 1, . . . , n. We claim that if p is the orthogonal projection onto an M-invariant closed subspace K ⊂ H, then p ∈ M ′ . To see this, let x ∈ M. Then (1 − p)xpξ ∈ (1 − p)(K) = {0}, for all

46 � A. Ioana ξ ∈ H. Hence (1 − p)xp = 0 and so xp = pxp. By taking adjoints, we get that px ∗ = px ∗ p and hence px = pxp, for all x ∈ M. This shows that p commutes with x, as claimed. Next, assume first that n = 1 and let p be the orthogonal projection onto Mξ1 = {xξ1 | x ∈ M}. Since Mξ1 is M-invariant, our claim gives p ∈ M ′ . Thus, xp = px and xξ1 = xpξ1 = pxξ1 ∈ Mξ1 . Therefore, there is y ∈ M such that ‖xξ1 − yξ1 ‖ < ε. In general, we use a “matrix trick.” Let H n = ⨁ni=1 H and identify 𝔹(H n ) = 𝕄n (𝔹(H)). Let π : M → 𝔹(H n ) be the “diagonal” ∗-homomorphism given by π(x)(ξ1 ⊕ ⋅ ⋅ ⋅ ⊕ ξn ) = xξ1 ⊕ ⋅ ⋅ ⋅ ⊕ xξn . If x ∈ M ′′ , then Exercise 1.12 below gives that π(x) ∈ π(M)′′ . Let ξ = (ξ1 , . . . , ξn ) ∈ H n . By applying the case n = 1, we conclude that there is y ∈ M such that ‖π(x)ξ − π(y)ξ‖ < ε. Since ‖π(x)ξ − π(y)ξ‖2 = ∑ni=1 ‖xξi − yξi ‖2 , we are done. Exercise 1.12. Prove that π(M ′′ ) ⊂ 𝕄n (M ′ )′ and π(M)′ ⊂ 𝕄n (M ′ ). Definition 1.13. A probability space (X, μ) is called standard if X is a Polish space and μ is a Borel probability measure on X. Proposition 1.14. Let (X, μ) be a standard probability space. Define π : L∞ (X, μ) → 𝔹(L2 (X, μ)) by letting πf (ξ) = fξ, for all f ∈ L∞ (X) and ξ ∈ L2 (X). Then π(L∞ (X))′ = π(L∞ (X)). Therefore, π(L∞ (X)) ⊂ 𝔹(L2 (X)) is a maximal abelian von Neumann subalgebra. Proof. Let T ∈ π(L∞ (X))′ and put g = T(1). Then fg = πf T(1) = Tπf (1) = T(f ) and hence 󵄩 󵄩 ‖fg‖2 = 󵄩󵄩󵄩T(f )󵄩󵄩󵄩2 ⩽ ‖T‖ ‖f ‖2 ,

∀f ∈ L∞ (X).

(1.1)

Let ε > 0 and f = 1{x∈X : |g(x)|⩾‖T‖+ε} . Then it is clear that ‖fg‖2 ⩾ (‖T‖ + ε)‖f ‖2 . In combination with inequality (1.1), we get that (‖T‖ + ε)‖f ‖2 ⩽ ‖T‖‖f ‖2 , and so f = 0, almost everywhere. Thus, we conclude that g ∈ L∞ (X). Since T(f ) = fg = πg (f ), for all f ∈ L∞ (X), and L∞ (X) is ‖ ⋅ ‖2 -dense in L2 (X), it follows that T = πg ∈ L∞ (X). This proves that π(L∞ (X))′ = π(L∞ (X)). If A ⊂ 𝔹(L2 (X)) is an abelian algebra which contains π(L∞ (X)), then A commutes with π(L∞ (X)). The previous paragraph implies that A ⊂ π(L∞ (X)) which proves that A = π(L∞ (X)). Exercise 1.15. Let I be a set. Define π : ℓ∞ (I) → 𝔹(ℓ2 (I)) by letting πf (ξ) = fξ, for all f ∈ ℓ∞ (I) and ξ ∈ ℓ2 (I). Prove that π(ℓ∞ (I))′ = π(ℓ∞ (I)). Therefore, π(ℓ∞ (I)) ⊂ 𝔹(ℓ2 (I)) is a maximal abelian von Neumann subalgebra.

2 The spectral theorem The spectral theorem for normal matrices a ∈ 𝕄n (C) implies that a = ∑z∈σ(a) zpz , where pz is the orthogonal projection onto the eigenspace corresponding an eigenvalue z ∈ σ(a). Given Δ ⊂ σ(a), put E(Δ) = ∑z∈Δ pz . Thus, informally, we have that a =

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“ ∫σ(a) z dE(z)”. The spectral theorem proves such a statement for normal operators a on possibly infinite-dimensional Hilbert spaces, where E is a so-called spectral measure defined on the spectrum of a. We start this section by recalling several fundamental facts concerning C∗ -algebras. We then discuss the expression of representations of abelian C∗ -algebras in terms of spectral measures, and use this to derive the spectral theorem for normal operators and classify abelian von Neumann algebras.

2.1 C∗ -algebras For proofs of the facts presented below, we refer the reader to the introduction to C∗ -algebras in this volume [28] or [11, Chapter 1]. Definition 2.1. A C∗ -algebra is a Banach algebra (A, ‖ ⋅ ‖) together with an adjoint operation ∗ : A → A such that ∀a, b ∈ A and λ ∈ ℂ we have (a + b)∗ = a∗ + b∗ ,

(λa)∗ = λa∗ ,

∗

(a∗ ) = a,

(ab)∗ = b∗ a∗

and

󵄩󵄩 ∗ 󵄩󵄩 2 󵄩󵄩a a󵄩󵄩 = ‖a‖ .

Such C∗ -algebras are called abstract because, in contrast to concrete C∗ -algebras, they are not a priori represented on a Hilbert space. However, as we will see in Theorem 2.8, any abstract C∗ -algebra is isomorphic to a concrete one. If A is a unital C∗ -algebra and a ∈ A, then the spectrum of a is a nonempty compact subset of ℂ defined by σ(a) = {λ ∈ ℂ | λ ⋅ 1 − a is not invertible}. Theorem 2.2. Let A and B be unital C∗ -algebras. Then any unital ∗-homomorphism π : A → B is contractive: ‖π(a)‖ ≤ ‖a‖, ∀a ∈ A. If π is injective, then it is isometric: ‖π(a)‖ = ‖a‖, ∀a ∈ A. In particular, any ∗-isomorphism π : A → B is automatically isometric. Definition 2.3. Let A a unital C∗ -algebra. A linear functional φ : A → ℂ is called positive if φ(a∗ a) ≥ 0, ∀a ∈ A. A positive linear functional φ : A → ℂ is called a state if φ(1) = 1 and faithful if φ(a∗ a) = 0 ⇒ a = 0. Exercise 2.4. Let A be a unital C∗ -algebra and φ : A → ℂ be a positive linear functional. (1) (The Cauchy–Schwarz inequality) Prove that |φ(y∗ x)|2 ≤ φ(x ∗ x)φ(y∗ y), ∀x, y ∈ A. (2) Prove that φ is bounded and ‖φ‖ = φ(1). Exercise 2.5. Let X be a compact Hausdorff space. Prove that C(X) = {f : X → ℂ continuous function} is an abstract C∗ -algebra, with the norm ‖f ‖∞ = supx∈X |f (x)| and adjoint f ∗ (x) = f (x). The following result shows that every abstract unital abelian C∗ -algebra A arises ̂ the set of nonzero homomorphisms φ : A → ℂ. Then A ̂ ⊂ {φ ∈ this way. We denote by A

48 � A. Ioana ̂ is a compact Hausdorff space with respect to the weak∗ -topology A∗ | ‖φ‖ = 1} and A ∗ inherited from A . Here, A∗ denotes the dual Banach space of A, which consists of all bounded linear functionals φ : A → ℂ. Theorem 2.6 (Gelfand–Naimark). Let A be a unital abelian C∗ -algebra. Then the Gelfand ̂ given by â(φ) = φ(a), ∀a ∈ A, φ ∈ A, ̂ is a ∗-isomorphism. transform ∧ : A → C(A) Theorem 2.6 implies the following: Theorem 2.7 (Continuous functional calculus). Let A be a unital C∗ -algebra and a ∈ A be a normal element. Then there exists an isometric unital ∗-homomorphism C(σ(a)) ∋ f 󳨃→ f (a) ∈ A which maps the identity function on σ(a) to a. By Exercise 1.1, any concrete C∗ -algebra is an abstract C∗ -algebra. The converse is also true: Theorem 2.8 (Gelfand–Naimark–Segal). Every abstract C∗ -algebra is ∗-isomorphic to a concrete C∗ -algebra.

2.2 Representations of abelian C∗ -algebras Let A ⊂ 𝔹(H) be a concrete abelian C∗ -algebra (e. g., the C∗ -algebra generated by a nor̂ This result, however, does mal operator). By Theorem 2.6, A is ∗-isomorphic to C(A). not explain how A “acts” on H. The next theorem gives a description of all representations of C(X), where X is a compact Hausdorff space. We denote by ℬ the σ-algebra of Borel subsets of X, by B(X) the C∗ -algebra of bounded Borel functions f : X → ℂ, and by ℳ(X) the space of complex-valued regular measures on X endowed with the norm ‖μ‖ = sup{∫X f dμ | f ∈ C(X), ‖f ‖∞ ≤ 1}. In this and the next subsection, we follow the presentation from [11, Sections 2.9 and 2.10]. Theorem 2.9. Let π : C(X) → 𝔹(H) be a ∗-homomorphism. Then there exists a spectral measure E : ℬ → 𝔹(H) such that π(f ) = ∫ f dE,

∀f ∈ C(X).

X

Definition 2.10. A spectral measure for (X, ℬ) is a map E : ℬ → 𝔹(H) that satisfies the following: (1) E(Δ) is a projection, ∀Δ ∈ ℬ. (2) E(0) = 0 and E(X) = 1. (3) E(Δ1 ∩ Δ2 ) = E(Δ1 )E(Δ2 ), ∀Δ1 , Δ2 ∈ ℬ. (4) The map ℬ ∋ Δ 󳨃→ Eξ,η (Δ) := ⟨E(Δ)ξ, η⟩ belongs to ℳ(X), ∀ξ, η ∈ H.

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Lemma 2.11. Let E : ℬ → 𝔹(H) be a spectral measure. If ξ, η ∈ H, then ‖Eξ,η ‖ ⩽ ‖ξ‖ ‖η‖. Proof. Let Δ1 , . . . , Δn ∈ ℬ be pairwise disjoint sets. Let αi ∈ 𝕋 such that |Eξ,η (Δi )| = αi Eξ,η (Δi ). Then ∑ni=1 |Eξ,η (Δi )| = ⟨∑ni=1 αi E(Δi )ξ, η⟩ ⩽ ‖ ∑ni=1 αi E(Δi )ξ‖ ‖η‖. Since we also have that 󵄩󵄩2 󵄩󵄩 n 󵄩 󵄩󵄩 󵄩󵄩∑ αi E(Δi )ξ 󵄩󵄩󵄩 = ∑⟨E(Δi )ξ, ξ⟩ = ⟨E(∪n Δi )ξ, ξ⟩ ⩽ ‖ξ‖2 , i=1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩i=1 i=1 we conclude that ∑ni=1 |Eξ,η (Δi )| ⩽ ‖ξ‖ ‖η‖, as desired. Lemma 2.12. Let E : ℬ → 𝔹(H) be a spectral measure. Then for every f ∈ B(X), there exists an operator π(f ) ∈ 𝔹(H) such that ‖π(f )‖ ⩽ ‖f ‖∞ and ⟨π(f )ξ, η⟩ = ∫X f dEξ,η , for all ξ, η ∈ H. Moreover, the map π : B(X) → 𝔹(H) is a ∗-homomorphism. Proof. Let f ∈ B(X). Since the map H × H ∋ (ξ, η) → ∫X f dEξ,η is sesquilinear and satisfies | ∫X f dEξ,η | ⩽ ‖f ‖∞ ‖Eξ,η ‖ ⩽ ‖f ‖∞ ‖ξ‖‖η‖, the existence of π(f ) is a consequence of Riesz’s representation theorem. Secondly, ⟨π(1Δ )ξ, η⟩ = ∫X 1Δ dEξ,η = Eξ,η (Δ) = ⟨E(Δ)ξ, η⟩ and so π(1Δ ) = E(Δ). We get that π(1Δ1 ∩Δ2 ) = π(1Δ1 )π(1Δ2 ), for every Δ1 , Δ2 ∈ ℬ. Thus, π(f1 f2 ) = π(f1 )π(f2 ), for simple functions f1 , f2 ∈ B(X). Since ‖π(f )‖ ⩽ ‖f ‖∞ , for all f ∈ B(X), approximating bounded Borel functions by simple functions gives that π is multiplicative. It follows that π is a ∗-homomorphism. Before proving the spectral theorem, we need one additional result. Lemma 2.13. Let π : C(X) → 𝔹(H) be a ∗-homomorphism. Then there exists a ∗-homomorphism π̃ : B(X) → 𝔹(H) such that π̃ |C(X) = π. Moreover, if f ∈ B(X) and (fi ) ⊂ B(X) is ̃ i ) → π(f ̃ ) in the WOT. a net such that ∫X fi dμ → ∫X f dμ, for every μ ∈ ℳ(X), then π(f Proof. Let ξ, η ∈ H. Note that C(X) ∋ f → ⟨π(f )ξ, η⟩ ∈ ℂ is a linear functional such that |⟨π(f )ξ, η⟩| ⩽ ‖π(f )‖ ‖ξ‖ ‖η‖ ⩽ ‖f ‖∞ ‖ξ‖ ‖η‖. Riesz’s representation theorem implies that there exists μξ,η ∈ ℳ(X) such that ∫X f dμξ,η = ⟨π(f )ξ, η⟩, for all f ∈ C(X), and ‖μξ,η ‖ ⩽ ‖ξ‖ ‖η‖. Note that μξ,η = μη,ξ , so the map (ξ, η) → μξ,η is sesquilinear. Next, let f ∈ B(X). Repeating the argument from the proof of Lemma 2.12 shows that ̃ ) ∈ 𝔹(H) such that ‖π(f ̃ )‖ ⩽ ‖f ‖∞ and ⟨π(f ̃ )ξ, η⟩ = ∫ f dμξ,η , there exists an operator π(f X ̃ ) = π(f ), if f ∈ C(X). It is also easy to see that π̃ is linear for all ξ, η ∈ H. It is clear that π(f and ∗-preserving, so it remains to argue that π̃ is multiplicative. Let f ∈ B(X) and g ∈ C(X). Then we can find a net (fi ) ⊂ C(X) such that ‖fi ‖∞ ⩽ ‖f ‖∞ , for all i, and ∫X fi dμ → ∫X f dμ, for every μ ∈ ℳ(X) (see [11, Lemma 9.7]). Since ̃ )ξ, η⟩, for all μξ,η ∈ ℳ(X), it follows that ⟨π(fi )ξ, η⟩ = ∫X fi dμξ,η → ∫X f dμξ,η = ⟨π(f ̃ ) in the WOT. Similarly, π(fi g) → π(fg) ̃ ξ, η ∈ H. Thus, π(fi ) → π(f in the WOT. Since ̃ ̃ )π(g), for all f ∈ B(X) and g ∈ C(X). π(fi g) = π(fi )π(g), for all i, we deduce that π(fg) = π(f Finally, let f , g ∈ B(X). By approximating g with continuous functions as above and ̃ ̃ )π(g). ̃ using the last identity, it follows similarly that π(fg) = π(f Thus, π̃ is multiplicative.

50 � A. Ioana For the moreover assertion, let f , fi ∈ B(X) as in the hypothesis. Then for every ξ, η ∈ ̃ i )ξ, η⟩ = ∫ fi dμξ,η → ∫ f dμξ,η = ⟨π(f ̃ )ξ, η⟩. Therefore, π(f ̃ i ) → π(f ̃ ) H we have that ⟨π(f X X in the WOT. We are now ready to sketch the proof of the spectral theorem, leaving some details to the reader. Proof of Theorem 2.9. By Lemma 2.13, π extends to a ∗-homomorphism π̃ : B(X) → ̃ Δ ). Then one checks that E is a spectral 𝔹(H). Define E : ℬ → 𝔹(H) by letting E(Δ) = π(1 measure. By Lemma 2.12, ρ : B(X) → 𝔹(H) given by ρ(f ) = ∫X f dE is a ∗-homomorphism. ̃ Δ ), for every Δ ∈ ℬ. Consequently, ρ(f ) = π(f ̃ ), Then we have ρ(1Δ ) = ∫X 1Δ dE = E(Δ) = π(1 for every simple function f ∈ B(X). Since simple functions are ‖ ⋅ ‖∞ -dense in B(X) and ρ, π̃ are contractive, we get that ρ = π.̃ In particular, π(f ) = ∫X f dE, for every f ∈ C(X). This finishes the proof.

2.3 The spectral theorem Theorem 2.14. Let a ∈ 𝔹(H) be a normal operator and ℬ the σ-algebra of Borel subsets of σ(a). (1) (The spectral theorem) There is a spectral measure E : ℬ → 𝔹(H) such that a = ∫σ(a) z dE. (2) (Borel functional calculus) The map B(σ(a)) ∋ f → f (a) := ∫σ(a) f (z) dE ∈ 𝔹(H) is a ∗-homomorphism. Moreover, if f ∈ B(σ(a)) and (fi ) ⊂ B(σ(a)) is a net such that ∫σ(a) fi dμ → ∫σ(a) f dμ, for every μ ∈ ℳ(σ(a)), then fi (a) → f (a) in the WOT.

Proof. By Theorem 2.7, there exists a ∗-homomorphism π : C(σ(a)) → 𝔹(H) such that π(z) = a. The conclusion now follows directly from Theorem 2.9. Corollary 2.15. Let M ⊂ 𝔹(H) be a von Neumann algebra. (1) If a ∈ M is normal, then f (a) ∈ M, for every f ∈ B(σ(a)). (2) M is equal to the norm closure of the linear span of its projections. Proof. (1) Let f ∈ B(σ(a)). Let fi ∈ C(σ(a)) be a net such that ‖fi ‖∞ ⩽ ‖f ‖∞ , for all i, and ∫σ(a) fi dμ → ∫σ(a) f dμ, for every μ ∈ ℳ(σ(a)). By Theorem 2.14, we have that fi (a) → f (a) in the WOT. Since fi (a) ∈ C ∗ (a) ⊂ M, we conclude that f (a) ∈ M. (2) If a ∈ M, then we can write a = b+ic,where b, c ∈ M are self-adjoint. So it suffices to show that any self-adjoint a ∈ M belongs to the norm closure of the linear span of projections of M. To this end, let ε > 0 and write a = ∫σ(a) z dE. Then we can find α1 , . . . , αn ∈

ℝ and Borel sets Δ1 , . . . , Δn ⊂ σ(a) such that ‖z − ∑ni=1 αi 1Δi ‖∞ ⩽ ε. It follows that ‖a − ∑ni=1 αi 1Δi (a)‖ ⩽ ε. Since the projections 1Δi (a) belong to M by part (1), we are done.

Exercise 2.16. Let M ⊂ 𝔹(H) be a von Neumann algebra and a ∈ M with a ⩾ 0. Prove ∞ −n that there exist projections {pn }∞ n=1 ⊂ M such that a = ‖a‖ ∑n=1 2 pn .

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2.4 Abelian von Neumann algebras By Theorem 1.14, L∞ (X) is a von Neumann algebra, for any standard probability space (X, μ). Conversely, we have: Theorem 2.17. Let H be a separable Hilbert space and M ⊂ 𝔹(H) be an abelian von Neumann algebra. Then M is ∗-isomorphic to L∞ (X), where (X, μ) is a standard probability space. For a von Neumann algebra M, we denote by (M)1 = {x ∈ M | ‖x‖ ≤ 1} its closed unit ball and by M+ = {x ∈ M | x ≥ 0} the set of its positive elements. Proof. For simplicity, we only prove this theorem under the additional assumption that there is a vector (called M-cyclic) ξ ∈ H such that Mξ = H. Since H is separable, (𝔹(H)1 , WOT) is a compact metrizable space (see Exercise 1.6(5)) and hence so is ((M)1 , WOT). Let {an } ⊂ (M)1 be a WOT-dense sequence. Let A be the ̂ is compact and C∗ -algebra generated by {an }. Then A is SOT-dense in M and X = A ∞ 1 ′ ′ metrizable. Specifically, we have that d(φ, φ ) = ∑n=1 2n |φ(an ) − φ (an )|, for φ, φ′ ∈ X, defines a compatible metric on X. Let π : C(X) → A ⊂ 𝔹(H) be the inverse of the Gelfand transform (see Theorem 2.6). By applying Theorem 2.9, we get a spectral measure E on X such that π(f ) = ∫X f dE, for every f ∈ C(X). Then μ(Δ) = ⟨E(Δ)ξ, ξ⟩ defines a measure μ ∈ ℳ(X) such that ∫X f dμ = ⟨π(f )ξ, ξ⟩, for every f ∈ C(X). Thus, for every f ∈ C(X) we get that 󵄩󵄩 󵄩2 ∗ 2 2 󵄩󵄩π(f )ξ 󵄩󵄩󵄩 = ⟨π(f )ξ, π(f )ξ⟩ = ⟨π(f ) π(f )ξ, ξ⟩ = ⟨π(|f | )ξ, ξ⟩ = ∫ |f | dμ, X

so ‖π(f )ξ‖ = ‖f ‖L2 (X) . As π(C(X)) = A is SOT-dense in M, {π(f )ξ | f ∈ C(X)} is dense in Mξ = H. Since C(X) is dense in L2 (X), we can define a unitary operator U : L2 (X) → H by U(f ) = π(f )ξ,

∀f ∈ C(X).

Let ρ : L∞ (X) → 𝔹(L2 (X)) be the ∗-homomorphism given by ρf (η) = fη. Then for all f , g ∈ C(X) we have Uρf (g) = U(fg) = π(fg)ξ = π(f )π(g)ξ = π(f )U(g). As C(X) is dense in L2 (X), we get that Uρf = π(f )U and thus π(f ) = Uρf U ∗ , for all f ∈ C(X). Hence, π(C(X)) = Uρ(C(X))U ∗ . Since ρ(C(X)) is WOT-dense in L∞ (X), we conclude that M = UL∞ (X)U ∗ . We recall the isomorphism theorem for standard probability spaces (see [18, Theorem 17.41]). An isomorphism between two standard probability spaces (X, μ) and (Y , ν) is a Borel isomorphism θ : X → Y (i. e., a bijection such that θ and θ−1 are Borel maps) such that θ∗ μ = ν, where θ∗ μ is the Borel probability measure on Y given by θ∗ μ(Z) = μ(θ−1 (Z)), for every Borel set Z ⊂ Y .

52 � A. Ioana Theorem 2.18. Let (X, μ) be a standard probability space. Assume that μ is nonatomic, i. e., μ({x}) = 0, for every x ∈ X. Then (X, μ) is isomorphic to ([0, 1], λ), where λ is the Lebesgue measure on [0, 1]. Definition 2.19. Let M be a von Neumann algebra. A projection p ∈ M is called minimal if every projection q ∈ M such that 0 ≤ q ≤ p satisfies q ∈ {0, p}. A von Neumann algebra M is called diffuse if it has no nonzero minimal projections. Corollary 2.20. Let H be a separable Hilbert space and M ⊂ 𝔹(H) be a diffuse abelian von Neumann algebra. Then M is ∗-isomorphic to L∞ ([0, 1], λ). Proof. By Theorem 2.17, M is ∗-isomorphic to L∞ (X), where (X, μ) is a standard probability space. Since M is diffuse, μ is nonatomic. Otherwise, if μ({x}) > 0, for x ∈ X, then 1{x} ∈ L∞ (X) would be a nonzero minimal projection. Theorem 2.18 thus implies the conclusion. Exercise 2.21. Let M be a von Neumann algebra and p ∈ M be a projection. Prove that p is minimal if and only if pMp = ℂp. Exercise 2.22. Let M be a diffuse von Neumann algebra. Prove that any maximal abelian von Neumann subalgebra A ⊂ M is diffuse. Hence, deduce that M contains a copy of L∞ ([0, 1], λ). Further, use this fact to conclude that there exists a sequence (uk ) ⊂ 𝒰 (M) such that uk → 0, in the weak operator topology. Exercise 2.23. Let M be a finite-dimensional abelian von Neumann algebra. Prove that M is ∗-isomorphic to ℓ∞ ({1, . . . , n}), for some n ∈ ℕ.

3 Decomposition into types for von Neumann algebras This section is devoted to the type decomposition for von Neumann algebras. By Corollary 2.15, any von Neumann algebra M ⊂ 𝔹(H) is generated by projections. To better understand the structure of M, it will be important to “compare” its projections. If H is finite dimensional, then the projections of M can be ordered using the dimension of their range space. A main goal of this section is to define a way to compare projections, in the absence of a suitable notion of dimension.

3.1 The polar decomposition We start this section by discussing the polar decomposition for bounded operators a ∈ 𝔹(H). This is an analogue of the decomposition of a complex number as the product of a number of absolute value 1 and its absolute value. The absolute value of a is given

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1

by |a| = (a∗ a) 2 . Thus, we would like to write a = v|a|, where v satisfies |v| = 1, i. e., is an isometry. It turns out that this is true, if we allow v to be a partial isometry, in the following sense: Definition 3.1. An operator v ∈ 𝔹(H) is called a partial isometry if ‖v(ξ)‖ = ‖ξ‖, ∀ξ ∈ (ker v)⊥ . In this case, (ker v)⊥ and the range ran(v) = vH are called the initial and final space of v, respectively. The orthogonal complement of a closed subspace K ⊂ H is K ⊥ = {ξ ∈ H | ⟨ξ, η⟩ = 0, ∀η ∈ K}. Exercise 3.2. Prove that v ∈ 𝔹(H) is a partial isometry if and only if v∗ v is a projection. Theorem 3.3 (Polar decomposition). If a ∈ 𝔹(H), then there exists a unique partial isometry v ∈ 𝔹(H) with initial space (ker a)⊥ and final space ran(a) such that a = v|a|, where 1 |a| = (a∗ a) 2 . Proof. If ξ ∈ H, then ‖aξ‖2 = ⟨aξ, aξ⟩ = ⟨a∗ aξ, ξ⟩ = ⟨|a|2 ξ, ξ⟩ = ‖ |a|ξ‖2 . Then the formula v(|a|ξ) = aξ defines a unitary operator v : ran(|a|) → ran(a). We extend v to H by letting v(η) = 0, for all η ∈ (ran(|a|))⊥ . Then v is a partial isometry such that v|a| = a. By definition, the final space of v is ran(a), while the initial space of v is ran(|a|) = (ker |a|)⊥ = (ker a)⊥ (where the second equality follows from the first line of the proof). The uniqueness of v is obvious. Exercise 3.4. Let M ⊂ 𝔹(H) be a von Neumann algebra and a ∈ M. Let v be the partial isometry provided by Theorem 3.3. Define l(a) to be the projection onto ran(a) (the left support of a) and r(a) to be the projection onto (ker a)⊥ (the right support of a). (1) Prove that v commutes with every unitary element u ∈ M ′ and deduce that v ∈ M. (2) Prove that l(a) = vv∗ and r(a) = v∗ v, and use (1) to deduce that l(a), r(a) ∈ M.

3.2 Projections For a von Neumann algebra M, we denote by 𝒫 (M) the set of its projections and by 𝒰 (M) the group of its unitaries. Definition 3.5. Let {pi }i∈I ∈ 𝔹(H) be a family of projections. We denote by – ⋁i∈I pi the smallest projection p ∈ 𝔹(H) such that p ⩾ pi , ∀i ∈ I; – ⋀i∈I pi the largest projection p ∈ 𝔹(H) such that p ⩽ pi , ∀i ∈ I. Proposition 3.6. If pi ∈ 𝒫 (M), ∀i ∈ I, then ⋁i∈I pi , ⋀i∈I pi ∈ M. For a proof of this result, see [1, Proposition 2.4.2]. We next use Proposition 3.6 to establish that every von Neumann algebra has a unit.

54 � A. Ioana Corollary 3.7. Let M be a von Neumann algebra and define p = ⋁q∈𝒫(M) q. Then p ∈ 𝒫 (M) is a multiplicative unit of M, i. e., a = pa = ap, ∀a ∈ M. Proof. By Proposition 3.6, we have that p ∈ M. If a ∈ M, then l(a), r(a) ∈ 𝒫 (M) by Exercise 3.4 and thus l(a), r(a) ≤ p. Since a = l(a)a = ar(a), we get that a = pa = ap. Definition 3.8. Let M ⊂ 𝔹(H) be a unital von Neumann algebra. – 𝒵 (M) = M ∩ M ′ is called the center of M; – M is called a factor if 𝒵 (M) = ℂ1; – the central support of p ∈ 𝒫 (M) is the smallest projection z(p) ∈ 𝒵 (M) with p ⩽ z(p). Lemma 3.9. Projection z(p) is the orthogonal projection onto MpH. Proof. Let z be an orthogonal projection onto MpH. Since pH ⊂ MpH, we have that p ⩽ z. Since MpH is both M and M ′ invariant, we get that z ∈ M ′ ∩ (M ′ )′ = 𝒵 (M). Finally, since p = z(p)p we have MpH = Mz(p)pH = z(p)MpH ⊂ z(p)H and hence z ⩽ z(p). Altogether, z = z(p). Exercise 3.10. Prove that z(p) = ⋁u∈𝒰 (M) upu∗ . Proposition 3.11. Let M ⊂ 𝔹(H) be a von Neumann algebra. Let p ∈ 𝒫 (M) and p′ ∈ 𝒫 (M ′ ). We denote pMp = {pxp | x ∈ M} and Mp′ = {xp′ | x ∈ M} and view them as algebras of operators on the Hilbert spaces pH and p′ H, respectively. Then we have the following: (1) Mp′ ⊂ 𝔹(p′ H) is a von Neumann algebra and (Mp′ )′ = p′ M ′ p′ . (2) pMp ⊂ 𝔹(pH) is a von Neumann algebra and (pMp)′ = M ′ p. (3) 𝒵 (Mp′ ) = 𝒵 (M)p′ and 𝒵 (pMp) = 𝒵 (M)p. For a proof of this result, see, e. g., [11, Proposition 43.8].

3.3 Equivalence of projections Definition 3.12. Let M be a von Neumann algebra. Two projections p, q ∈ M are called equivalent (in symbols, p ∼ q) if there exists a partial isometry v ∈ M such that p = v∗ v and q = vv∗ . We say that p is dominated by q (and write p ≺ q) if p ∼ q′ , for some projection q′ ∈ M with q′ ⩽ q. Exercise 3.13. Prove the following: (1) If p ∼ q, then z(p) = z(q). (2) If p ∼ q via a partial isometry v, then the map pMp ∋ x → vxv∗ ∈ qMq is a ∗-isomorphism. (3) If {pi }i∈I , {qi }i∈I are families of mutually orthogonal projections such that pi ∼ qi , ∀i ∈ I, then ∑i∈I pi ∼ ∑i∈I qi . Here, for orthogonal projections {pi }i∈I , we let ∑i∈I pi = ⋁i∈I pi . (4) If p ∼ q and z ∈ 𝒵 (M) is a projection, then zp ∼ zq.

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Lemma 3.14. If M is a von Neumann algebra and p, q ∈ 𝒫 (M), then the following are equivalent: (1) pMq ≠ {0}; (2) there exist nonzero projections p1 , q1 ∈ M such that p1 ⩽ p, q1 ⩽ q and p1 ∼ q1 ; (3) z(p)z(q) ≠ 0. Proof. (1) ⇒ (2) Let x ∈ M such that y = pxq ≠ 0. Let p1 = l(y) and q1 = r(y). Then 0 ≠ p1 ⩽ p, 0 ≠ q1 ⩽ q, p1 , q1 ∈ M and p1 ∼ q1 by Exercise 3.4. (2) ⇒ (1) If v ∈ M is such that p1 = vv∗ and q1 = v∗ v, then 0 ≠ v = pvq ∈ pMq. (1) ⇒ (3) If z(p)z(q) = 0, then pxq = pz(p)xz(q)q = pxz(p)z(q)q = 0, for all x ∈ M. (3) ⇒ (1) If pMq = {0}, then p(xqξ) = 0, for all x ∈ M. Since z(q) is the orthogonal projection onto MqH, we get that pz(q) = 0 and thus p ⩽ 1 − z(q). Since the projection 1 − z(q) belongs to the center of M, we get that z(q) ⩽ 1 − z(p), hence z(p)z(q) = 0. Exercise 3.15. Let M be a von Neumann algebra and p ∈ 𝒫 (M). Prove that there exist partial isometries {vi }i∈I such that vi v∗i ≤ p and ∑i∈I v∗i vi = z(p). Theorem 3.16 (The comparison theorem). Let M be a von Neumann algebra and p, q ∈ 𝒫 (M). Then there exists a projection z ∈ 𝒵 (M) such that pz ≺ qz and q(1 − z) ≺ p(1 − z). Proof. By Zorn’s lemma, there exist maximal families of mutually orthogonal projections {pi }i∈I , {qi }i∈I such that pi ⩽ p, qi ⩽ q and pi ∼ qi , for all i ∈ I. Put p1 = ∑i∈I pi , and q1 = ∑i∈I qi . Then p1 ∼ q1 . Also, let p2 = p − p1 and q2 = q − q1 . Since p2 , q2 do not have equivalent nonzero subprojections, Lemma 3.14 implies that z(p2 )z(q2 ) = 0. Thus, if we let z = z(q2 ), then p2 z = 0 and q2 (1 − z) = 0. The conclusion now follows since pz = ∑ pi z + p2 z = ∑ pi z ∼ ∑ qi z ≺ ∑ qi z + q2 z = qz, i∈I

i∈I

i∈I

i∈I

and similarly q(1 − z) ≺ p(1 − z). Corollary 3.17. If M is a factor and p, q ∈ 𝒫 (M), then p ≺ q or q ≺ p. Exercise 3.18. Let M be a finite-dimensional von Neumann algebra. (1) Assume that M is a factor. Prove that M has a minimal nonzero projection p. Deduce that there exist pairwise equivalent projections p1 , . . . , pn ∈ M such that p1 = p and ∑ni=1 pi = 1, for some n ≥ 1. Use this to conclude that M is ∗-isomorphic to 𝕄n (ℂ). (2) Prove that M is ∗-isomorphic to ⨁Kk=1 𝕄nk (ℂ), for some K, n1 , . . . , nK ≥ 1.

3.4 Classification into types Definition 3.19. Let M be a von Neumann algebra. A projection p ∈ M is called: (1) abelian if pMp is abelian; (2) finite if whenever q ∈ M is a projection such that q ⩽ p and q ∼ p, then q = p.

56 � A. Ioana Remark 3.20. Every abelian projection is finite. A subprojection of an abelian (resp. finite) projection is abelian (resp. finite). Definition 3.21. A unital von Neumann algebra M ⊂ 𝔹(H) is called – finite if 1 ∈ M is finite; – of type I if any nonzero central projection contains a nonzero abelian subprojection; – of type II if it has no abelian projections and any nonzero central projection contains a nonzero finite subprojection; – of type III if it contains no nonzero finite projection; – of type Ifin if it is of type I and finite; – of type I∞ if it is of type I and not finite; – of type II1 if it is of type II and finite; – of type II∞ if it is of type II and not finite. Remark 3.22. A unital von Neumann algebra M is finite if and only if any isometry v ∈ M is a unitary, i. e., v∗ v = 1 ⇒ vv∗ = 1. Theorem 3.23 (Decomposition into types). Let M ⊂ 𝔹(H) be a unital von Neumann algebra. Then there exist projections z1 , . . . , z5 ∈ 𝒵 (M) with ∑5i=1 zi = 1 and Mz1 , Mz2 , Mz3 , Mz4 , Mz5 are von Neumann algebras of type Ifin , I∞ , II 1 , II ∞ , III, respectively. Let p, q, r ∈ 𝒵 (M) be the maximal projections such that Mp is of type I, Mq is of type II, and r is a finite projection. Then z1 = pr, z2 = p(1 − r), z3 = qr, z4 = q(1 − r) and z5 = 1 − (p + q) satisfy the conclusion of Theorem 3.23, see, e. g., [11, Theorem 48.16]. Remark 3.24. Any factor M is of one of the types Ifin , I∞ , II 1 , II ∞ , or III.

3.5 von Neumann algebras of type I Definition 3.25. Let M ⊂ 𝔹(H) and N ⊂ 𝔹(K) be von Neumann algebras. For x ∈ M, y ∈ N we define x ⊗ y ∈ 𝔹(H ⊗ K) by letting (x ⊗ y)(ξ ⊗ η) = xξ ⊗ yη, for all ξ ∈ H, η ∈ K. The tensor product von Neumann algebra M⊗N ⊂ 𝔹(H ⊗ K) is defined as the WOT-closure of the linear span of {x ⊗ y | x ∈ M, y ∈ N}. Exercise 3.26. Let K be a Hilbert space and (X, μ) be a standard probability space. Prove that 𝔹(K)⊗L∞ (X) is a type Ifin von Neumann algebra if K is finite dimensional and a type I∞ von Neumann algebra if K is infinite dimensional. Any type I von Neumann algebra is isomorphic to ∏i∈I (𝔹(Ki )⊗L∞ (Xi )), where (Ki )i∈I are Hilbert spaces and {(Xi , μi )}i∈I are standard probability spaces, see [11, Section 50] for a proof of this fact. Here, we only prove this fact in the factorial case. Theorem 3.27. Any factor M of type I is ∗-isomorphic to 𝔹(K), for some Hilbert space K.

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Proof. Let p ∈ M be a nonzero abelian projection. Then pMp is both abelian and a factor. Therefore, pMp = ℂp. Let {pi }i∈I be a maximal family of pairwise orthogonal projections in M that are equivalent to p. Put q = 1 − ∑i∈I pi . We claim that q = 0. Indeed, if q ≠ 0, then by Corollary 3.17 we have that either (1) p ≺ q or (2) q ≺ p. Now, (1) contradicts the maximality of {pi }i∈I , while (2) implies that there exists a nonzero projection q′ ⩽ p such that q′ ∼ q. Since pMp = ℂp, it follows that q′ = p, contradicting again the maximality of {pi }i∈I . We will show that M ≅ 𝔹(ℓ2 (I)). Denote by {δi }i∈I the canonical orthonormal basis of 2 ℓ (I). For i, j ∈ I, we let ei,j ∈ 𝔹(ℓ2 (I)) be the “elementary” operator given by ei,j δk = δj,k δi , for all k ∈ I. For i ∈ I, let vi ∈ M be a partial isometry such that v∗i vi = p and vi v∗i = pi . Put vi0 = p. We define U : H → ℓ2 (I) ⊗ pH by letting U(ξ) = ∑i∈I δi ⊗ v∗i ξ. Since ∑i∈I ‖v∗i ξ‖2 = ∑i∈I ‖pi ξ‖2 = ‖ξ‖2 , for all ξ ∈ H, it follows that U is a unitary. We claim that UMU ∗ = 𝔹(ℓ2 (I))⊗ℂp, so M ≅ 𝔹(ℓ2 (I)). We note that Uvi U ∗ = ei,i0 ⊗ p, ∀i ∈ I. This implies that U 𝒜U ∗ = ℬ ⊗ ℂp, where 𝒜 ⊂ M and ℬ ⊂ 𝔹(ℓ2 (I)) are the ∗-algebras generated by {vi }i∈I and {ei,i0 }i∈I , respectively. Let x ∈ M and for F ⊂ I finite, put pF = ∑i∈F pi . As pF → 1, we get that pF xpF → x, in the SOT. Since pF xpF = ∑i,j∈F vi v∗i xvj v∗j = ∑i,j∈F vi (v∗i xvj )v∗j and v∗i xvj ∈ pMp = ℂp, we get that pF xpF ∈ 𝒜. This shows that 𝒜 is SOT-dense in M. Similarly, we get that ℬ is SOT-dense in 𝔹(ℓ2 (I)). The claim and the theorem are now proven.

3.6 von Neumann algebras of types II and III While finding examples of von Neumann algebras of type I is immediate, it is not obvious that type II or III algebras should exist. We next present Murray and von Neumann’s group measure space construction [21]. This connects von Neumann algebras with ergodic theory, and leads to examples of factors of types II and III. Moreover, group measure space factors are the subject of intense current research (see, e. g., [13]). Let Γ be a countable group and (X, μ) a σ-finite standard measure space. We say that an action Γ ↷ (X, μ) is nonsingular if for every g ∈ Γ and measurable set Y ⊂ X, the set gY is measurable and μ(Y ) = 0 ⇒ μ(gY ) = 0. We denote by g∗ μ the measure on X given by g∗ μ(Y ) = μ(g −1 Y ). Since g∗ μ ≺ μ, we have a Radon–Nikodym derivative dg∗ μ ∈ L1 (X, μ)+ such that dμ ∫f X

dg∗ μ dμ = ∫ f dg∗ μ = ∫ f ∘ g dμ, dμ X

∀f ∈ L∞ (X)+ .

X

1

This equation implies that the formula σg (f )(x) = ( dgdμ∗ μ (x)) 2 f (g −1 x), for x ∈ X, f ∈ L2 (X), defines a unitary operator σg on L2 (X). Let λ : Γ → 𝒰 (ℓ2 (Γ)) be the left regular representation given by λ(g)(δh ) = δgh . Denote H = L2 (X) ⊗ ℓ2 Γ and define a unitary

58 � A. Ioana representation u : Γ → 𝒰 (H) be letting ug = σg ⊗λ(g). We also define a ∗-homomorphism π : L∞ (X) → 𝔹(H) by letting π(f )(ξ ⊗ δg ) = fξ ⊗ δg , and view L∞ (X) ⊂ 𝔹(H), via π. Then ug fug∗ = σg (f ),

∀f ∈ L∞ (X),

g ∈ Γ.

Definition 3.28. The group measure space von Neumann algebra L∞ (X) ⋊ Γ ⊂ 𝔹(H) is defined as the WOT-closure of the linear span of {fug | f ∈ L∞ (X), g ∈ Γ}. Definition 3.29. A nonsingular action Γ ↷ (X, μ) is called: – ergodic if every Γ-invariant measurable set Y ⊂ X satisfies μ(Y ) = 0 or μ(X \ Y ) = 0; – (essentially) free if μ({x ∈ X | gx = x}) = 0, for every g ∈ Γ \ {e}. Theorem 3.30. Let Γ ↷ (X, μ) be a free ergodic nonsingular action of a countable group Γ. Then L∞ (X) ⋊ Γ is a factor of (1) type I if μ has atoms; (2) type II1 if μ is nonatomic and there is a finite Γ-invariant measure ν such that ν ∼ μ; (3) type II ∞ if μ is nonatomic and there is an infinite Γ-invariant measure ν such that ν ∼ μ; (4) type III if there is no (finite or infinite) Γ-invariant measure ν such that ν ∼ μ. For a proof of this theorem, see [29, Theorem 7.12, Chapter V]. We will prove item (2) of this result as part of Proposition 5.15. Example 3.31. Let Γ be a countable discrete group and μ be its counting measure. Then the left translation Γ ↷ (Γ, μ) is clearly free ergodic and measure preserving. The group measure space factor ℓ∞ (Γ) ⋊ Γ is of type I, and is in fact ∗-isomorphic to 𝔹(ℓ2 (Γ)). Example 3.32. Let G be a nondiscrete second countable locally compact group, mG be a left Haar measure of G and Γ < G a countable dense subgroup. For instance, we can take the inclusion Γ < G to be ℤ ≡ {exp(2πinα) | n ∈ ℤ} < 𝕋 = {z ∈ ℂ | |z| = 1}, for some α ∈ ℝ \ ℚ, or ℚ < ℝ. Then the left translation action Γ ↷ (G, mG ) is free, ergodic, and measure preserving. The group measure space factor L∞ (G) ⋊ Γ is of type II1 if G is compact and of type II∞ if G is noncompact. Example 3.33. In the setting of Example 3.32, let θ be a topological automorphism of G. Assume that G is noncompact and connected, θ(Γ) = Γ and θ∗ mG ≠ mG . View θ|Γ as an automorphism of Γ and define the semidirect product group Γ̃ = Γ ⋊θ ℤ. The action Γ̃ ↷ (G, mG ) given by (g, n) ⋅ x = gθn (x), for g ∈ Γ, n ∈ ℤ, x ∈ G, is free, ergodic, ̃ and nonsingular. Then there is no Γ-invariant measure ν on G with ν ∼ mG . Otherwise, dν = f dmG , for a Γ-invariant measurable function f : G → [0, ∞) and ergodicity of Γ ↷ (G, mG ) would imply that ν = λmG , for a λ > 0, contradicting that θ∗ mG ≠ mG . Thus, L∞ (G) ⋊ Γ̃ is a factor of type III. For a concrete example, one can take Γ = ℚ, G = ℝ, and θ(x) = rx, for any r ∈ ℚ \ {0, ±1}.

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4 Tracial von Neumann algebras In the rest of these notes, we will focus on the study of von Neumann subalgebras of II1 factors. These are exactly the von Neumann algebras which admit a trace. The existence of a trace is an extremely useful property in particular because it allows defining an equivalence-invariant notion of dimension for projections. For a comprehensive reference on tracial von Neumann algebras, we refer the reader to the book in preparation [1].

4.1 Tracial von Neumann algebras Let M, N be von Neumann algebras. A map Φ : M → N is called positive if Φ(M+ ) ⊂ N+ . A positive map Φ : M → N is called normal if Φ(xi ) → Φ(x), for any increasing net (xi ) ⊂ M+ such that xi → x (SOT). A positive linear functional φ : M → ℂ is called tracial if φ(xy) = φ(yx), ∀x, y ∈ M, and faithful if φ(x) = 0 ⇒ x = 0, ∀x ∈ M+ . Remark 4.1. A positive linear functional φ : M → ℂ is normal if and only if it is completely additive, φ(∑i∈I pi ) = ∑i∈I φ(pi ), for any family (pi )i∈I ⊂ M of mutually orthogonal projections. For a proof of this result, see [1, Theorem 2.5.5]. Definition 4.2. A von Neumann algebra M is called tracial if it admits a trace, i. e., a faithful normal tracial state τ : M → ℂ. (In short, we call the pair (M, τ) a tracial von Neumann algebra.) The following exercise provides a standard source of faithful states. Exercise 4.3. Let M ⊂ 𝔹(H) be a von Neumann algebra and ξ ∈ H such that M ′ ξ = H (ξ is called an M ′ -cyclic vector). Prove that the positive linear functional φ : M → ℂ given by φ(x) = ⟨xξ, ξ⟩ is faithful. Examples 4.4 (Tracial von Neumann algebras). (1) L∞ (X) is a tracial von Neumann algebra with the trace given by τ(f ) = ∫X f dμ. (2) 𝕄n (ℂ) is a tracial von Neumann algebra with the normalized trace τ([ai,j ]) = 1 n ∑ a . n i=1 i,i (3) More generally, 𝕄n (L∞ (X)) is a tracial von Neumann algebra, where τ([fi,j ]) =

1 n ∑ ∫ f dμ. n i=1 i,i X

Remark 4.5. Any tracial von Neumann algebra M is finite. If v ∈ M satisfies v∗ v = 1, then vv∗ is a projection, so 1 − vv∗ is a projection. As τ(1 − vv∗ ) = τ(v∗ v − vv∗ ) = 0 and τ is faithful, vv∗ = 1. Theorem 4.6 (Existence of the trace). Any finite von Neumann algebra M on a separable Hilbert space H is tracial. Any II1 factor is a tracial von Neumann algebra.

60 � A. Ioana Remark 4.7. Any finite von Neumann algebra M ⊂ 𝔹(H) admits a normal center-valued trace Ψ : M → 𝒵 (M) (see [16, Chapter 8], for a constructive proof, and [11, Section 55], for a proof based on the Ryll–Nardzewski fixed point theorem). In particular, any II1 factor M is tracial. If H is separable, then 𝒵 (M) is isomorphic to L∞ (X), for a standard probability space (X, μ), by Theorem 2.17. Then τ(T) = ∫X Ψ(T) dμ defines a trace on M. Exercise 4.8. Let M be a II1 factor with a faithful normal tracial state τ. Prove that two projections p, q ∈ M are equivalent if and only if τ(p) = τ(q). Exercise 4.9. Let (M, τ) be a diffuse tracial von Neumann algebra. Prove that for every t ∈ [0, 1], there exists a projection p ∈ M such that τ(p) = t. Exercise 4.10 (Uniqueness of the trace). Let M be a II1 factor with a faithful normal tracial state τ. Prove that any tracial state τ ′ : M → ℂ must be equal to τ.

4.2 The standard representation A von Neumann algebra can sit in many ways inside 𝔹(H). In this section, we show that any tracial von Neumann algebra (M, τ) has a canonical Hilbert space representation. This is a particular case of the GNS construction (see [28] or [11, Chapter 1]). Endow M with the scalar product ⟨x, y⟩ = τ(y∗ x). Define L2 (M) as the closure of M with respect to the norm ‖x‖2 = √τ(x ∗ x). Let M ∋ x → x̂ ∈ L2 (M) be the canonical embedding. Then 󵄩 󵄩 ‖xy‖22 = τ(y∗ x ∗ xy) ⩽ 󵄩󵄩󵄩x ∗ x 󵄩󵄩󵄩τ(y∗ y) = ‖x‖2 ‖y‖22 ,

∀x, y ∈ M.

Thus, letting π(x)(̂y) = x̂y, for all x, y ∈ M, defines a ∗-homomorphism π : M → 𝔹(L2 (M)) called the standard representation of M. Then π is isometric and thus π(M) is a C∗ -algebra. Moreover, π(M) is a von Neumann algebra (see [1, Theorem 2.6.1]). Hereafter, we view M ⊂ 𝔹(L2 (M)) by identifying M with π(M). We next show that the commutant of M in the standard representation is antiisomorphic to M. Define J : L2 (M) → L2 (M) by letting J(x̂) = x̂∗ . Then J is a conjugate linear unitary involution: J(αx̂ + β̂y) = αJ(x̂) + βJ(̂y), ⟨J(x̂), J(̂y)⟩ = ⟨̂y, x̂⟩, ∀α, β ∈ ℂ, x, y ∈ M, and J 2 = I. Theorem 4.11. One has M ′ = JMJ. Proof. Denote H = L2 (M). Notice that {x̂1 | x ∈ M} is dense in H and J(x̂1) = x ∗̂1, for all x ∈ M. Using these properties for every x, y, z ∈ M, we get that JxJy(ẑ1) = JxJ(yẑ1) = Jx(z∗ y∗̂1) = J(xz∗ y∗̂1) = yzx ∗̂1 = yJ(xz∗̂1) = yJxJ(ẑ1). Thus JMJ ⊂ M ′ and hence {x ′̂1 | x ′ ∈ M ′ } ⊃ {JxJ ̂1 | x ∈ M} = {x̂∗ | x ∈ M} = {x̂ | x ∈ M}. This implies that {x ′̂1 | x ′ ∈ M ′ } is dense in H. Further, if x ′ ∈ M ′ , then for all

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y ∈ M we have ⟨Jx̂1, ŷ1⟩ = ⟨Jŷ1, x̂1⟩ = ⟨x ∗ y∗̂1, ̂1⟩ = ⟨y∗ x ∗̂1, ̂1⟩ = ⟨x ∗̂1, ŷ1⟩. This shows that Jx̂1 = x ∗̂1, for all x ∈ M ′ . Altogether, have shown that the two properties satisfied by M are also satisfied by M ′ . Thus, we deduce that JM ′ J ⊂ M ′′ = M and hence JMJ = M ′ . Exercise 4.12. Let (M, τ) be a tracial von Neumann algebra. Let ξ ∈ L2 (M) and C > 0 be such that ‖xξ‖2 ≤ C‖x‖2 , ∀x ∈ M. Prove that ξ = ̂y, for some y ∈ M.

4.3 Hilbert modules Next, we address the following question: On what Hilbert spaces H other than L2 (M) can a tracial von Neumann algebra M be represented? The answer, showing that H is isomorphic to a direct sum of specific Hilbert subspaces of L2 (M), is important as it allows us to define a notion of dimension for H as an M-module. This is crucial in applications such as defining L2 -Betti numbers for groups and manifolds. Definition 4.13. Let M be a von Neumann algebra. A left Hilbert M-module is a Hilbert space H together with a unital normal ∗-homomorphism π : M → 𝔹(H). (Note that defining xξ := π(x)(ξ) makes H a left M-module.) For the rest of this subsection, we assume that (M, τ) is a tracial von Neumann algebra. Example 4.14. If p ∈ M is a projection, then L2 (M)p := JpJ(L2 (M)) is a left Hilbert M-module. Exercise 4.15. Let x ∈ M and denote by p ∈ M the right support projection of x. Prove that the left Hilbert M-module M x̂ ⊂ L2 (M) is isomorphic to L2 (M)p. Theorem 4.16. If H is a left Hilbert M-module, there exists a family of projections {pi }i∈I in M such that H ≅ ⨁i∈I L2 (M)pi . More precisely, there exists a unitary operator U : H → ⨁i∈I L2 (M)pi such that U(xξ) = xU(ξ), ∀x ∈ M, ξ ∈ H. The dimension of H is defined by letting dimM (H) := ∑ τ(pi ). i∈I

One can show that the dimension of H is independent of the choice of the projections {pi }i∈I . We omit the proof of this fact, and refer the reader to [1, Chapter 8]. A right Hilbert M-module is a Hilbert space H together with a unital normal ∗-homomorphism ρ : M op → 𝔹(H), where M op is the opposite von Neumann algebra of M. The dimension of H as a right M-module is denoted by dim(HM ) and satisfies an analogue of Theorem 4.16.

62 � A. Ioana The proof of Theorem 4.16 uses the next lemma and the exercise following it. Lemma 4.17 (Radon–Nikodym). Let φ : M → ℂ be a linear functional with 0 ⩽ φ(x) ⩽ τ(x), ∀x ∈ M+ . Then there is y ∈ M such that 0 ⩽ y ⩽ 1 and φ(x) = τ(xy), for all x ∈ M. Proof. The Cauchy–Schwarz inequality (see Exercise 2.4(1)) gives that 󵄨󵄨 ∗ 󵄨󵄨2 ∗ ∗ ∗ ∗ 2 2 󵄨󵄨φ(y x)󵄨󵄨 ⩽ φ(x x)φ(y y) ⩽ τ(x x)τ(y y) = ‖x‖2 ‖y‖2 ,

∀x, y ∈ M.

In particular, |φ(x)| ⩽ ‖x‖2 = ‖x̂‖2 , for all x ∈ M. By Riesz’s representation theorem, we find ξ ∈ L2 (M) such that φ(x) = ⟨x̂, ξ⟩, for all x ∈ M. Next, for y ∈ M, we get that ‖yξ‖2 =

sup

󵄨󵄨 ̂ 󵄨 󵄨󵄨⟨x , yξ⟩󵄨󵄨󵄨 =

x∈M,‖x‖2 ⩽1

sup

󵄨󵄨 ̂ 󵄨 󵄨󵄨⟨y∗ x, ξ⟩󵄨󵄨󵄨 =

x∈M,‖x‖2 ⩽1

sup

󵄨󵄨 ∗ 󵄨󵄨 󵄨󵄨φ(y x)󵄨󵄨 ⩽ ‖y‖2 .

x∈M,‖x‖2 ⩽1

By Exercise 4.12, we can find y ∈ M such that ξ = ŷ∗ . Thus, φ(x) = τ(xy), for all x ∈ M. It is left as an exercise to show that 0 ⩽ y ⩽ 1. Exercise 4.18. Let M be a von Neumann algebra and φ, ψ : M → ℂ be normal positive linear functionals such that φ(1) < ψ(1). Prove that there exists a nonzero projection q ∈ M such that φ(x) ≤ ψ(x), ∀x ∈ (qMq)+ . (Let (ri ) ⊂ M be a maximal family of mutually orthogonal projections such that φ(ri ) ≥ ψ(ri ), for every i. Then the projection q = 1−∑i ri has the desired property.) Proof of Theorem 4.16. The proof relies on the following claim: Claim. Let ξ ∈ H \ {0}. Then we can find nonzero projections q, p ∈ M such that Mqξ ≅ L2 (M)p. Proof of the claim. Define φ : M → ℂ by letting φ(x) = ⟨xξ, ξ⟩, for x ∈ M. Let c > 0 such that φ(1) < cτ(1). Since φ and cτ are normal positive linear functionals on M, by Exercise 4.18 we can find a projection q ∈ M such that φ(x) ⩽ cτ(x), for all x ∈ (qMq)+ . By applying Lemma 4.17, we can find y ∈ qMq such that 0 ⩽ y ⩽ c and φ(x) = τ(xy), for all x ∈ qMq. Let z ∈ (qMq)+ such that z2 = y. If x ∈ M, then since qx ∗ xq ∈ qMq, we get that 󵄩󵄩 󵄩2 ∗ ∗ ∗ ∗ 2 2 󵄩󵄩x(qξ)󵄩󵄩󵄩 = φ(qx xq) = τ(qx xqy) = τ(x xy) = τ(x xz ) = ‖x ̂z‖2 , and thus ‖x(qξ)‖ = ‖x ̂z‖2 . Thus, θ : Mqξ → M ̂z ⊂ L2 (M) given by θ(x(qξ)) = x ̂z extends to a unitary operator. It follows that Mqξ ≅ M ̂z. The claim now follows from Exercise 4.15. Finally, let {Hi }i∈I be a maximal family of mutually orthogonal left Hilbert M-submodules of H such that for every i ∈ I, there exists a projection pi ∈ ℳ with Hi ≅ L2 (M)pi . Then the above claim implies that H = ⨁i∈I Hi , which finishes the proof.

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4.4 Hilbert bimodules In the early 1980s, Connes discovered that Hilbert bimodules give an appropriate representation theory for tracial von Neumann algebras (see [9, 22] and [1, Chapter 13]). Definition 4.19. Let (M, τ) be a tracial von Neumann algebra. A Hilbert M-bimodule is a Hilbert space H equipped with commuting normal ∗-homomorphisms π : M → 𝔹(H), ρ : M op → 𝔹(H), where M op is the opposite von Neumann algebra of M. We write xξy = π(x)ρ(yop )ξ. Examples 4.20 (Hilbert bimodules). (1) The trivial bimodule L2 (M) with xξy = xJy∗ Jξ. (2) The coarse bimodule L2 (M)⊗L2 (M) with x(ξ ⊗ η)y = xξ ⊗ ηy. ̃ with xξy = α(x)Jβ(y)∗ Jξ, where (M, ̃ τ̃) is a tracial von Neumann algebra and (3) L2 (M) ̃ are ∗-homomorphisms such that τ̃ ∘ α = τ̃ ∘ β = τ. α, β : M → M

4.5 Jones’ basic construction Let (M, τ) be a tracial von Neumann algebra in its standard representation and B ⊂ M be a von Neumann subalgebra. Theorem 4.11 implies that B ⊂ M = JM ′ J ⊂ JB′ J ⊂ 𝔹(L2 (M)). The von Neumann algebra JB′ J is called the basic construction associated to B ⊂ M. As shown in Proposition 4.24, this is generated by M and the orthogonal projection from L2 (M) onto L2 (B). The basic construction was used by Jones (via an iteration argument) to prove his famous index theorem for subfactors [15]. It is now a key tool in the study of II1 factors. Definition 4.21. Let M be a von Neumann algebra and B ⊂ M be a von Neumann subalgebra. A positive linear map E : M → B is called a conditional expectation if it satisfies the following: (1) E(b) = b, ∀b ∈ B. (2) E(b1 xb2 ) = b1 E(x)b2 , ∀b1 , b2 ∈ B, x ∈ M. Proposition 4.22. Let (M, τ) be a tracial von Neumann algebra and B ⊂ M be a von Neumann subalgebra. Then there exists a unique conditional expectation E : M → B such that τ ∘ E = τ. Proof. Let eB : L2 (M) → L2 (B) be the orthogonal projection, where L2 (B) denotes the ̂ and hence ‖ ⋅ ‖2 -closure of {b̂ | b ∈ B}. If x ∈ M and b ∈ B, then beB (x)̂ = eB (bx) 󵄩󵄩 󵄩 󵄩 ̂ 󵄩󵄩 ̂ = ‖bx‖ ⩽ ‖x‖ ‖b‖ = ‖x‖ ‖b‖̂ . 󵄩󵄩beB (x)̂ 󵄩󵄩󵄩2 = 󵄩󵄩󵄩eB (bx) 󵄩󵄩2 ⩽ ‖bx‖ 2 2 2 2

64 � A. Ioana ̂ Since T ∈ B′ , we get that Thus, there is T ∈ 𝔹(L2 (B)) such that T(b)̂ = beB (x). ̂ satisfies T ∈ JBJ, and thus eB (x)̂ ∈ B.̂ One checks that EB : M → B given by E? B (x) = eB (x) the conclusion. Definition 4.23. Let (M, τ) be a tracial von Neumann algebra and B ⊂ M be a von Neumann subalgebra. The basic construction ⟨M, eB ⟩ is the von Neumann subalgebra of 𝔹(L2 (M)) generated by M and the orthogonal projection eB from L2 (M) onto L2 (B). Proposition 4.24. We have the following: (1) JeB = eB J, beB = eB b and eB xeB = EB (x)eB , ∀b ∈ B, x ∈ M. (2) ⟨M, eB ⟩ = (JBJ)′ = JB′ J. (3) eB ∈ ⟨M, eB ⟩ has central support 1. (4) The linear span of MeB M is an SOT-dense ∗-subalgebra of ⟨M, eB ⟩. (5) There exists a semifinite faithful normal trace Tr : ⟨M, eB ⟩ → ℂ such that Tr(xeB y) = τ(xy),

∀x, y ∈ M.

(6) If p ∈ ⟨M, eB ⟩ is a projection, then pL2 (M) is a right Hilbert B-module and dim(pL2 (M)B ) = Tr(p). For the notion of a semifinite faithful normal trace, see [1, Definition 8.3.1]. Proof. (1) The proof of this assertion is left as an exercise. (2) Since ⟨M, eB ⟩′ = M ′ ∩ {eB }′ = JMJ ∩ {eB }′ = J(M ∩ {eB }′ )J = JBJ, the double commutant theorem implies that ⟨M, eB ⟩ = (JBJ)′ . (3) Since ⟨M, eB ⟩eB L2 (M) ⊃ ⟨M, eB ⟩eB̂1 = ⟨M, eB ⟩̂1 ⊃ M ̂1, by Lemma 3.9 we deduce that eB ∈ ⟨M, eB ⟩ has central support 1. (4) Let ℳ be the SOT-closure of the linear span of MeB M. Then ℳ is a von Neumann algebra and a two sided ideal of ⟨M, eB ⟩. In particular, ueB u∗ ∈ ℳ, for every u ∈ 𝒰 (⟨M, eB ⟩). Thus, by (3) and Exercise 3.10 we get that 1 = ⋁u∈𝒰 (⟨M,eB ⟩) ueB u∗ ∈ ℳ. This implies that ℳ = ⟨M, eB ⟩. (5) Since the central support of eB in ⟨M, eB ⟩ is 1, there are partial isometries (vi ) ⊂ ⟨M, eB ⟩ such that v∗i vi ≤ eB and ∑i vi v∗i = 1 (see Exercise 3.15). We define Tr : ⟨M, eB ⟩+ → [0, +∞] by letting Tr(T) = ∑⟨Tvî1, vî1⟩. i

Let T ∈ ⟨M, eB ⟩ with Tr(T ∗ T) = 0. Then Tvî1 = 0 and thus Tvi b̂ = Tvi Jb∗ J ̂1 = Jb∗ JTvî1 = 0, for every b ∈ B. Hence, Tvi L2 (B) = {0} for every i. Since v∗i vi ≤ eB , we have Tvi v∗i L2 (M) ⊂ Tvi L2 (B) and thus Tvi v∗i = 0 for every i. Since ∑i vi v∗i = 1, we get that T = 0, so Tr is faithful. To show that Tr is a trace, note that since ⟨M, eB ⟩eB = MeB , we can find wi ∈ M such that vi = vi eB = wi eB . Let x, y ∈ M. By applying the identity ∑i wi eB wi∗ = ∑i vi v∗i = 1 to

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x̂ ∈ L2 (M), we derive that ∑i wi EB (wi∗ x) = x. Also, if a, b, c ∈ M, then ⟨aeB beB̂1, ceB̂1⟩ = τ(EB (c∗ a)b). By combining these identities we get that Tr(xeB y) = ∑⟨xeB ywi eB̂1, wi eB̂1⟩ = ∑ τ(EB (wi∗ x)ywi ) = ∑ τ(wi EB (wi∗ x)y) = τ(xy). i

i

i

Thus, if T = xeB y and T ′ = x ′ eB y′ , for some x, y, x ′ , y′ ∈ M, then

(4.1)

Tr(TT ′ ) = τ(xEB (yx ′ )y′ ) = τ(EB (y′ x)EB (yx ′ )) = τ(x ′ EB (y′ x)y) = Tr(T ′ T). This proves that Tr is a trace. (6) Equation (4.1) shows that Tr does not depend on the choice on {vi }i∈I . Thus, we may assume that there is a subset J ⊂ I such that p = ∑i∈J vi v∗i = ∑i∈J wi eB wi∗ . Thus, Tr(p) = ∑i∈J τ(wi wi∗ ). Since v∗i vi = EB (wi∗ wi )eB is a projection, EB (wi∗ wi ) is a projection. We leave it as an exercise to check that wi eB wi∗ (L2 (M)) is isomorphic to EB (wi∗ wi )L2 (B), as a right Hilbert B-module. Thus, pL2 (M) ≅ ⨁i∈J EB (wi∗ wi )L2 (B), as right Hilbert B-modules, hence dim(pL2 (M)B ) = ∑ τ(EB (wi∗ wi )) = ∑ τ(wi∗ wi ) = ∑ τ(wi wi∗ ) = Tr(p), i∈I

i∈J

i∈J

which finishes the proof. Remark 4.25. Let ℳ be a von Neumann algebra endowed with a normal, faithful, 1

semifinite trace Tr. For 1 ≤ p < ∞, define ‖x‖p = Tr(|x|p ) p , for every x ∈ ℳ. The Banach space Lp (ℳ) is defined as the closure of the set {x ∈ ℳ | ‖x‖p < ∞} with respect to ‖⋅‖p . If ℳ = ⟨M, eB ⟩, then Lp (ℳ) is equal to the closure of the span of MeB M with respect to ‖⋅‖p .

4.6 Popa’s intertwining-by-bimodules technique Given subalgebras A, B of a von Neumann algebra M, it is a natural question whether uAu∗ ⊂ B, for some u ∈ 𝒰 (M). To address this question, Popa developed a technique, called intertwining-by-bimodules. This technique has been instrumental in the progress made in the classification of II1 factors via Popa’s deformation/rigidity theory (see the surveys [25, 30, 13]) and is now a fundamental tool in the study of II1 factors. Theorem 4.26 (Popa, [24]). Let A, B be von Neumann subalgebras of a tracial von Neumann algebra (M, τ). Then the following conditions are equivalent: (1) There is no net (ui ) ⊂ 𝒰 (A) such that ‖EB (xui y)‖2 → 0, ∀x, y ∈ M. (2) There is a nonzero projection e ∈ A′ ∩ ⟨M, eB ⟩ such that Tr(e) < +∞. (3) There are nonzero projections p ∈ A, q ∈ B, a nonzero partial isometry v ∈ M, and a ∗-homomorphism θ : pAp → qBq such that v∗ v ≤ p, vv∗ ≤ q and θ(x)v = vx, ∀x ∈ pAp. (4) There exists an A-B-subbimodule ℋ of L2 (M) such that dim(ℋB ) < +∞.

66 � A. Ioana If conditions (1)–(4) hold, we write A ≺M B and say that a corner of A embeds into B inside M. Next, we mention two cases when A ≺M B implies the existence of u ∈ 𝒰 (M) such that uAu∗ ⊂ B. Definition 4.27. Let (M, τ) be a tracial von Neumann algebra. We say that a von Neumann subalgebra A ⊂ M is a Cartan subalgebra if it is maximal abelian and the normalizing group 𝒩M (A) = {u ∈ 𝒰 (M) | uAu∗ = A} satisfies 𝒩M (A)′′ = M. Theorem 4.28 (Popa, [23]). Let A, B be Cartan subalgebras of a II1 factor M. Then A ≺M B if and only if there exists u ∈ 𝒰 (M) such that uAu∗ = B. Remark 4.29. Let (M, τ) be a tracial von Neumann algebra and A, B ⊂ M be von Neumann subalgebras. Assume that A′ ∩ M = ℂ1 and the inclusion B ⊂ M is mixing: ‖EB (xbn y)‖2 → 0, for every x, y ∈ M with EB (x) = 0 and any sequence bn ⊂ (B)1 such that bn → 0 in the WOT. Then A ≺M B if and only if there exists u ∈ 𝒰 (M) such that uAu∗ ⊂ B [24].

5 Examples of tracial von Neumann algebras 5.1 The hyperfinite II1 factor For n ≥ 1, let An = 𝕄2n (ℂ) and τn : An → ℂ be the normalized trace. Consider the diagonal embedding An ⊂ An+1 given by x 0

x 󳨃→ (

0 ) x

Define A = ⋃n≥1 An and notice that A is a ∗-algebra which is equipped with a norm ‖ ⋅ ‖ which satisfies ‖x ∗ x‖ = ‖x‖2 , for all x ∈ A. Moreover, τ : A → ℂ defined by τ(x) = τn (x), if x ∈ An , is a faithful tracial linear functional which satisfies |τ(x)| ≤ ‖x‖, for all x ∈ A. We denote by H the closure of A with respect to the norm ‖x‖2 = √τ(x ∗ x), and consider the GNS ∗-homomorphism π : A → 𝔹(H) given by π(x)(̂y) = x̂y, for all x, y ∈ A. WOT

Theorem 5.1. The set R := π(A) is a II1 factor and the map φ : R → ℂ given by φ(x) = ⟨x̂1, ̂1⟩ is a normal faithful tracial state such that φ ∘ π = τ. ̂ Proof. Showing that φ is tracial on R is equivalent to proving that ⟨y1,̂ x ∗ 1⟩̂ = ⟨x 1,̂ y∗ 1⟩, for all x, y ∈ R. Since this holds for all x, y ∈ π(A) (⟨π(y)1,̂ π(x)∗ 1⟩̂ = ⟨y,̂ x ∗̂ ⟩ = τ(xy), for all x, y ∈ A) and π(A) is SOT-dense in R, we deduce that φ is tracial on R. Given z ∈ A, we have ‖yz‖22 = τ(z∗ y∗ yz) = τ(yzz∗ y∗ ) ≤ ‖zz∗ ‖τ(yy∗ ) = ‖z‖2 ‖y‖22 , for all y ∈ A. This implies the existence of ρ(z) ∈ 𝔹(H) such that ρ(z)(y)̂ = ŷz, for all y ∈ A.

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Since ρ(z) ∈ π(A)′ , we get that ρ(z) ∈ R′ . Thus, R′ 1̂ ⊃ {ρ(z)1̂ | z ∈ A} = {ẑ | z ∈ A} and since {ẑ | z ∈ A} is ‖ ⋅ ‖2 -dense in H, Exercise 4.3 gives that φ is faithful. Since φ is a normal state, the second assertion of the theorem is proven. Finally, let us show that R is a factor. To this end, let x ∈ 𝒵 (R) and put x0 = x −φ(x)⋅1. For n ≥ 1, let Rn = π(An ) ⊂ R and En : R → Rn be the unique φ-preserving conditional expectation. Then En (x) ∈ 𝒵 (Rn ). Since Rn ≅ 𝕄2n (ℂ) is factor and En is φ-preserving, we get that En (x) = φ(En (x)) ⋅ 1 = φ(x) ⋅ 1, or equivalently En (x0 ) = 0. Thus, for every n ≥ 1 and y ∈ Rn we have that φ(x0 y) = φ(En (x0 y)) = φ(En (x0 )y) = 0. Hence, φ(x0 y) = 0, for all y ∈ π(A). Since π(A) is SOT-dense in R, we conclude that this equality holds for every y ∈ R. In particular, we have that φ(x0 x0∗ ) = 0. Since φ is faithful we conclude that x0 = 0 and thus x = φ(x)1 ∈ ℂ1. Definition 5.2. A von Neumann algebra M is called hyperfinite if it admits an increasing sequence (Mn )n≥1 of finite-dimensional ∗-subalgebras such that ⋃n≥1 Mn is SOT-dense in M. The II1 factor R from Theorem 5.1 is hyperfinite by definition. Murray and von Neumann [20] proved that any hyperfinite II1 factor is isomorphic to R, which justifies the following: Definition 5.3. The II1 factor R is called the hyperfinite II1 factor. As it turns out, R is the smallest II1 factor: Exercise 5.4. Let M be a II1 factor and τ : M → ℂ be a faithful normal tracial state. (1) By Exercise 4.9, there exists a projection p ∈ M such that τ(p) = 21 . Use this fact to prove that there exists an injective unital ∗-homomorphism ρ : 𝕄2 (ℂ) → M. (2) Prove that there exists an injective unital ∗-homomorphism π : R → M.

5.2 Group von Neumann algebras Let Γ be a countable group. The left and right regular representations λ, ρ : Γ → 𝒰 (ℓ2 (Γ)) are given by λ(g)(δh ) = δgh and ρ(g)(δh ) = δhg −1 . The group von Neumann algebra L(Γ) ⊂ 𝔹(ℓ2 (Γ)) is the WOT-closure of the linear span of {λ(g) | g ∈ Γ} [20]. We denote by R(Γ) ⊂ 𝔹(ℓ2 (Γ)) the WOT-closure of the linear span of {ρ(g) | g ∈ Γ}. Convention. Following the tradition in the subject, we denote ug := λ(g), for g ∈ Γ. Proposition 5.5. The map τ : L(Γ) → ℂ given by τ(x) = ⟨xδe , δe ⟩ is a faithful normal tracial state. Moreover, L(Γ)′ = R(Γ). Proof. Since τ(1) = 1 and τ(x ∗ x) = ‖xδe ‖2 ⩾ 0, for all x ∈ M, we get that τ is a normal state. Since τ(ug uh ) = τ(ugh ) = δgh,e = δhg,e = τ(uhg ) = τ(uh ug ), we get that τ is a trace. If τ(x ∗ x) = 0, then the first line of the proof implies that xδe = 0. If g ∈ Γ, then since

68 � A. Ioana ρ(g −1 ) ∈ L(Γ)′ , we get that xδg = x(ρ(g −1 )δe ) = ρ(g −1 )(xδe ) = 0. This implies that x = 0, hence τ is faithful. We identify L2 (L(Γ)) with ℓ2 Γ via the unitary ug → δg . Under this identification, the involution J becomes J(δg ) = δg −1 . Now, if g, h ∈ Γ, then Jug J(δh ) = Jug δh−1 = Jδgh−1 = δhg −1 = ρ(g)(δh ). This shows that Jug J = ρ(g), for all g ∈ Γ, hence L(Γ)′ = JL(Γ)J = R(Γ). Notation 5.6. For x ∈ L(Γ), we write xδe = ∑g∈Γ xg δg ∈ ℓ2 Γ. Observe that in the above identification L2 (L(Γ)) = ℓ2 (Γ), we have that x̂ = xδe . The complex coefficients {xg }g∈Γ are called the Fourier coefficients of x and can be calculated as xg = ⟨xδe , δg ⟩ = τ(xug∗ ). We will write x = ∑g∈Γ xg ug , where the convergence holds in the ‖⋅‖2 (but not necessarily the WOT!). Exercise 5.7. Let x, y ∈ L(Γ) and let x = ∑g∈Γ xg ug , y = ∑g∈Γ yg ug be their Fourier expansions. Prove that x ∗ = ∑g∈Γ xg −1 ug and xy = ∑g∈Γ (∑h∈Γ xh yh−1 g )ug . Remark 5.8. Let Γ be a countable abelian group. The group of all homomorphisms η : ̂ Let μ be the Γ → 𝕋 is a compact abelian group, called the dual of Γ and denoted Γ. 2 ̂ ̂ ̂ Haar measure of Γ. For g ∈ Γ, let ĝ ∈ L (Γ) be given by g(η) = η(g). Then the map ̂ given by U(δg ) = ĝ extends to a unitary such that UL(Γ)U ∗ = L∞ (Γ). ̂ U : ℓ2 (Γ) → L2 (Γ) ∞ ̂ In particular, L(Γ) is ∗-isomorphic to L (Γ). If Γ is infinite, then μ has no atoms and so L(Γ) is ∗-isomorphic to L∞ ([0, 1], λ) by Corollary 2.20. The next result clarifies when L(Γ) is a II1 factor. Proposition 5.9. Let Γ be a countable group. Then L(Γ) is a factor if and only if Γ has infinite conjugacy classes (or, is icc): the conjugacy class {hgh−1 | h ∈ Γ} is infinite, for every g ∈ Γ \ {e}. Proof. (⇒) Assume that C = {hgh−1 | h ∈ Γ} is finite, for some g ≠ e. Then x = ∑k∈C uk belongs to the center of L(Γ) and x ∉ ℂ1. (⇐) Assume that Γ is icc and let x be an element in the center of L(Γ). Let x = ∑g∈Γ xg ug be the Fourier expansion of x and h ∈ Γ. Let y = ∑g∈Γ yg ug for the Fourier ∗ expansion of y = uh xuh∗ . Then yg = τ(yug∗ ) = τ(uh xuh∗ ug∗ ) = τ(xuhgh −1 ) = xhgh−1 . Since x commutes with uh , we get that y = x, and hence xhgh−1 = xg , for all g, h ∈ Γ. Since ∑g∈Γ |xg |2 = ‖x‖22 < ∞, and Γ is icc, we conclude that xg = 0, for all g ∈ Γ \ {e}. Thus, x ∈ ℂ1. Exercise 5.10. Prove that the following countable groups are icc: (1) The group S∞ of bijections π : ℕ → ℕ such that {n ∈ ℕ | π(n) ≠ n} is finite. (2) The free product group Γ = Γ1 ∗ Γ2 , where Γ1 , Γ2 are any groups with |Γ1 | > 1 and |Γ2 | > 2. In particular, the free group 𝔽n on n ≥ 2 generators is icc. (3) SLn (ℤ) := {A ∈ 𝕄n (ℤ) | det(A) = 1}, for every odd n ⩾ 3.

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5.3 Group measure space von Neumann algebras In this section, we study group measure space von Neumann algebras arising from pmp actions. Let Γ be a countable group and (X, μ) a standard probability space. An action Γ ↷ (X, μ) is called probability measure preserving (pmp) if for every g ∈ Γ and measurable set Y ⊂ X, the set gY is measurable and μ(gY ) = μ(Y ). Recall that the group measure space von Neumann algebra L∞ (X) ⋊ Γ is defined as the closure, in the WOT, of the linear span of {fug | f ∈ L∞ (X), g ∈ Γ} ⊂ 𝔹(H). Here, H = L2 (X) ⊗ ℓ2 Γ, we define a unitary representation u : Γ → 𝒰 (H) and view L∞ (X) ⊂ 𝔹(H) by ug (ξ ⊗ δh ) = σg (ξ) ⊗ δgh

and

f (ξ ⊗ δh ) = fξ ⊗ δh ,

∀f ∈ L∞ (X),

ξ ∈ L2 (X),

g, h ∈ Γ,

where σg (ξ)(x) = ξ(g −1 x). Also, we recall that ug fug∗ = σg (f ), for every f ∈ L∞ (X), g ∈ Γ. Proposition 5.11. The map τ : L∞ (X) ⋊ Γ → ℂ given by τ(x) = ⟨x(1 ⊗ δe ), 1 ⊗ δe ⟩ is a faithful normal tracial state. Proof. For all f ∈ L∞ (X) and g ∈ Γ, we have that τ(fug ) = ⟨fug (1 ⊗ δe ), 1 ⊗ δe ⟩ = ⟨f ⊗ δg , 1 ⊗ δe ⟩ = δg,e ∫ f dμ. X

If f1 , f2 ∈ L∞ (X) and g1 , g2 ∈ Γ, then f1 ug1 f2 ug2 = f1 σg1 (f2 )ug1 g2 and f2 ug2 f1 ug2 = f2 σg2 (f1 )ug2 g1 . Since τ(σg (f )) = τ(f ), for all f ∈ L∞ (X) and g ∈ Γ, we get that τ(f1 ug1 f2 ug2 ) = τ(f2 ug2 f1 ug2 ). This implies that τ is a trace. We leave the rest of the proof as an exercise. Proposition 5.12. Let Γ ↷ (X, μ) be a pmp action. Denote M = L∞ (X) ⋊ Γ and A = L∞ (X). Every a ∈ M has a unique Fourier expansion of the form a = ∑g∈Γ ag ug , where ag = EA (aug∗ ) ∈ A, for every g ∈ Γ, and the series converges in ‖ ⋅ ‖2 . Moreover, we have the following: – a∗ = ∑g∈Γ σg −1 (ag∗ )ug −1 ; – –

‖a‖22 = ∑g∈Γ ‖ag ‖22 ; ab = ∑g∈Γ (∑h∈Γ ah σh (bh−1 g ))ug .

Proof. The formula U(fug ) = f ⊗ δg defines a unitary operator U : L2 (M) → L2 (X) ⊗ ℓ2 Γ. Thus, every a ∈ M can be written as a = ∑g∈Γ ag ug , where ag ∈ L2 (X) satisfy ∑g∈Γ ‖ag ‖22 = ‖a‖22 . Moreover, we have that â e = eA (a)̂ and thus ae = EA (a). Since auh∗ = ∑g∈Γ agh ug , we get that ah = EA (auh∗ ), for every h ∈ Γ. We leave the rest of the proof as an exercise. Lemma 5.13. A pmp action Γ ↷ (X, μ) is ergodic if and only if any function f ∈ L2 (X) which satisfies that σg (f ) = f , for every g ∈ Γ, is essentially constant.

70 � A. Ioana Proof. (⇐) If Y is a Γ-invariant set, then f = 1Y ∈ L2 (X) is a Γ-invariant function. Thus, there is c ∈ ℂ such that f = c. As f 2 = f , we get that c ∈ {0, 1}, hence μ(Y ) = ∫X f dμ = c ∈ {0, 1}. (⇒) Let f ∈ L2 (X) be a σ(Γ)-invariant function. If f is not constant, then it admits at least two distinct essential values z, w ∈ ℂ. Let δ = |z − w|/2. Then Y = {x ∈ X| |f (x) − z| < δ} and Z = {x ∈ X| |f (x) − w| < δ} are disjoint, Γ-invariant, measurable sets. Since μ(Y ) > 0 and μ(Z) > 0, we get a contradiction with the ergodicity of the action. Exercise 5.14. Let Γ be an infinite group and (Y , ν) be a nontrivial standard probability space. Define (X, μ) = (Y Γ , ν⊗Γ ). Consider the Bernoulli action Γ ↷ (X, μ) given by gx = (xg −1 h )h∈Γ , for every g ∈ Γ and x = (xh )h∈Γ ∈ X. Prove that this action is pmp, essentially free, and ergodic. Moreover, prove that this action is mixing: limg→∞ μ(gY ∩ Z) = μ(Y )μ(Z), ∀Y , Z ⊂ X measurable. Proposition 5.15. Let Γ ↷ (X, μ) be a pmp action. Denote M = L∞ (X) ⋊ Γ and A = L∞ (X). (1) The action Γ ↷ (X, μ) is free if and only if A ⊂ M is maximal abelian, i. e., A′ ∩ M = A. (2) Assume that the action Γ ↷ (X, μ) is free. Then M is a factor if and only if the action Γ ↷ (X, μ) is ergodic. Proof. (1) Assume that A′ ∩ M = A. Let g ∈ Γ \ {e} and put Y = {x ∈ X | gx = x}. Since 1Y σg (f ) = 1Y f , for all f ∈ A, we get that a = 1Y ug ∈ A′ ∩ M. Hence a ∈ A and thus a = EA (a) = 0, showing that μ(Y ) = 0. This implies that the action is free. Conversely, assume that the action is free. Let a ∈ A′ ∩ M and a = ∑g∈Γ ag ug be its Fourier decomposition. If b ∈ A, then ∑g∈Γ bag ug = ba = ab = ∑g∈Γ ag σg (b)ug , thus bag = σg (b)ag , for all g ∈ Γ. Let g ∈ Γ \ {e} and put Yg = {x ∈ X | ag (x) ≠ 0}. The latter equality gives that b(g −1 x) = b(x), for almost every x ∈ Yg . Since (X, μ) is a standard probability space, we can find a sequence of measurable sets (Xn ) ⊂ X which separate points in X. By applying the latter identity to b = 1Xn , for all n ≥ 1, we deduce that g −1 x = x, for almost every x ∈ Yg . Since the action is free, we get that μ(Yg ) = 0, hence ag = 0. As this holds for all g ∈ Γ \ {e}, we conclude that a ∈ A. (2) Since the action is free, (1) implies that 𝒵 (M) = A ∩ M ′ = {a ∈ A | σg (a) = a, ∀g ∈ Γ}. By Lemma 5.13, the conclusion follows. Exercise 5.16. Let Γ be an icc group and Γ ↷ (X, μ) be a pmp action. Prove that L∞ (X)⋊Γ is a II1 factor if and only if the action Γ ↷ (X, μ) is ergodic.

5.4 Cartan subalgebras and orbit equivalence By Proposition 5.15, L∞ (X) ⊂ L∞ (X)⋊Γ is a Cartan subalgebra, for every free pmp action Γ ↷ (X, μ). It is a fundamental observation of Singer [27] (see also Feldman and Moore’s work [12]) that the isomorphism class of the inclusion L∞ (X) ⊂ L∞ (X) ⋊ Γ captures exactly the orbit equivalence class of the action Γ ↷ (X, μ).

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Proposition 5.17 (Singer, [27]). If Γ ↷ (X, μ) and Λ ↷ (Y , ν) are free pmp actions, then the following conditions are equivalent: (1) There exists a ∗-isomorphism π : L∞ (X) ⋊ Γ → L∞ (Y ) ⋊ Λ such that π(L∞ (X)) = L∞ (Y ). (2) The actions are orbit equivalent, i. e., there exists an isomorphism θ : (X, μ) → (Y , ν) (called an orbit equivalence between the actions) such that θ(Γx) = Λθ(x), for a. e. x ∈ X. Both implications of Proposition 5.17 are important. Thus, (1) ⇒ (2) reduces the classification of group measure space factors to the classification of actions up to orbit equivalence, provided that the Cartan subalgebras can be shown to be unique. Conversely, the implication (2) ⇒ (1) provides a von Neumann algebraic approach to the study of orbit equivalence of actions. Proof. Denote A = L∞ (X), B = L∞ (Y ), M = L∞ (X) ⋊ Γ, and N = L∞ (Y ) ⋊ Λ. (1) ⇒ (2) Since π|A : A → B is a ∗-isomorphism, we can find an isomorphism θ : (X, μ) → (Y , ν) such that π(a) = a ∘ θ−1 , for all a ∈ A (see [1, Theorem 3.3.4]). We will prove that θ is the desired orbit equivalence. To this end, fix g ∈ Γ and denote v = π(ug ). Then v normalizes B and thus we can find an isomorphism α : (Y , ν) → (Y , ν) such that vbv∗ = b ∘ α, for all b ∈ B. Claim. We have α(y) ∈ Λy, for almost every y ∈ Y . Consider the Fourier expansion v = ∑h∈Λ vh uh , where vh ∈ B for all h ∈ Λ. Since vb = (b ∘ α)v, we deduce that vh (b ∘ h−1 ) = vh (b ∘ α), for all h ∈ Λ and b ∈ B. If we let Yh = {y ∈ Y | vh (y) ≠ 0}, then the same argument as in the proof of Proposition 5.15 shows that α(y) = h−1 y, for almost every y ∈ Yh and all h ∈ Λ. Now, if we let Z = Y \ (⋃h∈Λ Yh ), then 1Z vh = 0, for all h ∈ Λ and thus 1Z v = ∑h∈Λ (1Z vh )uh = 0. Hence ν(Z)1/2 = ‖1Z ‖2 = ‖1Z v‖2 = 0, which implies that the set ⋃h∈Λ Yh is conull in Y . This clearly implies the claim. If a ∈ A, then a ∘ g −1 ∘ θ−1 = π(ug aug∗ ) = vπ(a)v∗ = π(a) ∘ α = a ∘ θ−1 ∘ α. Thus,

g −1 ∘ θ−1 = θ−1 ∘ α hence θ ∘ g −1 = α ∘ θ. Together with the claim, this implies that θ(g −1 x) = α(θ(x)) ∈ Λθ(x), for almost every x ∈ X. Since g ∈ Γ is arbitrary, we conclude that θ(Γx) ⊂ Λθ(x), for almost every x ∈ X. Since the reverse inclusion can be proved similarly, it follows that θ is an orbit equivalence. (2) ⇒ (1) Let θ : (X, μ) → (Y , ν) be an orbit equivalence. Define a ∗-isomorphism π : A → B by letting π(a) = a ∘θ−1 . Our goal is to show that π extends to a ∗-isomorphism π : M → N. To this end, fix g ∈ Γ. Then (θ∘g −1 ∘θ−1 )(y) ∈ Λ⋅y, for almost every y ∈ Y . For h ∈ Λ, put Yg,h = {y ∈ Y | (θ∘g −1 ∘θ−1 )(y) = h−1 y}. Then {Yg,h }h∈Λ is a measurable partition of Y . Since h−1 Yg,h = {y ∈ Y | (θ ∘ g ∘ θ−1 )(y) = hy}, we also have that {h−1 Yg,h }h∈Λ is a measurable partition of Y . Using the last two facts, one checks that the formula π(ug ) = ∑h∈Λ 1Yg,h uh defines a unitary in N such that π(ug )π(a)π(ug )∗ = π(a ∘ g −1 ), for all a ∈ A. This entails

72 � A. Ioana that π extends to a ∗-homomorphism from the linear span of {aug | a ∈ A, g ∈ Γ} to N. Moreover, π is trace preserving. We leave it as an exercise to show that π extends to a ∗-isomorphism π : M → N.

5.5 Tensor product von Neumann algebras We next establish that the class of tracial von Neumann algebras is closed under tensor products. Proposition 5.18. Let (M1 , τ1 ) and (M2 , τ2 ) be tracial von Neumann algebras. Then M1 ⊗M2 is a tracial von Neumann algebra. Moreover, if M1 and M2 are II1 factors, then so is M1 ⊗M2 . The proof of the last assertion of Proposition 5.18 relies on the following exercise: Exercise 5.19. Let (M, τ) be a tracial von Neumann algebra. For x ∈ M, let Kx ⊂ L2 (M) ?∗ | u ∈ 𝒰 (M)}. Then we have be the ‖ ⋅ ‖2 -closure of the convex hull of the set {uxu (1) Assume that M is a II1 factor and let x ∈ M. Prove that τ(x)1̂ is the unique element of minimal ‖ ⋅ ‖2 of Kx . Deduce in particular that τ(x)1̂ ∈ Kx . (2) Assume that the linear span of the set of x ∈ M such that τ(x)1̂ ∈ Kx is ‖ ⋅ ‖2 -dense in M. Prove that M is II1 factor. Proof of Proposition 5.18. Using that M1 ⊗M2 ⊂ 𝔹(L2 (M1 ) ⊗ L2 (M2 )), define τ : M1 ⊗M2 → ℂ by τ(x) = ⟨x(̂1 ⊗ ̂1), ̂1 ⊗ ̂1⟩. Then τ(x1 ⊗ x2 ) = τ1 (x1 )τ2 (x2 ) for all x1 ∈ M1 , x2 ∈ M2 , and τ is a trace. Since M1′ ⊗M2′ ⊂ (M1 ⊗M2 )′ and ̂1 ⊗ ̂1 is M1′ ⊗M2′ -cyclic, Exercise 4.3 implies that τ is faithful. For the last assertion, assume that M1 and M2 are II1 factors. If x1 ∈ M1 and x2 ∈ M2 , then by Exercise 5.19(1) we get that τ1 (x1 )1̂ ∈ Kx1 and τ2 (x2 )1̂ ∈ Kx2 . This easily implies that τ(x1 ⊗ x2 )(1̂ ⊗ 1)̂ = τ1 (x1 )1̂ ⊗ τ2 (x2 )1̂ ∈ Kx1 ⊗x2 . Since the linear span of {x1 ⊗ x2 | x1 ∈ M1 , x2 ∈ M2 } is SOT-dense and so ‖ ⋅ ‖2 -dense in M1 ⊗M2 , Exercise 5.19(2) gives that M1 ⊗M2 is a II1 factor. Exercise 5.20. Let (X1 , μ1 ) and (X2 , μ2 ) be standard probability spaces and consider the product probability space (X1 × X2 , μ1 ⊗ μ2 ). Prove that L∞ (X1 )⊗L∞ (X2 ) ≅ L∞ (X1 × X2 ). Exercise 5.21. Let Γ1 and Γ2 be countable groups. Prove that L(Γ1 )⊗L(Γ2 ) ≅ L(Γ1 × Γ2 ).

5.6 Free product von Neumann algebras We next recall the definition of the free product of two tracial von Neumann algebras (M1 , τ1 ) and (M2 , τ2 ). Denote Hi = L2 (Mi ) ⊖ ℂ̂1, for i ∈ {1, 2}, and define the Hilbert space H = ℂ̂1 ⊕ (⨁

⨁

n≥1 i1 =i̸ 2 =⋅⋅⋅ ̸ =i̸ n

Hi1 ⊗ Hi2 ⊗ ⋅ ⋅ ⋅ ⊗ Hin ).

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Also, for i ∈ {1, 2}, define the Hilbert space ℋi = ℂ̂1 ⊕ (⨁

⨁

n≥1 i=i̸ 1 =i̸ 2 =⋅⋅⋅ ̸ =i̸ n

Hi1 ⊗ Hi2 ⊗ ⋅ ⋅ ⋅ ⊗ Hin ).

Then we have a natural unitary identification H = L2 (Mi ) ⊗ ℋi which allows us to view Mi ⊂ 𝔹(H). Definition 5.22. The free product von Neumann algebra M1 ∗ M2 is defined as the von Neumann algebra generated by M1 , M2 ⊂ 𝔹(H). Proposition 5.23. Let (M1 , τ1 ) and (M2 , τ2 ) be tracial von Neumann algebras. Then the free product M = M1 ∗ M2 is a tracial von Neumann algebra. ̂ Let x = x1 x2 ⋅ ⋅ ⋅ xn , Proof. Let τ : M → ℂ be the normal state given by τ(x) = ⟨x 1,̂ 1⟩. where n ≥ 1, xj ∈ Mij and τ(xj ) = 0, for every 1 ≤ j ≤ n, and ij ≠ ij+1 , for every 1 ≤ j ≤ n − 1. Since x̂1 = x̂i ⊗ x̂i ⊗ ⋅ ⋅ ⋅ ⊗ x̂i ∈ Hi ⊗ Hi ⊗ ⋅ ⋅ ⋅ ⊗ Hi is orthogonal to 1,̂ we get that τ(x) = 0. 1

2

n

1

2

n

Let y = ym ⋅ ⋅ ⋅ y2 y1 , where m ≥ 1, yk ∈ Mlk and τ(yk ) = 0, for every 1 ≤ k ≤ m, and lk ≠ lk+1 , for every 1 ≤ k ≤ m − 1. By using the previous paragraph, it follows that τ(xy) = τ(yx) = 0, unless n = m and ij = lj , for every 1 ≤ j ≤ n, in which case we have that τ(xy) = τ(yx) = ∏j=1 τij (xj yj ). In either case, we get that τ(xy) = τ(yx) and since the linear span of x (respectively, y) of the form above is SOT-dense in {z ∈ M | τ(z) = 0}, we deduce that τ is tracial.

Exercise 5.24. Let (M1 , τ1 ) and (M2 , τ2 ) be tracial von Neumann algebras such that M1 is diffuse and M2 ≠ ℂ1. Let (uk ) ∈ 𝒰 (M1 ) be a sequence such that uk → 0, in the weak operator topology. Prove that uk xuk∗ → 0, in the weak operator topology, for every x ∈ M1 ∗ M2 with EM1 (x) = 0. Use this to prove that M1 ∗ M2 is a II1 factor. Exercise 5.25. Let Γ1 and Γ2 be countable groups. Prove that L(Γ1 ) ∗ L(Γ2 ) ≅ L(Γ1 ∗ Γ2 ).

5.7 Ultraproduct von Neumann algebras We end this section by defining ultraproducts of tracial von Neumann algebras. We start by reviewing the notion of free ultrafilters on ℕ. Definition 5.26. The Stone–Čech compactification of ℕ, denoted by βℕ, is defined as the Gelfand dual of the abelian C∗ -algebra ℓ∞ (ℕ). An ultrafilter of ℕ is an element of βℕ, i. e., a nonzero homomorphism ω : ℓ∞ (ℕ) → ℂ. For n ∈ ℕ, we denote by en ∈ βℕ the evaluation at n, i. e., en (f ) = f (n). An ultrafilter ω ∈ βℕ is free if it does not belong to ℕ ≡ {en }n∈ℕ . Remark 5.27. We have that βℕ \ ℕ ≠ 0. To see this, let Kn ⊂ βℕ be the weak∗ -closure of {ek | k > n}. Then Kn is weak∗ -compact by Alaoglu’s theorem and Kn+1 ⊂ Kn , for all n.

74 � A. Ioana Thus, ⋂n Kn ≠ 0. If ω ∈ ⋂n Kn , then ω ∈ Kn and thus ω(δn ) = 0, for all n ∈ ℕ. This shows that ω ∈ ̸ ℕ. Exercise 5.28. If ω ∈ βℕ, we denote limn→ω xn := ω((xn )n ), for every (xn )n ∈ ℓ∞ (ℕ). Prove that if ω ∈ βℕ \ ℕ and limn→∞ xn = x, then limn→ω xn = x. Warning. The notation limn→ω will be used to denote the limit of a sequence of complex numbers along an ultrafilter ω and should not be confused with the limit as n approaches an ordinal ω. Definition 5.29. Let ω ∈ βℕ \ ℕ and (Mn , τn ) be a sequence of tracial von Neumann algebras. Let ∏n∈ℕ Mn be the C ∗ -algebra of sequences (xn ) with xn ∈ Mn , ∀n ∈ ℕ, and ‖(xn )‖ := sup ‖xn ‖ < ∞. Let ℐω ⊂ ∏n∈ℕ Mn be the two-sided norm closed ideal consisting of sequences (xn ) ∈ ∏n∈ℕ Mn such that limn→ω ‖xn ‖2 = 0. The ultraproduct ∏n∈ω Mn is defined as the quotient ∏n∈ℕ Mn /ℐω . Then ∏n∈ω Mn is a tracial von Neumann algebra which has a canonical trace τω ((xn )) = limn→ω τn (xn ) (see [3, Appendix 4.A] or [1, Proposition 5.4.1] for a proof of this fact). If Mn = M, ∀n ∈ ℕ, we denote ∏n∈ω Mn by M ω and call it the ultrapower of M.

6 Properties of von Neumann algebras In the first two parts of this section we present two fundamental representationtheoretic properties (amenability and property (T)) of groups and von Neumann algebras. We end this section by briefly discussing two asymptotic properties of II1 factors (property Gamma and McDuff’s property).

6.1 Amenability Definition 6.1. A countable group Γ is called amenable if there exists a state φ : ℓ∞ (Γ) → ℂ which is invariant under the left translation action: φ(g ⋅ f ) = φ(f ), for all g ∈ Γ and f ∈ ℓ∞ (Γ). Here, g ⋅ f ∈ ℓ∞ (Γ) is defined as (g ⋅ f )(h) = f (g −1 h). Theorem 6.2. Let Γ be a countable group. Then the following conditions are equivalent: (1) Γ is amenable. (2) Γ satisfies the Reiter condition: there exists a sequence of nonnegative functions fn ∈ ℓ1 (Γ) such that ‖fn ‖1 = 1, for all n, and limn→∞ ‖g ⋅ fn − fn ‖1 = 0, for all g ∈ Γ. (3) Γ satisfies the Følner condition: there exists a sequence of finite subsets Fn ⊂ Γ such that limn→∞ |gFn \ Fn |/|Fn | = 0, for all g ∈ Γ. (4) The left regular representation of Γ has almost invariant vectors: there exists a sequence ξn ∈ ℓ2 (Γ) such that ‖ξn ‖2 = 1, for all n, and limn→∞ ‖λ(g)ξn − ξn ‖2 = 0, for all g ∈ Γ.

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Proof. The proof relies on two very useful tricks, due to Day (the proof of (1) ⇒ (2)) and Namioka (the proof of (2) ⇒ (3)). Enumerate Γ = {gn }n≥1 . (1) ⇒ (2) Fix n ≥ 1 and consider the convex subset C := {(g1 ⋅ f − f , g2 ⋅ f − f , . . . , gn ⋅ f − f ) | f ∈ ℓ1 (Γ), f ≥ 0, ‖f ‖1 = 1} of the Banach space ℓ1 (Γ)⊕n with the norm ‖(f1 , f2 , . . . , fn )‖ = ∑ni=1 ‖fi ‖1 . ‖⋅‖

We claim that 0 = (0, 0, . . . , 0) ∈ C . Assuming this claim, we can find fn ∈ ℓ1 (Γ) such that fn ≥ 0, ‖fn ‖1 = 1 and ∑ni=1 ‖gi ⋅ fn − fn ‖1 ≤ 1/n. This clearly implies (2). ‖⋅‖

If the claim were false, then since C ⊂ ℓ1 (Γ)⊕n is a closed convex set and (ℓ1 (Γ)⊕n )∗ = ℓ (Γ)⊕n , the Hahn–Banach separation theorem would imply the existence of F1 , F2 , . . . , Fn ∈ ℓ∞ (Γ) and α > 0 such that ∑ni=1 ℜ⟨gi ⋅ f − f , Fi ⟩ ≥ α, for any f ∈ ℓ1 (Γ) with f ≥ 0 and ‖f ‖1 = 1. If we put F = ∑ni=1 ℜ(gi−1 ⋅ Fi − Fi ), then the last inequality can be rewritten as ⟨f , F⟩ ≥ α, for any f ∈ ℓ1 (Γ) with f ≥ 0 and ‖f ‖1 = 1. For f = δg , this implies that F(g) ≥ α, for all g ∈ Γ. Thus, we get that φ(F) ≥ φ(α1) = α > 0. On the other hand, φ(F) = ∑ni=1 (φ(ℜ(gi−1 ⋅ Fi )) − φ(ℜFi )) = 0. This gives the desired contradiction. (2) ⇒ (3) If f1 , f2 ∈ ℓ1 (Γ) and f1 , f2 ≥ 0, then Fubini’s theorem implies that ∞

∞

∞

‖f1 − f2 ‖1 = ∫ ‖1{f1 >t} − 1{f2 >t} ‖1 dt 0

and

‖f1 ‖1 = ∫ ‖1{f1 >t} ‖1 dt.

(6.1)

0

By (2), for any n ≥ 1 we can find f ∈ ℓ1 (Γ) such that f ≥ 0, ‖f ‖1 = 1 and ∑ni=1 ‖gi ⋅f −f ‖1 < 1/n. For t > 0, let Kt = {f > t}. Since f ∈ ℓ1 (Γ), we get that Kt is a finite subset of Γ. Also, note that {g ⋅ f > t} = gKt and thus ‖1{g⋅f >t} − 1{f >t} ‖1 = |gKt △ Kt |, for all g ∈ Γ. By combining the last inequality with (6.1), we derive that ∞ n

∫ ∑ |gi Kt △ Kt | dt < 0 i=1

∞

|K | 1 1 = ‖f ‖ = ∫ t dt. n n 1 n 0

Hence, there is tn > 0 such that Fn := Ktn satisfies ∑ni=1 |gi Fn △Fn | < |Fn |/n. This proves (3). (3) ⇒ (4) Let ξn := 1Fn /√|Fn |. Then ‖ξn ‖2 = 1 and ‖λ(g)ξn − ξn ‖2 = √|gFn △ Fn |/|Fn |, for all n ≥ 1 and g ∈ Γ, which clearly implies (4). (4) ⇒ (1) Let ω be a free ultrafilter on ℕ. Define φ : ℓ∞ (Γ) → ℂ by letting φ(f ) = limn→ω ⟨fξn , ξn ⟩. Then φ is a state and φ(g ⋅ f ) = limn→ω ⟨f (g −1 ⋅ ξn ), g −1 ⋅ ξn ⟩ = φ(f ), for all f ∈ ℓ∞ (Γ), g ∈ Γ. Exercise 6.3. Let Γ be a countable group. Assume that (a) any finitely generated subgroup of Γ is amenable or (b) Γ is abelian. Prove that Γ is amenable. Proposition 6.4. Group 𝔽2 is not amenable.

76 � A. Ioana Proof. Assume by contradiction that there exists a left translation invariant state φ : ℓ∞ (𝔽2 ) → ℂ. Define m : 𝒫 (𝔽2 ) → [0, 1] by m(A) = φ(1A ). Then m is finitely additive (m(A ∪ B) = m(A) + m(B), for every disjoint A, B ⊂ 𝔽2 ) and left invariant (m(gA) = m(A), for every g ∈ 𝔽2 and A ⊂ 𝔽2 ). Let a and b be the free generators of 𝔽2 . Let S be the set of elements of 𝔽2 whose reduced form begins with a nonzero power of a, and put T = 𝔽2 \ S. Then aT ⊂ S, bS ∪ b2 S ⊂ T and bS ∩ b2 S = 0. Thus, we get m(S) ≥ m(aT) = m(T) ≥ m(bS ∪ b2 S) = m(bS) + m(b2 S) = 2m(S). This implies that m(S) = m(T) = 0. Since m(S) + m(T) = m(𝔽2 ) = 1, this provides a contradiction. Exercise 6.5. Let Γ1 and Γ2 be any countable groups such that |Γ1 | > 1 and |Γ2 | > 2. Prove that the free product group Γ = Γ1 ∗ Γ2 is not amenable. Definition 6.6. A tracial von Neumann algebra (M, τ) is called amenable if there exists a state Φ : 𝔹(L2 (M)) → ℂ such that Φ|M = τ and Φ(Tx) = Φ(xT), for all x ∈ M and T ∈ 𝔹(L2 (M)). Theorem 6.7. Let Γ be a countable group. Then Γ is amenable if and only if L(Γ) is amenable. Proof. Assume that Γ is amenable and let φ : ℓ∞ (Γ) → ℂ be a left translation invariant state. Define a state Φ : 𝔹(ℓ2 (Γ)) → ℂ by letting Φ(T) := φ(g 󳨃→ ⟨Tδg , δg ⟩). If T ∈ L(Γ), then for all g ∈ Γ we have ⟨Tδg , δg ⟩ = ⟨Tρ(g)δe , ρ(g)δe ⟩ = ⟨ρ(g)∗ Tρ(g)δe , δe ⟩ = ⟨Tδe , δe ⟩ = τ(T), and thus Φ(T) = τ(T). If T ∈ 𝔹(ℓ2 (Γ)) and h ∈ Γ, then the left invariance of φ gives that Φ(λ(h)Tλ(h)∗ ) = φ(g 󳨃→ ⟨λ(h)Tλ(h)∗ δg , δg ⟩) = φ(g 󳨃→ ⟨Tδh−1 g , δh−1 g ⟩) = Φ(T). Thus, if 𝒞 := {x ∈ L(Γ) | Φ(Tx) = Φ(xT), for all T ∈ 𝔹(ℓ2 (Γ))}, then λ(g) ∈ 𝒞 , for all g ∈ Γ. By Cauchy–Schwarz inequality, we have that |Φ(Tx)|2 ≤ Φ(TT ∗ )Φ(x ∗ x) ≤ ‖T‖2 Φ(x ∗ x) = ‖T‖2 ‖x‖22 and similarly |Φ(xT)|2 ≤ ‖T‖2 ‖x‖22 , for all x ∈ L(Γ) and T ∈ 𝔹(ℓ2 (Γ)). This implies that 𝒞 is ‖ ⋅ ‖2 -closed. Since 𝒞 contains the linear span of λ(Γ), we conclude that 𝒞 = L(Γ). This shows that L(Γ) is amenable. Conversely, assume that L(Γ) is amenable. Let Φ be a state on 𝔹(ℓ2 (Γ)) such that Φ(Tx) = Φ(xT), for all x ∈ L(Γ) and T ∈ 𝔹(ℓ2 (Γ)). Consider the natural embedding ℓ∞ (Γ) ⊂ 𝔹(ℓ2 (Γ)) and notice that λ(g)fλ(g)∗ = f ∘ g −1 = g ⋅ f , for all f ∈ ℓ∞ (Γ) and g ∈ Γ. Thus, for all f ∈ ℓ∞ (Γ) and g ∈ Γ, we have that Φ(g ⋅ f ) = Φ(λ(g)fλ(g)∗ ) = Φ(f ). This implies that Γ is amenable. Exercise 6.8. Prove that any hyperfinite tracial von Neumann algebra (M, τ) is amenable.

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It is a remarkable fact, proved by Connes in his celebrated work [7], that the converse is true. A tracial von Neumann algebra (M, τ) is called separable if the Hilbert space L2 (M) is separable. Theorem 6.9 (Connes, [7]). Let (M, τ) be a separable tracial von Neumann algebra. Then the following are equivalent: (1) M is amenable. (2) M is injective: there exists a conditional expectation E : 𝔹(L2 (M)) → M. (3) There exists a sequence ξn ∈ L2 (M)⊗L2 (M) such that ⟨xξn , ξn ⟩ = τ(x) for all n ≥ 1 and limn→∞ ‖xξn − ξn x‖ = 0, for all x ∈ M. (4) M is hyperfinite. Consequently, any separable amenable II1 factor M is isomorphic to R. In particular, L(Γ) is isomorphic to R, for any icc amenable group Γ. For a proof of this result, see [1, Chapters 11 and 13].

6.2 Property (T) Definition 6.10 ([17]). A countable group Γ has Kazhdan’s property (T) if any unitary representation π : Γ → 𝒰 (H) which has almost invariant vectors admits a nonzero invariant vector. Examples of countable groups with property (T) include SLn (ℤ), for n ≥ 3, and, more generally, any lattice in a simple Lie group of rank at least 2 [17]. For more on property (T), see [2]. Exercise 6.11. Let Γ be a countable group and (Γn )n∈Γ be an increasing sequence of subgroups with ⋃n∈ℕ Γn = Γ. Endow X = ⨆n∈ℕ Γ/Γn with the left multiplication action of Γ and denote by π : Γ → 𝒰 (ℓ2 (X)) the associated unitary representation. Prove that (1) π has almost invariant vectors. (2) π has nonzero invariant vectors if and only if Γn = Γ for some n ∈ ℕ. Deduce that if Γ has property (T), then it is finitely generated. Exercise 6.12. Prove that any countable amenable group Γ which has property (T) must be finite. We next recall Connes and Jones’ notion of property (T) for II1 factors. Definition 6.13 ([10]). A tracial factor M has property (T) if there exists F ⊂ M finite and δ > 0 such that whenever H is a Hilbert M-bimodule and ξ ∈ H is a unit vector with maxx∈F ‖xξ − ξx‖ < δ, there exists a nonzero vector η ∈ H such that xη = ηx, ∀x ∈ M.

78 � A. Ioana Exercise 6.14. Prove that any amenable tracial factor M which has property (T) must be finite dimensional. Proposition 6.15 (Connes and Jones, [10]). Let Γ be a non-trivial countable icc group. Then Γ has property (T) if and only if L(Γ) has property (T). Proof. We prove the “only if” assertion and refer the reader to [10] for the proof of the “if” assertion. Assume that Γ has property (T) and enumerate Γ = {gn }n∈ℕ . Then there are S ⊂ Γ finite and δ > 0 such that if π : Γ → 𝒰 (H) is a unitary representation and ξ ∈ H is a unit vector with maxg∈S ‖π(g)ξ − ξ‖ < δ, then H has a nonzero π(Γ)-invariant vector. Otherwise, for any n ∈ ℕ we find a unitary representation πn : Γ → 𝒰 (Hn ) without nonzero invariant vectors and a unit vector ξn ∈ Hn such that max1≤i≤n ‖πn (gi )ξn − ξn ‖ ≤ 1 . Then the representation π = ⨁n∈ℕ πn has almost invariant vectors but no nonzero n invariant vectors. Let H be a Hilbert L(Γ)-bimodule which has a unit vector ξ such that maxg∈S ‖ug ξ − ξug ‖ < δ. Define a unitary representation π : Γ → 𝒰 (H) by letting π(g)η = ug ηug∗ , for every g ∈ Γ, η ∈ H. Then ‖π(g)ξ − ξ‖ = ‖ug ξ − ξug ‖ < δ, for every g ∈ S. By the previous paragraph, we can find a nonzero vector η such that π(g)η = η and thus ug η = ηug , for all g ∈ Γ. Since the linear span of {ug | g ∈ Γ} is SOT-dense in L(Γ) we conclude that xη = ηx, for all x ∈ L(Γ). Definition 6.16. Let M be a II1 factor. We denote by Aut(M) be group of automorphisms of M and by Inn(M) = {Ad(u) | u ∈ 𝒰 (M)} the subgroup of inner automorphisms of M. We endow Aut(M) with the pointwise ‖ ⋅ ‖2 -topology: θi → θ ⇐⇒ ‖θi (x) − θ(x)‖2 → 0, ∀x ∈ M. The outer automorphism group of M is defined as the quotient group Out(M) = Aut(M)/Inn(M). The fundamental group of M is defined by ℱ (M) = {

τ(p) | p, q ∈ M nonzero projections such that pMp ≅ qMq}. τ(q)

The fundamental group was introduced by Murray and von Neumann in [20] who showed that it is a multiplicative subgroup of (0, +∞), and that ℱ (R) = (0, +∞). The outer automorphism group of R is also a very large group that contains every second countable locally compact group. In contrast, II1 factors with property (T) have “small” (countable) symmetry groups: Proposition 6.17. Let M be a II1 factor with property (T). Then Inn(M) is an open subgroup of Aut(M). Thus, Out(M) is countable. Proof. Let F ⊂ M, δ > 0 as in Definition 6.13. To prove that Inn(M) < Aut(M) is an open subgroup, it suffices to show that a neighborhood of IdM in Aut(M) is contained

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in Inn(M). Let θ ∈ Aut(M) such that maxx∈F ‖θ(x) − x‖2 < δ. Consider the Hilbert space H = L2 (M) with the Hilbert M-bimodule structure given by xξy = θ(x)Jy∗ Jξ. Then for every x ∈ F we have 󵄩 󵄩 ‖x̂1 − ̂1x‖2 = 󵄩󵄩󵄩θ(x) − x 󵄩󵄩󵄩2 < δ. Thus, we can find a unit vector η ∈ L2 (M) such that θ(x)η = Jx ∗ Jη, ∀x ∈ M. Then φ : M → ℂ given by φ(x) = ⟨xη, η⟩ is a tracial state. Exercise 4.10 gives that φ = τ. So ‖xη‖2 = ‖x‖2 , ∀x ∈ M, and Exercise 4.12 implies that η = û , for some u ∈ 𝒰 (M). Therefore, θ = Ad(u) belongs to Inn(M). Since Inn(M) < Aut(M) is an open subgroup, it is also closed. Moreover, the quotient group Out(M) = Aut(M)/Inn(M) is both discrete and separable, and thus must be countable. This result was proved by Connes in [8] when M = L(Γ), for an icc property (T) group Γ. Connes moreover established that ℱ (M) is countable, for such M. In a major breakthrough in [23], Popa gave the first examples of II1 factors M with trivial fundamental group, ℱ (M) = {1}. The existence of II1 factors M with trivial outer automorphism group, Out(M) = {e}, was proved later on in [14]. Only very recently, the first examples of II1 factors with property (T) that have trivial fundamental group and, respectively, trivial outer automorphism group were found in [4] and [5].

6.3 Property Gamma and McDuff’s property Definition 6.18. Let M be a II1 factor. A sequence (xn ) ⊂ M is called uniformly bounded if supn ‖xn ‖ < ∞ and almost central if ‖xn y − yxn ‖2 → 0, ∀y ∈ M. We say that M has property Gamma if it admits a uniformly bounded central sequence (xn ) such that infn ‖xn − τ(xn )1‖2 > 0. Property Gamma was introduced by Murray and von Neumann in [20] who used it to show that R ≇ L(𝔽2 ). Specifically, they proved that R has property Gamma while L(𝔽2 ) does not. Let M be a II1 factor with property Gamma and (xn ) ⊂ M be as in Definition 6.18. Consider the “diagonal” embedding M ⊂ M ω given by M ∋ y 󳨃→ (yn = y) ∈ M ω and view x = (xn ) ∈ M ω . Then x belongs to M ′ ∩ M ω = {a ∈ M ω | ab = ba, ∀b ∈ M}. Since ‖x − τ(x)1‖2 = limn→ω ‖xn − τ(xn )1‖2 > 0, we get that M ′ ∩ M ω ≠ ℂ1. Conversely, if M ′ ∩ M ω ≠ ℂ1, then M has property Gamma. Moreover, the following holds: Theorem 6.19. Let M be a separable II1 factor. Then the following are equivalent: (1) M has property Gamma. (2) M ′ ∩ M ω ≠ ℂ1, where ω is a free ultrafilter on ℕ. (3) Inn(M) is not a closed subgroup of Aut(M).

80 � A. Ioana (4) There exists a sequence of unit vectors ξn ∈ L2 (M) ⊖ ℂ̂1 such that ‖xξn − ξn x‖2 → 0, ∀x ∈ M. For a proof of this result, we refer the reader to [1, Chapter 15]. The above equivalences were obtained as follows: (1) ⇔ (2) in [6], (1) ⇔ (3) in [6, 26], and (1) ⇔ (4) in [7]. Exercise 6.20. A group Γ is called inner amenable if the representation π : Γ → 𝒰 (ℓ2 (Γ \ {e})) given by π(g)(δh ) = δghg −1 has almost invariant vectors. Assume that Γ ↷ (X, μ) is a pmp action of a non-trivial noninner amenable group Γ and let M = L∞ (X) ⋊ Γ. Prove that M ′ ∩ M ω ⊂ L∞ (X)ω . Deduce, by taking X to consist of one point, that L(Γ) is a II1 factor without property Gamma. Exercise 6.21. Let Γ be a nonamenable group such that the centralizer {h ∈ Γ | gh = hg} is amenable, ∀g ∈ Γ \ {e}. Prove that Γ is not inner amenable. Deduce that 𝔽2 is not inner amenable. Definition 6.22. Let M be a II1 factor. We say that M is McDuff if it admits two uniformly bounded central sequences (xn ), (yn ) such that infn ‖xn yn − yn xn ‖2 > 0. The following is the main result of [19]: Theorem 6.23. Let M be a separable II1 factor. Then the following are equivalent: (1) M is McDuff. (2) M ′ ∩ M ω is not abelian, where ω is a free ultrafilter on ℕ. (3) M is isomorphic to M⊗R. If M is McDuff then it has property Gamma. The converse is false, as the next example shows. Example 6.24. Recall that a pmp action Γ ↷ (X, μ) of a countable group Γ is strongly ergodic if for any sequence of measurable sets (An ) ⊂ X such that μ(gAn △ An ) → 0, for every g ∈ Γ, we must have that μ(An )μ(X \ An ) → 0. Let ℤ ↷σ (X, μ) be a free ergodic pmp action and δ : 𝔽2 → ℤ be an onto homomorphism. Consider the ergodic pmp action 𝔽2 ↷σ∘δ (X, μ). Since 𝔽2 is not inner amenable, Exercise 6.20 gives that M = L∞ (X) ⋊σ∘δ 𝔽2 is a II1 factor which satisfies M ′ ∩ M ω ⊂ L∞ (X)ω . Since being strongly ergodic is an orbit equivalence invariant and σ is orbit equivalent to any other free ergodic pmp action of ℤ by Dye’s theorem, σ is not strongly ergodic. Thus, σ ∘ δ is not strongly ergodic and therefore M ′ ∩ L∞ (X)ω ≠ ℂ1. Therefore, M ′ ∩ M ω is nontrivial and abelian. In other words, M has property Gamma but is not McDuff.

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Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

C. Anantharaman and S. Popa, An introduction to II1 factors. Available at https://www.math.ucla.edu/ ~popa/Books/IIunV15.pdf. B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. N. P. Brown and N. Ozawa, C∗ -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, AMS, Providence, RI. I. Chifan, S. Das, C. Houdayer and K. Khan, Examples of property (T) factors with trivial fundamental group, Amer. J. Math. in press. arXiv:2003.08857. I. Chifan, A. Ioana, D. Osin and B. Sun, Wreath-like product groups and rigidity of their von Neumann algebras. arXiv:2111.04708. A. Connes, Almost periodic states and factors of type III1 , J. Funct. Anal. 16 (1974), 415–445. A. Connes, Classification of injective factors, Ann. Math. 74 (1976), 73–115. A. Connes, A factor of type II1 with countable fundamental group, J. Oper. Theory 4 (1980), 151–153. A. Connes, Classification des facteurs, Operator algebras and applications (Kingston, Ont., 1980), Part 2, Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R, 1982, pp. 43–109. A. Connes and V. F. R. Jones, Property (T) for von Neumann algebras, Bull. Lond. Math. Soc. 17 (1985), 57–62. J. B. Conway, A course in operator theory, Graduate Studies in Mathematics, vol. 21, AMS, 1999. J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, I, II, Trans. Am. Math. Soc. 234 (1977), 289–324, 325–359. A. Ioana, Rigidity for von Neumann algebras, Proc. Int. Cong. of Math. (Rio de Janeiro, 2018), vol. 2, pp. 1635–1668. A. Ioana, J. Peterson and S. Popa, Amalgamated free products of w-rigid factors and calculation of their symmetry groups, Acta Math. 200 (2008), 85–153. V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. R. Kadison and J. R. Ringrose, Fundamental of the theory of operator algebras, Graduate Studies in Mathematics, vol. 16, AMS, 1997. D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. 1 (1967), 63–65. A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer, New York, 1995. xviii+402 pp. D. McDuff, Central sequences and the hyperfinite factor, Proc. Lond. Math. Soc. (3) 21 (1970), 443–461. F. Murray and J. von Neumann, Rings of operators, IV, Ann. Math. 44 (1943), 716–808. F. J. Murray and J. von Neumann, On rings of operators, Ann. Math. 37 (1936), 116–229. S. Popa, Correspondences, INCREST preprint 56 (1986). Available at https://www.math.ucla.edu/ ~popa/preprints.html. S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. Math. 163 (2006), 809–899. S. Popa, Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I, Invent. Math. 165 (2006), 369–408. S. Popa, Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians, Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 445–477. H. Sakai, On automorphism groups of II1 -factors, Tohoku Math. J. (2) 26 (1974), 423–430. I. M. Singer, Automorphisms of finite factors, Am. J. Math. 77 (1955), 117–133. G. Szabo, Introduction to C∗ -algebras, Model theory of operator algebras. M. Takesaki, Theory of operator algebras, I, Springer, New York–Heidelberg, 1979. vii+415 pp. S. Vaes, Rigidity for von Neumann algebras and their invariants, Proceedings of the ICM, Vol. III (Hyderabad, India, 2010), Hindustan Book Agency, 2010, pp. 1624–1650. J. von Neumann, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann. 102 (1930), no. 1, 370–427.

Bradd Hart

An introduction to continuous model theory Abstract: We present an introduction to modern continuous model theory with an emphasis on its interactions with topics covered in this volume such as C∗ -algebras and von Neumann algebras. The role of ultraproducts is highlighted and expositions of definable sets, imaginaries, quantifier elimination, and separable categoricity are included. Keywords: Continuous model theory, metric structures, Urysohn sphere, C∗ -algebras MSC 2020: 03C66, 03C98, 46L05

1 Introduction It is not a new idea to have more truth values than just true and false. For example, Łukasiewicz and Tarski suggested a real-valued logic in the 1930s [40]. The modern understanding of continuous model theory has other antecedents as well. Chang and Keisler wrote a book [13] in which they consider a logic with truth values in arbitrary compact Hausdorff spaces. This logic had many quantifiers and did not catch on; see Ben Yaacov [6] for further commentary. Henson introduced positive bounded logic in [33], mostly in the service of the model theory of normed vector spaces with additional structure. Ben Yaacov introduced the very general setting of cats (short for compact abstract theories) in [4] in order to capture expansions of first-order logic that would allow the seamless use of hyperimaginaries as introduced in [32]; this line of research built on earlier work around Robinson theories by Hrushovski, Pillay, and others. In particular, it was realized that Hausdorff cats interpreted a metric (see [5]). Ben Yaacov, Berenstein, Henson, and Usvyatsov then introduced a logic for metric structures which is the subject of this paper. This logic is first presented in [10], which strangely was published after the de facto standard [7]. The history of the ultraproduct is very interesting and more colorful than most model theorists are aware. A short history is contained in [43]. The thumbnail sketch is that an ultraproduct-like construction in the operator algebra context was introduced by Kaplansky and Wright in the early 1950s. Sakai also used an ultraproduct in the early 1960s, again in the service of operator algebras. McDuff’s systematic use of ultraproducts in her paper [41] highlighted how important the construction had become in the operator algebra world. In parallel, the seminal introduction of the ultraproduct in model Acknowledgement: The author was supported by the NSERC. Bradd Hart, Department of Mathematics and Statistics, McMaster University, 1280 Main St., Hamilton ON, Canada L8S 4K1, e-mail: [email protected]; URL: http://ms.mcmaster.ca/~bradd/ https://doi.org/10.1515/9783110768282-003

84 � B. Hart theory by Łoś [39] and its use by Robinson in the development of nonstandard analysis were key early moments in model theory. It does not seem that anyone at that time saw anything more than a formal connection between the two uses of ultraproducts. Keisler [37] provides a brief history on the model theory side in which he gives the nod for a precursor to the ultraproduct to Skolem and interestingly to Hewitt [35], who was working on rings of continuous functions. Although there was prior work done by Krivine, Stern, and others, the first systematic connection between the analytic ultraproduct construction and model theory, particularly for Banach spaces, was the work of Henson [33], in which he also introduces his positive bounded logic. A general exposition of this logic appears with Iovino in [34]. The ultraproduct plays a slightly different role in the development of continuous model theory than it does in classical logic. While the connection between the ultraproduct construction and the compactness theorem is the same for both logics, it is the exploratory nature of the ultraproduct which is different, by which we mean that there are classes of structures appearing in analysis where an ultraproduct is used but without a formal language being present. The fact that mathematicians are using ultraproducts in certain settings “in the wild” often leads one to examine whether the class can in fact be presented in a model-theoretic framework. Examples relevant to this volume where this has been done successfully include Banach spaces ([33]), C∗ -algebras ([18]), tracial von Neumann algebras ([17]), W∗ -probability spaces ([15], [28] and [3]) and quantum group actions ([21] and [22]). As we will see in Example 3.8, every classical elementary class can be seen as a continuous elementary class. Continuous logic, however, is a proper extension of classical first-order logic. While there are a number of ways to see this, here is one suggestion based on a Borel complexity argument due to Farah, Toms, and Törnquist. In [20], they show that the isomorphism relation on the class of separable C∗ -algebras is turbulent and not Borel-reducible to any class of countable classical structures (see their article for the relevant definitions). Of course, the isomorphism relation for every class of countable classical structures is Borel-reducible to itself, so the isomorphism relation for the class of separable C∗ -algebras is not equivalent to any class of countable classical structures. As we will see in this chapter, the class of C∗ -algebras is an elementary class (Example 5.18) and by this complexity argument, this class is not equivalent to any classical elementary class. We now give a brief outline of the paper. After a quick reminder of the basics of ultrafilters in Section 2, we introduce the syntax and semantics of continuous logic in Sections 3 and 4. Basic continuous model theory is then developed in Section 5, where a generalized notion of formula is introduced. As was indicated above, ultraproducts play a crucial role in continuous model theory and the ultraproduct of metric structures is introduced in Section 6. Section 7 introduces the notion of type in continuous model theory and the type space is used to give another characterization of general formulas. A critical section is Section 8, where definable sets are discussed and contrasted with the notion of definable set from classical logic. Section 9 highlights the formulation of quantifier

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elimination in continuous logic and gives several examples. In Section 10, imaginary elements in the continuous setting are considered and the conceptual completeness theorem is stated in this context. We end the article with a discussion of omitting types in Section 11 and separable categoricity in Section 12. Inevitably, there are topics that we left out. In particular, we do not include a discussion of a proof system for continuous logic. This topic is covered in [8] and is important for the topic of decidability in continuous logic (see, for example, Section 3 of Goldbring’s paper in this volume, [25]). Another topic not discussed here is stability theory in continuous logic; one can find the basics of this theory in [7] and [10]. I would like to thank Isaac Goldbring for our many discussions regarding this article and for his patience as an editor.

2 Preliminaries on ultrafilters and ultraproducts Ultrafilters and ultraproducts will play an important role in continuous model theory. We cover the requisite facts in this section. Definition 2.1. Suppose that X is a set and F ⊂ P(X), where P(X) is the power set of X. We say that (1) F is a (proper) filter on X if: (a) 0 ∈ ̸ F, (b) A, B ∈ F implies A ∩ B ∈ F, and (c) A ⊂ B ⊂ X and A ∈ F implies B ∈ F. (2) F is an ultrafilter on X if it is a filter on X and, for every A ⊂ X, either A ∈ F or X \ A ∈ F (but not both). The following is an important characterization of ultrafilters. Note that we are assuming the axiom of choice throughout this paper. Exercise 2.2. Suppose F ⊂ P(X). Then (1) F is an ultrafilter on X if and only if F is a maximal (under inclusion) filter on X. (2) F is contained in an ultrafilter on X if and only if F has the finite intersection property, that is, for any finitely many A1 , . . . , An ∈ F, ⋂ni=1 Ai ≠ 0. Example 2.3. (1) If X is any infinite set and F is the collection of cofinite subsets of X, then F is a filter on X, called the Fréchet filter on X. Note that the Fréchet filter is not an ultrafilter. (2) If X is any set, a ∈ X is any element, and F is the collection of subsets of X which contain a, then F is an ultrafilter on X, called the principal ultrafilter on X generated by a.

86 � B. Hart Exercise 2.4. If X is an infinite set, then an ultrafilter on X is nonprincipal if and only if it contains the Fréchet filter on X. Consequently, nonprincipal ultrafilters on X exist. The notion of an ultralimit will be important throughout this paper. Exercise 2.5. Suppose that (ri : i ∈ I) is a sequence of real numbers and 𝒰 is an ultrafilter on I. Further suppose that there is a real number B such that {i ∈ I : |ri | < B} ∈ 𝒰 . Prove that there is a unique real number r satisfying, for every ϵ > 0, that {i ∈ I : |ri − r| < ϵ} ∈ 𝒰 . Definition 2.6. In the context of the previous exercise, we say that r is the ultralimit of (ri : i ∈ I) along 𝒰 , written r = lim𝒰 ri . We now introduce the notion of the metric ultraproduct for bounded metric spaces. Suppose that (Xi , di ) is an I-indexed family of metric spaces that is uniformly bounded, that is, there is some B > 0 such that the diameter of each Xi is bounded by B (meaning for all i ∈ I and a, b ∈ Xi , di (a, b) ≤ B). Fix an ultrafilter 𝒰 on I. Set X = ∏i∈I Xi and define, for a,̄ b̄ ∈ X, d(a,̄ b)̄ = lim𝒰 d(ai , bi ). It is easy to see that d is a pseudometric on X. We define the ultraproduct of the family (Xi , di ) with respect to 𝒰 , denoted ∏𝒰 Xi , to be the metric space obtained from X by quotienting by the pseudometric d. For x̄ ∈ X, we let [x]̄ 𝒰 denote the equivalence class of x̄ in the quotient. If Xi is some fixed X for all i ∈ I, we write X 𝒰 for the ultraproduct and call it the ultrapower of X with respect to 𝒰 . There is a natural isometric embedding of X into X 𝒰 , called the diagonal embedding, which sends x ∈ X to the equivalence class of the constant sequence (x : i ∈ I). Exercise 2.7. (1) Suppose that (Xi : i ∈ I) is a collection of complete, uniformly bounded metric spaces and 𝒰 is an ultrafilter on I. Prove that ∏𝒰 Xi is complete. (2) Suppose that (Xn : n ∈ N) is a collection of uniformly bounded metric spaces (which are not necessarily assumed to be complete) and 𝒰 is a nonprincipal ultrafilter on N. Prove that ∏𝒰 Xn is complete. Exercise 2.8. Suppose that X is a compact bounded metric space. Show that for any index set I and ultrafilter 𝒰 on I, the diagonal embedding of X into X 𝒰 is surjective (and thus an isometry of metric spaces). Throughout this chapter, we will be dealing with various metric spaces equipped with additional structure. We highlight here a key point that will come up when we have to deal with such additional structure. To keep things simple, suppose we have a complete, bounded metric space (X, d) and, in addition, we have a unary function f on X. We can naturally define the product X N with the sup-metric and view f as being defined coordinatewise on X N . We ask: What constraints must there be on f if we wish to form an ultrapower of (X, d, f ) as a quotient of the product structure? Notice in the classical case, there are no restrictions (one can always form the discrete ultrapower).

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We claim that it is necessary and sufficient that f be uniformly continuous. Sufficiency is straightforward and is left as an exercise to the reader. To verify necessity, suppose that there is ϵ > 0 such that, for every n ∈ N (see the footnote1 on the use of N in this article), one can find an , bn ∈ X such that d(an , bn ) < 1/n and yet d(f (an ), f (bn )) ≥ ϵ. Set ā = (an : n ∈ N) and b̄ = (bn : n ∈ N). If 𝒰 is a nonprincipal ultrafilter on N, then lim𝒰 d(an , bn ) = 0 and so [a]̄ 𝒰 = [b]̄ 𝒰 as elements of the ultrapower X 𝒰 . However, ̄ , where f (a)̄ = (f (a ) : n ∈ N) and ̄ 𝒰 ≠ [f (b)] lim𝒰 d(f (an ), f (bn )) ≥ ϵ and so [f (a)] 𝒰 n ̄ likewise for f (b). It follows that f cannot be defined on the ultrapower as a quotient of the product structure. The conclusion we draw from the previous paragraph is that if we want to extend an ultraproduct construction to additional structure on a metric space, we are going to have to assume some form of uniform continuity. In fact, a similar argument to the one just given shows that if one wants to construct an ultraproduct of uniformly bounded metric spaces (Xn , dn ) for n ∈ N each of which is equipped with a uniformly continuous unary function fn , the uniform continuity must be uniform along the sequence of structures, that is, for every ϵ > 0, there is δ > 0 so that for any n ∈ N and any a, b ∈ Xn , if dn (a, b) < δ, then dn (fn (a), fn (b)) ≤ ϵ (see the discussion at the beginning of Section 3 where we address the asymmetry in the above inequalities).

3 Metric structures and languages 3.1 Continuous languages and their interpretation We now introduce the structures we are interested in studying and the languages of the intended logic at the same time. A metric structure ℳ, the continuous analog of a classical structure, will be multisorted. The language will have a collection of sorts S. For each sort S ∈ S, the language specifies a positive real number BS . The intended interpretation of S in a metric structure ℳ is a bounded, complete, nonempty metric space S ℳ whose diameter is at most BS . There is also a relation symbol dS in the language which is to be interpreted as the metric on the space S ℳ . A language also contains a set of function symbols F and relation symbols R (these are sometimes called predicate symbols in other presentations). Each of these symbols comes with some additional information; we first treat function symbols. If f ∈ F, then f has a domain and codomain as specified by the language. The domain of f , denoted dom(f ), is a product of sorts S1 ×⋅ ⋅ ⋅×Snf for some nf ∈ N, and the codomain of f , denoted cod(f ), is a sort Sf . In addition, the language specifies an increasing, continuous function δf : [0, 1] → [0, 1] satisfying limϵ→0+ δf (ϵ) = 0; δf is called the uniform continuity modu1 We will adopt the convention that when n ∈ N is being treated as a number and not an index then n ≠ 0. This will avoid issues with division by 0.

88 � B. Hart lus of f . The function δf is specified by the language (and motivated by the discussion in the previous section). Comment 3.1. It may seem strong to insist that the uniform continuity modulus above is a continuous function. As it turns out, it can always be arranged: Proposition 3.2 ([7, Proposition 2.10 and Remark 2.12]). Let F, G: X → [0, 1] be such that (∀ϵ > 0)(∃δ > 0)(∀x ∈ X)(F(x) < δ implies G(x) ≤ ϵ). Then there exists an increasing, continuous function α: [0, 1] → [0, 1] such that α(0) = 0 and such that, for all x ∈ X, we have G(x) ≤ α(F(x)). In fact, one may choose α so that it only depends on the map ϵ 󳨃→ δ and not on the particular functions F and G. Proposition 3.2 is quite useful throughout continuous model theory. For example, it is useful in expressing certain implications (which are otherwise difficult to express given that continuous logic is a “positive” logic); see [7, Section 7]. We shall have occasion to use this proposition in Section 8 below. In a metric structure ℳ for this language, the function symbol f is interpreted as a uniformly continuous function f ℳ : S1ℳ × ⋅ ⋅ ⋅ × Snℳ → Sfℳ f where the uniform continuity of f ℳ is witnessed by δf , that is, for all ϵ > 0 and ai , bi ∈ Siℳ satisfying dSi (ai , bi ) < δf (ϵ) for all i = 1, . . . , n, we have dSf (f ℳ (a1 , . . . , anf ), f ℳ (b1 , . . . , bnf )) ≤ ϵ. The careful reader will notice the asymmetry between strict inequality and weak inequality in this definition of uniform continuity. The short explanation for the asymmetry is that we want this definition to pass easily through the ultraproduct construction. Nevertheless, the definition here is equivalent to that where we have strict inequalities on both sides. Constant symbols will be thought of as function symbols with an empty sequence of sorts as domain and some sort as codomain. The interpretation in a metric structure will be a single element in the chosen codomain. For the relation symbols, if R ∈ R, the language specifies a domain, denoted dom(R), which is a product of sorts S1 × ⋅ ⋅ ⋅ × SnR for some nR ∈ N. The language also provides a bound BR > 0 and a uniform continuity modulus δR . In a metric structure ℳ for this language, R is interpreted as a uniformly continuous function Rℳ : S1ℳ × ⋅ ⋅ ⋅ × Snℳ → R [−BR , BR ] ⊆ ℝ with uniform continuity witnessed by δR as above. To summarize, a metric language ℒ will contain: – a set of sorts S together with, for every S ∈ S, a relation symbol dS and number BS > 0,

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a set of function symbols F together with, for every f ∈ F, a product of sorts dom(f ), a sort cod(f ), and a uniform continuity modulus δf , and a set of relation symbols R together with, for every R ∈ R, a product of sorts dom(R), a number BR > 0, and a uniform continuity modulus δR .

A metric structure ℳ for ℒ (also called an ℒ-structure) specifies the following: – for every sort S ∈ S, a complete nonempty metric space S ℳ , with metric dSℳ of diameter bounded by BS , – for every f ∈ F, a uniformly continuous function f ℳ whose uniform continuity is witnessed by δf and with domain S1ℳ ×⋅ ⋅ ⋅×Snℳ and codomain S ℳ , where dom(f ) = S1 × ⋅ ⋅ ⋅ × Sn and cod(f ) = S, and – for every R ∈ R, a real-valued, bounded, uniformly continuous function Rℳ with uniform continuity modulus δR , domain S1ℳ ×⋅ ⋅ ⋅×Snℳ , where dom(R) = S1 ×⋅ ⋅ ⋅×Sn , and codomain [−BR , BR ]. For ℒ-structures ℳ and 𝒩 , we say that 𝒩 is a substructure of ℳ if – for every S ∈ S, S 𝒩 is a closed subspace of S ℳ , – for every f ∈ F, f 𝒩 is the restriction of f ℳ to S1𝒩 × ⋅ ⋅ ⋅ × Sn𝒩 , where dom(f ) = S1 × ⋅ ⋅ ⋅ × Sn , and – for every R ∈ R, R𝒩 is the restriction of Rℳ to S1𝒩 × ⋅ ⋅ ⋅ × Sn𝒩 , where dom(R) = S1 × ⋅ ⋅ ⋅ × Sn . If ℒ′ ⊆ ℒ are two languages and ℳ is an ℒ-structure, then by ℳ ↾ ℒ′ we mean the ℒ′ -structure obtained from ℳ by only interpreting those sorts, relations, and functions arising from ℒ′ . We call ℳ ↾ ℒ′ the reduct of ℳ to ℒ′ and we refer to ℳ as an expansion of ℳ ↾ ℒ′ to ℒ. We want to consider several categories of ℒ-structures. All of these categories will have ℒ-structures themselves as objects; the difference between these categories will be the morphisms. For now, we will consider two choices for maps between two ℒ-structures, ρ : ℳ → 𝒩 , by which we mean a family of maps (ρS : S ∈ S) such that ρS : S ℳ → S 𝒩 . Below, we will write ā ∈ ℳ to mean that ā is a tuple from the union of the interpretations of the sorts of ℳ and by ρ(a)̄ we will mean ρℳ S applied to the elements of the tuple ā for the appropriate S’s. – ρ is a homomorphism if: – for all function symbols f of ℒ and appropriately sorted ā ∈ ℳ, we have ̄ = f 𝒩 (ρ(a)); ̄ ρ(f ℳ (a)) – for all relation symbols R in ℒ and appropriately sorted ā ∈ ℳ, we have ̄ Rℳ (a)̄ ≥ R𝒩 (ρ(a)). – ρ is an embedding if it is a homomorphism which furthermore satisfies, for all rē Nolation symbols R in ℒ and appropriately sorted ā ∈ ℳ, Rℳ (a)̄ = R𝒩 (ρ(a)). tice that if ρ is an embedding, then ρS is an isometric embedding for all sorts S of ℒ.

90 � B. Hart In all categories we consider, ρ is an isomorphism if it is an embedding and ρS is surjective for all S. If ρ : ℳ → 𝒩 is an embedding, then by ρ(ℳ) we will mean the substructure of 𝒩 induced on the image of the maps ρS for all sorts S.

3.2 Examples Example 3.3. A toy example of a metric structure is a bounded complete metric space. The language has one sort and one binary relation symbol for the metric. There are no uniform continuity requirements here. A slightly more involved situation arises if one wants to consider an unbounded complete metric space (X, d). While there is no canonical way to deal with this situation, the following approach is helpful, particularly when there is more structure as we will see later. Fix a basepoint ⋆ ∈ X. Introduce in the language sorts Sn for n ∈ N; {a ∈ X : d(a, ⋆) ≤ n} is the intended interpretation of the sort Sn . The metric on Sn will just be the restriction of d to this sort. For m < n, we also introduce function symbols im,n with domain Sm , codomain Sn , and the identity function for modulus of uniform continuity. The intended interpretation of im,n is the natural inclusion of Sm into Sn . Example 3.4. A particularly interesting bounded metric space from the model-theoretic perspective is the Urysohn sphere U. In [44], Urysohn constructed a separable complete 1-bounded metric space which is both universal, that is, every other separable 1-bounded metric space isometrically embeds into U, and ultrahomogeneous, that is, if A, B ⊂ U are finite and f : A → B is an isometric bijection, then f can be extended to an automorphism of U. Also U is the unique separable complete 1-bounded metric space which is both universal and ultrahomogeneous. Since U is bounded, it is automatically an example of a metric structure. One way to build U is via a Fraïssé-like construction using the class 𝒞 of finite metric spaces with rational distances at most 1. (To see how Fraïssé constructions are done in the continuous setting, see Vignati’s article in this volume [45].) There are only countably many objects in 𝒞 and so, besides bookkeeping, one is led to the following problem: Given A, B, C ∈ 𝒞 with A ⊂ B and A ⊂ C, how can one amalgamate B and C over A? To do this, one needs to decide the distance d(b, c) for b ∈ B\A and c ∈ C\A. We record the following lemma, whose proof is left as an exercise. Lemma 3.5. Suppose that A, B, C are all finite metric spaces with A ⊂ B and A ⊂ C and B and C are extensions of A. Then B and C can be amalgamated over A to form the metric space B ∪A C with the following properties: – the underlying set is B ∪ C, – the metric extends the metric on both B and C, and – the metric further satisfies

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d(b, c) := min(d(b, a) + d(a, c)) a∈A

for b ∈ B \ A and c ∈ C \ A. Exercise 3.6. Prove Lemma 3.5. Using Lemma 3.5, one can form a chain of finite metric spaces from 𝒞 in the style of a Fraïssé construction which yields a countable metric space U0 with the property that whenever A, B ∈ 𝒞 are such that A ⊂ B and A ⊂ U0 , then there is a B0 ⊂ U0 with A ⊂ B0 and an isometry i : B → B0 which is the identity on A. A mildly tedious exercise shows that the completion U of U0 satisfies the defining properties of the Urysohn sphere. Exercise 3.7. Verify the claim made at the end of the previous example. Example 3.8. A classical (discrete) structure can be thought of as a metric structure as follows. Suppose that ℳ is a classical structure in a classical language ℒ. Equip each sort of ℳ with the discrete metric (that is, d(a, b) = 1 if a ≠ b). Additionally, for each relation symbol R ∈ ℒ, reinterpret R by the function gR = 1 − char(RM ), where char(RM ) is the characteristic function of RM . Note that ā ∈ RM iff gR (a)̄ = 0.2 Moreover, gR is uniformly continuous as is the interpretation of every function symbol. For completeness, we should specify that the identity function is the modulus of continuity for all function and relation symbols. In this way, one can think of continuous logic as an extension of classical logic. Example 3.9. Recall that a Hilbert space is a complex inner product space which is complete with respect to the induced metric. We wish to capture the class of Hilbert spaces as an example of a class of metric structures. Here is the language ℒHS and standard interpretation we will work with: (1) For each n ∈ N, we have a sort Sn . The intended interpretation of Sn in a Hilbert space is the ball centered at 0 of radius n. On Sn , there is a metric symbol with bound 2n whose intended interpretation is the metric induced by the inner product. (2) For each n ∈ N, we have a binary function symbol +n with domain Sn × Sn and codomain S2n . The intended interpretation for a Hilbert space is addition on the ball of radius n. The uniform continuity modulus is ϵ 󳨃→ ϵ/2. (3) For each n ∈ N, we have a constant symbol 0n in sort Sn which will be interpreted as 0. (4) For each n ∈ N and λ ∈ C, there is a unary function symbol λn with domain Sn and codomain Sm , where m = ⌈|λ|⌉. The uniform continuity modulus can be taken to be ϵ 󳨃→ ϵ/m The intended interpretation is scalar multiplication by λ.

2 Although arbitrary, the convention in continuous logic is that 0 represents true; see Section 4 for further discussion.

92 � B. Hart (5) For each m, n ∈ N with m < n, we have unary function symbols im,n with domain Sm and codomain Sn . The intended interpretation is the inclusion map from the ball of radius m into the ball of radius n and the uniform continuity modulus is the identity function. (6) For each n ∈ N, we have two binary relation symbols Re(⟨⋅, ⋅⟩)n and Im(⟨⋅, ⋅⟩)n on Sn . The intended interpretation of these symbols is the restriction to Sn of the real and imaginary parts of the inner product. The bounds for each of these symbols are n2 and the moduli of uniform continuity can be take to be ϵ 󳨃→ ϵ/2n. If H is a Hilbert space, then the ℒHS -structure just described will be denoted by D(H) and referred to as the dissection of H. In practice, we will write formulas in this language which will omit the subscripts when the intention is clear. We may also write the inner product symbols without reference to the real and imaginary part if it is clear that an equivalent formula can be written adhering to the above formalism. One of the most important examples for the current volume is the class of C∗ -algebras. Let us look at this class in detail. Example 3.10. Given a C∗ -algebra A, the natural metric on A is that induced by the operator norm. As in the case of Hilbert spaces treated in the previous example, this metric is unbounded. Once again, we get around this issue by introducing a series of sorts to represent operator norm balls of various radii. As before, we fix sorts Sn for each n ∈ N with the intended interpretation being the operator norm ball of radius n in A. The intended metric on each sort is just the restriction of the metric induced by the operator norm to the given sort. The language will also have a number of function symbols. Since we will be dissecting C∗ -algebras, as in the case of Hilbert spaces, we will need to consider the restriction of functions like addition and multiplication to each sort and we need to connect the sorts via embedding maps. With that in mind, we introduce the language ℒC ∗ with its intended interpretation: (1) For each n ∈ N, there are binary function symbols +n and ⋅n with domain Sn ×Sn and codomains S2n and Sn2 , respectively. The intended interpretation is the restriction of addition and multiplication on A to the operator norm ball of radius n. The uniform continuity modulus for +n is ϵ 󳨃→ ϵ/2 and for ⋅n is ϵ 󳨃→ ϵ/2n. (2) For each n ∈ N, there is a unary function symbol ⋅∗n with domain and codomain Sn . The intended interpretation is the restriction of the adjoint to the operator norm ball of radius n. The uniform continuity modulus is the identity function. (3) For each n ∈ N, there is a constant symbol 0n in Sn . The intended interpretation is 0. (4) For each λ ∈ C and n ∈ N, we have a unary function symbol λn with domain Sn and codomain Sm , where m = ⌈|λ|⌉. The uniform continuity modulus of this symbol is ϵ 󳨃→ ϵ/m. (5) For each n ∈ N, the metric symbol on Sn , denoted dn , will be interpreted as ‖x − y‖ for x, y in the operator norm ball of A of radius n.

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(6) For each m, n ∈ N with m < n, there is a unary function symbol im,n with domain A Sm and codomain Sn . The intended interpretation is the inclusion map between Sm A and Sn . The uniform continuity modulus is the identity map. Given a C∗ -algebra A, the ℒC ∗ -structure considered above is denoted D(A) and is called the dissection of A. It is clear that if we start with A and consider D(A), we can then recover A from D(A) (simply take the union of the sorts and piece together the operations as given). Moreover, D(A) ≅ D(B) (as ℒC ∗ -structures) if and only if A ≅ B (as C∗ -algebras). We now address a subtle point. Suppose that we have a ℒC ∗ -substructure ℳ ⊆ D(A) for some C∗ -algebra A. Does ℳ arise as D(B) for some subalgebra B ⊆ A? To be clear, it is easy to see that if one considers the direct limit of the sorts Snℳ along the embeddings arising from the maps im,n , we do indeed obtain a C∗ -subalgebra B of A (the completeness of each individual sort guarantees that the union is complete). What is in question is whether each individual sort Sn captures exactly the elements of B which are of operator norm at most n. In order to see that we can arrange for this to happen, we need the following fact. The functional analytic argument that underlies this fact can be found in [18, Section 3.1 at the top of page 486]. Fact 3.11. For any n ∈ N, there are one-variable polynomials qkn (k ∈ N) such that if A is a C∗ -algebra and a ∈ A satisfies ‖a‖ ≤ n, then ‖qkn (a)‖ ≤ 1 for all k ∈ N. If, moreover, ‖a‖ ≤ 1, then qkn (a) → a as k → ∞. We henceforth assume that, for each k, n ∈ N, we have added function symbols qnk to the language ℒC ∗ with domain Sn and codomain S1 . The modulus of continuity can be taken to be the modulus of continuity of the polynomial qnk when restricted to the operator norm ball of radius n. In the dissection of a C∗ -algebra, we will interpret qnk by the value of the corresponding polynomial (taking values in S1 ). Since each polynomial qkn is naturally a composition of the functions in the language of C∗ -algebras, there is some m ≥ n such that Sm will be the (naïve) codomain of qkn when restricted to Sn . When qnk is interpreted in the dissection of a C∗ -algebra, we have i1,m (qnk (x)) = qkn (x) for all x in sort Sn . The justification for adding these function symbols is as follows. Suppose that ℳ and B are as above and ‖a‖ ≤ 1 for some a ∈ B. Take n ∈ N for which a ∈ Snℳ . Consider qkn for k ∈ N as in Fact 3.11. Then in B, qkn (a) tends to a in operator norm. On the other hand, since the codomain of qnk is S1 , (qnk )ℳ (a) converges ℳ n to something in S1ℳ , call it b. Since i1,n is an isometric embedding, i1,n (qk (a)) converges ℳ M to a in Sn and hence i1,n (b) = a, which means that a and b are identified in B. We conclude that D(B) ≅ M. The dissection functor D is in fact an equivalence of categories from the category of C∗ -algebras with homomorphisms as morphisms and ℒC ∗ -structures of the form D(A) for C∗ -algebras A, again with homomorphisms as morphisms. The equivalence between

94 � B. Hart these categories is actually stronger than this, but this stronger statement will have to wait until we define ultraproducts of structures.

4 Formulas 4.1 Basic formulas We now introduce the syntax of formulas for continuous logic. Fix a continuous language ℒ with sorts S, function symbols F, and relation symbols R. For each sort S ∈ S, we introduce variables xnS for n ∈ N. As in classical logic, we need to define terms before formulas. Terms in ℒ, along with their domain, codomain, and associated uniform continuity modulus, are defined inductively as follows: (1) If x is a variable of sort S, then x is a term. Its domain and codomain are S and its uniform continuity modulus is the identity function. (2) If f ∈ F is a function symbol with domain S1 × ⋅ ⋅ ⋅ × Sn and codomain S and τ1 , . . . , τn are terms with codomains S1 , . . . , Sn respectively, then f (τ1 , . . . , τn ) is a term with codomain S and domain determined by composition. The uniform continuity modulus of this term is also determined by composition; for instance, one can take ϵ 󳨃→ min{δ1 (δf (ϵ)/2), . . . , δn (δf (ϵ)/2)}, where δf is the uniform continuity modulus of f and δk is the uniform continuity modulus of τk for every k = 1, . . . , n. Exercise 4.1. Verify that the function displayed at the end of item (2) above is indeed a uniform continuity modulus for f (τ1 , . . . , τn ). The basic formulas of ℒ are now defined by induction. We use the term basic formula because, as we will see later in this article, we will want to introduce a more general notion of formula to accommodate the approximate nature of continuous logic.3 Each basic formula will have a domain, bound, and uniform continuity modulus as is the case for relation symbols. (1) Suppose that R ∈ R has domain S1 × ⋅ ⋅ ⋅ × Sn and bound BR , and τ1 , . . . , τn are terms with codomains S1 , . . . , Sn , respectively. Then R(τ1 , . . . , τn ) is a basic formula with bound BR , and domain and uniform continuity modulus determined by composition. In particular, since all the symbols for metrics on the sorts are included in R, if τ1 and τ2 are sorts with the same codomain S, then dS (τ1 , τ2 ) is a basic formula. 3 We hope there is no confusion with this use of the term basic formula and its various uses in classical logic.

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(2) If f : ℝn → ℝ is a continuous function and φ1 , . . . , φn are basic formulas, then ψ := f (φ1 , . . . , φn ) is also a basic formula. If B1 , . . . , Bn are the bounds of φ1 , . . . , φn , respectively, then the bound of ψ is the minimum B such that n

f (∏[−Bi , Bi ]) ⊆ [−B, B]. i=1

The domain of ψ is determined by composition. The uniform continuity modulus of ψ is also determined by composition from the uniform continuity moduli of the φi ’s and a fixed uniform continuity modulus of f restricted to ∏ni=1 [−Bi , Bi ]. (3) Suppose that x is a variable and φ is a formula. We can then form two quantified basic formulas supx φ and infx φ. The bound and uniform continuity modulus of these formulas is the same as φ. The domain is only effected by the possible removal of the sort associated with x. The basic formulas arising from the first clause are called atomic formulas. Basic formulas arising from the first two clauses are called quantifier-free basic formulas. Comment 4.2. In the second clause above, we have chosen to be very generous about the functions – the connectives – we allow. It is convenient theoretically to allow all continuous functions (which are actually uniformly continuous since we are only really interested in their values on some hypercube in ℝn ). It is sometimes worthwhile to work with a smaller collection of connectives. The expressive power of continuous logic would not change if we used a collection of connectives which were dense in the set of all continuous functions. For instance, we could restrict ourselves to using only polynomials with rational coefficients as connectives. Other restricted sets of connectives have been suggested (see [7, Section 6] for more details). As we will see, it is the density character of the set of formulas which matters the most in continuous logic and not the cardinality of the set of formulas. Another set of basic formulas to highlight are the prenex formulas, which are those basic formulas beginning with a block of quantifiers followed by a quantifier-free basic formula. In a natural (pseudo)metric defined on the space of all basic formulas (see Section 5.1), the prenex formulas are dense. We will come back to prenex formulas in Section 7.2. We now consider the interpretation of basic formulas in a metric structure. Fix an

ℒ-structure ℳ.

A term τ is interpreted in ℳ by τ ℳ exactly as it would be in classical logic by following the inductive definition and composing.

Exercise 4.3. Verify that the domain, codomain, and uniform continuity modulus of τ as known to the language are in fact the domain, codomain, and uniform continuity modulus of τ ℳ .

96 � B. Hart The interpretation of basic formulas is not much more complicated. Suppose that ā = (a1 , . . . , an ) is a sequence of sorted elements of ℳ and the free variables x̄ = (x1 , . . . , xn ) of a basic formula φ are similarly sorted. We wish to define the interpretation φℳ (a)̄ inductively: (1) If φ := R(τ1 , . . . , τk ) for terms τ1 , . . . , τk , then ̄ . . . , τkℳ (a)). ̄ φℳ (a)̄ := Rℳ (τ1ℳ (a), (2) If φ := f (ψ1 , . . . , ψk ) for basic formulas ψ1 , . . . , ψk and a continuous real-valued function f , then ℳ ̄ ̄ φℳ (a)̄ := f (ψℳ 1 (a), . . . , ψk (a)).

(3) If φ := supx ψ, then φℳ (a)̄ := sup{ψℳ (b, a)̄ : b ∈ S ℳ } where S is the sort of x. Notice that this is well defined since the codomain of ψℳ is bounded. Similarly, if φ := infx ψ, then φℳ (a)̄ := inf{ψℳ (b, a)̄ : b ∈ S ℳ } where again S is the sort of x. Exercise 4.4. Verify that in each case of the above definition, these formulas have bound, domain, and uniform continuity modulus known to the language.

4.2 Some examples of formulas Since it takes some getting used to writing expressions in continuous logic, let us look at some standard ways of expressing certain statements. Example 4.5. Suppose we have two terms τ(x)̄ and σ(x)̄ with the same codomain. How do we assert that they are equal when they are interpreted? If d is the metric symbol ̄ σ(x)) ̄ on the sort of the common codomain of the terms, then the sentence supx̄ d(τ(x), evaluates to 0 in a metric structure ℳ if and only if the functions τ ℳ and σ ℳ are equal. Example 4.6. We sometimes wish to assert that one formula is less than or equal to another in value. We do this with the help of the truncated subtraction function x ∸ y := max{0, x − y}. Note that ∸ is a continuous function from ℝ2 to ℝ, hence may be used as a connective. Suppose that φ(x)̄ and ψ(x)̄ are two formulas. Then the sentence

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̄ sup(φ(x)̄ ∸ ψ(x)) x̄

evaluates to 0 in a structure ℳ if and only if φℳ ≤ ψℳ as functions. Example 4.7. Another useful thing to be able to write out is the isomorphism type of some finite metric space. Suppose that A = {a1 , . . . , an } is a finite metric space with n distinct elements and metric δ. Set rij = δ(ai , aj ) for 1 ≤ i < j ≤ n. Consider the formula 󵄨 󵄨 DA (x1 , . . . , xn ) = max 󵄨󵄨󵄨d(xi , xj ) − rij 󵄨󵄨󵄨, 1≤i 0 such that φℳ takes values in [−B, B] for all models ℳ of T. Notice that for a theory T, the collection of T-formulas is closed under composition with continuous functions, i. e., if φ1 , . . . , φn are T-formulas and f : Rn → R is a continuous function then f (φ1 , . . . , φn ) is a T-formula. Moreover, the set of T-formulas is closed ̄ under sup and inf. That is, if φ(x, y)̄ is a T-formula then so is supx φ(x, y)̄ and infx φ(x, y).

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Example 5.2. Suppose that we have a theory T and T-formulas φn in variables x̄ with bounds Bn for n ∈ N. Then the following is a T-formula in the sense just defined: φ(x)̄ . B 2n n∈N n ∑

It is not hard to see that all T-formulas can be realized as weighted sums of basic formulas. The set Formx̄ (T, ℒ) has the structure of a real normed algebra: it contains the constant functions and is closed under addition and multiplication. It will be convenient to generalize this notion as follows. Definition 5.3. A real algebra system A of T-formulas consists of, for every sequence of variables x,̄ a closed subalgebra Ax̄ of Formx̄ (T, ℒ). We have three particular cases of real algebra systems in mind. Example 5.4. (1) Suppose that ℒ′ ⊆ ℒ. Given a sequence x̄ of variables, define Ax̄ to be the closure in Formx̄ (T, ℒ) of the set of basic ℒ′ -formulas; A is then a real algebra system of T-formulas. (2) Given a sequence x̄ of variables, let Ax̄ be the closure in Formx̄ (0, ℒ) of the set of quantifier-free basic ℒ-formulas. Then A is a real algebra system of ℒ-formulas whose elements are called quantifier-free formulas. (3) Let Ax̄ be the closure in Formx̄ (0, ℒ) of the set of prenex formulas in ℒ in the free variables x.̄ Then A is a real algebra system. For a topological space X, we let χ(X) denote the density character of X, namely the smallest cardinality of a dense subset of X. We will refer to the density character of a language ℒ with respect to a theory T, denoted by χ(T, ℒ), as ∑ χ(Formx̄ (T, ℒ)), x̄

where the sum is taken over all x̄ of variables. If T is the empty theory, we will omit it from the notation and just write χ(ℒ). If χ(ℒ) = ℵ0 , we say that ℒ is separable. We also define the density character of an ℒ-structure ℳ, denoted χ(ℳ), to be sup χ(S ℳ ).

S∈S

5.3 First model-theoretic results Proposition 5.5 (Tarski–Vaught test). Suppose that 𝒩 ⊆ ℳ. Then the following are equivalent:

100 � B. Hart (1) 𝒩 ⪯ ℳ. (2) For all basic ℒ-formulas φ(y, x)̄ and all ā ∈ 𝒩 , inf{φℳ (b, a)̄ : b ∈ 𝒩 } = inf{φℳ (b, a)̄ : b ∈ ℳ}. Proof. Statement (1) implies (2) by the definition of elementary substructure. To show that (2) implies (1), we argue by induction on the formation of basic formulas. The interesting case is that of an inf quantifier. Thus, suppose φ(x)̄ = infy ψ(y, x)̄ and ā ∈ 𝒩 . By ̄ To prove the reverse inequality, fix ϵ > 0 induction, we always have φ𝒩 (a)̄ ≥ φℳ (a). ℳ ℳ ̄ ̄ and take b ∈ ℳ such that ψ (b, a) < φ (a) + ϵ. By condition (2), we can find b′ ∈ 𝒩 such that ψℳ (b′ , a)̄ < φℳ (a)̄ + ϵ, and so by induction and letting ϵ tend to 0, we have ̄ φ𝒩 (a)̄ ≤ φℳ (a). Theorem 5.6 (Downward Löwenheim–Skolem). Given X ⊆ ℳ, there is 𝒩 ⪯ ℳ such that X ⊆ N and χ(𝒩 ) ≤ χ(X) + χ(ℒ). Proof. We accomplish the creation of 𝒩 by iteratively closing off in order to satisfy the Tarski–Vaught test; we only sketch of the details. Fix a set of formulas {φi (y, x̄i ) : i ∈ I} which is dense in Form(Th(ℳ), ℒ) := ⋃x̄ Formx̄ (Th(ℳ), ℒ) with |I| = χ(ℒ). We create an increasing sequence (Xn ) of subsets of ℳ such that: (1) X0 = X, (2) for each n ∈ N, i ∈ I, and ā ∈ Xn , there is a sequence (bn ) contained in Xn+1 such that ℳ

̄ (infy φ(y, a))

= inf{φℳ (bn , a)̄ : n ∈ N},

and (3) χ(Xn ) ≤ χ(X) + χ(ℒ). We let N be the closure (in M) of the union of the Xn ’s and leave it as an exercise to the reader to check that N is the universe of a substructure 𝒩 of ℳ which is in fact an elementary substructure of ℳ. Exercise 5.7. Verify the claim made at the end of the previous proof. Definition 5.8. Fix an ℒ-structure ℳ and a subset A ⊆ ℳ. The language ℒA is ℒ together with a new constant symbol ca for every a ∈ A (and where ca has the same sort as a). Then ℳ can be canonically expanded to an ℒA -structure ℳA by interpreting ca as a for every a ∈ A. We define two types of ℒM -theories (often referred to as diagrams): (1) The atomic diagram of ℳ, denoted Diag(ℳ), is the set {φ : φ is an an atomic ℒℳ -sentence and ℳℳ 󳀀󳨐 φ}.

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(2) The elementary diagram of ℳ, Elem(ℳ), is the set {φ : φ is an an basic ℒℳ -sentence and ℳℳ 󳀀󳨐 φ}. Proposition 5.9. For ℒ-structures ℳ and 𝒩 , we have that: (1) ℳ can be embedded into 𝒩 if and only if 𝒩 can be expanded to an ℒℳ -structure which satisfies the atomic diagram of ℳ; (2) ℳ can be elementarily embedded into 𝒩 if and only if 𝒩 can be expanded to an ℒℳ structure which satisfies the elementary diagram of ℳ. Proof. The proofs are similar so we will only prove (2). If f : ℳ → 𝒩 is an elementary embedding, then by interpreting cm in 𝒩 by f (m), it is straightforward to show this expansion of 𝒩 satisfies the elementary diagram of ℳ. In the other direction, if 𝒩̃ is an ℒℳ expansion of 𝒩 which satisfies the elementary diagram of ℳ, then define 𝒩 f : ℳ → 𝒩 by f (m) = cm for all m ∈ ℳ. Again it is straightforward to show that this is an elementary map. The following result is analogous to the classical result regarding unions of chains. Proposition 5.10. The category of ℒ-structures with elementary embeddings as morphisms has directed colimits. More precisely, if we are given: (1) a directed partial order (P, 0. Suppose ℳ and 𝒩 are two ℒ-structures. We describe a game, the EF(ℳ, 𝒩 , Δ, ϵ)-game, between two players, Player I and Player II. The game has n rounds. In the ith round, Player I chooses either ai ∈ ℳ or bi ∈ 𝒩 of whatever sort the variable xi belongs to and then Player II replies with bi ∈ 𝒩 or ai ∈ ℳ, respectively. After n rounds, there are

102 � B. Hart two sequences ā = (a1 , . . . , an ) and b̄ = (b1 , . . . , bn ). Player II wins the game if for every φ(x)̄ ∈ Δ, we have 󵄨 󵄨󵄨 ℳ ̄ 𝒩 󵄨󵄨φ (a) − φ (b)̄ 󵄨󵄨󵄨 < ϵ. Theorem 5.12. For two ℒ-structures ℳ and 𝒩 , the following are equivalent: (1) ℳ ≡ 𝒩 . (2) For every finite set Δ of atomic ℒ-formulas and every ϵ > 0, Player II has a winning strategy for the EF(ℳ, 𝒩 , Δ, ϵ)-game. (3) For every finite set Δ of basic ℒ-formulas and every ϵ > 0, Player II has a winning strategy for the EF(ℳ, 𝒩 , Δ, ϵ)-game. Exercise 5.13. Prove Theorem 5.12. (Hint: For the direction (3) implies (1), proceed by induction on the natural notion of quantifier depth of a basic ℒ-formula.) A reader looking for a solution to the previous exercise can consult [31].

5.5 Examples of continuous theories Example 5.14 (The Urysohn sphere). The theory of the Urysohn sphere is axiomatized by sentences which express an extension property characterizing the space we described in Example 3.4 We let the theory T ext consist of the sentences φA,B defined by (recalling the notation from Example 4.7) ̄ sup(inf DB (x,̄ y) ∸ DA (x)), x̄

y

where n ≥ 1, A = {a1 , . . . , an } and B = {a1 , . . . , an , b} are 1-bounded metric spaces, and x̄ = x1 , . . . , xn . From our original characterization of the Urysohn sphere, it is clear that all the sentences φA,B are satisfied in U. Before we show that T ext axiomatizes the theory of U, we first describe a different amalgamation construction than the one from Example 3.4. Suppose that A, B = A ∪ {b}, and C = A ∪ {c} are all finite metric spaces with b ≠ c and with metrics dA , dB , and dC , respectively, for which dB and dC extend the metric dA . We wish to put a metric d on B ∪ C which extends both dB and dC ; in other words, we must decide the value of d(b, c). For the purposes of the argument to follow, we set 󵄨 󵄨 d(b, c) = max󵄨󵄨󵄨dB (b, a) − dC (c, a)󵄨󵄨󵄨. a∈A

Exercise 5.15. Prove that d is indeed a metric on B ∪ C. We denote the resulting the resulting metric space B ⊔A C.

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Now to see that T ext in fact axiomatizes the theory of U, it suffices to prove the stronger statement that any separable model of T ext is isomorphic to U. So suppose that ℳ is a separable model of T ext . One proves that ℳ is isomorphic to U using a standard back-and-forth argument. We will concentrate on the inductive step and leave the rest of the details to the reader. Consider a finite metric space A = {a1 , . . . , an } ⊂ ℳ and an abstract one point extension B = A ∪ {b} of A. We wish to find an isomorphic copy of B extending A in ℳ. First of all, we use the fact that φℳ A,B = 0 to find some b0 ∈ ℳ such that Dℳ (a , . . . , a , b ) < 1/2. If we set r = d(b, a) for a ∈ A, then we have that 1 n 0 a B |d(b0 , a) − ra | < 1/2 for all a ∈ A. Set B0 := {a1 , . . . , an , b0 }. We now inductively build a sequence (bm ) from ℳ such that, setting Bm := {a1 , . . . , an , bm } and B̂ m := Bm ⊔A B, we have DB̂ (a1 , . . . , an , bm , bm+1 ) < 1/2m+1 . m

Note that we can indeed obtain bm+1 ∈ ℳ by invoking the fact that φℳ B ,B̂ m+1

m+2

m

m

= 0.

This construction yields |d(bm , a) − ra | < 1/2 and also d(bm , bm+1 ) < 3/2 . It is now clear that (bm : m ∈ N) is a Cauchy sequence in ℳ; letting b̃ ∈ ℳ be the limit of this sequence, we get that d(b,̃ a) = ra for all a ∈ A and so the map a1 , . . . , an , b 󳨃→ a1 , . . . , an , b̃ is an isometry, yielding the desired copy of B in ℳ. We now turn to some examples that have more structure than just the metric symbol and are key to many of the articles in this volume. Example 5.16 (Hilbert space). We now write sentences (or axioms) that are true in all (dissections of) Hilbert spaces using the language ℒHS . These sentences will in fact axiomatize the theory of (dissections of) Hilbert spaces. The first set of axioms will express that we are dealing with a complex vector space. Due to the presence of the sorts Sn , we will need infinitely many sentences to express the usual axioms. For instance, for each n ∈ N, we will have a sentence sup dn (λn (x +n y), λn (x) +m λn (y)),

x,y∈Sn

where λ ∈ C and Sm is the codomain of λn . Of course, if these sentences hold in our structure, then we will know that λ(x + y) = λx + λy for all x and y. We leave it as an exercise to write out in a similar fashion all other sentences needed to guarantee that we have a complex vector space. The second set of axioms expresses that we have an inner product and expresses its relationship with the metric on each sort. For instance, we have the sentences 󵄨 󵄨 sup󵄨󵄨󵄨dn (x, 0)2 − Re⟨x, x⟩󵄨󵄨󵄨.

x∈Sn

We leave it as an exercise to express the other axioms related to the inner product.

104 � B. Hart We need a third set of axioms which tell us that the inclusion maps between the sorts are isometric embeddings that preserve the algebraic operations. First, we need to express that the inclusion maps themselves are compatible: for n < m < k, we include sentences sup dk (im,k in,m (x), in,k (x)),

x∈Sn

which assert that im,k in,m = in,k An instance of the inclusion maps preserving the algebraic structure are the sentences, for n < m and λ ∈ C, sup dl (ik,l (λn x), λm in,m (x)),

x∈Sn

where Sk is the codomain of λn and Sl is the codomain of λm . Finally, we want to express that the sorts are interpreted correctly. That is, Sn should be the ball of radius n. We express this fact with two axioms: sup(d1 (x, 0) ∸ 1) x∈S1

and sup min{1 ∸ ‖x‖, inf dn (y, i1,n (y))}.

x∈Sn

y∈S1

The first axiom says that everything in S1 should have norm at most 1 and by scaling, everything in Sn should have norm at most n. The second axiom says that if something in Sn has norm less than 1, then it is in the closure of the image of the inclusion map from S1 . Using the completeness of the sorts, we obtain that anything of norm at most 1 in Sn arises from something in S1 . Consequently, if these two axioms are satisfied (along with all the others above), then S1 is the unit ball of a Hilbert space, and the direct limit of the sorts, via the embeddings, is a Hilbert space. If we denote by THS the set of sentences listed above, then we conclude: Theorem 5.17. The set THS axiomatizes the class of dissections of Hilbert spaces. Moreover, models of THS with homomorphisms as morphisms and the category of Hilbert spaces with homomorphisms as morphisms are equivalent via the dissection functor. Example 5.18 (C∗ -algebras). As in the previous example, we now write out sentences that axiomatize the class of (dissections of) C∗ -algebras in the language ℒC ∗ from Example 3.10. First of all we, need to express that the structure is a complex *-algebra. This requires writing out a number of equations in the manner described above in the Hilbert space example. We will not write out all the equations, but here is an example:

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sup dn ((xy)∗ , y∗ x ∗ ).

x,y∈Sn

There is one such sentence for every n ∈ N. Of course, this set of sentences is expressing the relationship between multiplication and the adjoint in a C∗ -algebra, namely that (xy)∗ = y∗ x ∗ for all x and y. We also want to say that the underlying structure is a normed algebra. Again, we only write out a sample and leave it as an exercise to write out the rest. It is tempting to write the triangle inequality as ∀x,y∈Sn ‖x + y‖ ≤ ‖x‖ + ‖y‖ but more formally it is written as sup (‖x + y‖ ∸ (‖x‖ + ‖y‖)),

x,y∈Sn

where ‖z‖ is shorthand for d(z, 0). The latter sentence evaluates to 0 in a ℒC ∗ -structure precisely when the triangle inequality holds. We also want our Banach *-algebra to be a C∗ -algebra. The C∗ -identity is written as 󵄨󵄩 󵄩 󵄨 sup󵄨󵄨󵄨󵄩󵄩󵄩x ∗ x 󵄩󵄩󵄩 − ‖x‖2 󵄨󵄨󵄨,

x∈Sn

where we use the above convention for ‖ ⋅ ‖. We include one such sentence for every n ∈ N although, in light of the other axioms, it would be enough to include this axiom only for n = 1. As in the case of Hilbert spaces, there are now some sentences which must be included in order to make the formalism of continuous logic match with the normal operator algebraic setting. To start with, we have the embedding maps im,n . We can write out sentences which will express that these maps are isometric embeddings and they preserve the relevant algebraic structure (addition, multiplication, and the adjoint). We leave the formulation of these sentences to the reader. We also want the sorts Sn to be interpreted as the operator norm balls of radius n. We need two sets of sentences. First, we include sentences sup(‖x‖ ∸ n),

x∈Sn

which says that every element of the sort Sn has norm at most n. Finally, we record sentences which will guarantee that all the elements of norm at most n lie in Sn . Recall from Example 3.10 the introduction of function symbols qnk with domain Sn and codomain S1 . They are to represent polynomials qkn and so we write sentences sup dm (i1,m (qnk (x)), qkn (x)),

x∈Sn

106 � B. Hart where Sm is the codomain of the term qkn . We will call the set of sentences that we have outlined in this example TC ∗ . By the argument after Lemma 3.11, we see that the ℒC ∗ structures which satisfy TC ∗ are precisely those of the form D(A) for some C∗ -algebra A. We conclude: Theorem 5.19. The set TC ∗ axiomatizes the class of dissections of C∗ -algebras. Moreover the category of C∗ -algebras and the category of models of TC ∗ (with homomorphisms as morphisms) are equivalent via the dissection functor.

6 Ultraproducts 6.1 Ultraproducts of metric structures Fix a metric language ℒ = (S, F, R), an index set I, an ultrafilter 𝒰 on I, and ℒ-structures ℳi for i ∈ I. We construct the (metric) ultraproduct 𝒩 := ∏𝒰 ℳi as follows: – For each sort S ∈ S, we let S 𝒩 be the metric space ∏𝒰 (S ℳi , di ), where di is the metric on S ℳi . We interpret the metric on S 𝒩 as lim𝒰 di . Recall that this is well defined since all the metric spaces involved are uniformly bounded by the bound BS . – For each f ∈ F whose domain is S1 × ⋅ ⋅ ⋅ × Sn , we define f 𝒩 by f 𝒩 ([(ai1 )]𝒰 , . . . , [(ain )]𝒰 ) = [(f (ai1 , . . . , ain ))]𝒰 , j

where [(ai )]𝒰 ∈ Sj𝒩 for all j = 1, . . . , n. Again, since all interpretations of f satisfy

–

the same uniform continuity modulus, this definition of f 𝒩 is well defined and f 𝒩 satisfies the uniform continuity modulus specified by the language. For each R ∈ R, we define R𝒩 by R𝒩 ([(ai1 )]𝒰 , . . . , [(ain )]𝒰 ) = lim Rℳi (ai1 , . . . , ain ), 𝒰

j

where [(ai )]𝒰 ∈ SjN for all j = 1, . . . , n. As in the case of function symbols, since the interpretation of R is uniformly bounded in all ℳi with the same uniform continuity modulus, this definition of R𝒩 is well defined and R𝒩 satisfies the uniform continuity modulus specified by the language. Exercise 6.1. Verify that ∏𝒰 ℳi is indeed an ℒ-structure. If all the ℳi are the same ℒ-structure ℳ, then we call the resulting ultraproduct the ultrapower and denote it by ℳ𝒰 . As in classical logic, it is important to be able to evaluate formulas in the ultraproduct. Here is the appropriate version of Łoś’ theorem. Theorem 6.2. Suppose that ℳi is an ℒ-structure for all i ∈ I and 𝒰 is an ultrafilter on I. Set 𝒩 = ∏𝒰 ℳi . For an ℒ-formula φ(x)̄ and ā = [(ā i )]𝒰 ∈ 𝒩 sorted the same as the

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variables x,̄ we have φ𝒩 (a)̄ = lim φℳi (ā i ). 𝒰

Proof. The proof is by induction on the formulation of basic formulas. We only treat the ̄ case of φ(x)̄ = infy ψ(y, x). Suppose that lim𝒰 φℳi (ā i ) > r for some r ∈ R. Set X = {i ∈ I : φMi (ā i ) > r} and note that X ∈ 𝒰 . Fix b = [(bi )]𝒰 ∈ 𝒩 . For each i ∈ X, we have ψℳi (bi , ā i ) > r. By induction, we have ψ𝒩 (b, a)̄ ≥ r. Since this was true for all b ∈ 𝒩 , we have φ𝒩 (a)̄ ≥ r and thus φ𝒩 (a)̄ ≥ lim𝒰 φℳi (ā i ). Now suppose that s = φ𝒩 (a)̄ > r but lim𝒰 φℳi (ā i ) ≤ r. Pick ϵ > 0 small enough so that r + ϵ < s and let X = {i ∈ I : φℳi (ā i ) < r + ϵ} ∈ 𝒰 . For i ∈ X, choose bi ∈ ℳi so that ψℳi (bi , ā i ) < r + ϵ. Set b = [(bi )]𝒰 . By induction, we have ψ𝒩 (b, a)̄ ≤ r + ϵ < s, which is a contradiction. Exercise 6.3. Verify the remaining cases in the proof of Łos’s theorem including generalizing to arbitrary formulas from basic formulas. The following is an immediate consequence of Łos’s theorem. Proposition 6.4. For any ℒ-structure ℳ and any ultrafilter 𝒰 , the diagaonal embedding of ℳ into ℳU is an elementary map.

6.2 Applications of the ultraproduct We say that a set of sentences Γ is finitely satisfiable if every finite subset of Γ is satisfiable. We say that Γ is approximately finitely satisfiable if, for every finite subset Γ0 ⊂ Γ and every ϵ > 0, there is an ℒ-structure ℳ so that |φℳ | ≤ ϵ for every φ ∈ Γ0 . Equivalently, Γ is approximately finitely satisfiable if the set of sentences {|φ| ∸ ϵ : φ ∈ Γ, ϵ > 0} is finitely satisfiable. Theorem 6.5 (Compactness theorem). For a metric language ℒ and a set of ℒ-sentences Γ, the following are equivalent: (1) Γ is satisfiable. (2) Γ is finitely satisfiable. (3) Γ is approximately finitely satisfiable. Proof. Clearly, the conditions decrease in strength. To show that (3) implies (1), consider I to be the set of finite subsets of Γ∪N (disjoint union) which have nonempty intersection with both Γ and N. If i = F ∪ X is an element of I with F ⊂ Γ and X ⊂ N, let Mi be an ℒ-structure such that |φMi | ≤ 1/m for all φ ∈ F, where m = max{n : n ∈ X}. For φ ∈ Γ and n ∈ N, let Oφ,n be the set of i ∈ I which contain both φ and n. Notice that Y = {Oφ,n : φ ∈ Γ, n ∈ N} has the finite intersection property. Let 𝒰 be an ultrafilter on I which contains Y . Consider the ultraproduct 𝒩 = ∏𝒰 Mi . For every φ ∈ Γ and every

108 � B. Hart n ∈ N, we have Oφ,n ∈ 𝒰 , so by the Łoś theorem, we have |φ𝒩 | ≤ φ𝒩 = 0. It follows that 𝒩 satisfies Γ.

1 n

for all n and hence

We can use the compactness theorem and the method of expansion by constants to give a version of the upward Löwenheim–Skolem theorem in the continuous setting. We say that a metric structure ℳ is compact if S ℳ is a compact metric space for each sort S in the language. Theorem 6.6. Suppose that ℳ is a noncompact ℒ-structure and χ(Th(ℳ), ℒ) ≤ λ. Then ℳ has an elementary extension 𝒩 such that χ(𝒩 ) = λ. Proof. Since ℳ is not compact, there is a sort S and an ϵ > 0 such that the collection of ϵ-balls of S ℳ does not have a finite subcover. Now consider the language ℒ′ obtained from ℒ by introducing new constant symbols cα of sort S for α < λ. Consider the ℒ′ theory T ′ made up of Elem(ℳ) together with {d(cα , cβ ) ≥ ϵ : α < β < λ}. Note that T ′ is finitely satisfiable and χ(T ′ , ℒ′ ) = λ. By the compactness theorem and the downward Löwenheim–Skolem theorem, T ′ has a model with density character λ. Taking the reduct of this model to ℒ, we get the required model of T of density character λ.

6.3 Elementary equivalence and ultraproducts We begin this section with an exercise. Exercise 6.7. For ℒ-structures ℳ and 𝒩 , we have that ℳ ≡ 𝒩 if and only if there is an ℒ-structure 𝒫 such that ℳ and 𝒩 both elementarily embed into 𝒫 . This exercise can be significantly strengthened: Theorem 6.8 (Keisler–Shelah). For ℒ-structures ℳ and 𝒩 , we have that ℳ ≡ 𝒩 if and only if there are ultrafilters 𝒰 and 𝒱 such that ℳ𝒰 ≅ 𝒩 𝒱 . The proof in the classical case can be found in [36, 42]. In the continuous case, a proof appears in [29] while a proof in the positive bounded case appears in [34]. It is important to notice that in the Keisler–Shelah theorem, the index sets for the ultrafilters need not be countable. In the continuous setting, it is often difficult to explicitly write out a sentence that distinguishes the theories of two structures. The approach of showing that no two ultrapowers are isomorphic has been used to good effect in separating theories of II1 factors (see [12, 14] and Section 4 in the author’s article with Goldbring in this volume, [27]). EF-games were discussed in the previous section. There is also an infinite version of an EF-game involving ultrapowers which is sometimes easier to work with. We describe

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this game only in the case where ℒ is separable. Given two ℒ-structures ℳ and 𝒩 , the ∞-EF(ℳ, 𝒩 )-game is a two player game between Player I and Player II which now goes on for ω many rounds. The players play as in a finite EF-game, this time creating two sequences (ai : i ∈ N) in ℳ and (bi : i ∈ N) in 𝒩 . Player II wins the game if the map ai 󳨃→ bi , for i ∈ N, extends to an isomorphism of ℒ-substructures of ℳ and 𝒩 , respectively. Theorem 6.9. Assume ℒ is separable. Then for two ℒ-structures ℳ and 𝒩 , the following are equivalent: (1) ℳ ≡ 𝒩 . (2) For some (equiv. any) nonprincipal ultrafilter 𝒰 on N, player II has a winning strategy for the ∞-EF(ℳ𝒰 , 𝒩 𝒰 )-game. Exercise 6.10. Prove the previous theorem.

6.4 Elementary classes We now give a characterization of elementary classes, paralleling a result from classical first order logic. Fix a class 𝒞 of ℒ-structures. We say that 𝒞 is closed under ultraroots if for any ultrafilter 𝒰 , whenever 𝒩 𝒰 ∈ 𝒞 , then 𝒩 ∈ 𝒞 . Theorem 6.11. Suppose that 𝒞 is a class of ℒ-structures. The following are equivalent: (1) 𝒞 is an elementary class. (2) 𝒞 is closed under isomorphisms, ultraproducts, and elementary submodels. (3) 𝒞 is closed under isomorphisms, ultraproducts, and ultraroots. Proof. Clearly, (1) implies both (2) and (3). We first prove (2) implies (1) and then modify the proof to get that (3) implies (1). Thus, assume (2) and let T be the theory of 𝒞 , that is, let T be the set of all sentences φ such that φ𝒩 = 0 for all 𝒩 ∈ 𝒞 . We show that 𝒞 is the collection of models of T. It is clear that all elements of 𝒞 model T. On the other hand, fix a model ℳ of T. By (2), it suffices to elementarily embed ℳ into an ultraproduct of structures from 𝒞 . To this end, suppose φ(cm̄ ) is in Elem(ℳ) for some sequence of ̄ an ℒ-sentence whose value in ℳ constants cm̄ with m̄ ∈ ℳ. Now consider infx̄ |φ(x)|, ̄ < ϵ (for otherwise the is 0. Fix ϵ > 0. Then there is 𝒩φ,ϵ ∈ 𝒞 which satisfies infx̄ |φ(x)| ̄ belongs to T, contradicting the fact that ℳ 󳀀󳨐 T). By taking sentence ϵ ∸ infx̄ |φ(x)| an ultraproduct over all 𝒩φ,ϵ as φ and ϵ vary, we can elementarily embed ℳ into an ultraproduct of elements of 𝒞 , as desired. To obtain (3) implies (1), notice that the previous argument showed (assuming that 𝒞 is merely closed under isomorphisms and ultraproducts) that ℳ ≡ 𝒩 for some 𝒩 in 𝒞 (namely an ultraproduct of structures from 𝒞 ). By the Keisler–Shelah theorem, ℳ𝒰 ≅ 𝒩 𝒱 for some ultrafilters 𝒰 and 𝒱 . Assuming (3), we have that ℳ ∈ 𝒞 .

110 � B. Hart Although the semantic characterization of elementary class is the same in continuous logic as in classical logic, it plays a somewhat bigger role in the continuous setting as it is often difficult to explicitly write out sentences that axiomatize a given class of metric structures. For instance, although the class of W∗ -probability spaces (see [2] for the definition) was axiomatized in a complicated language in [15], it was relatively easy to show using the above semantic characterization that this class was elementary in a more natural language [28]; an explicit axiomatization in this language will only finally appear in [3]. Exercise 6.12. Referring back to Example 3.10 of the class of C∗ -algebras and their dissections as metric structures, we want to see that the ultraproduct in the metric structure sense and the C∗ -algebra sense match up. Recall that if (Ai : i ∈ I) is a family of C∗ -algebras and 𝒰 is an ultrafilter on I, then we set ∏∞ I Ai to be {ā ∈ ∏ Ai : there is B > 0 so that ‖ai ‖ ≤ B for all i ∈ I} I

and ∞

J = {ā ∈ ∏ Ai : lim ‖ai ‖ = 0}. 𝒰

I

The C*-algebraic ultraproduct of (Ai ) with respect to 𝒰 is then defined to be ∞

∏ Ai = ∏ Ai /J. 𝒰

I

Prove that D(∏𝒰 Ai ) = ∏𝒰 D(Ai ), where the ultraproduct appearing on the right-hand side is the ultraproduct of metric structures. This exercise, together with Example 3.10, shows, by Theorem 6.11, that the class of dissections of C∗ -algebras is an elementary class, categorically equivalent to the class of C∗ -algebras via the dissection functor which even preserves ultraproducts. Of course, we already know that the class of dissections is an elementary class because they are exactly the ℒC ∗ -structures which satisfy the theory TC ∗ from Example 5.18.

7 Types 7.1 The type space Suppose that T is a theory in some language ℒ. Fix a sequence of sorted variables x.̄ The ̄ ̄ ̄ := space of x-types of T, denoted Sx̄ (T), is the set of functions of the form tpℳ (a)(φ( x)) ℳ ̄ φ (a), where ℳ is a model of T, ā ∈ ℳ (sorted in the same way as the variables ̄ and φ is an ℒ-formula whose free variables are among x.̄ Then tpℳ (a)̄ is called in x),

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the complete type of a in ℳ (complete meaning that the value of every formula φ is completely specified). We use letters such as p and q to denote complete types. If p ∈ Sx̄ (T) and ā ∈ ℳ is such that tpℳ (a)̄ = p, then we say that ā realizes p and that p is realized in ℳ. Just as with complete theories, complete types are determined by knowing the kernel of the type functional: ̄ ̄ ̄ = r iff φ(x)̄ − r ∈ ker(tpℳ (a)). tpℳ (a)(φ( x)) It is convenient to talk about partial types as well. Suppose Σ is a set of ℒ-formulas in free variables x̄ and p is a complete type in the variables x.̄ Then the corresponding partial (Σ)-type is the function p ↾ Σ. If Σ is the set of all quantifier-free formulas in the free variables x,̄ we call the Σ-type a quantifier-free type. As in the classical case, the type space Sx̄ (T) carries a natural topology. For φ ∈ Formx̄ (T, ℒ) and real numbers r < s, set OT (φ, (r, s)) = {p ∈ Sx̄ (T) : p(φ) ∈ (r, s)}. Exercise 7.1. (1) Prove that the set of all OT (φ, (r, s)) is a base for a Hausdorff topology on Sx̄ (T), called the logic topology. (2) Prove that the logic topology on Sx̄ (T) is compact. (Hint: use the compactness theorem.) (3) Prove that if π(x)̄ is a partial type, then the set {p ∈ Sx̄ (T) : p extends π} is a closed subset of Sx̄ (T). (4) In fact, show that every nonempty closed in the logic topology on Sx̄ (T) has the form {p ∈ Sx̄ (T) : p extends π}, where π is a partial type. (Hint: Show that those p ∈ Sx̄ (T) for which p(φ) ≤ r are the same as those which extend the partial type π where π(φ ∸ r) = 0.) If the theory T is complete, the type space also has a natural metric topology: for ̄ where p = tpℳ (a)̄ and q = tpℳ (b)̄ p, q ∈ Sx̄ (T), we set d(p, q) to be the infimum of d(a,̄ b), ̄ for some model ℳ of T and some a,̄ b ∈ ℳ. The completeness of the theory guarantees that d(p, q) is defined (not infinite) for every pair of types p and q. The symmetry and reflexivity of d(p, q) are clear. To see that the triangle inequality holds, fix p, q, r ∈ Sx̄ (T). By compactness, we can find models ℳ and 𝒩 of T and a,̄ b̄ ∈ ℳ, b̄ ′ , c̄ ∈ 𝒩 with p = ̄ and d(q, r) = d(b̄ ′ , c). ̄ q = tpℳ (b)̄ = tp𝒩 (b̄ ′ ), r = tp𝒩 (c), ̄ d(p, q) = d(a,̄ b), ̄ We tpℳ (a), leave it as an exercise that by taking a common elementary extension of ℳ and 𝒩 , we can assume that ℳ = 𝒩 and b̄ = b̄ ′ . By the triangle inequality (in ℳ), d(p, q) + d(q, r) ≥ d(a,̄ c)̄ ≥ d(p, r). Exercise 7.2. (1) Show that the metric topology refines the logic topology on Sx̄ (T). (2) Show that Sx̄ (T) is complete with respect to the metric topology.

112 � B. Hart We will see later that there are model theoretic consequences when these two topologies agree (even locally).

7.2 Some uses of the Stone–Weierstrass theorem in continuous logic To understand the extension from basic formulas to more general formulas in a more abstract fashion, we remind the reader of the background necessary for the Stone– Weierstrass theorem. Suppose that X is a topological space and set C(X, ℝ) to be the set of continuous functions from X to ℝ. The set C(X, ℝ) has a natural real algebra structure by considering pointwise addition and multiplication together with scalar multiplication by elements of ℝ. It is also naturally topologized by uniform convergence, that is, for f ∈ C(X, ℝ) and ϵ > 0, the set 󵄨 󵄨 {g ∈ C(X, ℝ) : 󵄨󵄨󵄨f (x) − g(x)󵄨󵄨󵄨 < ϵ for all x ∈ X} is a basic open set. When X is compact, C(X, ℝ) is metrizable by the sup-norm. For A ⊆ C(X, ℝ), we say that A separates points if for all x, y ∈ X, there is f ∈ A such that f (x) ≠ f (y). Theorem 7.3 (Stone–Weierstrass theorem). Suppose that X is a compact, Hausdorf space and A is a real subalgebra of C(X, ℝ). If the constant function 1 belongs to A, then A is dense in C(X, ℝ) if and only if A separates points. If φ(x)̄ is an ℒ-formula and T is an ℒ-theory, then there is a natural evaluation map fφ : Sx̄ (T) → ℝ given by fφ (p) = p(φ). With the logic topology on the type space Sx̄ (T), it is easy to see that fφ is continuous. We want to use the Stone–Weierstrass theorem to give another understanding of formulas in continuous model theory. In fact, this alternative perspective is really the reason behind extending the class of formulas. Theorem 7.4. Suppose that T is a theory, x̄ a sequence of variables, and X = Sx̄ (T). Then the evaluation map ev from Formx̄ (T, ℒ) to C(X, ℝ) given by ev(φ) = fφ is an isometry which commutes with all continuous real-valued functions, that is, if f : ℝn → ℝ is a continuous function and φ1 , . . . , φn ∈ Formx̄ (T, ℒ), then f ∘ (ev(φ1 ), . . . , ev(φn )) = ev(f (φ1 , . . . , φn )). Proof. Note that the range of the evaluation map restricted to basic formulas is a real subalgebra containing 1 and which separates points. It follows then from the Stone– Weierstrass theorem that the evaluation map is onto. The map is norm-preserving since if φ(x)̄ is a basic formula such that ‖φ‖T = r, then, on the one hand, ‖ev(φ)‖ ≤ r in C(X, ℝ), while by compactness, we can find some p ∈ Sx̄ (T) such that ev(|φ|)(p) = r, hence ‖ev(φ)‖ = r. Commutation with continuous functions is immediate.

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We need a slight strengthening of the previous theorem for real algebra systems (see Definition 5.3). The proof follows from compactness and the Stone–Weierstrass theorem. Proposition 7.5. Suppose that T is an ℒ-theory and A is real algebra system of T-formulas. Let X be the set of restrictions of elements in Sx̄ (T) to formulas in Ax̄ . Then (1) X is compact, and (2) if the range of the evaluation map from Ax̄ to C(X, ℝ) separates points then the evaluation map is an isometry that commutes with all continuous functions. Exercise 7.6. Prove the previous proposition. As mentioned before, three places where the previous proposition will be useful are for the following real algebra systems: (1) Ax̄ is the closure of the basic ℒ′ -formulas in Formx̄ (T, ℒ) for ℒ′ , a sublanguage of ℒ, and T, an ℒ-theory; (2) Ax̄ is the set of quantifier-free ℒ-formulas in the free-variables x;̄ (3) Ax̄ is the set of prenex formulas in ℒ in the free variables x.̄ Note that prenex formulas in ℒ are dense in the set of all ℒ-formulas. (See Section 6 of [7] for details.)

7.3 Saturation Suppose that ℳ is an ℒ-structure and A ⊂ ℳ. We say that p is a (partial) type over A if p is a (partial) type in ℒA with respect to the theory of ℳA . Consequently, if p is a partial type over A, there is an elementary extension 𝒩 of ℳ and b̄ ∈ 𝒩 realizing p, that is, if ̄ = φ𝒩 (b,̄ a). ̄ φ(x,̄ a)̄ is an ℒA -formula (in the domain of p), then p(φ(x,̄ a)) Definition 7.7. For κ an infinite cardinal, we say that an ℒ-structure ℳ is κ-saturated if, whenever A ⊆ ℳ is such that χ(A) < κ and p is a type over A, then p is realized in ℳ. Then ℳ is said to be saturated if it is χ(ℳ)-saturated. As in the case of classical first-order logic, saturated models typically only exist under additional set-theoretic assumptions. For instance, if 2κ = κ+ , then the following proposition yields that if χ(ℳ) = κ, then ℳ has a saturated elementary extension of density character κ+ . Proposition 7.8. Suppose that ℳ is an ℒ-structure and χ(ℒ) ≤ χ(ℳ) = κ. Then there is an elementary extension 𝒩 of ℳ such that 𝒩 is κ+ -saturated and χ(𝒩 ) ≤ 2κ . Proof. We construct 𝒩 as a union of an increasing chain (ℳα )α 0. We can thus find a model ℳ of T and a,̄ b̄ ∈ ℳ such that ā ̄ and b̄ have the same ℒ′ -type but φℳ (a)̄ ≠ φℳ (b). + Take an |M| -saturated elementary extension 𝒩 of ℳ. Let ℳ′ and 𝒩 ′ be the reducts of ℳ and N to ℒ′ . Since 𝒩 ′ is also |ℳ′ |+ -saturated, we can find an ℒ′ -elementary map ρ: ℳ′ → N ′ such that ρ(b)̄ = a.̄ Since F is an equivalence of categories, ρ is also an ̄ ℒ-elementary map, which contradicts that φℳ (a)̄ ≠ φℳ (b).

8 Definable Sets 8.1 Introducing definable sets We will now look at one of the biggest differences between continuous model theory and classical model theory. In the classical setting, it is often advantageous to be able to quantify over a subset of the universe which is described by a formula. For instance, if φ(x) is a formula expressing some property, it is straightforward to also express “for all x, if φ(x) then …” or “there exists x such that φ(x) and …”. In the continuous setting, where formulas take on real values, the subsets over which one can quantify is a delicate matter. Suppose that ℳ is an ℒ-structure and φ(x)̄ is an ℒ-formula. The zero-set of φ in ℳ, written Z ℳ (φ), is the set {ā ∈ ℳ : φℳ (a)̄ = 0}. Naively one might think that one can quantify over zero sets of formulas, but as we will soon see, that is not quite correct. Suppose that T is an ℒ-theory and S̄ is a tuple of sorts from ℒ. A T-functor (for S)̄ is a functor X: Mod(T) → Met, where Met is the category of metric spaces with isometric embeddings as morphisms, X(ℳ) is a closed subspace of S(̄ ℳ), and if ρ : ℳ → 𝒩 is an elementary map in Mod(T), then X(ρ) is the restriction of ρ to X(ℳ). Example 8.1. A natural example of a T-functor is Xφ for some formula φ, where Xφ (ℳ) = Z(φℳ ) = {ā ∈ ℳ : φℳ (a)̄ = 0}, the zero set of φ. Theorem 8.2. Suppose that T is an ℒ-theory and X is a T-functor for S.̄ The following are equivalent: (1) There is a T-formula φ(x)̄ such that for any model ℳ of T and any ā ∈ ℳ, φℳ (a)̄ = d(a,̄ X(ℳ)). ̄ there is a T-formula χ(y)̄ such that for any model ℳ of T (2) For any T-formula ψ(x,̄ y), and any ā ∈ ℳ, χ ℳ (a)̄ = inf{ψℳ (b,̄ a)̄ : b̄ ∈ X(ℳ)}

116 � B. Hart (or equivalently the analogous statement replacing inf by sup). (3) For every ϵ > 0, there is a T-formula φ(x)̄ and δ > 0 such that, for all models ℳ of T: – X(M) ⊆ Z(φℳ ), and – For all ā ∈ ℳ, if φℳ (a)̄ < δ, then d(a,̄ X(ℳ)) ≤ ϵ. (4) For every ϵ > 0, there is a basic ℒ-formula φ(x)̄ and δ > 0 such that, for all models ℳ of T: – X(ℳ) ⊆ Z(φℳ ), and – for all ā ∈ ℳ, if φℳ (a)̄ < δ, then d(a,̄ X(ℳ)) ≤ ϵ. (5) For any set I, family of ℒ-structures ℳi for i ∈ I, and ultrafilter 𝒰 on I, X(𝒩 ) = ∏ X(ℳi ), 𝒰

where 𝒩 = ∏ ℳi . 𝒰

Comment 8.3. Before we begin the proof of Theorem 8.2, we want to address one small point. What do we do with the empty functor, that is, the functor X which satisfies X(ℳ) = 0 for all models ℳ of T? The empty functor certainly satisfies condition (5) above; what about the other four conditions? In condition (1) we must decide what the distance is to the empty set – it should be the maximum possible value of the metric (and hence is given by a formula). In (2), the inf is taken over an empty set hence equals the maximum possible value of ψ – again, a formula. In (3) and (4), we are measuring the distance to the empty set and so we again take this to be the upper bound on the metric. Consequently, the empty functor satisfies the equivalent conditions of Theorem 8.2 when conditions (1)–(4) are suitably interpreted. Moreover, if X satisfies any of the conditions of the theorem and X(ℳ) is empty for some model ℳ of T, then it is empty for all models ℳ of T (and thus satisfies all of the other conditions in the theorem). Proof of Theorem 8.2. Based on the previous comment, we can assume, without loss of generality, that X(ℳ) ≠ 0 for all models ℳ of T. The proof of (1) implies (2) can be found in [7, Theorem 9.17]. We only remark that if dX (x) is a formula which expresses the distance from x to X in any model of T and α is a uniform continuity modulus for the formula ψ, then in any model ℳ of T (remembering that X(ℳ) is not empty), inf{ψℳ (b,̄ a)̄ : b̄ ∈ X(ℳ)} equals ̄ inf (ψℳ (b,̄ a)̄ + α(dXℳ (b))).

̄ b∈ℳ

It is easy to see that (2) implies (3). To show that (3) implies (4), it is enough to remember that T-formulas are uniformly approximated by basic ℒ-formulas. To see that (4) implies (5), we first show that ∏𝒰 X(ℳi ) ⊆ X(𝒩 ). To this end, fix ϵ > 0 and let φ and δ be as in (4). If [(ai )]𝒰 ∈ ∏𝒰 X(ℳi ), then φℳi (ai ) = 0 for all i ∈ I, hence φ𝒩 ([ai ]𝒰 ) = 0 and thus d([(ai )]𝒰 , X(𝒩 )) ≤ ϵ. Since ϵ > 0 was arbitrary, we have that [(ai )]𝒰 ∈ X(𝒩 ). Now suppose, towards a contradiction, that there is [(ai )]𝒰 ∈ X(𝒩 ) \ ∏𝒰 X(ℳi ). Since ∏𝒰 X(ℳi ) is closed in 𝒩 , there is ϵ > 0 such that d([(ai )]𝒰 , ∏𝒰 X(ℳi )) > ϵ. Let φ and

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δ > 0 be as in (4). Since X(𝒩 ) ⊆ Z(φ𝒩 ), we have that φℳi (ai ) < δ for 𝒰 -almost all i ∈ I, hence d(ai , X(ℳi )) ≤ ϵ for 𝒰 -almost all i ∈ I, which is a contradiction. It remains to show that (5) implies (1). Let 𝒞 be the class of structures {(ℳ, dXℳ ) : ℳ 󳀀󳨐 T}, where dXℳ measures the distance from an element to X(ℳ). Note that measuring the distance to a closed set is indeed an allowable relation in continuous logic: the bound is the maximum of the bounds on the metrics in S̄ and the modulus of uniform continuity is the identity function. By Theorem 6.11, condition (5) guarantees that 𝒞 is an elementary class. The forgetful functor from 𝒞 onto Mod(T) is an equivalence of categories since any model of T extends uniquely to an object in 𝒞 . By the Beth definability theorem, we conclude that dX is in fact equivalent to a T-formula. If X is a T-functor (for S)̄ satisfying any of the equivalent conditions of the previous theorem, then we say that X is a T-definable set. By condition (2) of the previous theorem, (T-)definable sets are exactly those sets in metric structures over which one can quantify.

8.2 Examples and nonexamples We begin with some examples of definable sets. Example 8.4. Any sort is a definable set relative to any theory as is any finite product of sorts. Perhaps less trivially, the range of any term is definable, since if we are given a term τ(x)̄ and we wish to quantify over the range of τ, for instance inf{φℳ (y) : y ∈ ̄ This fact already pays dividends in the case range(τ ℳ )}, we would evaluate infx̄ φ(τ(x)). of operator algebras, where we can express any self-adjoint element as (x + x ∗ )/2 and any positive element as x ∗ x, hence both of these sets of elements are definable relative to theory of C∗ -algebras. Example 8.5. A slightly less trivial example is that the set of projections is definable relative to the class of C∗ -algebras. Recall that the element p in a C∗ -algebra A is a projection if it is self-adjoint and satisfies p2 = p. To see that the set of projections is definable, we use some functional calculus. Suppose we have a self-adjoint element b in some C∗ -algebra A satisfying ‖b2 −b‖ < ϵ, where ϵ < 1/4. We know that the subalgebra B of A generated by b and 1A is isomorphic to C(σ(b)) by an isomorphism sending b to the identity function and 1A to the constant function 1, where σ(b) is the spectrum of b and C(σ(b)) is the C∗ algebra of complex-valued continuous functions on σ(b). By applying the isomorphism, in C(σ(b)), we have ‖x 2 −x‖ < ϵ and so σ(b) can be partitioned into two disjoint open sets O0 and O1 with i ∈ Oi for i = 0, 1. If we let π be the function which is constantly i on Oi , then this is a continuous function on σ(b). Additionally, π is a projection and ‖π − x‖ < ϵ. Pulling the isomorphism from C(σ(b)) back into B, we find a projection p ∈ B such that

118 � B. Hart ‖b − p‖ < ϵ. By Theorem 8.2(3), the set of projections is definable relative to the theory of C∗ -algebras. We now give two nonexamples to demonstrate that the notion of definable set is quite distinct from the notion of zero set. The first is a somewhat abstract example. Example 8.6. The structure ℳ has a single sort with underlying set [0, 1] equipped with the discrete metric and a single relation symbol R which is interpreted as R(x) = x. Let T = Th(ℳ) and consider the T-functor X, for 𝒩 ∈ Mod(T), defined by X(𝒩 ) = {a ∈ 𝒩 : R𝒩 (a) ≤ 1/2}. There are a variety of ways to see that X is not definable. We will use the ultraproduct characterization. Notice that if 𝒰 is a non-principal ultrafilter on N, then the sequence (1/2 + 1/n : n ∈ N) has R-value 1/2 in ℳ𝒰 but the distance from this element to every element in [0, 1/2]𝒰 is 1. This shows that X(ℳ)𝒰 ≠ X(M 𝒰 ) and so X is not T-definable. Example 8.7. An example that appears in [16] in the context of C∗ -algebras is the set of normal elements. In a C∗ -algebra, the element a is said to be normal if it commutes with its adjoint, hence we are looking at the zero set of ‖a∗ a − aa∗ ‖. The set of normal elements in a C∗ -algebra is not definable. (See [16, Proposition 3.2.0.8] for a proof.)

8.3 Principal types For emphasis, we record the following remark made during Comment 8.3. Proposition 8.8. If X is a T-definable set, then X(ℳ) is nonempty for some model ℳ of T if and only if X(ℳ) is nonempty for all models ℳ of T. The following proposition makes a connection between the logic topology and the metric on Sx̄ (T). We say that a (partial) type is principal relative to a theory T if the T-functor X which sends ℳ to the set of realizations of p in ℳ is a T-definable set. By Proposition 8.8, any principal type is realized in all models of T; the converse of this statement is the continuous form of the omitting types theorem (proven in Section 10 below), and uses the next result in a crucial way. Proposition 8.9. For p ∈ Sx̄ (T), the following are equivalent: (1) p is principal relative to T. (2) Every ϵ-ball around p contains a logical open neighborhood of p. (3) Every ϵ-ball around p has nonempty interior in the logic topology. Proof. If p is principal, then (2) follows from Theorem 8.2(4). We now prove that (2) implies (1). This argument is based on the proof of [7, Theorem 9.12]. Fix n ∈ N and consider a formula φn (x) such that p(φn ) = 0 and p ∈ OT (φn , (−∞, δn )) ⊆ B(p, 1/n). It follows that if ℳ is a model of T and a ∈ ℳ ℳ satisfies φℳ n (a) < δn , then d(tp (a), p) ≤ 1/n. Now consider the T-formula P(x) defined

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by ∑ n

|φn (x)| , Bn 2 n

where Bn is the bound for the formula φn (see Example 5.2). Notice that, for all ϵ > 0, there is δ > 0 so that, for all models ℳ of T and all a ∈ ℳ, if Pℳ (a) < δ, then d(tpℳ (a), p) < ϵ. Using Proposition 3.2, we can find an increasing continuous function α with α(0) = 0 such that d(tp(a), p) ≤ α(Pℳ (a)) holds for all models ℳ of T and all a ∈ ℳ. Setting D(x) = infy (α(P(y)) + d(x, y)), we claim that D(x) = d(x, p(ℳ)) for all models ℳ of T. To see this, first assume that ℳ is ω-saturated, which in particular implies that for all a ∈ ℳ, we have d(a, p(ℳ)) = d(tpℳ (a), p). Fix a ∈ ℳ; we first show that Dℳ (a) ≤ d(a, p(ℳ)). Recalling that P(y) = 0 whenever y realizes p, we see that Dℳ (a) = (inf(α(P(y)) + d(a, y)))

ℳ

y

≤

inf d(a, y) = d(a, p(ℳ)).

y∈p(ℳ)

Note that this inequality did not use the ω-saturation assumption. For the other inequality, given b ∈ ℳ, we have α(Pℳ (b)) ≥ d(tpℳ (b), p) = d(b, p(ℳ)), hence Dℳ (a) = (inf(α(P(y)) + d(a, y))) y

ℳ

≥ inf (d(y, p(ℳ)) + d(a, y)) ≥ d(a, p(ℳ)). y∈ℳ

Now suppose that 𝒩 is an arbitrary model of T and ℳ an ω-saturated elementary extension, hence Dℳ (x) = d(x, p(ℳ)) by the previous paragraph. We now show that D𝒩 (a) = d(a, p(𝒩 )) for all a ∈ 𝒩 . As noted above, the inequality D𝒩 (a) ≤ d(a, p(𝒩 )) holds without the ω-saturation assumption. For the other inequality, fix ϵ > 0; we will show d(a, p(𝒩 )) ≤ D𝒩 (a) + ϵ, which suffices by letting ϵ tend to 0. Towards this end, first notice that if a ∈ 𝒩 is such that P𝒩 (a) = 0, then a realizes p and so p(𝒩 ) = Z(P𝒩 ). Now choose a = a0 ∈ 𝒩 . We know that Dℳ (a) = d(a, p(ℳ)) which implies that there is c ∈ ℳ so that Dℳ (c) = 0 and d(a, c) = Dℳ (a). So ℳ satisfies infy max{D(y), |d(a, y)−D(a)|}. By elementarity, we can choose a1 ∈ 𝒩 so that D𝒩 (a1 ) ≤ 8ϵ and |D𝒩 (a0 )− d(a0 , a1 )| ≤ 8ϵ . We continue to create a Cauchy sequence (an ) in 𝒩 such that D𝒩 (an ) ≤

ϵ , 2n+2

ϵ 󵄨 󵄨 and 󵄨󵄨󵄨D𝒩 (an ) − d(an , an+1 )󵄨󵄨󵄨 ≤ n+2 . 2

ϵ Given that we have constructed an , we note that since D𝒩 (an ) ≤ 2n+2 , in ℳ we can find ℳ ℳ c realizing p so that D (c) = 0 and D (an ) = d(an , c). By elementarity of 𝒩 in ℳ, we can construct an+1 satisfying the properties listed above. By the completeness of 𝒩 , the sequence (an ) has a limit b ∈ 𝒩 and by the continuity of D𝒩 , we have that D𝒩 (b) = 0, ϵ hence b realizes p. We also conclude from our two items that d(an , an+1 ) ≤ 2n+1 for all n. We thus have

120 � B. Hart n−1

d(a, p(𝒩 )) ≤ d(a, b) = lim d(a, an ) ≤ lim(d(a0 , a1 ) + ∑ d(aj , aj+1 )) ≤ D𝒩 (a) + ϵ. n

n

j=1

We have thus finished the proof that (2) implies (1). The direction (2) implies (3) is trivial, so it remains to prove that (3) implies (2). Suppose that the nonempty logic open set OT (φ, (r, s)) is contained in the ϵ-ball around p. Choose r ′ and s′ so that r < r ′ < s′ < s and such that, setting ψ(x) = infy max(r ′ ∸ φ(y), φ(y) ∸ s′ , d(x, y) ∸ ϵ), we have p(ψ) = 0. It follows that p belongs to the logic open set OT (ψ, (−∞, s′′ )) for any s′′ > 0; if r < r ′ − s′′ , s′ + s′′ < s and s′′ < ϵ, then OT (ψ, (−∞, s′′ )) is contained in the 3ϵ-ball around p. Since ϵ is arbitrary, this proves (2).

9 Quantifier elimination We say that a theory T in a language ℒ has quantifier elimination if, for every ℒ-formula ̄ there is a quantifier-free ℒ-formula ψ(x)̄ which is T-equivalent to φ(x). ̄ In terms of φ(x), ̄ there is a basic ℒ-formulas, this says: for every ϵ > 0 and every basic ℒ-formula φ(x), ̄ T < ϵ. quantifier-free basic ℒ-formula ψ such that ‖φ(x)̄ − ψ(x)‖ We will give three characterizations of quantifier elimination. The first is syntactical. Proposition 9.1. Suppose that T is an ℒ-theory. Then T has quantifier-elimination if ̄ where φ(x, y)̄ is a basic ℒ-formula, is and only if every formula of the form infx φ(x, y), T-equivalent to a quantifier-free ℒ-formula. Proof. The forward direction is immediate and the backward direction is proven by induction on the formation of basic ℒ-formulas, the quantifier case being taken care of by the backwards assumption and the connective case being immediate. Proposition 9.2. Suppose that T is an ℒ-theory. Then T has quantifier-elimination if and only if whenever ℳ is a model of T and a,̄ b̄ ∈ ℳ have the same values for basic atomic formulas, then they satisfy the same complete type. Proof. This follows from Proposition 7.5 applied to the real algebra system of quantifierfree formulas. Example 9.3. We can use Proposition 9.2 to see that the theory T of the Urysohn sphere U has quantifier elimination. Indeed, suppose that 𝒩 is a model of T and a,̄ b̄ ∈ 𝒩 have the same quantifier-free type. Without loss of generality, we can assume that 𝒩 is separable and thus we may further assume that 𝒩 is U itself by the back-and-forth argument given in Example 5.14. Since ā and b̄ have the same quantifier-free type, the map f sending ā 󳨃→ b̄ is an isometry. Since U is ultrahomogeneous, f extends to an automorphism of U, which implies that ā and b̄ have the same complete type in U, as desired.

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A more semantic characterization of quantifier elimination is given by the follow-

Proposition 9.4. Suppose T is an ℒ-theory. Then T has quantifier elimination if and only if T has the property that whenever ℳ is a model of T, ā and b̄ ∈ ℳ have the same quantifier-free type, then there is 𝒩 with ℳ ≺ 𝒩 such that the map sending ā to b̄ can be extended to an embedding of ℳ into 𝒩 . Note that if ℳ in the proposition is separable then 𝒩 could be taken to be ℳ𝒰 where 𝒰 is a nonprincipal ultrafilter on N. Proof. We leave the proof of the forward direction as an exercise to the reader and sketch a proof of the backwards direction using Proposition 9.1. Fix a basic quantifier̄ Suppose that for some ϵ, the following set of sentences is satisfifree ℒ-formula φ(x, y). able: 󵄨 󵄨 {󵄨󵄨󵄨ψ(c)̄ − ψ(d)̄ 󵄨󵄨󵄨 : ψ(y)̄ is a basic quantifier-free ℒ-formula} 󵄨󵄨 󵄨󵄨 ∪{ϵ ∸ 󵄨󵄨󵄨inf φ(x, c)̄ − inf φ(x, d)̄ 󵄨󵄨󵄨}, x 󵄨x 󵄨 where c̄ and d̄ are sequences of new constants. It follows that there is a model ℳ of T and ̄ By a,̄ b̄ ∈ ℳ with the same quantifier-free type but (infx φ)ℳ (x, a)̄ < (infx φ)ℳ (x, b). ̄ assumption, there is an elementary extension 𝒩 of ℳ such that the map ā 󳨃→ b extends to an embedding f from ℳ to 𝒩 . However, if we choose c ∈ ℳ such that φℳ (c, a)̄ < ̄ then we have a contradiction when we consider the (infx φ)ℳ (x, b)̄ = (infx φ)𝒩 (x, b), 𝒩 ̄ value of φ (f (c), b). Consequently, the above set of sentences cannot be satisfiable for any ϵ. By compactness, for any ϵ > 0, we can find, after some manipulation, a quantifier-free basic formula ψ(x)̄ and δ > 0 such that, for any model ℳ of T and a,̄ b̄ ∈ ℳ, 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̄ 󵄨 󵄨󵄨ψ(a) − ψ(b)̄ 󵄨󵄨󵄨 < δ implies 󵄨󵄨󵄨inf φ(x, a)̄ − inf φ(x, b)̄ 󵄨󵄨󵄨 ≤ ϵ. x 󵄨x 󵄨 By Proposition 7.5 applied to the real algebra system of quantifier-free formulas, we have that infx φ(x, y)̄ is equivalent to a quantifier-free formula. By Proposition 9.1, T has quantifier elimination. Exercise 9.5. Prove the forward direction of the previous proposition. Example 9.6. We can use the previous proposition to show that the theory of infinitedimensional Hilbert spaces admits quantifier-elimination. (We leave it to the reader as an exercise to verify that infinite-dimensionality is expressible in continuous logic.) Indeed, suppose that ℋ is an infinite-dimensional Hilbert space and that a,̄ b̄ ∈ ℋ are tuples with the same quantifier-free type. Letting ℋ0 and ℋ1 denote the finite-dimensional subspaces of ℋ generated by ā and b,̄ respectively. It follows that the map ai 󳨃→ bi for i = 1, . . . , n extends to an isomorphism ℋ0 → ℋ1 . Since ℋ is infinite-dimensional, we

122 � B. Hart have that the dimension of the orthogonal complement ℋ0⊥ of ℋ0 in ℋ is the same as the dimension of the orthogonal complement ℋ1⊥ of ℋ1 in ℋ. Thus, mapping ℋ0⊥ isomorphically into ℋ1⊥ allows one to extend the original mapping to an automorphism of ℋ. Note that in this example, there is no need for an elementary extension; in more complicated examples, passing to an elementary extension is in fact necessary.

10 Imaginaries In classical model theory, imaginary elements play an important role. The construction of the theory T eq from T creates the largest expansion of T for which the category of models is equivalent. Although somewhat more complicated in the continuous setting, one can define imaginaries for metric structures which have similar properties. We consider them in the next three subsections.

10.1 Products Fix a continuous language ℒ and an ℒ-theory T. Suppose that S̄ = (Sn : n ∈ N) is a sequence of sorts from ℒ. We add one new sort S to ℒ to form ℒ′ . The intention is that we will expand models of T to include as the interpretation of S the product of the sorts Sn . We expand every model ℳ of T by interpreting S as ∏n∈N Snℳ . We need a metric for this sort. The choice of metric is not unique but for definiteness we use, for x,̄ ȳ ∈ ∏n∈N Snℳ , dnℳ (xn , yn ) , Bn 2n n∈N

d(x,̄ y)̄ = ∑

where dn is the metric symbol for Sn and Bn is the bound on that metric. The key property that d has is that for every ϵ > 0, there is an n ∈ N such that if x,̄ ȳ ∈ ∏n∈N Snℳ and xi = yi for every i < n, then d(x,̄ y)̄ < ϵ. The only other symbols we add to the language ℒ′ are maps πn from S to Sn , and in the canonical expansion of an ℒ-structure to an ℒ′ -structure, we interpret πn as the projection onto the nth coordinate. The uniform continuity modulus for πn can be taken to be ϵ/Bn 2n . We record here three families of sentences that hold in all expanded structures. First, for any n ∈ N, we want to state that diℳ (xi , yi ) diℳ (xi , yi ) 1 ≤ d(x, y) ≤ + n ∑ i i 2 Bi 2 Bi 2 i≤n i≤n ∑

holds for all x, y ∈ S; we can express this fact in continuous logic by requiring that the following two sets of sentences evaluates to 0:

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diℳ (πi (x), πi (y)) 1 + n )) i 2 B 2 i i≤n

sup (d(x, y) ∸ (∑

x,y∈S

and diℳ (πi (x), πi (y)) ∸ d(x, y)). Bi 2 i i≤n

sup (∑

x,y∈S

We would also like the finite products to be dense in S. We can achieve this by satisfying the following sentences, one for each n ∈ N: sup inf max{di (πi (x), xi )}.

xi ∈Si ,i 0 such that the open ϵ-ball around p does not contain any logical open set. Suppose that we are at some stage of the construction and we have constructed the logical open set On , which we may assume, after some manipulation, has ̄ (−∞, 0)), where c is the new constant we are currently interested the form OT (θ(ck , d), k in keeping away from p and d̄ is the rest of the new constants in θ. Now consider the ̄ (−∞, 0)). By the assumption that p is not principal, there logical open set OT (infȳ θ(x, y), ̄ (−∞, 0)) and d(p, q) > ϵ2 . By is some q ∈ Sx (T) such that q belongs to OT (infȳ θ(x, y), compactness, we can find some ℒ-formula ψ(x) such that: ̄ ℳ < 0, and – for all ℳ 󳀀󳨐 T and a ∈ ℳ with ψℳ (a) < 0, we have (infȳ θ(a, y)) – for any ℳ 󳀀󳨐 T and a, b ∈ ℳ such that a realizes p and ψℳ (b) < 0, we have d(a, b) > ϵ2 . We now let On+1 = On ∩ OT (ψ(ck ), (−∞, 0)) and note that this will guarantee that no interpretation of ck will be within ϵ2 of a realization of p in any model of T. Since we can achieve this for all of the new constants, no Cauchy sequence of the new constants will approach a realization of p and so the model we construct will omit p. The situation regarding omitting partial types in continuous logic is quite problematic as evidenced by the results of Farah and Magidor from [19]. There they show the following: (1) There is a complete theory T in a separable language and countably many partial types such that every finite subset is omitted in some model of T but no model of T simultaneously omits all of them. (2) There is a complete theory T in a separable language and partial types s and t, each one omitted in some model of T, but no model omits both. A variant of the omitting types theorem used to omit certain kinds of partial types has proven useful in the operator algebra context; for example, see [16], [23], and Goldbring’s article in this volume [25].

12 Separable categoricity For an infinite cardinal λ, we say that a theory T is λ-categorical if λ ≥ χ(T, ℒ) and whenever ℳ and 𝒩 are models of T such that χ(ℳ) = χ(𝒩 ) = λ, then ℳ ≅ 𝒩 . We will focus on the case where λ is ℵ0 and we will typically say separably categorical instead of ℵ0 -categorical. As in the classical case, we have the following theorem. Theorem 12.1. Suppose T is an ℒ-theory which has only models of density character at least χ(T, ℒ) and is λ-categorical. Then T is complete.

128 � B. Hart Proof. Suppose that ℳ and 𝒩 are models of T. If ℳ has density character at least λ, then by the downward Löwenheim–Skolem theorem, we can find ℳ0 ≺ ℳ with χ(ℳ0 ) = λ. If χ(ℳ) < λ, then by the upward Löwenheim–Skolem theorem applied to Elem(ℳ), we can find ℳ0 with χ(ℳ0 ) = λ and ℳ ≺ ℳ0 . Applying the same reasoning to 𝒩 , we can find ℳ0 and 𝒩0 so that ℳ ≡ ℳ0 ≅ 𝒩0 ≡ 𝒩 .

From this we conclude that T is complete. We now state the Ryll–Nardewski theorem for continuous logic; the proof, done originally in the positive bounded case, is due to Henson. Theorem 12.2. Suppose that ℒ is separable and T is an ℒ-theory with no compact models. Then T is separably categorical if and only T is complete and every complete type is principal. Proof. If T is separably categorical then it is complete. If there is a complete type p which is not principal, then by the omitting types theorem, there is some separable model of T which omits p. However, there is also a separable model of T which realizes p; since these two models of T cannot be isomorphic, T is not separably categorical; contradiction. Now suppose that T is complete and every complete type is principal. Consider two separable models ℳ and 𝒩 of T. We need to show that ℳ ≅ 𝒩 . We construct two sequences a00 , a01 a11 , a02 a12 a22 , . . . in ℳ and b00 , b10 b11 , b20 b21 b22 , . . . in 𝒩 such that all initials segments of the same length have the same type, that is, for any k, there is a fixed type for a0n . . . akn and bn0 . . . bnk independent of n ≥ k. We will also want to arrange that for every k, ⟨akn : n ≥ k⟩ and ⟨bnk : n ≥ k⟩ form Cauchy sequences converging to ak and bk , respectively, so that {ak : k ∈ N} and {bk : k ∈ N} are dense in ℳ and 𝒩 , respectively. If we can achieve this, then the map sending ak to bk extends to an isomorphism from ℳ to 𝒩 . To start, we enumerate countable dense subsets in ℳ and 𝒩 ; call them respectively ⟨ck : k ∈ N⟩ and ⟨dk : k ∈ N⟩. At stage 0, let a00 = c0 . By assumption, tpℳ (c0 ) is principal and hence realized in 𝒩 by some b00 . In general, at each step we alternate by either choosing a ck or dk and we revisit each ck and dk infinitely often in the construction. Assume we have chosen a0n . . . ann already and we consider whatever ck is given to us at this stage. Let p(x0 , . . . , xn ) be the type of a0n . . . ann in ℳ and q(x0 , . . . , xn+1 ) be the type of a0n . . . ann ck in ℳ. Since q is principal, we have a formula dq (x0 , . . . , xn+1 ) expressing the distance to realizations of q. Since dqℳ (a0n . . . ann , ck ) = 0, we have that infy dqM (a0n . . . ann , y) = 0. Since bn0 . . . bnn satisfies p by assumption, we have infy dq𝒩 (bn0 . . . bnn , y) = 0. This means we can

n+1 n n+1 n find bn+1 0 . . . bn+1 realizing q and such that d(bi , bi ) ≤ 1/2 for i = 0, . . . , n. This guarantees we have the required Cauchy sequences and we have the required density as well. To see the latter fact, fix ϵ > 0. Choose N large enough so that ∑n≥N 1/2n < ϵ. If we visit ck at stage t > N, then at is within ϵ of ck and so the ak ’s are dense in M. Similarly, the bk ’s are dense in N, finishing the proof.

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Note that the condition “every complete type is principal” in Theorem 12.1 is equivalent to the statement that the logic and metric topologies on Sx̄ (T) agree for all finite tuples x̄ of variables. Exercise 12.3. Prove that a complete theory T in a separable language with no compact models is separably categorical if and only if Sx̄ (T) is compact in the metric topology for all finite tuples x̄ of variables. There are a number of nontrivial examples of separably categorical continuous theories: Example 12.4. The following theories are separably categorical: (1) The theory of infinite-dimensional Hilbert spaces. (2) The theory of the Urysohn sphere. (3) The theory of atomless probability algebras. The fact that the theory of infinite-dimensional Hilbert spaces is separably categorical follows from the fact that any two separable infinite-dimensional Hilbert spaces are isomorphic to ℓ2 while the separable categoricity of the theory of the Urysohn sphere follows from the argument presented in Example 5.14. The separable categoricity of the theory of atomless probability algebras is discussed in Berenstein and Henson’s article in this volume [11]. In the context of operator algebras, there are very few examples of separably categorical theories. For instance, for any separable II1 factor N, there are continuum many nonisomorphic separable II1 factors elementarily equivalent to N (see Theorem 4.1 in the author’s article with Goldbring in this volume, [27]). One notable example of a separably categorical C∗ -algebra is the algebra C(X) for X a totally disconnected compact Hausdorff space; in this case, C(2N ) is the unique separable model. The proof of this fact follows from Stone duality together with the fact that there is a unique countable atomless Boolean algebra. We end this section with some related considerations. We say that a model ℳ of T is atomic if tpℳ (a)̄ is principal for every finite tuple ā ∈ ℳ. Theorem 12.1 (and its proof) yields the following: Theorem 12.5. (1) Any two separable atomic models of T are isomorphic. (2) If T is a complete theory in a separable language, then T is separably categorical if and only if all separable models of T are atomic. In the remainder of this section, for simplicity, we assume that T is a complete theory in a separable language. We say that a model of T is prime if it embeds elementarily into all other models of T.

130 � B. Hart Exercise 12.6. Prove that a model ℳ of T is prime if and only if ℳ is separable and atomic. Moreover, T can have at most one prime model up to isomorphism. (Hint: To show that a separable atomic model of T is prime, use a “forth only” version of the back and forth argument appearing in the proof of Theorem 12.1.) Exercise 12.7. Prove that T has a prime model if and only if the principal types are dense in Sx̄ (T) for all finite tuples x̄ of variables. We say that T is small if Sx̄ (T) is separable in the metric topology for all finite tuples x̄ of variables. The following is Proposition 1.17 in [9]: Proposition 12.8. If T is small, then T has a prime model.

Bibliography [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

J. M. Albert and B. Hart, Metric logical categories and conceptual completeness for first order continuous logic. arXiv:1607.03068. H. Ando, Introduction to non-tracial ultraproducts of von Neumann algebras, this volume. J. Arulseelan, I. Goldbring and B. Hart, The undecidability of having the QWEP. arXiv:2205.07102. I. Ben Yaacov, Positive model theory and compact abstract theories, J. Math. Log. 3 (2003), 85–118. I. Ben Yaacov, Uncountable dense categoricity in cats, J. Symb. Log. 70 (2005), 829–860. I. Ben Yaacov, On the expressive power of quantifiers in continuous logic. arXiv:2207.01863. I. Ben Yaacov, A. Berenstein, C. W. Henson and A. Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge University Press, Cambridge, 2008, pp. 315–427. I. Ben Yaacov and A. P. Pederson, A proof of completeness for continuous first order logic, J. Symb. Log. 75 (2010), no. 1, 168–190. I. Ben Yaacov and A. Usvyatsov, On d-finiteness in continuous structures, Fundam. Math. 194 (2007), 67–88. I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability, Trans. Am. Math. Soc. 362 (2010), no. 10, 5213–5259. A. Berenstein and C. W. Henson, Model Theory of Probability Spaces, this volume. R. Boutonnet, I. Chifan and A. Ioana, II1 factors with non-isomorphic ultrapowers, Duke Math. J. 166 (2017), 2023–2051. C. C. Chang and H. J. Keisler, Continuous model theory, Princeton University Press, 1966. I. Chifan, A. Ioana and S. Kunnawalkam Elayavalli, Two non-elementarily equivalent II1 factors without property Gamma. arXiv:2209.10645. Y. Dabrowski, Continuous model theories for von Neumann algebras, J. Funct. Anal. 277 (2019). I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati and W. Winter, Model Theory of C∗ -algebras, Mem. Am. Math. Soc. (2021). I. Farah, B. Hart and D. Sherman, Model theory of operator algebras I: Stability, Bull. London Math. Soc. 45 (2013), no. 4, 825–838. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras II: Model theory, Isr. J. Math. 201 (2014), 477–505. I. Farah and M. Magidor, Omitting types in logic of metric structures, J. Math. Log. (2018). I. Farah, A. Toms and A. Törnquist, Turbulence, orbit equivalence, and the classification of nuclear C∗ -algebras, J. Reine Angew. Math. 688 (2014), 101–146.

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[21] E. Gardella, M. Kalantar and M. Lupini, Model theory and Rokhlin dimension for compact quantum group actions, J. Noncommut. Geom. 13 (2019), 711–767. [22] E. Gardella and M. Lupini, Applications of model theory to C∗ -dynamics, J. Funct. Anal. 275 (2018), 1341–1354. [23] I. Goldbring, Enforceable operator algebras, J. Inst. Math. Jussieu 20 (2021), 31–63. [24] I. Goldbring, Ultrafilters throughout mathematics, American Mathematical Society’s Graduate Studies in Mathematics series, vol. 220, 2022. [25] I. Goldbring, Model theory and Ultrapower Embedding Problems in Operator Algebras, this volume. [26] I. Goldbring and B. Hart, On the theories of McDuff’s II1 factors, Int. Math. Res. Not. 27 (2017), 5609–5628. [27] I. Goldbring and B. Hart, A survey on the model theory of tracial von Neumann algebras, this volume. [28] I. Goldbring, B. Hart and T. Sinclair, Correspondences, ultraproducts and model theory. arXiv:1809.00049. [29] I. Goldbring and J. Keisler, Continuous Sentences Preserved under Reduced Products, J. Symb. Log. 87 (2022), no. 2, 649–681. [30] I. Goldbring and T. Sinclair, Games and elementary equivalence of II1 factors, Pac. J. Math. 278 (2015), 103–118. [31] B. Hart, EF games, unpublished notes. https://ms.mcmaster.ca/~bradd/EF-games.pdf. [32] B. Hart, B. Kim and A. Pillay, Coordinatisation and canonical bases in simple theories, J. Symb. Log. 65, 293–309. [33] C. W. Henson, Nonstandard hulls of Banach spaces, Isr. J. Math. 25 (1976), 108–144. [34] C. W. Henson and J. Iovino, Ultraproducts in analysis, Analysis and logic, London Mathematical Society Lecture Notes Series, vol. 262, 2002, pp. 1–113. [35] E. Hewitt, Rings of real-valued continuous functions, Trans. Am. Math. Soc. 64 (1948), 45–99. [36] H. J. Keisler, Ultraproducts and saturated models, Proc. K. Ned. Akad. Wet., Ser. A 67 (1964), 178–186. (= Indag. Math. 26). [37] J. Keisler, The ultraproduct construction, Ultrafilters across mathematics, ed. by V. Bergelson et al. Contemporary Mathematics, vol. 530, 2010, pp. 163–179. [38] K. Kunen, Ultrafilters and independent sets, Trans. Am. Math. Soc. 172 (1972), 199–206. [39] J. Łoś, Quelques remarques, théorèmes et problèmes sur les classes d’efinissables d’algébres, Mathematical interpretations of formal systems, North Holland, 1955, pp. 98–113. [40] J. Łukasiewicz and A. Tarski, Untersuchungen über den Aussagenkalkül, C. R. Séanses Soc. Sci. Lett. Vars. 23 (1930), 1–21. [41] D. McDuff, Central sequences and the hyperfinite factor, Proc. Lond. Math. Soc. 21 (1970), 443–461. [42] S. Shelah, Every two elementarily equivalent models have isomorphic ultrapowers, Isr. J. Math. 10 (1971), 224–233. [43] D. Sherman, Notes on automorphisms of ultrapowers of II1 factors, Stud. Math. 195 (2009), 201–217. [44] P. S. Urysohn, Sur un espace métrique universel, C. R. Acad. Sci. Paris 180 (1925), 803–806. [45] A. Vignati, Fraïssé theory in operator algebras, this volume.

Isaac Goldbring and Bradd Hart

A survey on the model theory of tracial von Neumann algebras Abstract: We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last 15 years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II1 factors and how modeltheoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters. We move on to more model-theoretic concerns, such as theories of II1 factors and existentially closed II1 factors, and conclude with two recent applications of model-theoretic ideas to questions around relative commutants. Keywords: Model theory of tracial von Neumann algebras, isomorphic ultrapowers, existentially closed factors, elementary equivalence of factors, relative commutants MSC 2020: 03C66, 46L10, 03C20, 03C45, 03C10, 03C65, 46L36

1 Introduction The tracial ultraproduct construction is an integral tool in the modern study of tracial von Neumann algebras. Implicitly present in the work of Sakai [40], this construction was of fundamental importance in both the work of McDuff [35] on central sequence algebras and in Connes’ landmark result that injective II1 factors are hyperfinite [12]. To a model theorist, the presence of an ultraproduct construction makes it natural to try to view tracial von Neumann algebras as structures in an appropriate logic. Discussions around this idea were initiated by Ben Yaacov, Henson, Junge, and Raynaud during a workshop at the American Institute of Mathematics in 2006 [6]. Farah, Sherman, and the second named author took up this line of inquiry motivated by a question of Popa about isomorphisms of matrix ultraproducts (see Section 3 below). In a series of three papers [15, 16, 17], they introduced a natural language for which the class of tracial von Neumann algebras (as well as the subclasses of tracial factors and II1 factors) become axiomatizable, used model-theoretic ideas to settle a variety of questions about isomorAcknowledgement: Goldbring was partially supported by NSF grant DMS-2054477. Hart was supported by the NSERC. Isaac Goldbring, Department of Mathematics University of California, 340 Rowland Hall (Bldg.# 400), Irvine, CA 92697-3875, USA, e-mail: [email protected]; URL: http://www.math.uci.edu/~isaac Bradd Hart, Department of Mathematics and Statistics, McMaster University, 1280 Main St., Hamilton ON, Canada L8S 4K1, e-mail: [email protected]; URL: http://ms.mcmaster.ca/~bradd/ https://doi.org/10.1515/9783110768282-004

134 � I. Goldbring and B. Hart phism of tracial ultraproducts, and initiated the study of elementary equivalence of II1 factors (see Section 4 for a summary of the results proven there). After these papers, the model theory of operator algebras became a vibrant area of research. The current authors and Thomas Sinclair [25] proved that the class of tracial von Neumann algebras does not admit a model companion. Later, Farah, Sherman, and the current authors studied the class of existentially closed II1 factors in greater detail [14]; this work was later complemented by the first author’s work on model-theoretic forcing [20] (see also his article in this volume). These topics are discussed in greater length in Sections 4 and 5. The general impression is that tracial von Neumann algebras lie on the “wild” side of the model-theoretic spectrum. Indeed, in Section 3 it will be shown that they are almost always unstable and these arguments can be elaborated upon to establish other kinds of unclassifiable Shelah-style model-theoretic behavior. On the other hand, modeltheoretic techniques can be used to prove facts about tracial von Neumann algebras that purely operator-algebraic techniques have been unable to prove. The last section of this chapter exhibits a number of examples of this latter line of research. We assume that the reader is familiar with the requisite material on tracial von Neumann algebras presented in Adrian Ioana’s article in this volume.

2 The language of tracial von Neumann algebras In this section, we describe the language ℒtr for tracial von Neumann algebras together with its intended interpretation. For each n ∈ ℕ, there will be a sort Bn together with a symbol dn for a metric on Bn . If (M, τ) is a tracial von Neumann algebra, then the intended interpretation of Bn will be the set of all x ∈ M with ‖x‖ ≤ n (the closed operator norm ball of radius n) and for x, y ∈ Bn (M), the intended interpretation of dn (x, y) will be ‖x −y‖2 = √tr((x − y)∗ (x − y)); based on this interpretation, the bound on dn provided by ℒtr is 2n. As we will see, the operator norm will not be a symbol in our language and so there will be work to be done in order to guarantee that the sorts Bn have the intended interpretation in all models of the theory we are describing. Regarding the function symbols in ℒtr , for every n ∈ ℕ, we will have: (1) binary function symbols +n and −n with domain Bn2 and range B2n . The intended interpretation is simply addition and subtraction restricted to the operator norm ball of radius n. The modulus of uniform continuity in both cases is the identity function. (2) a binary function ⋅n with domain Bn2 and range Bn2 . The intended interpretation is the restriction of multiplication to the operator norm ball of radius n. The modulus of uniform continuity of this function symbol is ϵ 󳨃→ ϵ/n. (3) two constant symbols 0n and 1n which lie in the sort Bn . The intended interpretation of these symbols are the elements 0 and 1.

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(4) for every λ ∈ ℂ, there is a unary function symbol λn whose domain is Bn and range is Bmn , where m = ⌈|λ|⌉. The intended interpretation is scalar multiplication by λ restricted to the operator norm ball of radius n. The modulus of uniform continuity is ϵ 󳨃→ ϵ/|λ|. (5) a unary function symbol ∗n with domain and range Bn . The intended interpretation is the restriction of the adjoint to the operator norm ball of radius n. The modulus of uniform continuity is the identity function. (6) for every m > n, we include unary function symbols in,m with domain Bn and range Bm . The intended interpretation is the inclusion map between the balls of the given radii. The modulus of continuity is the identity map. For each n ∈ ℕ, there are formally two relation symbols (besides the metric symbol) τnre and τnim whose intended interpretations are the restriction to the operator norm unit ball of radius n of the real and imaginary parts of the given tracial state on the von Neumann algebra. Their domains are Bn with bound n and the modulus of uniform continuity is the identity function. Given a tracial von Neumann algebra M, we view it is an ℒtr -structure as above (interpreting each symbol with its intended meaning); we denote the corresponding ℒtr structure by D(M) and refer to it as the dissection of M. Remarks 2.1. (1) We have introduced the language of tracial von Neumann algebras formally with a subscript for every symbol in a given sort. In practice, these subscripts will not be used. (2) In a similar vein, we will use the symbol τ in sentences for the trace as if this complex-valued predicate were a part of the language. In practice, one should formally rewrite any such expression using the real and imaginary parts of the trace, which are actually symbols in the language. (3) The presentation of tracial von Neumann algebras in a continuous logic was superficially different in [16]. There, sorts were referred to as “domains of quantification” and one did not need connecting maps as everything lay in one common universe. The multisorted presentation is more in keeping with other presentations in continuous logic. (4) It is possible to give a presentation of tracial von Neumann algebras with a single sort; for instance, one could use only the operator norm ball of radius 1. The language would have to change somewhat as, for example, the operator norm unit ball is not closed under addition. (5) It is tempting to consider whether we could present tracial von Neumann algebras in unbounded continuous logic say as presented in [4]. Although the natural choice of gauge would appear to be that given by the operator norm, since the operator norm is not uniformly continuous with respect to the 2-norm in the class of tracial von Neumann algebras, this naïve approach does not appear to work.

136 � I. Goldbring and B. Hart We now list a set of ℒtr -conditions, denoted Ttr , which we will show axiomatize (dissections of) tracial von Neumann algebras. We will write these axioms informally and leave it to the reader to write them down as official continuous logic sentences. (1) We have equations which tell us that we are dealing with a complex ∗-algebra. Because the language is sorted, we need to have an equation for every sort. (2) There are axioms that express that τ is a trace. This involves writing axioms for both the real and imaginary parts of the trace. (3) There are axioms that say that the connecting maps im,n preserve addition, multiplication, the adjoint and the trace. (4) dn (x, y) = √τ2n ((x − y)∗ (x − y)). These axioms connect the 2-norm coming from the trace with the metric on each sort and also guarantees that the trace is faithful. (5) supx∈Bn supy∈B1 (τn (y∗ x ∗ xy) ∸ nτ1 (y∗ y)). These axioms say that elements of the sort Bn have operator norm at most n in the standard representation. Theorem 2.2. The theory Ttr axiomatizes the class of (dissections of) tracial von Neumann algebras. To prove this theorem, we first note that, given a tracial von Neumann algebra M, the dissection D(M) of M is easily seen to be a model of Ttr . Suppose, conversely, that we have a model A of the theory Ttr . We begin by forming the direct limit M of the sorts Bn (A) for n ∈ ℕ via the embeddings im,n . Using the interpretation of the function symbols on each sort, we see that M is naturally a complex ∗-algebra. Furthermore, using the trace on each sort, one can define an inner product ⟨x, y⟩ := τ(y∗ x) on M. We let H be the Hilbert space completion of M with respect to this inner product. Then M acts naturally on H by left multiplication. By axiom 5, left multiplication by any element of Bn (A) has operator norm at most n. This allows us to faithfully represent M as a *-subalgebra of B(H). We will show that M is actually SOT-closed in B(H) (and is thus a von Neumann algebra) and that D(M) = A. Towards this end, we first observe that any element of M that has operator norm at most n actually belongs to Bn (A); this follows from the same “functional calculus trick” that was used in the axiomatization of C∗ -algebras presented in the second author’s article in this volume. Note also that, once we show in the next paragraph that M is SOT-closed, this observation also shows that D(M) = A. SOT

We now show that M is SOT-closed. To see this, suppose that y ∈ M ; we aim to show that y ∈ M. Suppose, without loss of generality, that ‖y‖ ≤ 1. By the Kaplansky density theorem, there is a sequence (xn )n∈ℕ from M with each ‖xn ‖ ≤ 1 such that xn SOT-converges to y. By the previous paragraph, we have that each xn ∈ B1 (A). Now note that (xn ) is Cauchy with respect to the metric d1 on B1 (A). Indeed, this follows from the fact that 󵄩 󵄩 󵄩 󵄩 d1 (xm , xn ) = ‖xm − xn ‖2 = 󵄩󵄩󵄩xm (1) − xn (1)󵄩󵄩󵄩H → 󵄩󵄩󵄩y(1) − y(1)󵄩󵄩󵄩H = 0.

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Since B1 (A) is d1 -complete, it follows that (xn )n∈ℕ d1 -converges to some element z ∈ B1 (A). Now given any w ∈ M, we have that ‖xn w − zw‖H = ‖xn w − zw‖2 → 0; since M is dense in H, we have that xn SOT-converges to z, whence y = z ∈ M, as desired. One concludes the proof of the theorem by noting that the dissection procedure is an equivalence of categories between tracial von Neumann algebras with embeddings as morphisms and the category of models of Ttr again with embeddings as morphisms. Exercise 2.3. Suppose that (Mi )i∈I is a family of tracial von Neumann algebras and 𝒰 is an ultrafilter on I. Prove that D(∏ Mi ) = ∏ D(Mi ), 𝒰

𝒰

where the ultraproduct on the left-hand side of the equation is the tracial ultraproduct while the ultraproduct on the right-hand side of the equation is the model-theoretic ultraproduct of ℒtr -structures. It is known that both the classes of tracial factors and the subclass of II1 factors are closed under (tracial) ultraproducts and ultraroots, and thus form axiomatizable classes. We can actually write down concrete axioms for these classes. First, we need the following exercise. (See the second author’s article in this volume for the definition of definable set.) Exercise 2.4. Show that the following are definable sets relative to the theory Ttr : – the set U of unitaries; – the set P of projections; – the set P2 of pairs of projections of the same trace. The original axiomatization for the class of factors appearing in [16, Proposition 3.4(1)] used a Dixmier averaging-type argument. Here, we will instead use the following characterization of factors amongst tracial von Neumann algebras: Exercise 2.5. Suppose that M is a tracial von Neumann algebra. Then M is a factor if and only if for any (p, q) ∈ P2 (M), there is u ∈ U(M) such that upu∗ = q. Consider now the ℒtr -sentence σfactor given by sup inf d(upu∗ , q).

(p,q)∈P2 u∈U

M M If M is a tracial factor, then σfactor = 0 by Exercise 2.5. Conversely, if σfactor = 0, then given any two projections in M of the same trace, we can a priori only conclude that these projections are approximately unitarily conjugate, meaning that they can be unitarily conjugated arbitrarily close to each other (in 2-norm). However, by ℵ1 -saturation, we see that any two projections in M 𝒰 of the same trace are unitarily conjugate, whence M 𝒰 is a factor by Exercise 2.5, and, consequently M is also a factor (for otherwise any

138 � I. Goldbring and B. Hart nontrivial element of the center of M would also be a nontrivial element of the center of M 𝒰 ). As a result, we see that Tfactor := Ttr ∪ {σfactor = 0} axiomatizes the class of tracial factors. Now let σII1 denote the ℒtr -sentence infp∈P |τ(p) − π1 | and consider the ℒtr -theory TII1 := Tfactor ∪ {σII1 = 0}. If M is a II1 factor, then M has a projection of trace π1 , whence M 󳀀󳨐 TII1 . Conversely, if M 󳀀󳨐 TII1 , then M is a tracial factor that has projections of trace arbitrarily close to π1 ; this precludes M from being type In for any n ≥ 1, whence M is a type II1 factor. (Of course, nothing is special about π1 except that it is irrational.)

3 Instability and nonisomorphic ultrapowers One of the first applications of model theoretic ideas to tracial von Neumann algebras was in addressing the question of the dependence on the ultrafilter of an ultrapower of a given separable tracial von Neumann algebra; this was a question first raised by McDuff in [35]. To a model theorist, the question of isomorphic ultrapowers is tied up with (model-theoretic) stability. We give a precise definition of stability below (Definition 3.4), but the key point is: if M is a separable metric structure in a separable language, then M is stable if and only if all ultrapowers of M with respect to nonprincipal ultrafilters on ℕ are saturated (see [16, Theorem 5.6]). Since any two elementarily equivalent saturated structures are isomorphic (by an easy back-and-forth argument), it follows that all such ultrapowers of a stable structure are isomorphic. That being said, as we shall soon see, most tracial von Neumann algebras (and, in particular, all II1 factors) are unstable. In the setting of the previous paragraph, when M is unstable, one only knows that the ultrapowers M 𝒰 are ℵ1 -saturated and of density character 2ℵ0 . Consequently, in models of set theory where the continuum hypothesis holds, such ultrapowers are in fact saturated, the above back-and-forth can still be carried out, and once again one sees that all such ultrapowers are isomorphic. However, in models of set theory where the continuum hypothesis fails, some ultrapowers will necessarily not be saturated and thus it is possible that there can exist nonisomorphic such ultrapowers. This possibility is in fact always realized, as we will soon see. The question of whether a given tracial von Neumann algebra is stable is a function of the type decomposition of the tracial von Neumann algebra. We recall that from Theorem 3.23 in Ioana’s article in this volume that any tracial von Neumann algebra (M, τ) decomposes as a direct sum (M, τ) = (M1 , τ1 ) ⊕ (M2 , τ2 ), where M1 is a type I tracial von Neumann algebra and M2 is a type II tracial von Neumann algebra. Moreover, for each i = 1, 2, we can write Mi = Mzi and τi = τ(z1 ) τ, where z1 , z2 are central projections in M i such that z1 + z2 = 1. It is straightforward to check that, for any ultrafilter 𝒰 , one has (M, τ)𝒰 = (M1 , τ1 )𝒰 ⊕ (M2 , τ2 )𝒰 (see [15, Lemma 4.1]). Let us focus on the type I summand first. Towards that end, let us suppose that our original (M, τ) is itself of type I. In that case, by the discussion in Section 3.5 of Ioana’s

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article in this volume, M further decomposes as M ≅ ⨁n Mn (An ) with each An an abelian tracial von Neumann algebra and where Mn (An ) is equipped with its obvious trace. Once again appealing to [15, Lemma 4.1], one has M 𝒰 ≅ ⨁n (Mn (An ))𝒰 . It is immediate to check that (Mn (An ))𝒰 ≅ Mn (A𝒰 n ). This calculation thus reduces the type I situation to the following: Proposition 3.1. Suppose that (A, τ) is a separable abelian tracial von Neumann algebra. Then all ultrapowers of (A, τ) with respect to nonprincipal ultrafilters on ℕ are isomorphic. To see this, note that the equivalence of categories between abelian tracial von Neumann algebras and probability algebras is “ultrapower preserving,” meaning that it suffices to show that all nonprincipal ultrapowers of a given separable probability algebra X are isomorphic. One could prove this result using the discussion above together with the fact that any probability algebra is stable (see Berenstein and Henson’s article in this volume for a proof of this fact). However, it will behoove us to give a more direct argument that works in this special case that will also prove useful in our discussion of relative commutants below. Since the set of atoms do not grow in ultrapowers, one may further assume that X is atomless. As discussed in Berenstein and Henson’s article in this volume, it suffices to show that any nonprincipal ultrapower X 𝒰 of X is Maharam homogeneous of density 2ℵ0 . We first show that the top element 1 has density 2ℵ0 . Since it clearly has density at most 2ℵ0 , it suffices to show that it has density at least 2ℵ0 . Towards this end, note that one has a countable family of independent events in X, each of measure 21 . It will be beneficial to enumerate these elements by as as s ranges over finite subsets of ℕ. Now, given f : ℕ → ℕ, we can consider the element af = (af ↾n )𝒰 of X 𝒰 . If f , g : ℕ → ℕ are distinct, then μ(af ∩ ag ) = 41 , whence d(af , ag ) ≥ 41 and the density of 1 is at least 2ℵ0 , as desired. To obtain that X 𝒰 is itself homogeneous of density 2ℵ0 , we have to show that b has density 2ℵ0 for any nonzero b ∈ X 𝒰 . To see this, write b = (bn )𝒰 where we may assume each μ(bn ) = μ(b) > 0. By quantifier-elimination and the strong ω-homogeneity of the theory of atomless probability algebras, there are automorphisms σn of X for which σ(bn ) = b0 , yielding an automorphism σ of X 𝒰 for which σ(b) = b0 , where we identify b0 with its diagonal image in X 𝒰 . Now we may repeat the argument from the previous paragraph, this time taking a countable family of independent subsets of b0 . By the discussion preceding Proposition 3.1, we immediately obtain: Corollary 3.2. Suppose that (M, τ) is a type I tracial von Neumann algebra. Then all ultrapowers of (M, τ) with respect to nonprincipal ultrafilters on ℕ are isomorphic. So what if (M, τ) is not of type I? This is when things get interesting and (in)stability begins to creep in. We begin with the following simple calculation, which is [15, Lemma 3.2(1)]. In the remainder of this discussion, set φ(x1 , y1 , x2 , y2 ) = ‖[x1 , y2 ]‖2 , a quantifier-free formula with variables ranging over the unit ball.

140 � I. Goldbring and B. Hart Lemma 3.3. In M2n (ℂ), there are pairs of contractions (ai , bi ), i = 1, . . . , n, for which φ(ai , bi , aj , bj ) = 0 if i ≤ j while φ(ai , bi , aj , bj ) = 2 if i > j. To see that this lemma holds, write M2n (ℂ) = ⨂ni=1 M2 (ℂ) and set i

n

k=1

k=i+1

ai = ⨂ x ⊗ ⨂ 1,

i

n

k=1

k=i+2

bi = ⨂ 1 ⊗ y ⊗ ⨂ 1,

where x = ( 00 √02 ) and y = x ∗ = ( √02 00 ). The behavior exhibited in the previous lemma is precisely the definition of instability: Definition 3.4. A metric structure M is unstable (or has the order property) if there is a formula ψ(x, y) and numbers r < s such that for every n ∈ ℕ, there are tuples ci ∈ M for i < n such that ψM (ci , cj ) ≤ r for i ≤ j while ψ(ci , cj ) ≥ s for i > j. Using this terminology, Lemma 3.3 implies: Corollary 3.5. Any tracial von Neumann algebra that is not of type I is unstable. The previous corollary is indeed a consequence of Lemma 3.3 and the fact that any tracial von Neumann algebra that is not of type I embeds a copy of M2n (ℂ) for all n ≥ 1 (see Exercise 5.4 in Ioana’s article). Returning to the discussion of nonisomorphic ultrapowers, one now quotes the following model-theoretic fact: Theorem 3.6. Suppose that the continuum hypothesis fails and M is an unstable separable metric structure. Then there are nonprincipal ultrafilters 𝒰 and 𝒱 on ℕ such that M 𝒰 ≇ M 𝒱 . The ideas behind this proof are present in Shelah’s treatise [41] on classification theory but were elucidated in [15, Proposition 2.6] (in the special case of tracial von Neumann algebras and related objects) and then in [16, Theorem 5.6(2)] (in the general situation). The basic idea is as follows. Let ≺ψ denote the partial order associated with some unstable formula ψ, that is, a ≺ψ b if and only if ψ(a, b) ≤ r and ψ(b, a) ≥ s (where r and s are as in Definition 3.4). For an infinite regular cardinal λ, we define an (ℵ0 , λ)-gap in M 𝒰 to be a pair of sequences (ci )i0 (whence the name). Let N be a II1 factor whose fundamental group is a countable dense subgroup of ℝ>0 (such a II1 factor was first constructed by Golodets–Nessonov in [27]). Then the index of ℱ (N) in ℝ>0 is 2ℵ0 and for s, t ∈ ℝ>0 in distinct cosets of ℱ (N), we have that Ns ≇ Nt . However, as shown in [23], the first-order fundamental group ℱfo (N) := {t ∈ ℝ>0 : N ≡ Nt } is a closed subgroup of ℝ>0 containing ℱ (N), whence, in our case, must be all of ℝ>0 , that is, each Nt is elementarily equivalent to N. We now turn our attention towards finding different theories of II1 factors. Interestingly enough, the history of finding nonelementarily equivalent II1 factors followed the history of finding nonisomorphic separable II1 factors. The first example of nonisomorphic separable II1 factors was in Murray and von Neumann’s paper [36], where they showed that ℛ has property Gamma while L(𝔽2 ) does not. In [17, Section 3.2.2], it was shown that property Gamma is an ∀∃-axiomatizable property of II1 factors, whence ℛ and L(𝔽2 ) are not even elementarily equivalent. In connection with the previous paragraph, it is somewhat surprising that the following question (first raised in [17, Question 3.5]) is still open: Question 4.2. Is the class of II1 factors that do not have property Gamma an elementary class? We digress for a moment to discuss the general progress in understanding firstorder theories of II1 factors without property Gamma. In general, this line of inquiry is not well understood. In fact, the problem of whether or not there exist two nonelementarily equivalent II1 factors without property Gamma has been open since at least 2015. By the recent refutation of the Connes Embedding Problem [29], one can see that this latter question indeed has a positive answer. More precisely, if M is a II1 factor that does not embed into ℛ𝒰 , then the free product M ∗ ℛ is a II1 factor without property Gamma which also does not embed into ℛ𝒰 , and thus cannot have the same theory (even universal theory) as, say, L(𝔽2 ). Using model-theoretic techniques, one can leverage the proof of the failure of the Connes Embedding Problem from the results of [29] to find infinitely many distinct universal theories of II1 factors without property Gamma (see Corollary 3.14 in the first author’s article in this volume).

144 � I. Goldbring and B. Hart More recently, in [11], Chifan, Ioana, and Kunnawalkam Elayavalli gave an explicit construction (not reliant on the resolution of the Connes Embedding Problem) of a II1 factor N without property Gamma which is not elementarily equivalent to L(𝔽2 ). In fact, they proved the following more general statement: Theorem 4.3 (Chifan, Ioana, and Kunnawalkam Elayavalli). There exists a separable II1 factor N without property Gamma which is not elementarily equivalent to (in fact, cannot have the same ∀∃-theory as) any II1 factor with positive 1-bounded entropy. Here, 1-bounded entropy is a particular quantity associated to any finitely generated tracial von Neumann algebra which has its roots in free probability theory. An example of a II1 factor N without property Gamma but with positive 1-bounded entropy is L(𝔽2 ); it follows that N must have a different ∀∃-theory from L(𝔽2 ). It is unknown whether or not the N constructed in Theorem 4.3 embeds in ℛ𝒰 . Consequently, the following question is still open: Question 4.4. Are there II1 factors M and N, neither of which having property Gamma, both of which embed into ℛ𝒰 , for which M ≢ N? One might attempt to settle the previous question by considering the following natural model-theoretic variant of the well-known free group factor problem: Question 4.5. Are L(𝔽m ) and L(𝔽n ) elementarily equivalent for distinct m, n ≥ 2? In [17, Theorem 5.1], it is shown that any nonprincipal ultraproduct of matrix algebras does not have property Gamma. In connection with Question 4.4, it is thus interesting to ask: Question 4.6. Are there n ≥ 2 and a nonprincipal ultrafilter 𝒰 on ℕ for which L(𝔽n ) ≡ ∏𝒰 Mn (ℂ)? We point out that, using recent work of Jekel [28], Chifan, Ioana, and Kunnawalkam Elayavalli show that the II1 factor N from Theorem 4.3 provides the first example of a II1 factor without property Gamma that is not elementarily equivalent to ∏𝒰 Mn (ℂ) for any nonprincipal ultrafilter on ℕ. We now return to the main thread concerning new theories of II1 factors with property Gamma. The next example of a “new” II1 factor used the property of being McDuff. It is trivial to see that ℛ is a McDuff II1 factor. Dixmier and Lance [13] produced an example of a II1 factor M that had property Gamma (so M ≇ L(𝔽2 )) but was not McDuff (whence M ≇ ℛ). In [17, Proposition 3.9], it was shown that being McDuff is an ∀∃-axiomatizable property of separable II1 factors (and these axioms can be used to define McDuff II1 factors of arbitrary density character), whence the Dixmier and Lance factor represents a third elementary equivalence class of II1 factors. Throughout the years, a handful of new examples of separable II1 factors appeared, albeit at a fairly slow pace. (It is worth mentioning that one of these new examples, introduced by Zeller-Meier in [43], also represented a new, that is, fourth, elementary

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equivalence class, as pointed out by the authors in [23].) A breakthrough finally occurred in 1969, where in a pair of papers [33] and [34], McDuff constructed first countably many and then continuum many nonisomorphic separable II1 factors. It is worth pointing out that McDuff’s construction is quite explicit. Indeed, she constructs two (very concrete) functors T0 and T1 from the category of countable groups into itself, which can then be composed, yielding functors Tα for every α ∈ 2 0, there is u ∈ U(N) such that ‖ρ1 (x) − uρ2 (x)u∗ ‖2 < ϵ for all x ∈ F. If N is additionally assumed to be ℵ1 -saturated (e. g., when N is an ultraproduct), then this approximate unitary conjugacy can be upgraded to actual unitary conjugacy. A model-theoretic proof of Fact 5.5 was given by the first author in [22]. Lemma 5.4 might lead one to believe that Th(ℛ) is model-complete and is the model companion (but not model completion by Theorem 5.1) of its universal theory. As alluded to above, this is in fact not the case: Theorem 5.6 (Farah, Goldbring, Hart, and Sherman [14]). The theory Th(ℛ) is not modelcomplete. In other words, Th∀ (ℛ) does not have a model companion. We note that one cannot simply prove this theorem as in the unrestricted case above as the following question is currently open: Question 5.7. Does Th∀ (ℛ) have the amalgamation property? In particular, if M and N are models of Th∀ (ℛ) with a common subalgebra Q, is the amalgamated free product M ∗Q N once again a model of Th∀ (ℛ)? Surprisingly, the state of knowledge of the latter question is that the answer is positive if one assumes that the algebra Q is amenable (a very serious restriction). The referee pointed out to us that the work of Bowen and Burton, specifically [8, Theorem 2.1 and Lemma 2.1], might be viewed as evidence towards Question 5.7 having a negative answer. We also mention in passing that a positive answer to Question 5.7 yields a positive answer to Question 4.4 above as witnessed by the II1 factor from Theorem 4.3. The proof of Theorem 5.6 rests on the following fundamental result of Jung, which gives a converse to Fact 5.5: Fact 5.8 (Jung [31]). Suppose that M is a finitely generated II1 factor that is a model of Th∀ (ℛ). Further suppose that any two embeddings M 󳨅→ ℛ𝒰 are unitarily conjugate. Then M ≅ ℛ.

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The proof of Theorem 5.6 proceeds by showing that, under the assumption that Th(ℛ) is model-complete, any “nonstandard” model of Th(ℛ), that is, any ℛ′ ≡ ℛ with ℛ′ ≇ ℛ (which exists by Theorem 4.1) satisfies the assumption of Fact 5.8, yielding the desired contradiction. To see this, we first note that ℛ′ is finitely generated (in fact, singly generated) since it is a McDuff II1 factor; fix a generator a of ℛ′ . Now fix embeddings j1 , j2 : ℛ′ 󳨅→ ℛ𝒰 ; we aim to show that j1 and j2 are unitarily conjugate, that is, we wish to find u ∈ U(ℛ𝒰 ) such that j2 (a) = uj1 (a)u∗ . Since Th(ℛ) is model-complete, j1 and j2 𝒰 𝒰 are elementary embeddings, whence tpℛ (j1 (a)) = tpℛ (j2 (a)). Take an elementary extension ℛ̄ of ℛ𝒰 and an automorphism α of R̄ such that α(j1 (a)) = j2 (a). By considering N := ℛ̄ ⋊α ℤ, which contains a unitary implementing α, the fact that ℛ𝒰 is existentially closed (which follows from Th(ℛ) being model-complete) and ℵ1 -saturated implies the existence of the desired unitary. An alternative proof of Theorem 5.6, showing directly that ℛ𝒰 is not an existentially closed model of Th∀ (ℛ), is given in [2, Theorem 2.3.4]; the proof there relies on deeper results but still is of interest in its own right. Despite these negative results, understanding properties of e. c. II1 factors has been important in applications of model-theoretic techniques in studying purely operatoralgebraic problems, as we will see in the next section. We are still far from having a great understanding of the class of e. c. II1 factors. In fact, the following question is still open: Question 5.9. Do there exist nonelementarily equivalent e. c. II1 factors? Do there exist nonelementarily equivalent e. c. models of Th∀ (ℛ)? The analogous question for e. c. groups was famously solved by Macintyre in [32] using model-theoretic forcing; it is our hopes that such techniques might also prove useful in settling the previous question. Some further properties of e. c. factors can be found in Chifan, Drimbe, and Ioana’s article in this volume.

6 Two recent applications In this section we discuss two recent applications of model-theoretic techniques in von Neumann algebra theory.

6.1 Popa’s factorial commutant embedding problem We will say that a separable II1 N that embeds into ℛ𝒰 satisfies the factorial commutant embedding condition (FCEC) if there is an embedding i : N 󳨅→ ℛ𝒰 such that the relative commutant i(N)′ ∩ R𝒰 is a factor. Popa formulated the following problem in connection with the CEP:

150 � I. Goldbring and B. Hart Conjecture 6.1 (Popa’s factorial commutant embedding problem (FCEP)). Every separable II1 factor that embeds into ℛ𝒰 satisfies the FCEC. We will also concern ourselves with the following special case of Conjecture 6.1: Conjecture 6.2 ( FCEP-property (T) version). Every separable property (T) II1 factor that embeds into ℛ𝒰 satisfies the FCEC. Here, property (T) is a certain rigidity property of a von Neumann algebra (see Section 6.2 in Ioana’s article in this volume). A large class of examples of von Neumann algebras with property (T) are the group von Neumann algebras L(G) corresponding to a discrete property (T) group G, for example, G = SL3 (ℤ). The FCEC holds for N = L(SL3 (ℤ)) (as Popa himself showed in [38, Section 1.7]). Besides this result, very little progress had been made on this problem. The FCEP has a nice geometric interpretation and connects with the work of Brown mentioned in the previous section. Indeed, for any model M of Th∀ (ℛ), Brown considers the set ℍom(M, ℛ𝒰 ) of embeddings of M into ℛ𝒰 modulo unitary conjugacy. He endows this set with a natural “convex-like structure” and then shows that the extreme points of this space are precisely those embeddings whose image has factorial commutant. Thus, the FCEP asks if every such space ℍom(M, ℛ𝒰 ) has extreme points. In [19], the first author made some progress on Conjecture 6.2 by proving the following theorem: Theorem 6.3. There is a locally universal II1 factor M such that every separable property (T) factor N admits an embedding i : N 󳨅→ M 𝒰 for which i(N)′ ∩ M 𝒰 is a factor. Here, the fact that M is locally universal means every II1 factor admits an embedding into an ultrapower of M. By the negative solution of the CEP referred to above, we know that M cannot be ℛ as in the original version Conjecture 6.2 (nor can it even embed in ℛ𝒰 ); nevertheless, as we will discuss momentarily, the proof of Theorem 6.3 may indicate how one might settle Conjecture 6.2 in its entirety. In what follows, the only particular aspect of property (T) von Neumann algebras we will use is the following: Definition 6.4. If N is a subalgebra of M, we say that N has w-spectral gap in M if N ′ ∩ M 𝒰 = (N ′ ∩ M)𝒰 . It is interesting to note that a recent result of Tan [42] shows that a separable II1 factor has property (T) if and only if it has w-spectral gap in every extension, answering a question raised by the first author in [19]. (The original definition of property (T) is more complicated, using the notion of a bimodule over a II1 factor.) w-spectral gap is a definability property, as was made explicit by the first author in [21]. Note indeed that we can reformulate the above definition by saying that N has w-spectral gap in M if and only if for every ϵ > 0, there is a finite subset F ⊆ N and δ > 0 such that, for all x ∈ B1 (M), if ‖xy − yx‖2 < δ for all y ∈ F, then there is x ′ ∈ N ′ ∩ M such

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that ‖x − x ′ ‖2 ≤ ϵ. By replacing the condition “for all x ∈ B1 (M)” with the stronger condition “for all x ∈ M” in the previous sentence, we arrive at the definition of a spectral gap subalgebra; the reason for the terminology can also be found in [21]. In [21], the following fact was established about w-spectral gap subalgebras of e. c. II1 factors: Theorem 6.5. Suppose that N is a w-spectral gap subalgebra of the e. c. II1 factor M. Then (N ′ ∩ M)′ ∩ M = N. As alluded to in the previous section, Theorem 6.5 can be used to give an alternate proof that there is no model companion for the theory of tracial von Neumann algebras. Using the previous “bicommutant” theorem, it is easily verified that if N is a w-spectral gap subfactor of the e. c. II1 factor M, then N ′ ∩ M is itself a factor, whence so is N ′ ∩ M 𝒰 (as it coincides with (N ′ ∩ M)𝒰 ). Since every II1 factor embeds into some e. c. factor and e. c. factors are locally universal (see the introduction to the first author’s article in this volume for a proof of this latter fact), we have arrived at a weak version of Theorem 6.3 above, namely every property (T) factor embeds in some locally universal II1 factor with a factorial relative commutant. In order to complete the proof of Theorem 6.3 above, we need to find a common locally universal II1 factor which works for all separable property (T) factors. If Question 5.9 above has a positive solution, that is, if all e. c. II1 factors are elementarily equivalent, then we would be finished with the proof. Indeed, under this assumption, any e. c. II1 factor M would satisfy the conclusion of Theorem 6.3. To see this, fix a property (T) factor N and embed it in an e. c. factor Q. By assumption, Q𝒰 ≅ M 𝒰 , whence composing with any isomorphism Q𝒰 → M 𝒰 , the embedding N ⊆ Q 󳨅→ Q𝒰 yields an embedding N 󳨅→ M 𝒰 with factorial relative commutant. (We have assumed the continuum hypothesis here to ensure that Q𝒰 ≅ M 𝒰 ; it would be interesting to see if this dependence on the continuum hypothesis can be removed.) Whether or not Question 5.9 has a positive solution, there is a certain subclass of the e. c. factors, namely the class of infinitely generic factors (see Chifan, Drimbe, and Ioana’s article in this volume for a definition), which are all pairwise elementarily equivalent. The class of infinitely generic structures was introduced by Abraham Robinson in [39] and imported to the continuous setting in [14, Section 5]. Since every separable tracial von Neumann algebra embeds in a separable infinitely generic factor, running the above argument, but replacing e. c. factors by infinitely generic factors, yields the desired proof of Theorem 6.3. We point out that recent work of Chifan, Drimbe, and Ioana [10, Theorem 6.2] extended Theorem 6.3 by showing that, for any infinitely generic II1 factor M and any separable II1 factor N for which N ′ ∩ N 𝒰 is a factor (in particular, when N does not have property Gamma), there is an embedding i : N 󳨅→ M 𝒰 for which i(N)′ ∩ M 𝒰 is a factor. The proof of Theorem 6.3 given above points to how one might resolve Conjecture 6.2. Indeed, if one can solve two particular problems in the affirmative, then one can indeed accomplish this goal.

152 � I. Goldbring and B. Hart The first problem is purely model-theoretic: Question 6.6. Is ℛ an infinitely generic model of Th∀ (ℛ)? A positive answer to this question was claimed in [14, Proposition 5.21], but the proof there is incorrect. We believe this question is of independent interest, for a negative solution implies a negative solution to Question 5.9 above as it would show that ℛ is not elementarily equivalent to any infinitely generic model of Th∀ (ℛ). Indeed, if M is an infinitely generic model of Th∀ (ℛ) and ℛ ≡ M, then since ℛ is a prime model of its theory (as any embedding of it into its ultrapower is elementary, as discussed above), we have that ℛ itself is an infinitely generic model of Th∀ (R) (as elementary substructures of infinitely generic structures are themselves infinitely generic [14, Proposition 5.17]). The second problem is purely operator-algebraic and has already been mentioned above in Question 5.7: if M and N are both models of Th∀ (ℛ) with a common subalgebra Q, is the amalgamated free product M ∗Q N also a model of Th∀ (ℛ)? How does this question arise in connection with adapting our proof of Theorem 6.3 to settle Conjecture 6.2? Well, the proof of Theorem 6.5 uses amalgamated free products in a seemingly essential way. More precisely, if one assumes, towards a contradiction, that there is b ∈ ((N ′ ∩ M)′ ∩ M) \ N, then one obtains a contradiction by considering the embedding M ⊆ M ∗N (N ⊗ L(ℤ)) 󳨅→ M 𝒰 one gets from the fact that M is e. c. and by considering the fact that b commutes with the image of the canonical unitary generator of L(ℤ) under this embedding. To run this argument in the relative setting of models of Th∀ (ℛ), one could only quote the fact that M is an e. c. model of Th∀ (ℛ) if the amalgamated free product is itself a model of Th∀ (ℛ). Note that this argument shows that it suffices to settle this amalgamated free product question in the affirmative when the algebra being amalgamated over has property (T). In [24], the authors showed that a positive solution to this amalgamated free product question can be combined with the weaker statement that ℛ and the infinitely generic models of Th∀ (R) have the same three-quantifier theory in order to obtain a positive solution to Conjecture 6.2. One can rephrase the previous discussion in the following enticing manner: if ℛ is an infinitely generic model of Th∀ (ℛ) (or even has the same three-quantifier theory as an infinitely generic model of Th∀ (ℛ)) and yet Conjecture 6.2 has a negative answer, then one obtains an alternate refutation of the CEP and in fact can deduce the stronger statement that the theory Th∀ (ℛ) is not closed under amalgamated free products. In a similar spirit to the FCEP but in a somewhat different direction, in Jekel’s article in this volume, a free probability-theoretic criterion for a separable tracial von Neumann algebra M to admit an embedding into a matrix ultraproduct ∏𝒰 Mn (ℂ) with trivial (and thus factorial) relative commutant is established.

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6.2 II1 factors with the generalized Jung property The starting point for the main result of this section is Jung’s theorem (Fact 5.8 from above). Jung’s theorem asserts that whenever N is a separable II1 factor that is a model of Th∀ (ℛ) and for which all of its embeddings into ℛ𝒰 are unitarily conjugate, then N must be isomorphic to ℛ. One might also ask what happens instead if one assumes that all embeddings of N into its own ultrapower N 𝒰 are unitarily conjugate? It turns out that, for models of Th∀ (ℛ), this once again characterizes ℛ, as shown by Atkinson and Kunnawalkam Elayavalli in [3, Corollary 2.7]: Theorem 6.7 (Atkinson and Kunnawalkam Elayavalli). If N is a separable II1 factor that is a model of Th∀ (ℛ) for which any two embeddings N 󳨅→ N 𝒰 are unitarily conjugate, then N ≅ ℛ. The idea of the proof of the previous result is as follows. Fix an embedding σ : N 󳨅→ ℛ and view σ as an embedding σ : N 󳨅→ N 𝒰 by composing it with the natural inclusion ℛ𝒰 ⊆ N 𝒰 . By assumption, there is a unitary u = (un )𝒰 ∈ U(N 𝒰 ) conjugating σ to the diagonal embedding. Let Eℛ : N → ℛ denote the canonical conditional expectation map. Then the map x 󳨃→ Eℛ (un xun∗ ) : N → ℓ∞ (ℛ) is a ucp lift of σ. Since ℓ∞ (ℛ) is injective, it follows that σ(N) and hence N is injective. By Connes’ fundamental theorem [12], this implies that N ≅ ℛ. Using model-theoretic techniques, Atkinson, Kunnawalkam Elayavalli, and the first author were able to generalize the previous theorem by merely assuming that any two embeddings of N into its ultrapower N 𝒰 were conjugate by some automorphism of N 𝒰 . 𝒰

Theorem 6.8 (Atkinson, Goldbring, and Kunnawalkam Elayavalli). If N is a separable II1 factor that is a model of Th∀ (ℛ) for which any two embeddings N 󳨅→ N 𝒰 are conjugate by some (not necessarily inner) automorphism, then N ≅ ℛ. The proof proceeds in two steps. The first step is to show that any N satisfying the hypotheses of the theorem must be elementarily equivalent to ℛ. To see this, first note that the assumption on N implies that any embedding N 󳨅→ N 𝒰 is elementary. Now consider embeddings i : ℛ 󳨅→ N and j : N 󳨅→ ℛ𝒰 . Note then that j ∘ i : ℛ 󳨅→ ℛ𝒰 and i𝒰 ∘ j : N 󳨅→ N 𝒰 are elementary as are j𝒰 ∘ i𝒰 : ℛ𝒰 → (ℛ𝒰 )𝒰 and (i𝒰 )𝒰 ∘ j𝒰 : N 𝒰 󳨅→ (N 𝒰 )𝒰 . Continuing in this way, we obtain a sequence ℛ 󳨅→ N 󳨅→ ℛ 󳨅→ N 𝒰

𝒰

𝒰

󳨅→ (ℛ𝒰 ) 󳨅→ ⋅ ⋅ ⋅

such that the composition of any two embeddings is an elementary map ℛk𝒰 󳨅→ ℛ(k+1)𝒰 or N k𝒰 󳨅→ N (k+1)𝒰 , where ℛk𝒰 denotes the kth iterated ultrapower of ℛ and similarly for N k𝒰 . Letting N∞ denote the limit of this chain, we see that ℛ ⪯ N∞ and N ⪯ N∞ , whence ℛ ≡ N, as desired. The second step is to establish the following fact, which is of independent interest: Theorem 6.9. If N ≡ ℛ is separable, then N satisfies the FCEC.

154 � I. Goldbring and B. Hart Before proving Theorem 6.9, we make three remarks. First, this theorem, combined with the first part of the proof, does indeed establish Theorem 6.8. Indeed, if N satisfies the hypotheses of Theorem 6.8, then N ≡ ℛ, whence admits an embedding N 󳨅→ ℛ𝒰 with factorial relative commutant. However, since any two such embeddings are conjugate by an automorphism, we see that all embeddings of N 󳨅→ ℛ𝒰 have factorial relative commutant. As mentioned in the previous subsection, the embeddings of N into ℛ𝒰 with factorial relative commutant are the extreme points of Brown’s space ℍom(N, ℛ𝒰 ), whence, under the present assumptions, every point is extreme. Consequently, there is a single element of ℍom(N, ℛ𝒰 ), whence N ≅ ℛ by Fact 5.8. The second remark is that Theorem 6.9, together with Theorem 4.1 above, gives continuum many nonisomorphic separable examples of II1 factors satisfying Conjecture 6.1. The final remark is that the authors showed in [24], using Ehrenfeucht–Fraïsse games, that if N is a II1 factor with the same four-quantifier theory as ℛ, then N satisfies the conclusion of Conjecture 6.1. Since we conjecture that ℛ does not satisfy any sort of quantifier simplification, this should lead to further examples. We now turn to the proof of Theorem 6.9. The main interest in the proof of this result is that it appears to be the first time that the model-theoretic notion of heir has been used in applications of continuous model theory. Definition 6.10. Suppose that T is an L-theory, M 󳀀󳨐 T, and A ⊆ B ⊆ M are parameter sets. We say that q ∈ S(B) is an heir of p ∈ S(A) if p ⊆ q and for all L-formulae φ(x, y), parameters b ∈ B, and ϵ > 0, there are parameters a ∈ A such that |φ(x, a)p −φ(x, b)q | < ϵ. In the above definition, recall that φ(x, a)p is the value the type p assigns to the formula φ(x, a) and likewise for φ(x, b)q . A slightly more complicated version of the usual existence of heir argument can be used to show the following (which was first claimed by Ben Yaacov in [5] but proven in detail in [2, Fact 2.2.6]): Lemma 6.11. Suppose that 𝕄 󳀀󳨐 T is ℵ1 -saturated, N ⪯ 𝕄 is a separable elementary substructure, and N ⊆ B ⊆ 𝕄 is a separable parameter set. Then every p ∈ S(N) has an extension to an element of S(B) that is an heir of p. Continuing the proof of Theorem 6.9, suppose that N ⊆ ℛ𝒰 is an arbitrary separable subalgebra. By yet another theorem of Brown [9, Theorem 6.9], there is a separable subfactor N ⊆ M ⊆ ℛ𝒰 such that M ′ ∩ ℛ𝒰 is a factor. Ordinarily, this would not imply that N ′ ∩ ℛ𝒰 is a factor. However, if N is an elementary substructure of ℛ𝒰 , then one can indeed conclude that N ′ ∩ ℛ𝒰 is a factor from the fact that M ′ ∩ ℛ𝒰 is a factor. The key observation is the following: Proposition 6.12. Suppose that N ⊆ M ⊆ ℛ𝒰 are separable subfactors and consider types p ∈ S(N) and q ∈ S(M) with q an heir of p. If p(ℛ𝒰 ) ⊆ N ′ ∩ ℛ𝒰 , then q(ℛ𝒰 ) ⊆ M ′ ∩ ℛ𝒰 .

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The proof of the proposition is routine: arguing by contrapositive, if there is b ∈ M q p such that ‖[x, b]‖2 ≥ ϵ, then there is a ∈ N such that ‖[x, a]‖2 ≥ ϵ2 . It remains to note how the previous proposition establishes Theorem 6.9 above. Indeed, we may assume that N ⪯ ℛ𝒰 and fix a separable subfactor N ⊆ M ⊆ ℛ𝒰 such that M ′ ∩ ℛ𝒰 is a factor. We wish to show that N ′ ∩ ℛ𝒰 is a factor. Take a ∈ Z(N ′ ∩ ℛ𝒰 ); we wish to show that a ∈ ℂ. Let p := tp(a/N) and let q ∈ S(M) be an heir of p to M. Then q(ℛ𝒰 ) ⊆ M ′ ∩ ℛ𝒰 by the previous proposition and q(ℛ𝒰 ) ⊆ p(ℛ𝒰 ) ⊆ Z(N ′ ∩ ℛ𝒰 ) (the latter inclusion follows by noting that any two realizations of p are conjugate by an automorphism of ℛ𝒰 fixing N pointwise). Consequently, q(ℛ𝒰 ) ⊆ Z(M ′ ∩ ℛ𝒰 ) = ℂ, whence d(x, λ)q = 0 for some λ ∈ ℂ. It follows that a = λ, as desired. We remark that Theorem 6.9 above yields yet another alternate proof (assuming the continuum hypothesis) that Th∀ (ℛ) has no model companion. Indeed, suppose towards a contradiction that Th(ℛ) is the model companion of its universal theory. Take a nonstandard model N of Th(ℛ). By assumption, any embedding N 󳨅→ N 𝒰 is elementary. A standard back-and-forth argument then shows that all embeddings N 󳨅→ N 𝒰 are conjugate by an automorphism, contradicting Theorem 6.9. By the negative solution to the CEP, there are II1 factors N that are counterexamples to CEP such that any two embeddings N 󳨅→ N 𝒰 are conjugate by an automorphism. Indeed, any finitely generic II1 factor will have this property (see [20]). However, the unitary conjugacy version of this fact (that is, the generalization of Theorem 6.7 above to the unrestricted situation) is still open: Question 6.13. Is there a separable II1 factor N that is not isomorphic to ℛ with the property that any two embeddings N 󳨅→ N 𝒰 are unitarily conjugate? Recently, it was shown by Chifan, Drimbe, and Ioana [10, Theorem 6.4] that if M is an infinitely generic factor, then for any elementary embedding i : M 󳨅→ M 𝒰 , one has that i(M)′ ∩M 𝒰 is a factor. If there was an infinitely generic factor M for which all embeddings M 󳨅→ M 𝒰 are elementary, then by the result mentioned in the previous sentence and [1, Proposition 5.3] (a generalization of a result of Brown referred to above), one would have that any two embeddings of M into M 𝒰 are unitarily conjugate, yielding a positive answer to Question 6.13. Recalling that all embeddings of a finitely generic factor into its ultrapower are elementary, a positive answer to the following question (which is the unrestricted version of Question 6.6) would consequently lead to a positive answer to Question 6.13: Question 6.14. Do the finitely generic and infinitely generic factors have the same theory? In other words, is every finitely generic factor also infinitely generic? While we are on the topic of finitely generic factors, since Theorem 6.8 above implies that ℛ is the unique finitely generic separable embeddable factor, it makes sense to ask if this is true in the unrestricted case: Question 6.15. Is there a unique finitely generic separable factor?

156 � I. Goldbring and B. Hart The following general automorphism version of Jung’s theorem is also still open: Question 6.16. Suppose that N is a separable II1 factor that is a model of Th∀ (ℛ) such that any two embeddings N 󳨅→ ℛ𝒰 are conjugate by some (not necessarily inner) automorphism of ℛ𝒰 . Must we have N ≅ ℛ? Note that, in model-theoretic terms, the assumption on N in the previous question is simply that the quantifier-free type of N in ℛ𝒰 determines its complete type.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

S. Atkinson, Convex sets associated to C∗ -algebras, J. Funct. Anal. 271 (2016), 1604–1651. S. Atkinson, I. Goldbring and S. Kunnawalkam Elayavalli, Factorial relative commutants and II1 factors with the generalized Jung property, Adv. Math. in press. S. Atkinson and S. Kunnawalkam Elayavalli, On ultraproduct embeddings and amenability for tracial von Neumann algebras, Int. Math. Res. Not. in press. I. BenYaacov, Continuous first order logic for unbounded metric structures, J. Math. Log. 8 (2008), 197–223. I. BenYaacov, Stability and stable groups in continuous logic, J. Symb. Log. 75 (2010), 1111–1136. I. BenYaacov, C. W. Henson, M. Junge and Y. Raynaud, Report on non-commutative probabilities and von Neumann algebras, AIM workshop, 2006. R. Boutonnet, I. Chifan and A. Ioana, II1 factors with non-isomorphic ultrapowers, Duke Math. J. 166 (2017), 2023–2051. L. Bowen and P. Burton, Flexible stability and nonsoficity, Trans. Am. Math. Soc. 373 (2020), 4469–4481. N. Brown, Topological dynamical systems associated to II1 factors, Adv. Math. 227 (2011), 1665–1699. With an appendix by N. Ozawa. I. Chifan, D. Drimbe and A. Ioana, Embedding universality for II1 factors with property (T). Preprint. I. Chifan, A. Ioana and S. Kunnawalkam Elayavalli, Two non-elementarily equivalent II1 factors without property Gamma. Preprint. A. Connes, Classification of injective factors. Cases II1 , II∞ , IIIλ , λ ≠ 1, Ann. Math. 104 (1976), 73–115. J. Dixmier and E. C. Lance, Deux nouveaux facteurs de type II1 , Invent. Math. 7 (1969), 226–234. I. Farah, I. Goldbring, B. Hart and D. Sherman, Existentially closed II1 factors, Fundam. Math. 233 (2016), 73–196. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras I: Stability, Bull. Lond. Math. Soc. 45 (2013), 825–838. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras II: Model theory, Isr. J. Math. 201 (2014), 477–505. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras III: Elementary equivalence and II1 factors, Bull. Lond. Math. Soc. 46 (2014), 609–628. I. Farah and S. Shelah, A dichotomy for the number of ultrapowers, J. Math. Log. 10 (2010), 45–81. I. Goldbring, Proc. Am. Math. Soc. 148 (2020), 5007–5012. I. Goldbring, Enforceable operator algebras, J. Inst. Math. Jussieu 20 (2021), 31–63. I. Goldbring, Spectral gap and definability, Beyond first-order model theory, Vol. II. in press. I. Goldbring, Nonembeddable II1 factors resembling the hyperfinite II1 factor, J. Noncommut. Geom. in press. I. Goldbring and B. Hart, On the theories of McDuffs II1 factors, Int. Math. Res. Not. 27 (2017), 5609–5628.

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[24] I. Goldbring and B. Hart, Properties expressible in small fragments of the theory of the hyperfinite II1 factor, Confluentes Math. 12 (2020), 37–47. [25] I. Goldbring, B. Hart and T. Sinclair, The theory of tracial von Neumann algebras does not have a model companion, J. Symb. Log. 78 (2013), 1000–1004. [26] I. Goldbring, B. Hart and H. Towsner, Explicit sentences distinguishing McDuff’s II1 factors, Isr. J. Math. 227 (2018), 365–377. [27] V. Ya. Golodets and N. I. Nessonov, Free groups, the T-property, and II1 -factors with different countable fundamental groups, Dokl. Akad. Nauk Ukr. SSR, Ser. A (1983), 7–10. (Russian). [28] D. Jekel, Covering entropy for types in tracial W*-algebras. arXiv:2204.02582. [29] Z. Ji, A. Natarajan, T. Vidick, J. Wright and H. Yuen. MIP∗ = RE. arxiv:2001.04383. [30] N. Jing, Unitary and orthogonal equivalence of sets of matrices, Linear Algebra Appl. 481 (2015), 235–242. [31] K. Jung, Amenability, tubularity, and embeddings into Rω , Math. Ann. 338 (2007), 241–248. [32] A. Macintyre, On algebraically closed groups, Ann. Math. 96 (1972), 53–97. [33] D. McDuff, A countable infinity of II1 factors, Ann. Math. 90 (1969), 361–371. [34] D. McDuff, Uncountably many II1 factors, Ann. Math. 90 (1969), 372–377. [35] D. McDuff, Central sequences and the hyperfinite factor, Proc. Lond. Math. Soc. 21 (1970), 443–461. [36] F. J. Murray and J. von Neumann, On rings of operators IV, Ann. Math. 44 (1943), 716–808. [37] R. Nicoara, S. Popa and R. Sasyk, On II1 factors arising from 2-cocycles of w-rigid groups, J. Funct. Anal. 242 (2007), 230–246. [38] S. Popa, Independence properties in subalgebras of ultraproduct II1 factors, J. Funct. Anal. 266 (2014), 5818–5846. [39] A. Robinson, Infinite forcing in model theory, Proc. second Scand. logic symposium, Oslo, 1970, pp. 317–340. [40] S. Sakai, The Theory of W∗ -algebras, lecture notes, Yale University, 1962. [41] S. Shelah, Classification theory, Studies in logic and the foundations of mathematics, North Holland, 1990. [42] H. Tan, Spectral gap characterizations of property (T) for II1 factors. arXiv:2202.06089. [43] G. Zeller-Meier, Deux nouveaux facteurs de type II1 , Invent. Math. 7 (1969), 235–242.

Alexander Berenstein and C. Ward Henson

Model theory of probability spaces Abstract: This expository paper treats the model theory of probability spaces using the framework of continuous [0, 1]-valued first-order logic. The metric structures discussed, which we call probability algebras, are obtained from probability spaces by identifying two measurable sets if they differ by a set of measure zero. The class of probability algebras is axiomatizable in continuous first-order logic; we denote its theory by Pr. We show that the existentially closed structures in this class are exactly those in which the underlying probability space is atomless. This subclass is also axiomatizable; its theory APA is the model companion of Pr. We show that APA is separably categorical (hence complete), has quantifier elimination, is ω-stable, and has built-in canonical bases, and we give a natural characterization of its independence relation. For general probability algebras, we prove that the set of atoms (enlarged by adding 0) is a definable set, uniformly in models of Pr. We use this fact as a basis for giving a complete treatment of the model theory of arbitrary probability spaces. The core of this paper is an extensive presentation of the main model-theoretic properties of APA. We discuss Maharam’s structure theorem for probability algebras, and indicate the close connections between the ideas behind it and model theory. We show how probabilistic entropy provides a rank connected to model-theoretic forking in probability algebras. In the final section we mention some open problems. Keywords: Measure algebras, continuous logic, model theory, metric structures, quantifier elimination, ω-stability, metric imaginaries, Maharam’s theorem, Maharam invariants, probabilistic independence, model-theoretic independence MSC 2020: 03C66, 03C10, 03C45, 28A60

1 Introduction In this paper we use the continuous version of first-order logic to investigate probability spaces (X, ℬ, μ). Here ℬ is a σ-algebra of subsets of X (requiring 0, X ∈ ℬ) and μ is a σ-additive probability measure on ℬ. There is a canonical pseudometric d on ℬ, obAcknowledgement: The authors are grateful to Itaï Ben Yaacov for helpful conversations. Research for this paper was partially supported by NSF grants and by grants from the Simons Foundation (202251 and 422088, to the second author). Alexander Berenstein, Universidad de los Andes, Cra 1 No 18A-12, Bogotá, Colombia, URL: http://www.pentagono.uniandes.edu.co/~aberenst C. Ward Henson, University of Illinois, Urbana-Champaign; Urbana, Illinois 61801, USA, URL: https://faculty.math.illinois.edu/~henson https://doi.org/10.1515/9783110768282-005

160 � A. Berenstein and C. W. Henson tained by taking the distance between sets to be given by d(A, B) := μ(A △ B). (Here △ denotes the symmetric difference operation on sets.) This gives rise to a prestructure (ℬ, 0, 1, ⋅c , ∩, ∪, μ, d) (which we often write as (ℬ, μ, d), regarding ℬ as a boolean algebra but suppressing the constants and the operations from our notation). We obtain a structure in the usual way by turning d into a metric. (Usually, we would also need to take the metric completion, but the metric quotient is automatically complete here, as we indicate below.) This ̂ where ℬ̂ is the quotient of ℬ by the equivalence relation ̂ , d), yields the structure (ℬ̂, μ ̂ ̂ , d are the canonical measure and metric induced on ℬ̂. That is, for μ(A △ B) = 0 and μ ̂ ([A]μ ) = μ(A); similarly each A ∈ ℬ and [A]μ := {B ∈ ℬ | μ(A △ B) = 0} ∈ ℬ̂, we have μ ̂ ̂ ([A △ B]μ ) = μ ̂ ([A]μ △ [B]μ )). d([A]μ , [B]μ ) = d(A, B) (= μ(A △ B) = μ ̂ is a strictly One sees that ℬ̂ is a complete (in the sense of order) boolean algebra, μ ̂ . The positive σ-additive measure on ℬ̂, and d̂ is the metric defined canonically from μ ̂ ̂ ̂ metric structures (ℬ, μ, d) are the principal objects of study in this paper. We use standard background from measure theory and analysis (which we summarize in Section 2) and from continuous first-order logic. The model-theoretic background for this paper comes from [7] and [8], as well as from Bradd Hart’s chapter in this volume, which present the [0, 1]-valued continuous version of first-order logic. In Section 3 we give references for some additional concepts and tools from continuous logic that we need here. The main content of this paper is in Sections 4, 5, 6, and 8. Sections 4 and 5 present the model theory of arbitrary probability spaces in the framework of continuous firstorder logic. Our results show that everything model-theoretic about arbitrary probability spaces can be systematically reduced to the atomless case, which is given a full treatment in Sections 6 and 8. Atomless probability spaces were studied by Ben Yaacov [1] using the framework of compact abstract theories, with an emphasis on issues around model-theoretic stability. In Sections 6 and 8 we study atomless probability spaces in the context of continuous logic and present analogues of results from [1], as well as additional results that are specific to the continuous first-order setting. In Section 6 we give axioms for the class of atomless probability algebras, and show that the theory of these structures (denoted by APA) is well behaved from the model-theoretic point of view: in particular, it is complete, has quantifier elimination, is separably categorical, and is ω-stable. We characterize (up to equivalence) the induced metric on type spaces of APA. In Section 8 we focus on features of APA that are connected to its stability. Following the work of Ben Yaacov [1], we give an intrinsic characterization of the independence relation of APA, and show that it has built-in canonical bases. We give a direct, elementary proof that APA is strongly finitely based, a fact originally proved in [6] using lovely pairs of APA models. We also look at APA from the point of view of Shelah’s classification program, and show that APA

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is nonmultidimensional but not unidimensional, using natural translations of those concepts into continuous model theory. Section 7 is devoted to Maharam’s structure theorem for probability algebras and its connections with model theory. In Section 9 we show how model-theoretic forking in probability algebras is related to probabilistic entropy. In the last section we identify some open problems that seem worth investigating further.

2 Probability spaces In this section we present basic information about probability spaces (X, ℬ, μ) and their measure algebra quotients. We recall that A ∈ ℬ is atomless if for every B ∈ ℬ with B ⊆ A and μ(B) > 0, there are B1 , B2 ∈ ℬ such that B = B1 ∪ B2 , B1 and B2 are disjoint, and μ(B1 ) > 0, μ(B2 ) > 0. We say that (X, ℬ, μ) is atomless if X is atomless in ℬ. We recall that A ∈ B is an atom if μ(A) > 0, and for every B ∈ ℬ with B ⊆ A one has μ(B) = 0 or μ(A \ B) = 0. Evidently, if A1 , A2 ∈ ℬ are atoms then either μ(A1 ∩ A2 ) = 0 or μ(A1 △ A2 ) = 0. Furthermore, there exists a finite or countable family 𝒜 ⊆ ℬ such that each A ∈ 𝒜 is an atom and such that whenever A ∈ ℬ is an atom, there exists A′ ∈ 𝒜 such that μ(A △ A′ ) = 0. The atomic part of X is the join (union) of the sets in 𝒜, and its complement is the atomless part of X; this partition of X is well defined up to a set of measure 0. The atomic part is atomic, in the sense that whenever A ∈ ℬ is contained in the atomic part of X, then A is (up to a set of measure 0) the union of the atoms it contains; equivalently, all atomless subsets of such an A have measure 0. Likewise, the atomless part of X is an atomless member of ℬ. We regard 0 (which is equal to 0 here) as atomic, since it is contained in the atomic part, and 0 is atomless by definition. Further, if A is atomless and B is atomic, then μ(A ∩ B) = 0. We say (A1 , . . . , An ) is a partition in ℬ if Ai ∩ Aj = 0 whenever i ≠ j. If, in addition, A1 ∪ ⋅ ⋅ ⋅ ∪ An = B, then we say (A1 , . . . , An ) is a partition of B in ℬ. Note that we allow Ai to be 0 in such a situation. We say that A1 , A2 ∈ ℬ determine the same event, and write A1 ∼μ A2 if the symmetric difference of the sets has μ-measure zero. Clearly, ∼μ is an equivalence relation. We denote the equivalence class of A ∈ ℬ by [A]μ . The collection of equivalence classes of ℬ modulo ∼μ is denoted by ℬ̂. The operations of complement, union and intersection are well defined for events and they make ℬ̂ a boolean algebra. Moreover, μ induces on ̂ a σ-additive, strictly positive probability measure μ ̂ . As noted above, we denote the ℬ ̂ ̂ ̂ canonical metric on ℬ by d and recall that it is defined by d([A] μ , [B]μ ) = d(A, B). It is ̂ important in this paper that (ℬ̂, d) is a complete metric space (see the calculation in [15, Lemma 323F]). ̂ as the probability algebra of (X, ℬ, μ). ̂ , d) We refer to (ℬ̂, μ

162 � A. Berenstein and C. W. Henson Notation 2.1. (a) If C is a subset of a boolean algebra, we denote the boolean subalgebra generated by C by C # . ̂ be its probability algebra. ̂ , d) Let (X, ℬ, μ) be a probability space and let (ℬ̂, μ (b) If S is a subset of ℬ, we let ⟨S⟩ denote the σ-subalgebra of ℬ generated by S. ̂ (c) If S is a subset of ℬ̂, we let ⟨S⟩ denote the d-closure of S # . Note that ⟨S⟩ is equal to the σ-subalgebra of ℬ̂ generated by S, by [15, Lemma 323F], and it is also equal to the ̂ ̂ d-closed boolean subalgebra generated by S. In other words, ⟨S⟩ is d-closed and has S # ̂ as a d-dense subset. (d) Throughout this paper, we use upper case letters such as A, B for elements of the σ-algebra ℬ and lower case letters such as a, b for elements of ℬ̂. If S is a subset of ℬ, we denote by Ŝ the set of events determined by the elements of S; i. e., Ŝ = {[A]μ | A ∈ S} ⊆ ℬ̂. Whenever 𝒞 ⊆ ℬ is a σ-subalgebra, (X, 𝒞 , μ↾𝒞 ) is a probability space in its own right, and we have defined 𝒞̂ to be the probability algebra of 𝒞 , and also (in Notation 2.1(d)) to be a certain subset of ℬ̂. There is no real ambiguity here; indeed, the inclusion map j of 𝒞 into ℬ induces a measure-preserving boolean isomorphism ̂j (which thus also preserves the metric) between the two versions of 𝒞̂. (The function ̂j maps [A]μ↾𝒞 in the sense of (X, 𝒞 , μ↾𝒞 ) to [A]μ in the sense of (X, ℬ, μ), for each A ∈ 𝒞 .) With this identifî is the probability algebra of the probability space (X, 𝒞 , μ↾𝒞 ), and it is ̂ , d) cation, (𝒞̂, μ ̂ ̂ , d). (canonically isomorphic to) a substructure of (ℬ̂, μ

In the next result, we record for later use that the converse of the preceding comment is also true.

̂ be its probability algebra. ̂ , d) Lemma 2.2. Let (X, ℬ, μ) be a probability space and let (ℬ̂, μ Let S be a subset of ℬ̂ and consider ⟨S⟩ ⊆ ℬ̂ as in Notation 2.1(c). Let 𝒮 := {A ∈ ℬ | [A]μ ∈ ⟨S⟩}.

Then 𝒮 is a σ-subalgebra of ℬ and ⟨S⟩ = 𝒮̂. ̂ ̂ , d) In particular, every substructure (⟨S⟩, μ↾⟨S⟩, d↾⟨S⟩) of the probability algebra (ℬ̂, μ of a probability space (X, ℬ, μ) is (isomorphic to) the probability algebra of a probability space (X, 𝒮 , μ↾𝒮 ), with 𝒮 a σ-subalgebra of ℬ. Proof. For A, B ∈ 𝒮 ⊆ ℬ, we have [A ∪ B]μ = [A]μ ∪ [B]μ ∈ ⟨S⟩, so 𝒮 is closed under the union operation of ℬ. Similar calculations show that 𝒮 is closed under ∩ and ⋅c . Also, note that 𝒮 contains every element of ℬ that has μ-measure 0. To finish the proof, we need to consider an increasing sequence (An ) in 𝒮 and show that the union of (An ) in ℬ is an element of 𝒮 . Given such an (An ), the sequence ([An ]μ ) ⊆ ⟨S⟩ must be increasing in ℬ̂, so it converges in the sense of the metric d̂ to an element [B]μ ∈ ⟨S⟩, with B ∈ 𝒮 . This means that μ(An △ B) → 0 in ℬ. Since (An ) is increasing, this implies μ(An \ B) = 0 for all n. It also implies μ(B \ (⋃ An )) = 0, and therefore B differs from ⋃ An by a set of μ-measure 0 in ℬ. Hence ⋃ An ∈ 𝒮 . We need the following familiar special case of the Radon–Nikodym theorem:

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Theorem 2.3 ([13, Theorem 3.8]). Let (X, ℬ, μ) be a probability space, let 𝒞 ⊆ ℬ be a σ-subalgebra, and consider A ∈ ℬ. Then there exists g ∈ L1 (X, 𝒞 , μ) such that for every B ∈ 𝒞 , one has ∫B gdμ = ∫B χA dμ. The function g is determined by a = [A]μ up to equality μ-almost everywhere; it is called the conditional probability of a with respect to 𝒞 and we denote it by ℙ(a|𝒞 ) or, equivalently, by ℙ(A|𝒞 ). We also refer to g as representing ℙ(a|𝒞 ) and ℙ(A|𝒞 ). More generally, for f ∈ L1 (X, ℬ, μ) there exists 𝔼(f |𝒞 ) ∈ L1 (X, 𝒞 , μ) such that for every B ∈ 𝒞 , one has ∫B 𝔼(f |𝒞 )dμ = ∫B fdμ. The function 𝔼(f |𝒞 ) is unique in the sense that the operation f 󳨃→ 𝔼(f |𝒞 ) preserves the equivalence relation of equality μ-almost everywhere. The element 𝔼(f |𝒞 ) is called the conditional expectation of f with respect to 𝒞 and we also denote it by 𝔼𝒞 (f ). Note that for any A ∈ ℬ, the function ℙ(A|𝒞 ) must have its values in [0, 1] μ-a. e. Notation 2.4. Let (X, ℬ, μ) be a probability space, and consider a ∈ ℬ̂. Suppose D is a ̂ and let 𝒟 = {A ∈ boolean subalgebra of ℬ̂ that is closed (with respect to the metric d), ̂ = D. We ℬ | [A]μ ∈ D} be the σ-subalgebra discussed in the proof of Lemma 2.2, so 𝒟 write ℙ(a|D) to denote ℙ(χA |𝒟), where a = [A]μ . Similarly, for a ∈ ℬ̂ we write χa to denote one of the characteristic functions χA where A ∈ ℬ and a = [A]μ . Note that if A1 , A2 are two such sets for a, then χA1 = χA2 holds μ-almost everywhere, and hence the same is true of ℙ(χA1 |𝒟) and ℙ(χA2 |𝒟). When we are given a probability algebra (ℬ, μ, d) without specifying the underlying probability space, and 𝒜 is a closed subalgebra of ℬ, we refer to ℙ(b|𝒜) as being an 𝒜-measurable function in order to avoid explicitly introducing the probability space representing (ℬ, μ, d) and the σ-subalgebra representing 𝒜 (as described in Lemma 2.2). Similarly we refer to χa as being 𝒜-measurable, when a ∈ 𝒜. As is customary, when a, b ∈ ℬ̂, we write ℙ(a|D) = ℙ(b|D) to mean that the functions ℙ(a|D) and ℙ(b|D) are equal μ-almost everywhere, and we give a similar interpretation to ℙ(a|D) ≤ ℙ(b|D). Therefore, the associated strict partial ordering ℙ(a|D) < ℙ(b|D) (meaning that ℙ(a|D) ≤ ℙ(b|D) is true while ℙ(a|D) = ℙ(b|D) is false) is true if and only if ℙ(a|D) ≤ ℙ(b|D) holds μ-almost everywhere and ℙ(a|D) < ℙ(b|D) holds on a set of positive μ-measure. Similar remarks apply to these relations between other measurable real-valued functions (such as χa ). When ℰ = {0, X} is the trivial subalgebra, and A ∈ ℬ, then ℙ(A|ℰ ) is the constant function f (x) := μ(A) for all x ∈ X. Indeed, this f is ℰ -measurable and ∫E fdμ = μ(A ∩ E) = ∫E χA dμ for all E ∈ ℰ , namely for E = 0 and E = X. More generally, for finite subalgebras ℰ ⊆ ℬ we have the following formula for ℙ(A|ℰ ), which is useful in many places below. Lemma 2.5. Let ℰ ⊆ ℬ be a finite subalgebra and A ∈ ℬ. Suppose E1 , . . . , En are the atoms in ℰ . Then ℙ(A|ℰ ) = ∑ j

μ(A ∩ Ej ) μ(Ej )

χEj .

164 � A. Berenstein and C. W. Henson Proof. By additivity of the integral, it suffices to prove that the integral of the displayed function over each atom Ej is equal to ∫E χA dμ, which equals μ(A ∩ Ej ). Since the sets Ej are pairwise disjoint, this is clear.

j

Fact 2.6. Let (X, ℬ, μ) be a probability space and let 𝒞 ⊆ ℬ be a σ-subalgebra. The conditional expectation operator 𝔼𝒞 restricted to L2 (X, ℬ, μ) is the Hilbert space orthogonal projection of L2 (X, ℬ, μ) onto the subspace L2 (X, 𝒞 , μ). (See [10, Proposition 4.2].) Let 𝒟 ⊆ 𝒞 ⊆ ℬ be σ-subalgebras and A ∈ ℬ. Then ℙ(A|𝒟) = 𝔼𝒟 (χA ) = 𝔼𝒟 (𝔼𝒞 (χA )) is the orthogonal projection of ℙ(A|𝒞 ) = 𝔼𝒞 (χA ) into L2 (X, 𝒟, μ), so 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩ℙ(A|𝒞 )󵄩󵄩󵄩2 − 󵄩󵄩󵄩ℙ(A|𝒟)󵄩󵄩󵄩2 = 󵄩󵄩󵄩ℙ(A|𝒞 ) − ℙ(A|𝒟)󵄩󵄩󵄩2 . Further, since we are working over a probability space, we have ‖f ‖2 = ‖f ‖2 ‖1‖2 ≥ |⟨f , 1⟩| = ‖f ‖1 for all L2 functions f , by the Cauchy–Schwartz inequality, and therefore 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ℙ(A|𝒞 ) − ℙ(A|𝒟)󵄩󵄩󵄩2 ≥ 󵄩󵄩󵄩ℙ(A|𝒞 ) − ℙ(A|𝒟)󵄩󵄩󵄩1 . These give useful quantitative conditions for ℙ(A|𝒞 ) ≠ ℙ(A|𝒟). They are used in proving Remark 8.2, Fact 9.4(5), and Corollary 9.5. If ℰ ⊆ 𝒟 ⊆ 𝒞 ⊆ ℬ and A ∈ ℬ, the preceding discussion yields 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ℙ(A|𝒞 ) − ℙ(A|𝒟)󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩ℙ(A|𝒞 ) − ℙ(A|𝒟)󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩ℙ(A|𝒞 ) − ℙ(A|ℰ )󵄩󵄩󵄩2 , which can be useful in working with approximations to ℙ(A|𝒞 ), as we illustrate next. Lemma 2.7. Let (X, ℬ, μ) be a probability space, 𝒞 ⊆ ℬ a σ-subalgebra, and A ∈ ℬ. For each k ≥ 1 there exists (E1 , . . . , Ek ) ∈ 𝒞 k , a partition of X, such that for any σ-subalgebra 𝒟 with {E1 , . . . , Ek } ⊆ 𝒟 ⊆ 𝒞 one has ‖ℙ(A|𝒞 ) − ℙ(A|𝒟)‖1 ≤ 1/k. Proof. Let f = ℙ(χA |𝒞 ), so f is a 𝒞 -measurable [0, 1]-valued function. Let I1 , . . . , Ik be the , j ) for j = 1, . . . , k − 1 and Ik = [ k−1 , 1]. So the intervals are pairwise intervals Ij = [ j−1 k k k disjoint and their union is [0, 1]. For each j let Ej = {x ∈ X | f (x) ∈ Ij }. Then (E1 , . . . , Ek ) is a partition of X in 𝒞 . For any sequence (r1 , . . . , rk ) such that rj ∈ Ij for all j, we have |f − ∑j rj χEj | ≤ 1/k pointwise on X, and therefore ‖f − ∑j rj χEj ‖2 ≤ 1/k. Now set rj =

μ(A∩Ej ) . μ(Ej )

Since f (x) is in Ij for x ∈ Ej , we have

j−1 μ(Ej ) k

≤ ∫E fdμ ≤ kj μ(Ej ) j

for all j. Noting that ∫E fdμ = ∫E χA dμ = μ(A ∩ Ej ) we see that rj ∈ Ij for all j = 1, . . . , k. It j

follows using Lemma 2.5 that

j

󵄩󵄩 μ(A ∩ Ej ) 󵄩󵄩󵄩 󵄩󵄩 󵄩 # 󵄩 χ 󵄩󵄩 ≤ 1/k. 󵄩󵄩f − ℙ(χA |{E1 , . . . , Ek } )󵄩󵄩󵄩2 = 󵄩󵄩󵄩f − ∑ 󵄩󵄩 μ(Ej ) Ej 󵄩󵄩󵄩2 j Further, if 𝒟 is any σ-subalgebra of ℬ that contains {E1 , . . . , Ek }, then ‖f −ℙ(χA |𝒟)‖1 ≤ 1/k, by the last statement in Fact 2.6, so (E1 , . . . , Ek ) satisfies the stated conditions.

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Probabilistic independence is very important in this paper. For A, B ∈ ℬ, we say A and B are (probabilistically) independent if μ(A∩B) = μ(A)μ(B), and write A B. Further, if S, T are subsets of ℬ, we say S and T are (probabilistically) independent and write S T if A B holds for every A ∈ ⟨S⟩ and B ∈ ⟨T⟩. Not surprisingly to model theorists, we need a more general version of independence that is relative to a set of parameters: 󳀀󳨐

Definition 2.8. If ℰ ⊆ ℬ is a σ-subalgebra of ℬ, we say A and B are (conditionally) independent over ℰ , and write A ℰ B if ℙ(A ∩ B|ℰ ) = ℙ(A|ℰ ) ⋅ ℙ(B|ℰ ).

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More generally, if S, T, W are subsets of ℬ, we say S and T are (conditionally) independent over W and write S W T if A ⟨W ⟩ B holds for every A ∈ ⟨S⟩ and B ∈ ⟨T⟩. 󳀀󳨐

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when W = 0, since ⟨0⟩ = {0, X}. Also, S W T if and Note that ⟨W ⟩ reduces to only if A ⟨W ⟩ B holds for every A ∈ S # and B ∈ T # . For us the following characterization of conditional independence is fundamental. As usual in model theory, if Y , Z are sets of parameters, we denote Y ∪ Z by YZ. 󳀀󳨐

Lemma 2.9. If S, T, W are subsets of ℬ, then the following statements are equivalent: (i) S W T; (ii) ℙ(A|⟨WT⟩) = ℙ(A|⟨W ⟩) for all A ∈ S # ; (iii) ℙ(A|⟨WT⟩) is ⟨W ⟩-measurable, for all A ∈ S # ; (iv) ‖ℙ(A|⟨WT⟩)‖2 = ‖ℙ(A|⟨W ⟩)‖2 for all A ∈ S # . Proof. (i) ⇔ (ii) Apply [18, Theorem 8.9], noting that (ii) is equivalent to the same statement with S # replaced by ⟨S⟩, since S # is dense in ⟨S⟩. (ii) ⇔ (iii) This is immediate. (iv) ⇔ (ii) Let A ∈ ℬ. Since ⟨W ⟩ ⊆ ⟨WT⟩, Fact 2.6 gives us 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩ℙ(A|⟨WT⟩)󵄩󵄩󵄩2 − 󵄩󵄩󵄩ℙ(A|⟨W ⟩)󵄩󵄩󵄩2 = 󵄩󵄩󵄩ℙ(A|⟨WT⟩) − ℙ(A|⟨W ⟩)󵄩󵄩󵄩2 , from which follows 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ℙ(A|⟨WT⟩)󵄩󵄩󵄩2 = 󵄩󵄩󵄩ℙ(A|⟨W ⟩)󵄩󵄩󵄩2

if and only if ℙ(A|⟨WT⟩) = ℙ(A|⟨W ⟩).

The next result shows that several different definitions of literature are equivalent.

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Applying the quantifier “for all A ∈ S # ” yields the desired equivalence. that one finds in the

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󳀀󳨐 󳀀󳨐

Corollary 2.10. If S, T, W are subsets of ℬ, then the following statements are equivalent: (i) S W T (ii) S W ⟨WT⟩ (iii) ⟨WS⟩ W ⟨WT⟩

166 � A. Berenstein and C. W. Henson

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Proof. (ii) ⇒ (i) is clear. Assume now that S W T holds and prove (ii). By Lemma 2.9(ii), we have that ℙ(A|⟨WT⟩) = ℙ(A|⟨W ⟩) for all A ∈ S # and thus S W ⟨WT⟩ holds, again using Lemma 2.9(ii). The proof of (ii) ⇔ (iii) is similar, since the definition of independence is a symmetric condition on the left and right families. 󳀀󳨐

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Remark 2.11. When W ⊆ ℬ is finite, we have the following simple characterization of . Namely, S W T if and only if μ(A ∩ B ∩ C)μ(C) = μ(A ∩ C)μ(B ∩ C) for every A ∈ S # , and B ∈ T # and every atom C ∈ W # . This is easily proved by comparing coefficients in the expressions for ℙ(A ∩ B|W # ) and ℙ(A|W # ) ⋅ ℙ(B|W # ) given by Lemma 2.5. E

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Notation 2.12. Suppose C, D, E are subsets of ℬ̂, and S, T, W are subsets of ℬ such that ̂ ⟨D⟩ = ⟨T⟩, ̂ and ⟨E⟩ = ⟨W ̂⟩. We write C ⟨C⟩ = ⟨S⟩, D to mean the same as S T. W

Next we prove a lemma that will be used in the proof of Theorem 8.1.

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Lemma 2.13 (Extension). Let (X, ℬ, μ) be a probability space. Let 𝒜 be a finite subalgebra of ℬ, with atoms A1 , . . . , Am , and let 𝒞 ⊆ 𝒟 be closed subalgebras of ℬ. Then there exists a probability space (X ′ , ℬ′ , μ′ ) and a boolean, measure-preserving embedding B 󳨃→ B′ of ℬ into ℬ′ , together with a finite subalgebra ℰ of ℬ′ whose atoms E1 , . . . , Em satisfy ℙ(Ej |𝒞 ′ ) = ℙ(A′j |𝒞 ′ ) for all i = 1, . . . , m and ℰ 𝒞 ′ 𝒟′ . (Here for 𝒵 = 𝒞 or 𝒟 we write 𝒵 ′ for {B′ | B ∈ 𝒵 }.) Proof. Let ([0, 1], ℱ , λ) be the Lebesgue measure space on [0, 1] and take (X ×[0, 1], ℬ′ , μ′ ) to be the product measure space, with ℬ′ = ℬ ⊗ ℱ and μ′ = μ ⊗ λ. For each B ∈ ℬ, let B′ := B×[0, 1]. The correspondence B 󳨃→ B′ is obviously a boolean, measure-preserving embedding of ℬ into ℬ′ . Also, for any ℬ-measurable function f : X → [0, 1], we let f ′ denote the ℬ′ -measurable function (x, y) 󳨃→ f (x). We see easily that the embedding preserves integration; namely for any B ∈ ℬ and ℬ-measurable f : X → [0, 1], we have by Fubini’s theorem ∫B′ f ′ d(μ ⊗ λ) = ∫B (∫[0,1] f ′ dλ) dμ = ∫B f dμ. We also note

that ℙ(B′ |𝒞 ′ ) = ℙ(B|𝒞 )′ for any B ∈ ℬ and σ-subalgebra 𝒞 of ℬ. (Using Lemma 2.7 it suffices to prove this when 𝒞 is finite; this is done by applying Lemma 2.5. Note that if C1 , . . . , Ck are the atoms of 𝒞 , then C1′ , . . . , Ck′ are the atoms of 𝒞 ′ , and for each j = 1, . . . , k we have (μ ⊗ λ)(B′ ∩ Cj′ ) = μ(B ∩ Cj ), (μ ⊗ λ)(Cj′ ) = μ(Cj ), and χC ′ = (χCj )′ .) j For each i = 1, . . . , m, let fi := ℙ(Ai |𝒞 ) and note that since A1 , . . . , Am is a partition of 1, we have f1 (x) + ⋅ ⋅ ⋅ + fm (x) = 1 on X μ-a. e. Now let E1 = {(x, y) ∈ X × [0, 1] : 0 ≤ y ≤ f1 (x)} and for 1 < i ≤ m let Ei = {(x, y) ∈ X × [0, 1] : f1 (x) + ⋅ ⋅ ⋅ + fi−1 (x) < y ≤ f1 (x) + ⋅ ⋅ ⋅ + fi (x)}. The sets {Ei }i≤m are ℬ′ -measurable and pairwise disjoint, and X ′ = X × [0, 1] = ⋃i≤m Ei (except possibly for a set of μ′ -measure zero). Note that for any B ∈ ℬ and i = 1, . . . , m, we have (μ ⊗ λ)(B′ ∩ Ei ) = ∫ χEi d(μ ⊗ λ) = ∫( ∫ χEi dλ) dμ = ∫ fi dμ B′

B

[0,1]

B

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using the definitions and Fubini’s theorem. If C ∈ 𝒞 , this gives ∫ χEi d(μ ⊗ λ) = ∫ fi dμ = ∫ ℙ(Ai |𝒞 )′ d(μ ⊗ λ) = ∫ ℙ(A′i |𝒞 ′ ) d(μ ⊗ λ) C

C′

C′

C′

and therefore ℙ(Ei |𝒞 ′ ) = ℙ(A′i |𝒞 ′ ) for all i = 1, . . . , m. For D ∈ 𝒟, we get (μ ⊗ λ)(D′ ∩ Ei ) = ∫ fi′ d(μ ⊗ λ) = ∫ ℙ(Ai |𝒞 )′ d(μ ⊗ λ) = ∫ ℙ(A′i |𝒞 ′ ) d(μ ⊗ λ), D′

D′

so ℙ(Ei |𝒟′ ) = ℙ(Ei |𝒞 ′ ) for all i = 1, . . . , m. Therefore ℰ

󳀀󳨐

D′

𝒞′

𝒟′ .

3 Some continuous model theory In this paper we use the setting of continuous logic to discuss the model theory of probability algebras. The fundamental ideas of continuous logic are presented in [7, 8] and in Bradd Hart’s article in this volume. We assume familiarity with the material in these sources, and often use it without specific reference. In addition, we need some background concerning metric imaginaries in continuous logic, and their role in some topics within stability theory, especially when dealing with canonical parameters for definable predicates and definability of types. Here we give pointers to published sources for this background, and a very brief summary of the topics we use. There is some treatment of imaginary sorts (i. e., of interpretations) in our key references [7, 8]. In [7, Section 11] only finitary imaginaries are presented; these are quotients of finite products of sorts modulo a definable pseudometric. However, in our Section 5 and later in the chapter, in connection with certain concepts in stability theory, we need more general, infinitary imaginaries. These are quotients of the product of a countably infinite family of sorts modulo a definable pseudometric; they are connected to the existing structure by their projection maps onto the sorts from which they come. A central example of these imaginaries is given by canonical parameters for a definable predicate relative to the (possibly infinite) sequence of parameters used in defining it. These are treated in detail in [8, Section 5]. Given a continuous theory T, the many sorted theory obtained by adding to T all possible metric imaginary sorts is called the meq expansion of T, and it is denoted here by T meq . Likewise, given ℳ 󳀀󳨐 T, the corresponding expansion of ℳ to a model of T meq is denoted ℳmeq . Presentations of the full construction of T meq and some of its properties are in [12, Section 3.3] and [5, Section 1], as well as in Section 10 of Bradd Hart’s article in this volume. In Section 8 we also use concepts and tools from stability theory in the setting of continuous model theory. Many of these, including canonical parameters for formulas,

168 � A. Berenstein and C. W. Henson definability of types, and canonical bases for stationary types, are developed in [8, Sections 7 and 8]. Beyond these, we use concepts such as types being parallel, the parallelism class of a stationary type, Morley sequences, orthogonal types, and nonmultidimensional theories. While these concepts lack a thorough exposition in the continuous model theory literature, it is not difficult to formulate and understand them based on how they are treated in the main references for stability theory in classical model theory, especially given the tools provided in [8]. An example needed here of such a fact is that the canonical base of a stationary type is contained in the meq definable closure of a Morley sequence of that type. For this material in the classical discrete setting, we follow closely the presentation in [11].

4 The model theory of probability spaces We deal here with structures of the form c

̂ ̂, 0, 1, ⋅ , ∩, ∪, μ ̂ , d) ℳ = (ℬ ̂ ) is the probability algebra of a probability space (X, ℬ, μ), 0 is the event where (ℬ̂, μ corresponding to 0 and 1 is the event corresponding to X; ⋅c is the complement operation and ∩, ∪ are the intersection and union operations on ℬ̂; and d̂ is the canonical metric on ̂ b) = μ ̂ take their ̂ (defined for a, b ∈ ℬ̂ by d(a, ̂ (a △ b)). The predicates, namely μ ̂ and d, ℬ values in the interval [0, 1]. The modulus of uniform continuity for the unary operation ⋅c ̂ is given by Δ(ϵ) = ϵ; for the binary operations ∩ and ∪ the and the unary predicate μ modulus is given by Δ(ϵ) = ϵ/2. For the rest of this paper, we take Lpr to be the continuous signature indicated in the previous paragraph. Note that every probability algebra of a probability space is indeed an Lpr -structure. (This requires, in particular, that it is complete as a metric space, which we noted in Section 2.) Notation 4.1. For any Lpr -prestructure ℳ and a ∈ M, we write a−1 for ac and a+1 for a. The following Lpr -conditions are easily seen to be true in every probability algebra of a probability space. (1) Boolean algebra axioms: Each of the usual axioms for a boolean algebra is the ∀-closure of an equation between terms (see [17, p. 38]) and thus it can be expressed in continuous logic as a condition. For example, the axiom ∀x∀y(x∪y = y∪x) is equivalent to supx supy (d(x∪ y, y ∪ x)) = 0. (2) Measure axioms: – μ(0) = 0 and μ(1) = 1, – supx supy (μ(x ∩ y) ∸ μ(x)) = 0,

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– supx supy (μ(x) ∸ μ(x ∪ y)) = 0, – supx supy |(μ(x) ∸ μ(x ∩ y)) − (μ(x ∪ y) ∸ μ(y))| = 0. The last three axioms express that μ(x ∪ y) + μ(x ∩ y) = μ(y) + μ(x) for all x, y. (3) Connections between d and μ: supx supy |d(x, y) − μ(xΔy)| = 0 where xΔy denotes the boolean term giving the symmetric difference, xΔy = (x ∩ yc ) ∪ (x c ∩ y). We denote the set of Lpr -conditions above by Pr. Notation 4.2. For the rest of this paper, we use the notation ℳ = (ℬ, μ, d) for a general model of Pr. We take care that this notation is not confused with our usual notation ̂ for its associated probability algebra. ̂ , d) (X, ℬ, μ) for a probability space and (ℬ̂, μ Theorem 4.3. The models ℳ = (ℬ, μ, d) of Pr are exactly the (abstract) probability algebras. That is, ℬ is a σ-order complete Boolean algebra, and μ is a strictly positive, σ-additive probability measure on ℬ; further, d is defined on ℬ by d(a, b) := μ(a △ b). Proof. If ℳ = (ℬ, μ, d) is indeed a probability algebra as described in the statement, then it is clear that it satisfies all conditions in Pr. Moreover, the metric space (ℬ, d) is complete, as shown by the calculation in [15, Lemma 323F]. Conversely, suppose ℳ is a model of Pr. It is clear from the axioms that ℳ consists of a boolean algebra ℬ with a finitely additive probability measure μ such that ℬ is a complete metric space under the metric d(a, b) = μ(a △ b). Moreover, μ must be continuous on ℬ with respect to d; indeed, μ is 1-Lipschitz with respect to d, as is dictated by the signature Lpr . Any increasing sequence in ℬ is necessarily a Cauchy sequence with respect to d, so it converges. This and the continuity of μ ensure that ℬ is σ-order complete as a boolean algebra and μ is σ-additive on ℬ. It follows from Theorem 4.3 that the models of Pr are (up to isomorphism) exactly the probability algebras of probability spaces. This is proved in [15, Theorem 321J]; a key ingredient in the proof of that result is the Loomis–Sikorski representation theorem for σ-order complete boolean algebras; see [15, Theorem 314M]. In Theorem 4.5 we give a proof of this fact about the models of Pr using tools from model theory. Example 4.4. Suppose 𝒜 is a boolean algebra and μ is a finitely additive probability measure on 𝒜. We may define a distance d on 𝒜 in the familiar way, by setting d(a, b) equal to μ(a △ b), where △ denotes the symmetric difference in 𝒜. Then (𝒜, μ, d) is an Lpr -prestructure, and it satisfies all of the axioms of Pr. Therefore we may obtain a model ̂ of Pr by first taking the quotient of (𝒜, μ, d) by the ideal of elements of μ-measure ̂, μ ̂ , d) (𝒜 0, and then taking the metric completion of the resulting quotient (as discussed in the middle of [7, pages 329–331]). For those readers who are familiar with Abraham Robinson’s nonstandard analysis (NSA), we note how this construction relates to the Loeb measure construction [20],

170 � A. Berenstein and C. W. Henson which has been one of the most important tools for applications of NSA. For that construction, we begin with 𝒜 being an internal boolean algebra of subsets of an internal set X, and μ being obtained from an internal finitely additive ∗ [0, 1]-valued measure ̂ be ̂, μ ̂ , d) ν on 𝒜, by taking μ(a) to be the standard part of ν(a) for each a ∈ 𝒜. Let (𝒜 constructed as above from (𝒜, μ, d) as in the preceding paragraph. In that setting, the quotient algebra of 𝒜 by the ideal of μ-null sets is already complete with respect to the quotient metric obtained from d, owing to the assumption of ω1 -saturation that is part of the basic NSA framework. Moreover, the saturation assumption also implies that μ has a natural and unique extension to a σ-additive probability measure on the σ-algebra of ̂ as its ̂, μ ̂ , d) subsets of X that is generated by 𝒜. The resulting probability space has (𝒜 probability algebra. See [19, Section II.2] and [22, Section 2.1] for elementary discussions of the Loeb construction and its basic properties. The metric ultraproduct of a family of probability algebras of probability spaces is an example of the Loeb construction. In that case, the internal measure space is the discrete ultraproduct of the family of probability spaces. This approach gives an alternative way of proving that every model of Pr is the probability algebra of some probability space, as we show next. Theorem 4.5. Let ℳ be a Lpr -structure. The following are equivalent: (1) ℳ is a model of Pr. (2) ℳ is isomorphic to the probability algebra of a probability space. Proof. (2) ⇒ (1) See the first paragraph of the proof of Theorem 4.3. (1) ⇒ (2) Let ℳ be a model of Pr. Let I be the set of all finite subsets of M. For each τ ∈ I, let ℳτ be the subalgebra τ # of ℳ, which is finite. Each ℳτ is the probability algebra of a finite probability space (Xτ , 𝒜τ , μτ ). Here Xτ is the set of atoms in ℳτ , 𝒜τ is the boolean algebra of all subsets of Xτ , and μτ ({a}) = μ(a) for each element a of Xτ . There exists an ultrafilter U on I such that for each a ∈ M the set {τ ∈ I | a ∈ τ} is an element of U. As discussed in the preceding example, the U-ultraproduct of the family (ℳτ | τ ∈ I) is the probability algebra of a probability space, by the Loeb measure construction. Moreover, ℳ is isomorphic to a substructure of this ultraproduct; the embedding maps a ∈ M to the equivalence class of the family (aτ | τ ∈ I) where we define aτ as follows: (i) if a ∈ ̸ τ # we take aτ = 0; (ii) if a ∈ τ # , we take aτ to be the subset of Xτ consisting of all atoms of ℳτ that are contained in a (so a is the join of aτ in ℳ). Therefore we have embedded ℳ into the probability algebra of a probability space. The proof is completed by applying Lemma 2.2. In the rest of this section we aim to discuss elementary equivalence of probability algebras and to characterize (axiomatize) the complete extensions of Pr. This depends on studying the definability in continuous logic of the set of atoms (and some related sets) in models of Pr. (See [7, Section 9] for a discussion of definable predicates and definable sets.)

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In the rest of this section, ℳ denotes a model of Pr, with underlying boolean algebra ℬ, measure μ, and metric d. We let Aℳ 1 denote the set of atoms of ℬ together with 0. Note

ℳ that for each r > 0 there are only finitely many a ∈ Aℳ 1 such that μ(a) ≥ r. Therefore A1 is finite or countable; the join (union) of Aℳ 1 is therefore in ℬ and provides a measurable splitting of 1 in ℬ between its atomic and atomless parts, either of which may be 0. Also, Aℳ 1 is a closed set with respect to the metric d. We consider the following formulas in the signature of Pr:

󵄨 󵄨 χ(x) := inf󵄨󵄨󵄨μ(x ∩ y) − μ(x ∩ yc )󵄨󵄨󵄨, y

ψ(x) := μ(x) ∸ χ(x), φ1 (x) := inf(d(x, z) ∔ ψ(z)), z

󵄨 󵄨 θ(x) := sup inf󵄨󵄨󵄨μ(x ∩ y ∩ z) − μ(x ∩ y ∩ zc )󵄨󵄨󵄨. y

z

To understand the meanings of these formulas in models of Pr, the next result is needed. The elementary argument needed for the proof is given in [16, Section 41, Theorem A]. Lemma 4.6. Suppose ℳ = (ℬ, μ, d) 󳀀󳨐 Pr. If b ∈ ℬ is atomless, then for every δ > 0 there is a partition of 1 in ℬ, say u = (u1 , . . . , un ), such that μ(b ∩ ui ) ≤ δ for all i = 1, . . . , n. Proof. It is sufficient to prove the result assuming b = 1 in ℬ. By the downward Löwenheim–Skolem theorem, ℳ has a separable elementary substructure ℳ′ , which is necessarily also atomless, and it obviously suffices to prove the Lemma for ℳ′ . Suppose ℳ′ is based on the algebra ℬ′ , which is a closed subalgebra of ℬ, and the predicates of ℳ′ are the restrictions of μ and d to ℬ′ . By separability of ℳ′ , we may take (𝒜n | n ≥ 1) to be an increasing family of finite boolean subalgebras of ℬ′ such that ⋃(𝒜n | n ≥ 1) is a dense subset of ℬ′ . For each n ≥ 1, let πn be the partition of 1 in 𝒜n that consists of the atoms of 𝒜n . The argument for Theorem A in [16, Section 41] shows that these partitions satisfy the conclusion of the lemma. Proposition 4.7. Let ℳ = (ℬ, μ, d) 󳀀󳨐 Pr and b ∈ ℬ. (a) If b is atomless, then χ ℳ (b) = 0. (b) Event b is an atom or 0 if and only if χ ℳ (b) = μ(b). (c) If b is not atomless and a is an atom of largest measure contained in b, then χ ℳ (b) ≤ μ(a). ℳ (d) dist(b, Aℳ 1 ) = φ1 (b). (e) Event b is atomless in ℬ if and only if θℳ (b) = 0. Proof. (a) Fix δ > 0 and use Lemma 4.6 to obtain a partition of 1 in ℬ, say u = (u1 , . . . , un ), such that μ(b ∩ ui ) ≤ δ for all i = 1, . . . , n. Let ai = b ∩ ui for all i, so b = a1 ∪ ⋅ ⋅ ⋅ ∪ an . There exists i such that μ(a1 ∪ ⋅ ⋅ ⋅ ∪ ai ) ≤ 21 μ(b) ≤ μ(a1 ∪ ⋅ ⋅ ⋅ ∪ ai+1 ). Then y = a1 ∪ ⋅ ⋅ ⋅ ∪ ai witnesses χ ℳ (b) ≤ δ.

172 � A. Berenstein and C. W. Henson (b) If b is an atom and a is arbitrary, then one of the events b ∩ a, b ∩ ac equals b and the other is 0. In that case |μ(b ∩ a) − μ(b ∩ ac )| = μ(b) for all a, so indeed χ ℳ (b) = μ(b). If b is not an atom, there exists a ∈ ℬ such that μ(b) > μ(b ∩ a) > 0 and μ(b) > μ(b ∩ ac ) > 0, from which it follows that |μ(b ∩ a) − μ(b ∩ ac )| < μ(b). (c) Suppose b is not atomless and let a1 , a2 , . . . be a listing of all the (finitely or countably many) distinct atoms of ℬ contained in b, arranged so that μ(a1 ) ≥ μ(a2 ) ≥ . . . . Take u ⊆ b to be the union of all aj such that j is odd and v ⊆ b to be the union of all aj such that j is even. Then u, v are disjoint and b \ (u ∪ v) is atomless. One checks easily that χ ℳ (b) ≤ μ(u) − μ(v) ≤ μ(a1 ). (d) The key idea is this: if b is atomless, then dist(b, Aℳ 1 ) = μ(b); if b is not atomless and a is an atom of largest measure contained in b, then dist(b, Aℳ 1 ) = μ(b) − μ(a). ℳ Therefore, from (a) and (c) we conclude that dist(b, A1 ) ≤ ψℳ (b) for all b. From ℳ (b) we see that ψℳ (b) = 0 when b is an atom, and therefore Aℳ 1 is the zero set of ψ . ℳ ℳ This makes it clear that dist(b, A1 ) ≥ φ1 (b). Conversely, for every b we have ℳ ℳ φℳ 1 (b) ≥ inf(d(b, z) ∔ dist(z, A1 )) ≥ dist(b, A1 ) z

which completes the proof. (e) This follows from (a) and (b). Note that θ(b) = 0 is equivalent to saying χ(u) = 0 holds for every u ≤ b. Proposition 4.7(d) shows that Aℳ 1 is a definable set, uniformly in all models ℳ of Pr. (See [7, Definition 9.16].) It is useful to introduce for each n > 1 the further set ℳ Aℳ n = {x1 ∪ ⋅ ⋅ ⋅ ∪ xn | x1 , . . . , xn ∈ A1 }. ℳ Note that Aℳ ⊆ Aℳ 1 2 ⊆ ⋅ ⋅ ⋅ ⊆ An ⊆ ⋅ ⋅ ⋅. Using [7, Theorem 9.17] and the definability of ℳ the set Aℳ 1 , we may conclude that An is a definable set in all models ℳ of Pr, for all n ≥ 1. Indeed, as we show next, the distance to Aℳ n is given explicitly by the following formula in the signature of Pr (where we define the formulas for n > 1 by induction on n):

φn (x) = inf(φn−1 (x ∩ w) ∔ φ1 (x ∩ wc )). w

Proposition 4.8. Let ℳ = (ℬ, μ, d) 󳀀󳨐 Pr and a ∈ ℬ. Then for each n ≥ 1, ℳ dist(a, Aℳ n ) = φn (a).

Proof. Let a ∈ ℬ and a1 , a2 , . . . be a listing of all distinct atoms contained in a, arranged so that μ(a1 ) ≥ μ(a2 ) ≥ ⋅ ⋅ ⋅, and extended to an infinite sequence by taking ak = 0 for larger k, if necessary. We note that dist(a, Aℳ n ) = d(a, w) where w = a1 ∪ ⋅ ⋅ ⋅ ∪ an , and for this w we have dist(a, w) = μ(a) − μ(a1 ∪ ⋅ ⋅ ⋅ ∪ an ) = μ(a) − (μ(a1 ) + ⋅ ⋅ ⋅ + μ(an )). To prove the lemma, we argue by induction on n ≥ 1. The n = 1 case is Proposition 4.7(d). Taking n > 1, it remains to prove the induction step from n − 1 to n.

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First note that if we take w = a1 ∪ ⋅ ⋅ ⋅ ∪ an−1 then we have φℳ n−1 (a ∩ w) = 0 (by c c the induction hypothesis) and φℳ (a ∩ w ) = μ(a ∩ w ) − μ(a ) (by Proposition 4.7(d)). n 1 Therefore, for this w we have ℳ c c φℳ n−1 (a ∩ w) + φ1 (a ∩ w ) = μ(a ∩ w ) − μ(an )

= (μ(a) − μ(a1 ∪ ⋅ ⋅ ⋅ ∪ an−1 )) − μ(an )

= μ(a) − (μ(a1 ) + ⋅ ⋅ ⋅ + μ(an−1 ) + μ(an )) = dist(a, Aℳ n ). To finish the argument, it suffices to prove that for any other w we have ℳ c (φℳ n−1 (a ∩ w) ∔ φ1 (a ∩ w )) ≥ μ(a) − (μ(a1 ) + ⋅ ⋅ ⋅ + μ(an )).

So fix w ∈ M and let a11 , a21 , . . . be a listing of all distinct atoms contained in a∩w, arranged so that μ(a11 ) ≥ μ(a21 ) ≥ ⋅ ⋅ ⋅, and extended to an infinite sequence by taking ak1 = 0 if necessary. Also, let a12 be one of the largest atoms contained in a ∩ wc (which can be 0). 1 Note that the nonzero elements among a12 , a11 , . . . , an−1 are distinct atoms, and all are ≤ a. ℳ 1 By the induction hypothesis, φn−1 (a∩w) = μ(a∩w)−∑n−1 i=1 μ(ai ) and, by Proposition 4.7(d), ℳ c c 2 φ1 (a ∩ w ) = μ(a ∩ w ) − μ(a1 ). Thus n−1

1 c 2 ℳ c φℳ n−1 (a ∩ w) ∔ φ1 (a ∩ w ) = (μ(a ∩ w) − ∑ μ(ai )) + (μ(a ∩ w ) − μ(a1 )) n−1

i=1

= μ(a) − ( ∑ μ(ai1 ) + μ(a12 )). i=1

1 The smallest possible value of this last expression occurs when a21 , a11 , . . . , an−1 have the 1 1 1 largest possible measures, which happens when the sequence μ(a2 ), μ(a1 ), . . . , μ(an−1 ) is a permutation of μ(a1 ), . . . , μ(an ).

For a similar treatment of atoms in the setting of random variable structures see [3, Lemma 2.16]. Remark 4.9. Let ℳ = (ℬ, μ, d) 󳀀󳨐 Pr and a ∈ ℬ. Propositions 4.7(d) and 4.8 make it clear that ℳ ℳ μ(a) ≥ φℳ 1 (a) ≥ φ2 (a) ≥ ⋅ ⋅ ⋅ ≥ φn (a) ≥ ⋅ ⋅ ⋅ .

Notation 4.10. Let ℳ = (ℬ, μ, d) 󳀀󳨐 Pr and a ∈ ℬ; let a1 , a2 , . . . be a listing of all distinct atoms contained in a, arranged so that μ(a1 ) ≥ μ(a2 ) ≥ ⋅ ⋅ ⋅, and extended to an infinite sequence by taking ak = 0 for larger k, if necessary. For each n ≥ 1, we refer to μ(an ) as the nth largest measure of an atom contained in a, and we denote this number as atℳ n (a). Note that the nonzero elements of (an | n ∈ ℕ) are distinct, whereas the measure values (μ(an ) | n ∈ ℕ) may contain repetitions.

174 � A. Berenstein and C. W. Henson Corollary 4.11. For each n ≥ 1, the predicate atn is definable in all models of Pr. Indeed, if ℳ = (ℬ, μ, d) 󳀀󳨐 Pr and a ∈ ℬ, then ℳ atℳ 1 (a) = μ(a) ∸ φ1 (a)

and, for each n > 1, ℳ ℳ atℳ n (a) = φn−1 (a) ∸ φn (a).

Proof. This is immediate from Proposition 4.8. We now consider an extension by definitions of Pr obtained by adding unary predicate symbols (Pn | n ≥ 1) to the signature and by adding as axioms the conditions 󵄨 󵄨 sup󵄨󵄨󵄨P1 (x) − (μ(x) ∸ φ1 (x))󵄨󵄨󵄨 = 0 x

and, for n > 1, 󵄨 󵄨 sup󵄨󵄨󵄨Pn (x) − (φn−1 (x) ∸ φn (x))󵄨󵄨󵄨 = 0. x This extension of Pr is denoted by Pr∗ . Note that each model ℳ of Pr has a unique expansion, which we denote by ℳ∗ , that is a model of Pr∗ . This expansion is given by ∗ interpreting each Pn so that, for each a ∈ M, one takes Pnℳ (a) to be the nth largest measure of an atom contained in a. Notation 4.12. Suppose ℳ = (ℬ, μ, d) 󳀀󳨐 Pr. Fix n ≥ 1 and consider any tuple a = (a1 , . . . , an ) ∈ M n = ℬn . Let e = (e1 , . . . , e2n ) be the partition of 1 in the boolean algebra ℬ generated by a1 , . . . , an . By this we mean that the elements of e are all possible interseck k tions of the form a1 1 ∩⋅ ⋅ ⋅∩ann , where each ki comes from {−1, +1}, and we list these intersections in order according to lexicographic order on the tuples of superscripts k1 , . . . , kn . We refer to e as the partition of 1 in ℬ associated to a. Further, note that when (a1 , . . . , an ) and (e1 , . . . , e2n ) are as above, then each ai is the union of the coordinates ej of e that k

k

are intersections a1 1 ∩ ⋅ ⋅ ⋅ ∩ ann in which ki = +1. That is, the correspondence between coordinates of (a1 , . . . , an ) and coordinates of (e1 , . . . , e2n ) is given, in both directions, by simple boolean terms that depend only on n. (In particular, these tuples are uniformly interdefinable in models of Pr.) k k For simplicity of notation, we write as for a1 1 ∩ ⋅ ⋅ ⋅ ∩ ann when s = (k1 , . . . , kn ) is an arbitrary element of {−1, +1}n and a = (a1 , . . . , an ) ∈ M n . Likewise we write x s for the k k boolean term x1 1 ∩ ⋅ ⋅ ⋅ ∩ xnn when s = (k1 , . . . , kn ) and x stands for the tuple (x1 , . . . , xn ) of variables. As indicated above, the identity xi = ⋃(x s | s = (k1 , . . . , kn ) and ki = +1) s

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is true in all models of Pr. Frequently when we use this notation, as here, we omit the standard specifications that x = (x1 , . . . , xn ) and s = (k1 , . . . , kn ) ∈ {−1, +1}n . In particular, we view ki as a function of s when s ∈ {−1, +1}n . When it is needed, we list the elements of {−1, +1}n in lexicographical order. Remark 4.13. For future use we note that for every Lpr -formula φ(x) there exists an Lpr -formula ψ(ys | s ∈ {−1, +1}n ), such that φ(x) is equivalent to ψ(x s | s ∈ {−1, +1}n ) in all models of Pr. Indeed, it suffices to take ψ(ys | s ∈ {−1, +1}n ) to be the result of substituting the boolean term ⋃(ys | s = (k1 , . . . , kn ) ∈ {−1, +1}n and ki = +1) for the variable xi in φ(x), for i = 1, . . . , n. Further, for every Lpr -formula φ(x) there exists an Lpr -formula ψ(x) such that ψ(x) is Pr-equivalent to φ(x) and every atomic formula occurring in ψ(x) is of the form μ(x s ) for some s ∈ {−1, +1}n . Moreover, ψ(x) can be chosen so that it is obtained from such atomic formulas using the restricted connectives 0, 1, t 󳨃→ t/2, and (t, u) 󳨃→ t ∸ u. Proof. A general atomic formula α(x) in Lpr can be taken to be one of the form μ(t(x)) where t(x) is a boolean term in x. If α(x) = d(t1 (x), t2 (x)), then α(x) can be replaced by μ(t1 (x) △ t2 (x)). For each such t(x), there is a subset S ⊆ {−1, +1} such that the equation t(x) = ∪(x s | s ∈ S) is true in all models of Pr. (If S is empty, then t(x) = 0 is true in all s ̂ ) | s ∈ S) is true in all models of Pr, where by models of Pr.) Moreover, μ(t(x)) = ∑(μ(x ̂ we mean the connective (u , . . . , u n ) 󳨃→ min(∑(u | s ∈ S), 1). For the last statement, ∑ 1 2 s ̂ see [7, Chapter 6]. including treatment of the connectives ∑, A consequence of the preceding observation is that when C is a subalgebra of a model ℳ of Pr, then for any a ∈ M n the type tpℳ (a/C) is determined by the values ψℳ (a) of Lpr -formulas ψ(x) over C in which all atomic formulas are of the form μ(x s ∩c) for some c ∈ C. (As above, x = (x1 , . . . , xn ) and s ∈ {−1, +1}n .) Theorem 4.14. The theory Pr∗ admits quantifier elimination. Proof. We use [8, Theorem 4.16], so we need to show that Pr∗ has the back-and-forth property given in [8, Definition 4.15]. Therefore, consider two ω-saturated models ℳ∗ , 𝒩 ∗ of Pr∗ and tuples (a1 , . . . , an ) in ℳ∗ ; (b1 , . . . , bn ) in 𝒩 ∗ such that the quantifier-free type of (a1 , . . . , an ) in ℳ∗ is the same as the quantifier-free type of (b1 , . . . , bn ) in 𝒩 ∗ . Given any u in ℳ∗ , we need to find v in 𝒩 ∗ such that (a1 , . . . , an , u) and (b1 , . . . , bn , v) have the same quantifier-free type in the language of Pr∗ . It suffices to do this for the case in which (a1 , . . . , an ) and (b1 , . . . , bn ) are partitions of 1, by the discussion in Subsection 4.12. Using Lemma 4.6 and the fact that ℳ is ω-saturated, for each atomless c ∈ M and 0 < r < 1 there exists a ≤ c in M such that μ(a) = rμ(c), and hence μ(c ∩ ac ) = (μ(c) − r)μ(c). Indeed, x = a can be taken to satisfy all of the conditions |μ(x) − rμ(c)| ≤ δ for δ > 0, which we just showed were finitely satisfiable in ℳ. This is used in the next paragraph. In the assumed situation we know that for each j = 1, . . . , n we have μ(aj ) = μ(bj ) and, for all k ≥ 1, we also have Pkℳ (aj ) = Pk𝒩 (bj ). For each j, let aj0 be the atomic ∗

∗

part of aj (i. e., the union of the atoms of ℳ that are ≤ aj ), so aj1 := aj ∩ (aj0 )c is the

176 � A. Berenstein and C. W. Henson atomless part of aj . Define b0j , b1j from bj similarly. Our assumptions yield that μ(aj0 ) = μ(b0j ) (and indeed, that the atoms below aj and bj are in a bijective, measure-preserving

correspondence). Hence also μ(aj1 ) = μ(b1j ).

Take any u in ℳ∗ and fix j = 1, . . . , n. Define v0j ≤ b0j to be the union of the atoms

below bj that correspond to atoms below aj ∩ u. Further, choose v1j ≤ b1j so that μ(v1j ) =

μ(aj ∩ u) − μ(v0j ), and let vj = v0j ∪ v1j . We obtain μ(vj ) = μ(aj ∩ u), ∗ Pk𝒩 (vj )

=

∗ Pkℳ (aj

and

∩ u)

for all k ≥ 1.

Then let v = v1 ∪ ⋅ ⋅ ⋅ ∪ vn , and note that vj = bj ∩ v for all j. It follows that the quantifierfree type of (a1 , . . . , an , u) in ℳ∗ is the same as the quantifier-free type of (b1 , . . . , bn , v) in 𝒩 ∗ , as desired. Theorem 4.14 allows us to characterize (and axiomatize) the complete extensions of Pr. Definition 4.15. For any ℳ = (ℬ, μ, d) 󳀀󳨐 Pr, let Φℳ denote the sequence (atℳ n (1) | n ≥ 1), which lists the sizes of the atoms of ℬ in decreasing order (and then has a tail of 0s if there are only finitely many atoms in ℬ). Note that the range of the operator Φ consists of all the sequences (tn | n ≥ 1) such that 1 ≥ t1 ≥ t2 ≥ ⋅ ⋅ ⋅ ≥ 0 and ∑∞ n=1 tn ≤ 1. Corollary 4.16. For models ℳ, 𝒩 of Pr, we have that ℳ ≡ 𝒩 if and only if Φℳ = Φ𝒩 . Therefore, any complete extension T of Pr (in the same signature) can be axiomatized by adding to Pr the conditions φ1 (1) = 1 − t1 and φn−1 (1) ∸ φn (1) = tn (for n > 1), where (tn | n ≥ 1) is the common value of Φℳ for ℳ 󳀀󳨐 T. Proof. Let ℳ, 𝒩 be models of Pr such that Φℳ = Φ𝒩 . From the definition of the operator Φ, we see that 1 has the same quantifier-free type in ℳ∗ as in 𝒩 ∗ . Theorem 4.14 yields that ℳ∗ ≡ 𝒩 ∗ , from which it follows that ℳ ≡ 𝒩 . The converse and the rest of the Corollary follow using the definition of Φ. Corollary 4.17. Every completion T of Pr (in Lpr ) is separably categorical, and the unique separable model of T is strongly ω-homogeneous. Proof. Let T be a completion of Pr and let ℳ be a separable model of T. A strengthening of Lemma 4.6 that is proved in [16, Section 41] says that the key property used in the proof of Theorem 4.14 is actually true in all models, without assuming they are ω-saturated. That is, for each atomless c ∈ M and 0 < r < 1 there exists a ≤ c in M such that μ(a) = rμ(c), and hence μ(c ∩ ac ) = (μ(c) − r)μ(c). So the proof of Theorem 4.14 not only shows that Pr∗ admits quantifier elimination, but shows further that for each ℳ∗ 󳀀󳨐 Pr∗ and every (a1 , . . . , an ) in ℳ∗ , every 1-type over (a1 , . . . , an ) for the theory of ℳ∗ is realized in ℳ∗ . In other words, every model of

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Pr∗ is ω-saturated. It follows trivially from the definition that the same is true of every structure (ℳ, a1 , . . . , an ) where ℳ is a model of Pr and a1 , . . . , an ∈ M. It is routine to show that if ℳ1 , ℳ2 are ω-saturated separable metric structures for the same language, and ℳ1 , ℳ2 are elementarily equivalent, then ℳ1 and ℳ2 are isomorphic. One uses the usual inductive back-and-forth argument to produce an elementary bijection f : S1 → S2 , where Si is a dense subset of Mi for both values of i. Then f extends to a map on M1 that is an isomorphism from ℳ1 onto ℳ2 . The proof of the corollary is completed by applying this construction to models of the form (ℳ, a1 , . . . , an ) discussed above. Remark 4.18. Corollary 4.17 yields the following well-known fact due to Carathéodory: if (X, ℬ, μ) and (Y , 𝒞 , ν) are atomless, countably generated measure spaces with μ(X) = ν(Y ) < ∞, then the measured algebras of (X, ℬ, μ) and (Y , 𝒞 , ν) are isomorphic. Proof. Without loss of generality, we may take μ(X) = ν(Y ) = 1. In that case, the measured algebras of these two probability spaces are atomless, separable models of Pr. Using Theorem 4.14, we see that these probability algebras are elementarily equivalent (since in an atomless probability algebra the predicates interpreting Pn are identically 0) and hence, by Corollary 4.17, we get the desired result. For proofs in analysis, see [16, Section 41] and [23, Theorem 4, p. 399]. Corollary 4.19. Let T be any complete Lpr -theory that extends Pr and let (tn | n ≥ 1) be the common value of Φℳ for ℳ 󳀀󳨐 T. (1) If ∑∞ n=1 tn = 1, then T has a unique model, which consists of an atomic probability algebra having atoms (an | n ≥ 1 and tn > 0) with μ(an ) = tn for all n. (2) If ∑∞ n=1 tn < 1, then the models of T are exactly the probability algebras with atoms as described in (1) together with an atomless part of measure 1 − ∑∞ n=1 tn . Proof. These statements are immediate from Corollaries 4.16 and 4.17. Remark 4.20. Let ℳ be any model of Pr. From the previous results it follows that aclℳ (0) is the σ-subalgebra generated by the atoms in ℳ. Remark 4.21. If ℳ 󳀀󳨐 Pr, let a0 ∈ M be the join of the atoms of ℳ, and let a1 = a0c . Further, let 𝒜i = {a ∈ M | a ≤ ai } for each i = 0, 1. Thus 𝒜0 is the set of atomic elements of ℳ and 𝒜1 is the set of atomless elements, and a1 is the largest atomless element. The partition {a0 , a1 } of 1 is important because it splits every element a of M into its atomic part a ∩ a0 and its atomless part a ∩ a1 . We note the following facts concerning the definability of these elements and sets: (i) Relative to all models ℳ of Pr: 𝒜0 is not a zeroset; 𝒜1 is a zeroset but is not a definable set; a0 , a1 are not definable elements. (ii) Fix a complete extension T of Pr. Relative to all models of T: a0 , a1 are definable elements and 𝒜0 , 𝒜1 are definable sets.

178 � A. Berenstein and C. W. Henson Proof. (i) First we show 𝒜0 is not a zero set over Pr. Suppose otherwise, so there exists a definable predicate R(x) over Pr such that for all ℳ 󳀀󳨐 Pr and all a ∈ M, we have Rℳ (a) = 0 if and only if a is an atomic element in ℳ. For each n ≥ 1, let ℳn be the probability algebra of the probability space having n points, each of which has measure 1/n, and let ℳ be the metric ultraproduct of (ℳn | n ≥ 1) with respect to a nonprincipal ultrafilter. Then 1 is atomic in every ℳn while it is not atomic in ℳ; indeed, ℳ is atomless, so 0 is its only atomic element. This means Rℳ (1) ≠ 0 whereas Rℳn (1) = 0 for all n. This violates the fundamental theorem of ultraproducts for the definable predicate R. (See Theorem 5.4 and the discussion of extensions by definition in Section 9 in [7].) Proposition 4.7(e) shows that 𝒜1 is the zero set of the formula θ(x) in all models of Pr. Further, 𝒜1 cannot be a definable set uniformly in all models of Pr, since otherwise there would be a definable predicate R(x) over Pr such that for all ℳ 󳀀󳨐 Pr and all a ∈ M, Rℳ (a) = sup{μ(a ∩ y) | y ∈ 𝒜1 }. But then the zero set of R(x) would be 𝒜0 in all models of Pr, which we just proved is not possible. Since a0c = a1 , they are either both definable or both undefinable. We work with a0 . If a0 were definable over Pr, then the operation x 󳨃→ a0 ∩ x would be definable, so its image, which is 𝒜0 , would be a definable set, but it isn’t. (ii) Consider a complete extension T of Pr and let 𝒰 be an ω1 -universal domain for T. By Corollary 4.19, we see that a0 and a1 are fixed by every automorphism of 𝒰 . Using [7, Exercise 10.7 or Theorem 9.32], it follows that a0 and a1 are each definable uniformly in all models of T. Finally, note that for each i = 0, 1, the set 𝒜i is the image of the definable function fi defined by fi (x) = x ∩ai . Using [7, Theorem 9.17], we infer that 𝒜i is a definable set uniformly in all models of T. Exercise 4.22. By Remark 4.9, for any ℳ 󳀀󳨐 Pr and any a ∈ M, we have that (φℳ n (a) | n ≥ 1) is a decreasing sequence from [0, 1], so it converges, and its limit must be the distance from a to the set of atomic elements of ℳ, which is the closure of ⋃n Aℳ n . Show that if we restrict attention to models of a completion T of Pr, this convergence is uniform (in ℳ as well as a), so its limit is a T-definable predicate. This gives an alternative proof of Remark 4.21(ii).

5 Random variables Here we discuss how to represent the space RV of [0, 1]-valued random variables (modulo pointwise equality a. e.), equipped with the L1 -distance, as a metric imaginary sort for the theory Pr. We focus on the measure-theoretic aspects of the matter, and avoid many technical details of the meq construction, for which we refer to various articles for

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the details (see Section 3). An interpretation of random variables in atomless probability algebras was originally given by Ben Yaacov in [1] in the CAT setting, and extended to an interpretation of RV in Pr by him in [3]. In [1] certain other classes of random variables are also treated, and the approach used here can easily be extended to apply to them. When ℳ is an arbitrary model of Pr, we let (X, ℬ, μ) denote a probability space ̂ is (isomorphic to) ℳ. In much of this section, we ̂ , d) whose probability algebra (ℬ̂, μ argue in (X, ℬ, μ) using measure theory. We denote by RV = RV(ℬ, μ) := L1 (ℬ, μ; [0, 1]) the space of all ℬ-measurable functions f : X → [0, 1]. We equip this space with the L1 (pseudo)metric, for which the distance between f and g is ‖f − g‖1 = ∫X |f − g| dμ. As done here for probability spaces, we consider the quotient metric space obtained by identifying two random variables if they are equal pointwise μ-a. e., equipped with the distance induced by ‖f − g‖1 (which we call the L1 -distance). The goal of this section is to explain how this quotient can be seen as a metric imagî of Pr. One value of doing so is that it allows seeing ℙ(a|𝒜) ̂ , d) nary sort for the model (ℬ̂, μ meq ̂ ̂ ̂ as an imaginary in (ℬ, μ, d) , for any a ∈ ℬ̂ and any closed subalgebra 𝒜 of ℬ̂. For n ≥ 1, consider the subset RVn = RVn (ℬ, μ) of L1 (ℬ, μ; [0, 1]) consisting of those functions that can be written as ∑ni=1 ni χEi where E = (E1 , . . . , En ) is a partition of X from n

ℬ. Below we denote ∑i=1 ni χEi by fE . On RVn (ℬ, μ) we take the L1 -distance. If fE ∈ RVm and fF ∈ RVn , we have fE , fF both in L1 (ℬ, μ; [0, 1]), so we can compare them, compute

the L1 -distance between them, etc., even if m ≠ n. Of course we may identify RVn with the set of E = (E1 , . . . , En ) that are partitions of X in ℬ. For two such partitions E, F, we define ρn (E, F) to be the L1 -distance, 󵄩 󵄩󵄩 n n 󵄩󵄩 i i 󵄩󵄩󵄩 ρn (E, F) := ‖fE − fF ‖1 = 󵄩󵄩󵄩∑ χEi − ∑ χFi 󵄩󵄩󵄩 󵄩󵄩 n n 󵄩󵄩󵄩1 i=1 󵄩i=1 1 = ∑ |i − j|μ(Ei ∩ Fj ). n i=j̸

(A)

By analyzing the expressions in (A), we obtain (in the following lemma) a Lipschitz equivalence between ρn (E, F) and dP (E, F) := 21 ∑ni=1 d(Ei , Fi ) = 21 ∑ni=1 μ(Ei △ Fi ). Note that when n = 2 we have dP (E, F) = 21 (d(E1 , F1 ) + d(E1c , F1c )) = d(E1 , F1 ), since d(E1c , F1c ) = d(E1 , F1 ); this explains the factor 21 . Also, dP is equivalent to the usual pseudometric d(E, F) := maxi d(Ei , Fi ), since 1 n d(E, F) ≤ dP (E, F) ≤ d(E, F) 2 2 holds for all E, F ∈ RVn . Lemma 5.1. For all E, F ∈ RVn we have 1 d (E, F) ≤ ρn (E, F) = ‖fE − fF ‖1 ≤ dP (E, F). n P

180 � A. Berenstein and C. W. Henson Proof. First note that for each i the family (Ei ∩ Fj | j ≠ i) is a partition of Ei \ Fi , and the same with E and F interchanged. Therefore n

n

∑ μ(Fi \ Ei ) = ∑ μ(Ei ∩ Fj ) = ∑ μ(Ei \ Fi ), i=1

i=j̸

i=1

from which follows ∑ μ(Ei ∩ Fj ) = i=j̸

1 n ∑ μ(Ei △ Fi ) = dP (E, F). 2 i=1

Then we get the desired inequality using (A) together with 1 1 ∑ μ(Ei ∩ Fj ) ≤ ∑ |i − j|μ(Ei ∩ Fj ) ≤ ∑ μ(Ei ∩ Fj ). n i=j̸ n i=j̸ i=j̸ ̂ n = RV ̂ n (ℬ, μ) for the image of RVn under the quotient map from ℬ onto ℬ̂. Write RV ̂ we also denote RV ̂ n by RV ̂ ℳ , and note that it satisfies ̂ , d), If ℳ is the model (ℬ̂, μ n

̂ ℳ = {(e1 , . . . , en ) ∈ M n | (e1 , . . . , en ) is a partition of 1 in ℳ}. RV n ̂ ℳ the pseudometric ρ ̂ℳ We put on RV n n obtained canonically from ρn on RVn ; that is, for ℳ ̂ , we have e, f ∈ RV n

󵄨󵄨 i j 󵄨󵄨󵄨󵄨 󵄨󵄨 ̂ℳ ρ (e, f ) = − ∑ ∑ 󵄨 󵄨μ(e ∩ f ) n 󵄨󵄨 n n 󵄨󵄨󵄨 i j i j 󵄨 =

1 ∑ ∑ |i − j|μ(ei ∩ fj ). n i j

(B)

̂ℳ ̂ℳ ̂ℳ Lemma 5.2. Let ℳ 󳀀󳨐 Pr. Then ρ n is a complete metric on RVn . Also, RVn is a definable ̂ℳ ̂ℳ set, and ρ n is a definable predicate on RVn , uniformly in all models of Pr. ̂ℳ Proof. Obviously, ρ n is a pseudometric. The fact that it is a metric follows from Lemma 5.1 and the definition of dP on RVℳ n , which implies that dP (E, F) = 0 iff μ(Ei △ Fi ) for all i = 1, . . . , n. ̂ ℳ is uniformly a definable subset of M n , it is sufficient to show that To show that RV n it is the image of a definable function on a definable set. To do this, consider the function c defined on M n−1 by (a1 , . . . , an−1 ) 󳨃→ (e1 , . . . , en ) where e1 = a1 , ej = aj ∩ a1c ∩ ⋅ ⋅ ⋅ ∩ aj−1 for c c 2 ≤ j ≤ n − 1, and en = a1 ∩ ⋅ ⋅ ⋅ ∩ an−1 . Obviously, this is a definable function, since the coordinates ej are given by boolean terms. Note that ei ∩ej = 0 whenever i ≠ j. Moreover, by induction on j < n, we can show a1 ∪ ⋅ ⋅ ⋅ ∪ aj = e1 ∪ ⋅ ⋅ ⋅ ∪ ej . Therefore (e1 , . . . , en−1 ) is a partition of a1 ∪ ⋅ ⋅ ⋅ ∪ an−1 . Since en = (a1 ∪ ⋅ ⋅ ⋅ ∪ an−1 )c , we have that (e1 , . . . , en ) is always

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̂ ℳ , note that whenever a partition of 1 in ℳ. To see that this map is surjective onto RV n (e1 , . . . , en ) is a partition of 1, then (e1 , . . . , en−1 ) 󳨃→ (e1 , . . . , en ). ̂ℳ Equation (B) shows that ρ n is a definable predicate, uniformly on all models ℳ of Pr. ̂ ℳ is complete. Because M = ℬ̂ is complete with respect It remains to show that RV n ̂ ℳ ⊆ (ℬ̂)n is closed, we see that RV ̂ ℳ is complete ̂ (a △ b) and RV to the metric d(a, b) := μ n

n

̂ ℳ is complete with with respect to the metric dP (e, f ) = 21 ∑ni=1 d(ei , fi ). Therefore RV n ̂ℳ ̂ℳ respect to ρ n , since ρn is uniformly equivalent to dP by Lemma 5.1.

Remark 5.3. Similar reasoning to that in the preceding proof shows that the map E 󳨃→ ℳ fE for E ∈ RVℳ n is an isometric map from RVn into L1 (ℬ , μ; [0, 1]) whose range is the collection of all random variables whose values are in {1/n, . . . , n/n}. (Here isometric means with respect to ρℳ n and the L1 -distance.) Hence this map induces an isometry ℳ ̂ from RV onto the set of = a. e. equivalence classes of those random variables. n

Fix ℳ 󳀀󳨐 Pr and, as above, let (X, ℬ, μ) be a probability space whose probability ℳ ̂ℳ the inverse limit of the spaces (RV n |

̂ algebra is (isomorphic to) ℳ. We denote by RV

2

̂ℳ ̂ℳ n ≥ 1) equipped with a suitable family of maps π̂ : RV 2n+1 → RV2n that we now define. ℳ ̂ n+1 , we set For (e1 , . . . , e n+1 ) ∈ RV 2

2

π̂ (e1 , . . . , e2n+1 ) := (e1 ∪ e2 , . . . , e2n+1 −1 ∪ e2n+1 ). ̂ℳ ̂ℳ This clearly makes π̂ : RV 2n+1 → RV2n a definable map, uniformly for all models of Pr. ℳ ̂ is given by a sequence (e(n))n = (e(n) | n ≥ 1) where Note that an element of RV ̂ℳ ̂ (e(n + 1)) = e(n) for all n ≥ 1. In what follows, we e(n) = (e1 (n), . . . , e2n (n)) ∈ RV 2n and π refer to such a sequence as coherent. ̂ ℳ we want to define the inverse limit pseudometric, denoted by ρ ̂ ℳ , by On RV ̂ ℳ ((e(n))n , (f (n))n ) := lim ρ ̂ n (e(n), f (n)) ρ n

̂ ℳ . The fact that the limit in this definition of ρ ̂ℳ for any elements (e(n))n , (f (n))n of RV exists (with a rate of convergence that can be taken to be uniform over all sequences ̂ ℳ and all ℳ 󳀀󳨐 Pr) and that the resulting quotient corresponds to L1 (ℬ, μ; [0, 1]) in RV

modulo the L1 -distance follows from the results in Lemma 5.5 below. For proving such results, it is useful to pull the objects involved back to the probability space (X, ℬ, μ) of which ℳ is the probability algebra. We write π for the corresponding maps from RV2n+1 to RV2n , namely π(E1 , . . . , E2n+1 ) := (E1 ∪ E2 , . . . , E2n+1 −1 ∪ E2n+1 ).

̂ ℳ described above corVia the quotient map from ℬ to M = ℬ̂, the inverse limit RV responds to the inverse limit of the spaces (RV2n (ℬ, μ) | n ≥ 1) equipped with their

182 � A. Berenstein and C. W. Henson L1 -pseudometrics and the connecting maps π: RV2n+1 → RV2n . An element of this inverse limit consists of a sequence (E(n) | n ≥ 1) such that E(n) ∈ RV2n and π(E(n + 1)) = E(n), for all n ≥ 1. As above, we refer to such a sequence as coherent. Fact 5.4. It is evident that the quotient map from ℬ to ℬ̂ induces a map from coherent ̂ sequences (E(n) | n ≥ 1) of measurable partitions of X to coherent sequences (E(n) | ? ? ̂ n ≥ 1) of partitions of 1 in ℳ, where E(n) := (E1 (n), . . . , E2n (n)) for all n ≥ 1. In fact, ̂ ℳ, this map is surjective. That is, suppose (e(n) | n ≥ 1) is a coherent sequence in RV with e(n) = (e1 (n), . . . , e2n (n)) for all n ≥ 1. It is easy to show, working by induction on n, that there exists a coherent sequence (E(n) | n ≥ 1) with E(n) ∈ RV2n for all n such that n ej (n) = E? j (n) for all 1 ≤ j ≤ 2 and all n ≥ 1.

Lemma 5.5. (a) For all m ≥ 1 and E ∈ RV2m , we have fE ≤ fπ(E) ≤ fE + m1 pointwise; therefore ‖fE − fπ(E) ‖1 ≤ m1 . (b) For every coherent sequence (E(n) | n ≥ 1), with E(n) ∈ RV2n for all n ≥ 1, the sequence (fE(n) | n ≥ 1) is monotone decreasing pointwise and has ‖fE(n+1) − fE(n) ‖1 ≤ 2−n for all n ≥ 1. Therefore (fE(n) | n ≥ 1) converges to its pointwise infimum, in L1 -distance, in L1 (ℬ, μ; [0, 1]), and does so with a rate of convergence that can be taken to be uniform over all sequences in RVℳ and all probability spaces (X, ℬ, μ). (c) For every f ∈ RV = L1 (ℬ, μ; [0, 1]) there is a coherent sequence (E(n) | n ≥ 1), with E(n) ∈ RV2n for all n ≥ 1 such that (fE(n) | n ≥ 1) converges to f in L1 -distance. Proof. (a) This follows immediately from the definitions. (b) From the definition of fE(n) we have 0 ≤ fE(n) (x) ≤ 1 for all x ∈ X. Further, (a) implies that χX − fE(n) is pointwise monotone increasing on X and that it converges in L1 . Since χX is integrable, the monotone convergence theorem implies that (χX −fE(n) | n ≥ 1) converges in L1 to its a. e. pointwise supremum, and therefore (fE(n) | n ≥ 1) converges in L1 to its a. e. pointwise infimum. Uniformity of the rate of convergence follows from the uniformity of the estimates in (a). (c) For each n ≥ 1, consider the dyadic intervals I1 , . . . , I2n defined by I1 := [0, 2−n ] and for 1 < j ≤ 2n by Ij := ((j − 1)2−n , j2−n ]. Let E(n) ∈ RV2n be defined by E(n) := (E1 (n), . . . , E2n (n)) where Ej (n) := f −1 (Ij ) for j = 1, . . . , 2n . Note that the sequence (E(n)|n ≥ 1) is coherent. Let 𝒞 ⊆ ℬ be the σ-subalgebra generated by {Ej (n) | n ≥ 1 and 1 ≤ j ≤ 2n }; clearly 𝒞 is the smallest σ-subalgebra of ℬ such that f is 𝒞 -measurable. In particular, 𝔼(f |𝒞 ) = f . The proof of Lemma 2.7 shows that (fE(n) | n ≥ 1) converges to f relative to the L1 -distance. The rate of convergence is as indicated in (b). ̂ℳ ̂ℳ ̂ℳ ̂ : RV Corollary 5.6. The inverse system (RV 2n , ρ2n )(n≥1) equipped with the maps π 2n+1 → ℳ ℳ ̂ ̂ℳ ̂ ̂ ̂ ̂ , d) RV has an inverse limit pseudometric space ( RV , ρ ) for every model ℳ = ( ℬ ,μ n 2

of Pr. Its metric quotient corresponds to L1 (ℬ, μ; [0, 1]) modulo the L1 -distance. Moreover, this quotient is a metric imaginary sort for ℳ, uniformly over all models ℳ of Pr. Proof. The first two sentences follow from Lemma 5.5. For the third sentence, we apply [5, Lemma 1.5].

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̂ ℳ are metric spaces (as shown in the Remark 5.7. Note that although the spaces RV n ̂ ℳ is not a metric. That is, there exist proof of Lemma 5.2), the inverse limit distance ρ ̂ℳ distinct coherent sequences (e(n))n , (f (n))n such that limn ρ n (e(n), f (n)) = 0. Therefore, to obtain the imaginary sort described in Corollary 5.6, it is necessary to form the metric quotient. However, in contrast to the general case of metric imaginary sorts, the metric ̂ ℳ, ρ ̂ ℳ ) is complete no matter which model ℳ of Pr is being conspace quotient of (RV sidered. (In general one needs to take the metric completion of such a quotient for some models.) Indeed, by the Riesz–Fischer theorem, the quotient of L1 (ℬ, μ) modulo the L1 distance is complete, no matter which probability space (X, ℬ, μ) is considered, and the image of L1 (ℬ, μ; [0, 1]) in that quotient is a norm-closed subset. (See [23, Theorem 6.6, pp. 124–125].) We next discuss some definable operations on the metric imaginary sort just described; they correspond to the operations taken to be basic (or proved to be definable) in [3] and thus show that our imaginary sort does indeed provide a model of the theory RV. (See [3, Lemma 2.13] for a discussion of the corresponding operations on models of RV.) Consider any continuous function θ: [0, 1]m → [0, 1]. This function induces an operation on L1 (ℬ, μ; [0, 1]) by composition, which we also denote by θ. Namely, we define θ(f1 , . . . , fm )(x) := θ(f1 (x), . . . , fm (x)) for all x ∈ X, where f1 , . . . , fm : X → [0, 1] are ℬ-measurable. ̂ m to RV. ̂ Let us here We show below that θ induces a definable operation from RV restrict its domain to RVn (ℬ, μ). We consider the case m = 2 to reduce the complexity of notation. For E, F ∈ RVn , we have n n i j θ(fE , fF ) = ∑ ∑ θ( , )χEi ∩Fj , n n i=1 j=1

which induces a definable function of the events of (E, F), uniformly over all probability spaces. ̂ n (ℬ, μ) converge to a definable funcMoreover, the restrictions of θ to the spaces RV tion on their inverse limit. Again restricting attention to the case m = 2, suppose (E(n))n and (F(n))n are two coherent families representing elements of the inverse limit, and suppose f , g ∈ L1 (ℬ, μ; [0, 1]) are their limits, f = limn fE(n) and g = limn fF(n) . Then θ(fE(n) , fF(n) ) converges in L1 -distance to θ(f , g), and does so at a uniform rate which is determined by the modulus of uniform continuity of θ and the exponential rates of convergence of limn fE(n) and limn fF(n) , as given by Lemma 5.5(b). We leave details to the reader. It is a general fact that every automorphism τ of a metric structure has a unique extension to an automorphism of its meq-expansion. We illustrate this in the present context: given ℳ 󳀀󳨐 Pr the probability algebra of (X, ℬ, μ) as above, consider an auto-

184 � A. Berenstein and C. W. Henson n

̂ ℳ is a definable subset of M 2 , we see that τ induces a natumorphism τ of ℳ. Since RV n ̂ ℳ onto itself, by the coordinatewise action, namely e = (e1 , . . . , e2n ) 󳨃→ ral bijection of RV n

τ(e) = (τ(e1 ), . . . , τ(e2n )). Furthermore, if (e(n) | n ≥ 1) is a coherent sequence in the ̂ ℳ , convergent to the element [f ]μ of the quotient of RV ̂ℳ inverse limit of the spaces RV n

modulo the L1 -distance, then (τ(e(n)) | n ≥ 1) is also coherent; the image of [f ]μ under the desired extension of τ is defined to be the (equivalence class of the) limit of (τ(e(n)))n . Another way of looking at this extension process concerns the situation where a ∈ ℬ̂ and C is a closed subalgebra of ℬ̂, and f ∈ L1 (ℬ, μ; [0, 1]) is a C-measurable random variable representing ℙ(a|C). When τ is an automorphism of ℳ, then τ(f ) as defined above represents ℙ(τ(a)|τ(C)), as can be shown by a routine argument based on the details presented in this section. Indeed, this fact is easy to show when C is finite (use Lemma 2.5) and the general case follows using the proof approach for Lemma 2.7 by taking limits. Finally, we use the operations θ defined above to prove a definability relationship between each random variable f ∈ RV(ℬ, μ) and the smallest σ-subalgebra of ℬ with respect to which f is measurable, which we will denote by σ(f ). Note that σ(f ) is generated as a σ-subalgebra by the measurable sets of the form f −1 (r, 1] for r ∈ [0, 1]. This implies that dclmeq (σ(f )) = dclmeq ({f −1 (r, 1] | r ∈ [0, 1]}). Lemma 5.8. Let ℳ 󳀀󳨐 Pr be the probability algebra of the probability space (X, ℬ, μ). For every f ∈ RV(ℬ, μ), we have dclmeq (f ) = dclmeq (σ(f )). Proof. First we note that the proof of Lemma 5.5(c) shows that f ∈ dclmeq (σ(f )). Thus it remains to show f −1 (r, 1] ∈ dclmeq (f ) for every r ∈ [0, 1]. For r = 1, this is 1 trivial. Fix 0 ≤ r < 1 and for each n ≥ 1−r let θn : [0, 1] → [0, 1] be the continuous function defined by: θn (t) = 0 when 0 ≤ t ≤ r, θn (t) = n(t − r) when r < t < r + n1 , and θn (t) = 1 for r + n1 ≤ t ≤ 1. For each t ∈ [0, 1], the sequence (θn (t))n is monotone increasing in n, and its supremum is the function with values 0 for t ≤ r and 1 for t > r. Therefore the sequence (θn (f ))n converges in L1 to the characteristic function of f −1 (r, 1], which shows that f −1 (r, 1] ∈ dclmeq (f ), as desired.

6 Atomless probability spaces By Proposition 4.7(e), the fact that a model of Pr is atomless is expressed by the condition θ(1) = 0, which is equivalent in Pr to the condition 󵄨 󵄨 sup inf󵄨󵄨󵄨μ(x ∩ y) − μ(x ∩ yc )󵄨󵄨󵄨 = 0. x

y

(A)

We denote by APA the set of axioms Pr together with (A). Corollary 6.1. Let ℳ be an Lpr -structure. Then ℳ is a model of APA if and only if ℳ is isomorphic to the probability algebra of an atomless probability space.

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Proof. Immediate from the discussion above. The main purpose of this section is to give a basic model-theoretic analysis of APA. Many of the results correspond to things about atomless probability algebras that were proved by Ben Yaacov in the framework of compact abstract theories [1]. We bring these results into the setting of continuous first-order logic and give proofs expressed in familiar language of measure theory and analysis. In terms of the invariants introduced in Definition 4.15, a model ℳ of Pr is a model of APA if and only if Φℳ consists of the constant sequence with every entry equal to 0. Therefore, using results in Section 4 we get the following basic properties of APA: Corollary 6.2. The theory APA admits quantifier elimination, is separably categorical, and is complete, and the unique separable model of APA is strongly ω-homogeneous. Further, APA is the model companion of Pr. Proof. Let APA∗ denote the theory of all structures ℳ∗ , where ℳ is a model of APA (i. e., ℳ is an atomless model of Pr). By Theorem 4.14, we have that APA∗ admits quantifier elimination. However, in models of APA∗ , all of the extra predicates Pn have the trivial value 0, and thus can be eliminated from any formula. That is, every formula is equivalent in APA∗ to a quantifier-free formula in the language of Pr. It follows that APA admits quantifier elimination. Separable categoricity of APA and strong ω-homogeneity of the separable model follow from Corollary 4.17. Completeness of APA follows from separable categoricity and also from quantifier elimination (because every model of APA contains the trivial probability algebra {0, 1} as a substructure). Since APA admits quantifier elimination, it is model complete; also, it is an extension of Pr. Therefore, to show that APA is the model companion of Pr it remains only to show that every model of Pr has an extension that is a model of APA. Let ℳ 󳀀󳨐 Pr be the measured algebra of the probability space (X, ℬ, μ) and let (Y , 𝒞 , ν) be any atomless probability space. Then the product measure space (X × Y , ℬ ⊗ 𝒞 , μ ⊗ ν) is atomless and it can be seen as an extension of (X, ℬ, μ) by the embedding that takes B ∈ ℬ to B×Y ∈ ℬ ⊗ 𝒞 . Therefore the probability algebra of (X × Y , ℬ ⊗ 𝒞 , μ ⊗ ν) is a model of APA into which ℳ can be embedded. Remark 6.3. Let ℳ be the unique separable model of APA, and let a = (a1 , . . . , an ), b = (b1 , . . . , bn ) ∈ M n . From Corollary 6.2, if tp(a) = tp(b), then there exists an automorphism σ of ℳ such that σ(ai ) = bi for all i = 1, . . . , n. However, this result gives no information about the behavior of σ on the rest of ℳ. In [9] we proved a stronger form of homogeneity for ℳ. In order to state that result, we need to bring in another natural metric on M n , defined by dP (a, b) :=

1 ∑ d(as , bs ). 2 s

(The version of this distance that is defined on measurable partitions of 1 in a probability space was used in Section 5; see Lemma 5.1. It also appears below in Corollary 6.13.) Here

186 � A. Berenstein and C. W. Henson we are using the notation introduced in Subsection 4.12 for the partitions of 1 associated to the tuples a and b, and s ranges over {−1, +1}n . Note that when a, b ∈ M are single elements, dP (a, b) = d(a, b), since μ(ac △ bc ) = μ(a △ b); this is the reason for the 21 factor in the definition of dP . Further, when a and b are partitions of 1, then dP (a, b) agrees with the definition of dP (a, b) given just before Lemma 5.1, since as , bs will both be 0 unless s contains exactly one occurrence of +1; further, if the unique occurrence of +1 is at place i, then as = ai and bs = bi . The homogeneity result from [9] is the following: Lemma (Lemma 5.6 in [9]). Let a, b ∈ M n be tuples with tp(a) = tp(b). Then there is an automorphism σ of ℳ such that σ(ai ) = bi for all i and, for every finite tuple c ∈ M k , we have dP (ac, bσ(c)) = dP (a, b). In particular, for every c ∈ M we have d(σ(c), c) ≤ dP (a, b). The following lemma appears in [1, Section 2.1], in the framework of compact abstract theories. To make our paper more self-contained and because our setting is different, and in order to make clear the elementary tools from analysis from which these facts can be derived, we give complete proofs. Lemma 6.4. Let ℳ = (ℬ, μ, d) 󳀀󳨐 APA. Let C ⊆ M = ℬ and a, b ∈ M n . Recall that ⟨C⟩ is the σ-subalgebra of ℬ generated by C. The following conditions are equivalent: (1) tp(a/C) = tp(b/C); (2) μ(as ∩ c) = μ(bs ∩ c) for all s = (k1 , . . . , kn ) ∈ {−1, +1}n and all c ∈ ⟨C⟩; (3) ℙ(as |⟨C⟩) = ℙ(bs |⟨C⟩) for all s = (k1 , . . . , kn ) ∈ {−1, +1}n . Proof. (1) ⇔ (2) First, we deal with the case C = 0. Only the right-to-left direction needs to be proved. By QE for APA (Corollary 6.2), showing tp(a1 , . . . , an ) = tp(b1 , . . . , bn ) is equivalent to proving μ(t(a1 , . . . , an )) = μ(t(b1 , . . . , bn )) holds for every boolean term t(x1 , . . . , xn ). By the discussion in Subsection 4.12, this holds whenever μ(as ) = μ(bs ) for all s ∈ {−1, +1}. Now consider arbitrary C. From the discussion before Theorem 2.3, we see C ⊆ ⟨C⟩ ⊆ dcl(C), so each type over ⟨C⟩ is determined by its restriction to a type over C. Since ⟨C⟩ is closed under boolean combinations, the equivalence of (1) and (2) follows immediately from the first part of this proof. (2) ⇔ (3) By Corollary 6.1, there is an atomless probability space (X, 𝒜, ν) whose probability algebra is (ℬ, μ, d). Using Lemma 2.2, there exists a σ-subalgebra 𝒟 of 𝒜 for ̂ = ⟨C⟩. According to 2.4, for any a ∈ ℬ = 𝒜 ̂ we have ℙ(a|⟨C⟩) = ℙ(a|𝒟). That is, which 𝒟 (3) is equivalent to the version of (3) in which we have replaced ⟨C⟩ by 𝒟. The equivalence of (2) and this new version of (3) follows from Theorem 2.3; specif̂, the function ℙ(a|𝒟) is 𝒟-measurable and ically, from the fact that for any a = [A]μ ∈ 𝒜

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is determined, up to equality μ-almost everywhere, by the values of ∫E χA dμ = μ(A ∩ E) as E ranges over 𝒟. Remark 6.5. The preceding result takes an especially simple form when a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) are partitions of 1 in ℳ. For example, condition (2) reduces to μ(ai ∩ c) = μ(bi ∩ c) for all i = 1, . . . , n and all c ∈ ⟨C⟩. This means in particular that when a = (a1 , . . . , an ) is a partition of 1, the n-type of a over C is determined by the 1-types tp(ai /C) for i = 1, . . . , n. Lemma 6.4 characterizes equality of types. Next we complete the description of the type space Sn (C). First, we need a definition: Definition 6.6. Let ℳ = (ℬ, μ, d) 󳀀󳨐 APA and C ⊆ M = ℬ. An additive functional on ⟨C⟩ is a finitely additive function λ: ⟨C⟩ → [0, 1]. Note 6.7. Let ℳ = (ℬ, μ, d) 󳀀󳨐 APA. Let C ⊆ M = ℬ and a ∈ M n . In Lemma 6.4(2), each of the functions λs : ⟨C⟩ → [0, 1] defined by c 󳨃→ μ(as ∩ c) is an additive functional on ⟨C⟩ with λs (1) = μ(as ). Furthermore, for every c ∈ ⟨C⟩ we have ∑{λs (c) | s ∈ {−1, +1}n } = μ(c), since (as ∩ c | s ∈ {−1, +1}n ) is a partition of c. Our next result shows that any such family of additive functionals arises from a type in Sn (⟨C⟩). Lemma 6.8. Let ℳ = (ℬ, μ, d) 󳀀󳨐 APA and C ⊆ M = ℬ. Assume ℳ is κ-saturated, where κ > card(⟨C⟩). Let (λs | s ∈ {−1, +1}n ) be a family of additive functionals on ⟨C⟩ such that ∑{λs (c) | s ∈ {−1, +1}n } = μ(c) for all c ∈ ⟨C⟩. Then there exists a = (a1 , . . . , an ) ∈ M n such that for every s ∈ {−1, +1}n and c ∈ ⟨C⟩ we have μ(as ∩ c) = λs (c). Moreover, tp(a/C) is determined by these conditions. Proof. Let x = (x1 , . . . , xn ) be a tuple of distinct variables, and let Σ(x) be the set of all conditions of the form |μ(x s ∩ c) − λs (c)| = 0 as c varies over ⟨C⟩ and s varies over {−1, +1}n . We must show that Σ is satisfiable in ℳ, and by saturation it suffices to show that Σ(x) is finitely satisfiable. So let F be any finite subset of ⟨C⟩ and let f1 , . . . , fk be the atoms of F # . For each i = 1, . . . , n, let (ai,s | s ∈ {−1, +1}n ) be a partition of fi in ℬ such that μ(ai,s ) = λs (fi ) for all s. This is possible because μ(fi ) = ∑{λs (fi ) | s ∈ {−1, +1}n }. Note that the family (ai,s ) is a partition of 1 in ℬ. Finally, for each i set ai = ∪{ai,s | si = +1}, and set a = (a1 , . . . , an ). An easy calculation shows that for all i, s we have as ∩ fi = ai,s and hence μ(as ∩ fi ) = λs (fi ). Additivity implies that μ(as ∩ f ) = λs (f ) for every f ∈ F. This shows that Σ(x) is finitely satisfiable in ℳ and completes the proof (when combined with Lemma 6.4 to provide uniqueness). Remark 6.9. The preceding lemma takes an especially simple form for 1-types over C. Namely, suppose λ is an additive functional on ⟨C⟩ that satisfies λ(c) ≤ μ(c) for all c ∈ ⟨C⟩. Define λ′ on ⟨C⟩ by λ′ (c) = μ(c) − λ(c). Then λ′ is also an additive functional on ⟨C⟩, and the pair λ, λ′ satisfies the assumptions in Lemma 6.8. Therefore, λ determines a 1-type pλ ∈ S1 (C), and every element of S1 (C) can be described in this way. Specifically, a realizes pλ if and only if μ(a ∩ c) = λ(c) for all c ∈ ⟨C⟩ (since this implies μ(ac ∩ c) = λ′ (c)). This

188 � A. Berenstein and C. W. Henson observation is especially useful when considering n-types of partitions of 1, as discussed in Remark 6.5. We also use this description of 1-types in discussing the model theoretic content of Maharam’s lemma in Section 7 (Lemma 7.16). An equivalent approach to 1-types over C in terms of ⟨C⟩-measurable, [0, 1]-valued functions (i. e., random variables), corresponding to clause (3) in Lemma 6.4, is the following: let f be any ⟨C⟩-measurable, [0, 1]-valued function. For ℳ 󳀀󳨐 APA and C ⊆ M, the condition ℙ(a|⟨C⟩) = f on a ∈ M is type-definable over C; indeed, it precisely determines tp(a/C). To see this, consider λf defined for c ∈ ⟨C⟩ by λf (c) := ∫c f dμ. Then λf is an additive functional on ⟨C⟩ that satisfies λ(c) ≤ μ(c) for all c ∈ ⟨C⟩. Therefore, as discussed in the preceding paragraph, λf exactly determines a 1-type in S1 (C). Moreover, the condition μ(a ∩ c) = λf (c) for all c ∈ ⟨C⟩ is equivalent to ℙ(a|⟨C⟩) = f . Lemma 6.10. Let ℳ = (ℬ, μ, d) 󳀀󳨐 APA and C ⊆ M. Then dcl(C) = acl(C) = ⟨C⟩. Proof. Recall that ⟨C⟩ is the σ-subalgebra of ℬ generated by C. As noted at the beginning of the previous proof, C ⊆ ⟨C⟩ ⊆ dcl(C). Therefore, to complete the proof it suffices to prove acl(C) ⊆ ⟨C⟩. Now let a ∈ M \ ⟨C⟩. To show a ∈ ̸ acl(C), by [7, Exercise 10.8] it suffices to prove that for some 𝒩 ⪰ ℳ, there is a realization of tp(a/C) in 𝒩 that is not in ℳ. Let ℳ be the probability algebra of the probability space (X, 𝒜, ν). Consider the standard probability space ([0, 1], ℒ, m) of Lebesgue measure and form the product space (X × [0, 1], 𝒜 ⊗ ℒ, ν ⊗ m); let 𝒩 be the probability algebra of this product space. There is a canonical embedding J: 𝒜 → 𝒜 ⊗ ℒ defined by J(A) = A × [0, 1] for A ∈ 𝒜; this map gives rise to an embedding ̂J of ℳ into 𝒩 . Since APA admits quantifier elimination, ̂J is an elementary embedding. As in the previous proof, there is a σ-subalgebra 𝒟 of 𝒜 for which ̂ = ⟨C⟩. Since ℙ(a|𝒟) is 𝒟-measurable, the set A′ = {(x, s) ∈ X × [0, 1] | s ≤ ℙ(a|𝒟)(x)} is 𝒟 ? ? 𝒟 ⊗ ℒ-measurable. We let a′ = [A′ ]ν⊗m ∈ 𝒟 ⊗ℒ⊆𝒜 ⊗ ℒ = N. We complete the proof by showing that a′ is not in the image of ℳ under ̂J and that a′ is a realization of tp(a/C) in 𝒩 . For the first of these statements, we note that 0 < ℙ(a|𝒟) < 1 holds on a set of positive measure. Otherwise ℙ(a|𝒟) = χB for some B ∈ 𝒟; this would imply a = [B]ν ∈ ⟨C⟩, which would contradict our assumptions. It follows that A′ is not of the form A×[0, 1] where A ∈ 𝒜, and thus a′ is not in the image of ℳ under ̂J. ̂′ = ̂J(⟨C⟩) = Finally, let 𝒟′ = {B × [0, 1] | B ∈ 𝒟} = J(𝒟), so 𝒟′ is a σ-algebra and 𝒟 ′ ′ ′ ⟨̂J(C)⟩. Fubini’s theorem shows that ℙ(a |𝒟 ) = ℙ(̂J(a)|𝒟 ), which implies by Lemma 6.4 that tp𝒩 (a′ /̂J(C)) = tpℳ (a/C). (Here we mean, of course, that the parameters in ̂J(C) are identified with those in C via the bijection ̂J.) In several results in the rest of this section it is convenient to work in a κ-universal domain for APA, where κ is uncountable. For the rest of the section, we denote such a model of APA as 𝒰 . Recall that a subset C of U is called small if card(C) < κ. In this situation, every type in Sn (C) is realized in 𝒰 . Furthermore, 𝒰 is strongly κ-homogeneous; i. e., every elementary map between small subsets of U extends to an automorphism

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of 𝒰 . (In applications, 𝒰 and the size of κ may need to be changed in order to insure that specific parameter sets are small.) Recall that the metric d on 𝒰 yields an induced metric on each space of types (see [7, Section 8]), as follows: when C ⊆ U is small and p, q are n-types over C, the distance between p and q is defined by d(p, q) = inf{max d(ai , bi ) : (a1 , . . . , an ) 󳀀󳨐 p, (b1 , . . . , bn ) 󳀀󳨐 q}. 1≤i≤n

The next result provides an explicit formula for the induced metric on types of partitions of 1 in atomless probability algebras. (When the parameter set C is empty, this formula occurs as (6.2) in the proof of [25, Lemma 6.3].) Theorem 6.11. Let C ⊆ U be small and let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) be partitions of 1 in 𝒰 . Then 󵄩 󵄩 d(tp(a/C), tp(b/C)) = max󵄩󵄩󵄩ℙ(ai |⟨C⟩) − ℙ(bi |⟨C⟩)󵄩󵄩󵄩1 1≤i≤n

where ‖ ⋅ ‖1 is the L1 -norm. Moreover, there exists b′ = (b′1 , . . . , b′n ), a partition of 1 in U, such that tp(b′ /C) = tp(b/C) and for all i = 1, . . . , n 󵄩 󵄩 d(ai , b′i ) = 󵄩󵄩󵄩ℙ(ai |⟨C⟩) − ℙ(bi |⟨C⟩)󵄩󵄩󵄩1 . Proof. Replacing C by C # (which is still small) we may assume throughout this proof that C is a boolean subalgebra of 𝒰 . Since C ⊆ C # ⊆ dcl(C), this does not change the types being considered nor the distance between them. Furthermore, it is obvious that ⟨C # ⟩ = ⟨C⟩, so the right-hand side of the equality to be proved is also not changed by this move. We begin the proof by noting that 󵄩󵄩 󵄩 󵄩󵄩ℙ(u|⟨C⟩) − ℙ(v|⟨C⟩)󵄩󵄩󵄩1 ≤ μ(u △ v) for any u, v ∈ U. Indeed, linearity of the conditional expectation yields 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ℙ(u|⟨C⟩) − ℙ(v|⟨C⟩)󵄩󵄩󵄩1 = 󵄩󵄩󵄩ℙ(u\v|⟨C⟩) − ℙ(v\u|⟨C⟩)󵄩󵄩󵄩1 ≤ μ(u △ v) where the last step uses the triangle inequality for the L1 -norm and the fact that ‖ℙ(w|C)‖1 = μ(w) for any w ∈ U. By Lemma 6.4, ‖ℙ(u|⟨C⟩) − ℙ(v|⟨C⟩)‖1 only depends on tp(u/C) and tp(v/C). Fixing i ∈ {1, . . . , n} and letting u, v range over realizations of tp(ai /C), tp(bi /C) respectively, and taking the infimums, we obtain 󵄩󵄩 󵄩 󵄩󵄩ℙ(ai |⟨C⟩) − ℙ(bi |⟨C⟩)󵄩󵄩󵄩1 ≤ d(tp(ai /C), tp(bi /C)). Taking the maximum over i yields

190 � A. Berenstein and C. W. Henson 󵄩 󵄩 max󵄩󵄩󵄩ℙ(ai |⟨C⟩) − ℙ(bi |⟨C⟩)󵄩󵄩󵄩1 ≤ d(tp(a/C), tp(b/C)). 1≤i≤n

Therefore it remains to show 󵄩 󵄩 d(tp(a/C), tp(b/C)) ≤ max󵄩󵄩󵄩ℙ(ai |⟨C⟩) − ℙ(bi |⟨C⟩)󵄩󵄩󵄩1 , 1≤i≤n

given that a, b ∈ U n are partitions of 1. We do this in the remainder of the proof. We first prove this inequality when C = 0, noting that the right-hand side of the latter inequality is equal to max1≤i≤n |μ(ai ) − μ(bi )| in this situation. Let I = {i | μ(ai ) ≥ μ(bi )} and J = {i | μ(ai ) < μ(bi )}. Note that I ≠ 0; also, we may assume J ≠ 0, since otherwise tp(a) = tp(b) and so the inequality to be proved is trivial. Since 𝒰 is atomless, we may choose b′i ≤ ai in U satisfying μ(b′i ) = μ(bi ), for each i ∈ I. For each such i, let ui = ai \ b′i , and set u = ∪{ui | i ∈ I}. Note that μ(u) = ∑{μ(ai ) − μ(bi ) | i ∈ I}. Because (a1 , . . . , an ) and (b1 , . . . , bn ) are partitions of 1 in 𝒰 , it follows that μ(u) = ∑{μ(bj ) − μ(aj ) | j ∈ J}. Hence we may partition u into {uj | j ∈ J} ⊆ U such that μ(uj ) = μ(bj ) − μ(aj ) for all j ∈ J. Finally, for j ∈ J we set b′j = aj ∪ uj and note that (b′1 , . . . , b′n ) is a measurable partition of 1 in 𝒰 satisfying μ(b′j ) = μ(bj ) for all j ∈ J; in other words, (b′1 , . . . , b′n ) realizes the same type as (b1 , . . . , bn ). Moreover, for all j ∈ J we have 󵄨 󵄨 󵄨 󵄨 μ(aj △ b′j ) = μ(uj ) = 󵄨󵄨󵄨μ(aj ) − μ(b′j )󵄨󵄨󵄨 = 󵄨󵄨󵄨μ(aj ) − μ(bj )󵄨󵄨󵄨, which justifies the desired inequality. Now assume that C is a finite boolean subalgebra of 𝒰 and let the atoms of C be c1 , . . . , cp . For each i ≤ n and j ≤ p, let aij = ai ∩ cj and bij = bi ∩ cj . We argue as in the previous paragraph within each cj . This yields b′ij for i ≤ n, j ≤ p with the following properties: (a) μ(b′ij ) = μ(bij ) for all i, j; (b) for each j ≤ p, the tuple (b′1j , . . . , b′nj ) is a partition of cj ; and (c) μ(aij △b′ij ) = |μ(aij )−μ(bij )| for all i, j. For each i ≤ n, let b′i = ∪j≤m b′ij . Then tp(b′1 , . . . , b′n /C) = tp(b1 , . . . , bn /C) and 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 μ(ai △ b′i ) = ∑ 󵄨󵄨󵄨μ(aij ) − μ(bij )󵄨󵄨󵄨 = ∑ 󵄨󵄨󵄨μ(ai ∩ cj ) − μ(bi ∩ cj )󵄨󵄨󵄨 = 󵄩󵄩󵄩ℙ(ai |⟨C⟩) − ℙ(bi |⟨C⟩)󵄩󵄩󵄩1 . j≤m

j≤m

Finally, consider a general algebra C. For each k ≥ 1, use Lemma 2.7 applied to C and a1 , . . . , an , b1 , . . . , bn to obtain a finite subalgebra Ck ⊆ C such that for all closed subalgebras D ⊆ C that contain Ck we have ‖ℙ(u|C) − ℙ(u|D)‖1 ≤ 1/k for all u = ai and u = bi with 1 ≤ i ≤ n. We may assume Ck ⊆ Ck+1 for all k ≥ 1. Further, we may use properties of type spaces to enlarge each Ck to ensure additionally for k ≥ 1 that 󵄨󵄨 󵄨 󵄨󵄨d(tp(a/C), tp(b/C)) − d(tp(a/Ck ), tp(b/Ck ))󵄨󵄨󵄨 ≤ 1/k. Indeed, note that if E ⊆ D ⊆ C, then

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d(tp(a/E), tp(b/E)) ≤ d(tp(a/D), tp(b/D)) ≤ d(tp(a/C), tp(b/C)). Moreover, d(tp(a/C), tp(b/C)) is the supremum of d(tp(a/D), tp(b/D)) as D varies over finite subsets of C. (Otherwise there would exist r < d(tp(a/C), tp(b/C)) such that d(tp(a/D), tp(b/D)) ≤ r for all finite D ⊆ C. Thus the following set of conditions would be finitely satisfiable in 𝒰 : Σ := {φ(x) = 0 | φ(x) ∈ tp(a/C)} ∪ {ψ(y) = 0 | ψ(x) ∈ tp(b/C)} ∪ {max d(xi , yi ) ≤ r}. i

Since C is a small set, we may choose x = a′ and y = b′ that realize Σ in 𝒰 . But then we would have tp(a′ /C) = tp(a/C), tp(b′ /C) = tp(b/C), and d(a′ , b′ ) ≤ r < d(tp(a/C), tp(b/C)), which is impossible.) Putting these two arguments together, we have an increasing family (Ck | k ≥ 1) of finite subalgebras of C such that for all k ≥ 1, 󵄩󵄩 󵄩 󵄩󵄩ℙ(u|C) − ℙ(u|Ck )󵄩󵄩󵄩1 ≤ 1/k for all u = ai and u = bi with 1 ≤ i ≤ n, and 󵄨󵄨 󵄨 󵄨󵄨d(tp(a/C), tp(b/C)) − d(tp(a/Ck ), tp(b/Ck ))󵄨󵄨󵄨 ≤ 1/k. From what is proved earlier for n-types over finite algebras, for all k ≥ 1 we have 󵄩 󵄩 d(tp(a/Ck ), tp(b/Ck )) = max󵄩󵄩󵄩ℙ(ai |⟨Ck ⟩) − ℙ(bi |⟨Ck ⟩)󵄩󵄩󵄩1 . 1≤i≤n

Taking limits as k → ∞ yields 󵄩 󵄩 d(tp(a/C), tp(b/C)) = max󵄩󵄩󵄩ℙ(ai |⟨C⟩) − ℙ(bi |⟨C⟩)󵄩󵄩󵄩1 , 1≤i≤n

completing the proof. Corollary 6.12. Let C ⊆ U be small and let a, b be elements of U. Then 󵄩 󵄩 d(tp(a/C), tp(b/C)) = 󵄩󵄩󵄩ℙ(a|⟨C⟩) − ℙ(b|⟨C⟩)󵄩󵄩󵄩1 . Proof. Apply the preceding lemma to (a, ac ) and (b, bc ), and use the fact that each side of the equation to be proved is unchanged if we replace a, b by ac , bc The definition of the metric dP on U n that is given in Remark 6.3 says for a, b ∈ U n dP (a, b) :=

1 ∑ d(as , bs ), 2 s

where s ranges over {−1, +1}. Following the established pattern, we can define dP on Sn (C) by

192 � A. Berenstein and C. W. Henson dP (p, q) := inf{dP (a, b) | a 󳀀󳨐 p and b 󳀀󳨐 q}. From Theorem 6.11 we get immediately Corollary 6.13. Let C ⊆ U be small and let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) be in U n . Then dP (tp(a/C), tp(b/C)) =

1 󵄩 󵄩 inf{∑󵄩󵄩󵄩ℙ(as |⟨C⟩) − ℙ(bs |⟨C⟩)󵄩󵄩󵄩1 | a 󳀀󳨐 p and b 󳀀󳨐 q}. 2 s

where ‖ ⋅ ‖1 is the L1 -norm. Proof. This follows from the “Moreover” statement in Theorem 6.11. Moving beyond types of partitions of 1, we now discuss the induced metric on the full type space Sn (C) for APA. For r ≥ 1, let Sr∗ (C) denote the space of r-types for APA that are realized by partitions (a1 , . . . , ar ) of 1 in the κ-universal domain 𝒰 for APA, where C is a small subset of U. Theorem 6.11 gives an explicit formula for the induced metric on Sr∗ (C). Since Sr∗ (C) is a proper, metrically closed subset of the full space of r-types Sr (C), this does not immediately characterize the metric on all of Sr (C). However, by looking at types for APA in the right way, and taking r = 2n , we can use this lemma to characterize the induced metric on Sn (C) up to equivalence of metrics, which is enough for most purposes. To accomplish this, consider the map Πn : Sn (C) → S2∗n (C) on types that is induced by mapping the type of an arbitrary n-tuple (a1 , . . . , an ) to the type of its associated partition (as | s ∈ {−1, +1}) (as discussed in Subsection 4.12). Since APA admits quantifier elimination, Πn is a bijection from Sn (C) onto S2∗n (C). The discussion in Remark 4.13 shows that Πn is also a homeomorphism for the (logic) topologies. In what follows, we often drop the subscript n when doing so will not cause confusion. Lemma 6.14. Let C ⊆ U be small and let p, q ∈ Sn (C). Then (2−n+1 ) ⋅ dn (p, q) ≤ d2n (Πn (p), Πn (q)) ≤ n ⋅ dn (p, q), where dn , d2n denote the induced metrics on the type spaces Sn (C), S2n (C), respectively (usually denoted simply by d, but here given a subscript to indicate the type space on which the metric is defined). Proof. This uses an easy calculation based on the description of the bijection between n-tuples (a1 , . . . , an ) and partitions (as | s ∈ {−1, +1}) that is given in Subsection 4.12. Thus Πn is a bi-Lipschitz homeomorphism from Sn (C) onto S2∗n (C) with respect to the two induced metrics. Since an explicit formula for the induced metric on S2∗n (C) is given by Theorem 6.11, this gives us considerable information about the induced metric topology on all of Sn (C). Note that this observation strengthens Lemma 6.4.

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We next prove that the theory APA is stable; indeed, we simply count types, and show that APA is ω-stable: Proposition 6.15 ([1, Proposition 4.4]). The theory APA is ω-stable. Proof. We may take 𝒰 to be the probability algebra of an atomless probability space (X, 𝒜, ν). Let C ⊆ U be countable. For each a ∈ C chose a set Aa ∈ 𝒜 satisfying a = [Aa ]ν and let 𝒞 be the boolean subalgebra of 𝒜 generated by {Aa | a ∈ C}. Then 𝒞 is countable and 𝒞̂ = C # . Let 𝒮 (C) be the set of 𝒞 -measurable simple functions with coefficients in ℚ ∩ [0, 1], and let ℱ = {tp(a/C) | ℙ(a|𝒞 ) ∈ 𝒮 (C)}.

Then ℱ is a countable set of types. By Lemmas 2.7 and 6.12, ℱ is a metrically dense subset of the space of 1-types over C.

7 Maharam’s theorem Maharam’s theorem [21] is a structure theorem for probability algebras. It says that a model ℳ = (ℬ, μ, d) of Pr is determined up to isomorphism by the information Φℳ given in Section 4 about the atomic part of ℬ together with a countable set 𝒦ℳ of infinite cardinal numbers and a function Ψ: 𝒦ℳ → (0, 1] whose sum equals the μ-measure of the atomless part of ℬ. Note that 1 is atomic in ℬ if and only if 𝒦ℳ = 0. In general, we know that the atomic part of ℳ is determined up to isomorphism by Φℳ , as is the measure of the atomless part of ℬ. (See Corollary 4.19.) Therefore we may focus our attention on the atomless part of ℳ. When it is nonzero, it can be considered as a model of APA by rescaling the measure and the metric. That is, to prove Maharam’s theorem, we may focus on models of APA. In this section we give a full discussion of Maharam’s theorem for models of APA, to make clear the ways in which its proof resonates with ideas from model theory. Definition 7.1. Let ℳ = (ℬ, μ, d) 󳀀󳨐 APA and 0 ≠ b ∈ ℬ. Define ℬ ↾ b to be the ideal of all a ≤ b in ℬ. Note 7.2. Since we require b ≠ 0 in the preceding definition, we may regard ℬ ↾ b as a boolean algebra; the interpretations of ∩ and ∪, as well as of 0, are inherited from ℬ, while 1 is interpreted as b and the complement operation is taken to be a 󳨃→ ac ∩ b. Note that with this understanding of the structure of ℬ ↾ b, the map a 󳨃→ a ∩ b is a boolean morphism from ℬ onto ℬ ↾ b. We equip ℬ ↾ b with the measure and distance obtained from ℳ by restriction to ℬ ↾ b; for convenience we continue to denote these restrictions by μ and d.

194 � A. Berenstein and C. W. Henson It is clear that (ℬ ↾ b, μ, d) is a measured algebra, and that it becomes a model of APA if we rescale μ and d appropriately (namely, multiply by 1/μ(b)). We systematically use this point of view below. Notation 7.3. Unless otherwise specified, in the rest of this section we take ℳ = (ℬ, μ, d) to be a model of APA. When we refer to the density of a subset of ℬ, we mean the metric density. A key quantity for the arguments behind Maharam’s theorem is the density of ℬ ↾ b; for brevity we also refer to this as the density of b. When b is atomless, this density is an infinite cardinal number. Definition 7.4. For b ∈ ℬ, we say ℬ ↾ b is homogeneous and (alternatively) b is homogeneous if b ≠ 0 and ℬ ↾ a has the same density as ℬ ↾ b for every 0 ≠ a ≤ b. We are now in position to define the Maharam invariants (𝒦ℳ , Ψℳ ) for a model ℳ = (ℬ, μ, d) of APA. Definition 7.5. Define 𝒦ℳ to be the set of all infinite cardinal numbers κ for which there exists b ∈ ℬ such that b is homogeneous and the density of ℬ ↾ b is κ. For each κ ∈ 𝒦ℳ define Ψℳ (κ) := sup{μ(b) | b is homogeneous and the density of ℬ ↾ b = κ}. We call b ∈ ℬ maximal homogeneous if b is homogeneous and μ(b) = Ψℳ (κ), where κ = density of ℬ ↾ b. We call ℬ homogeneous if 1 is homogeneous in ℬ. We say ℳ realizes its Maharam invariants if there exists a family (bκ | κ ∈ 𝒦ℳ ) of pairwise disjoint maximal homogeneous elements of ℬ such that bκ has density κ for every κ ∈ 𝒦ℳ and ∑{μ(bκ ) | κ ∈ 𝒦ℳ } exists and equals 1. We show below that every model ℳ of APA realizes its Maharam invariants. In particular, this means that 𝒦ℳ is nonempty and countable. Note 7.6. If ℳ 󳀀󳨐 APA realizes its Maharam invariants, then the density of ℳ is the supremum of 𝒦ℳ (taken in the cardinal numbers). Example 7.7. Obviously the unique separable model ℳ is homogeneous of density ℵ0 . Let (X, ℬ, μ) be any countably generated, atomless probability space, and let κ be any uncountable cardinal number. Let 𝒜 be the probability algebra of the product space X κ with the product probability measure obtained by taking μ as the measure on each factor. Then 𝒜 is homogeneous and has density κ. Proof. Let 𝒮 be a countable dense subset of ℬ. For each α < κ let πα be the coordinate projection from X κ onto X. The σ-algebra of product-measurable subsets of X κ is generated by the sets of the form πα−1 (Q), where α < κ and Q ∈ 𝒮 . Therefore 𝒜 has density at most κ. Also, if Q ∈ ℬ has μ(Q) = r ∈ (0, 1) and α, β are distinct, then d(πα−1 (Q), πβ−1 (Q)) = 2r(1 − r) > 0, so 𝒜 has density at least κ.

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If V is any product-measurable subset of X κ , then V only depends on countably many ordinals α < κ, in the sense that there is a countable set S of such ordinals such that for any u, v: κ → X, if u ∈ V and u(α) = v(α) for all α ∈ S, then also v ∈ V . When V , S satisfy this condition, we say V depends only on the coordinates in S. (Note that the collection of product measurable V ⊆ X κ that only depend on countably many α < κ is a σ-algebra, and it contains all sets of the form πα−1 (Q), where α < κ and Q ∈ 𝒮 .) A variant of the argument in the first paragraph shows that the restriction of 𝒜 to the event determined by any product-measurable set V also has density equal to κ. (Just work on the coordinates in κ \ S, where S is countable and V depends only on the coordinates in S.) Therefore 𝒜 is homogeneous of density κ. Remark 7.8. It is now clear that for every nonempty countable set 𝒦 of infinite cardinal numbers and every function Ψ: 𝒦 → (0, 1] whose sum equals 1, we can construct an atomless probability space (X, ℬ, μ) whose probability algebra ℳ realizes its Maharam invariants and such that 𝒦ℳ = 𝒦 and Ψℳ = Ψ. For each κ ∈ 𝒦, let (Xκ , ℬκ , μκ ) be a probability space whose probability algebra is homogeneous of density κ; take the sets Xκ to be pairwise disjoint. For each κ ∈ 𝒦, let μ′κ be Ψ(κ)μ. Then take X to be the union of (Xκ | κ ∈ 𝒦) and let ℬ be the σ-algebra of subsets of X generated by ⋃{ℬκ | κ ∈ 𝒦}. Note that each Q ∈ ℬ is equal to ⋃{Q ∩ Xκ | κ ∈ 𝒦}, and set μ(Q) := ∑{μ′κ (Q ∩ Xκ ) | κ ∈ 𝒦}. Then it is clear that (X, ℬ, μ) is an atomless probability space and that its probability algebra ℳ satisfies (𝒦ℳ , Ψℳ ) = (𝒦, ℳ). Next we state a lemma giving properties of homogeneous elements. Lemma 7.9. Let ℳ = (ℬ, μ, d) 󳀀󳨐 APA. (a) If b1 , b2 are homogeneous elements of ℬ, and if b1 , b2 have different densities, then b1 ∩ b2 = 0. (b) If bn is a homogeneous element of ℬ for n ≥ 1, and the density of bn is κ for all n, then b = ⋃{bn | n ≥ 1} is also homogeneous in ℬ and b has density κ. (c) If there exists a homogeneous element b ∈ ℬ of density κ, then there exists a maximal homogeneous element b′ such that b′ also has density κ. (d) If b′ is a maximal homogeneous element of density κ, then every homogeneous element b of density κ satisfies b ≤ b′ . Proof. Left as exercises for the reader. Proposition 7.10. Every model ℳ = (ℬ, μ, d) of APA realizes its system of Maharam invariants. Proof. We refer to the items in Lemma 7.9 by their letters. Let 𝒦ℳ be defined as in Definition 7.5. Note that 𝒦ℳ is nonempty, since taking 0 ≠ b ∈ ℬ such that b has the least possible density implies that ℬ ↾ b is homogeneous. For each κ ∈ 𝒦ℳ , let bκ be a maximal homogeneous element of ℬ that has density κ, which exists by (c). By (a), the

196 � A. Berenstein and C. W. Henson elements (bκ | κ ∈ 𝒦ℳ ) are pairwise disjoint in ℬ and by (d) we have μ(bκ ) = Ψℳ (κ) for every κ. Note that this implies that 𝒦ℳ is countable. It remains to show that ∑{μ(bκ ) | κ ∈ 𝒦ℳ } = 1. If not, let b be the complement in ℬ of ⋃{bκ | κ ∈ 𝒦ℳ }, so b > 0. Let b′ be a nonzero element of ℬ ↾ b of least possible density, so ℬ ↾ b′ is homogeneous. If κ is the density of b′ , then κ ∈ 𝒦ℳ by definition, and we have that b′ ∩ bκ = 0. This contradicts the maximality of bκ . Note 7.11. It remains to show that a model ℳ of APA is determined up to isomorphism by its Maharam invariants. Evidently, it suffices to prove the special case that when ℳ, 𝒩 are homogeneous models and have the same density, then ℳ ≅ 𝒩 . Indeed, if ℳ is any model of APA and the family (bκ | κ ∈ 𝒦ℳ ) witnesses that ℳ realizes its Maharam invariants (as in the proof of Proposition 7.10), then the isomorphism type of each ℬ ↾ bκ (as a measured algebra) would be determined by κ and μ(bκ ) = Ψℳ (κ). The isomorphism type of ℳ is easily reconstructed from this data, since 𝒦 is countable, the elements (bκ | κ ∈ 𝒦ℳ ) are pairwise disjoint, and ∑{μ(bκ ) | κ ∈ 𝒦ℳ } = 1. A similar discussion applies to arbitrary models ℳ = (ℬ, μ, d) of Pr. In this case the necessary decomposition of ℬ consists of a family (bi | i ∈ I) of elements of ℬ and a family (κi | i ∈ I) of cardinal numbers satisfying the following conditions: (i) the elements bi are pairwise disjoint and nonzero; (ii) ∑{μ(bi ) | i ∈ I} = 1; (iii) if κi is finite, it equals 1 and bi is an atom in ℬ; (iv) if κi is infinite, then bi is a maximal homogeneous component of the atomless part of ℬ of density κi ; and (v) if κi , κj are infinite with i ≠ j, they are distinct. As we show now, the additional information needed to determine ℳ up to isomorphism is the family (μ(bi ) | i ∈ I) of real numbers, which all come from (0, 1] and whose sum is 1. What remains to be proved is that every homogeneous model of APA is determined up to isomorphism by its density. It is in this proof where model-theoretic ideas come into play, as we explain next. Indeed, the homogeneous models of APA are the same as the saturated models (i. e., the models that have density κ and are κ-saturated, for some κ). Making this connection precise requires the introduction of the following notion. Definition 7.12. Let (ℬ, μ, d) be a probability algebra and let 𝒜 be a σ-subalgebra of ℬ. A nonzero element b ∈ ℬ is called an atom relative to 𝒜 if for all b′ ≤ b in ℬ there is a ∈ 𝒜 such that b′ = a ∩ b. We say that ℬ is atomless over 𝒜 if no nonzero element b ∈ ℬ is an atom relative to 𝒜. Remark 7.13. Consider the setting of Definition 7.12 and let b be a nonzero element of ℬ. Then b is an atom relative to 𝒜 if and only if ℙ(b′ | 𝒜) is equal to a restriction of ℙ(b | 𝒜), for every b′ ≤ b in ℬ. Here we are considering each ℙ(⋅ | 𝒜) as a μ-a. e. equivalence class of 𝒜-measurable [0, 1]-valued functions, and “restriction” means to multiply by the characteristic function of an 𝒜-measurable set. (See Subsection 2.4.) Note that if ℬ is a probability algebra and 0 ≠ b ∈ ℬ, then b is an atom in ℬ if and only if b is an atom relative to the trivial subalgebra {0, 1}.

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Lemma 7.14. Suppose ℳ = (ℬ, μ, d) is a homogeneous model of APA and its density is κ, and 𝒩 = (𝒜, μ, d) is a substructure of ℳ of density < κ. Then ℬ is atomless over 𝒜. Proof. For each nonzero b ∈ ℬ, the density of ℬ ↾ b is κ, whereas the density of {a ∩ b | a ∈ 𝒜} is at most the density of 𝒜. Lemma 7.15. Suppose ℳ = (ℬ, μ, d) is a model of APA, and 𝒩 = (𝒜, μ, d) is a substructure of ℳ. If ℬ is atomless over 𝒜, then ℬ is atomless over ⟨𝒜 ∪ F⟩ for every finite set F ⊆ ℬ. Proof. Using induction, it suffices to consider the case F = {b}. We prove the contrapositive. Suppose there is a nonzero b′ in ℬ that is an atom relative to ⟨𝒜 ∪ {b}⟩. We show that b′ ∩ b is either 0 or an atom relative to 𝒜, and the same for b′ ∩ bc . Since b′ ≠ 0, at least one of them must be an atom relative to 𝒜. Consider b′′ ≤ b′ ∩ b (the case of b′ ∩ bc is similar). Note that b′′ ≤ b, so b′′ ∩ bc = 0. Since b′ is an atom relative to ⟨𝒜 ∪ {b}⟩, there exists x ∈ ⟨𝒜 ∪ {b}⟩ with b′′ = x ∩ b′ . There exist a1 , a2 ∈ 𝒜 such that x = (a1 ∩ b) ∪ (a2 ∩ bc ), and therefore b′′ = x ∩ b′ = (a1 ∩ b ∩ b′ ) ∪ (a2 ∩ bc ∩ b′ ) = a1 ∩ (b′ ∩ b). The last equality is because b′′ and a2 ∩ bc ∩ b′ are disjoint, so a2 ∩ bc ∩ b′ = 0. It follows that b′ ∩ b is either 0 or an atom relative to 𝒜. Lemma 7.16 (Maharam’s lemma). Let ℳ = (ℬ, μ, d) be a model of APA and let 𝒩 = (𝒜, μ, d) be a substructure of ℳ. If ℬ is atomless over 𝒜, then ℳ realizes every n-type over 𝒜. Proof. Using Lemma 7.15 and the fact that it allows us to realize n-types over 𝒜 “coordinate by coordinate,” it suffices to prove the result for 1-types. Remark 6.9 implies that proving ℳ realizes every 1-type over 𝒜 is equivalent to proving the following statement: Suppose λ: 𝒜 → [0, 1] is an additive functional over 𝒜 such that λ(a) ≤ μ(a) holds for every a ∈ 𝒜. Then there exists b ∈ ℬ such that λ(a) = μ(a ∩ b) for every a ∈ 𝒜. A proof of exactly this statement is given as Lemma 3.2 in Fremlin’s chapter [14] on measure algebras, and also as Lemma 331B in volume 3 [15] of his multivolume treatise on measure theory. Corollary 7.17. Every homogeneous model of APA is determined up to isomorphism by its density. Proof. Suppose ℳ = (ℬ, μ, d) 󳀀󳨐 APA has density κ and is homogeneous. Since ℬ is homogeneous, it is atomless over ⟨C⟩ for every C ⊆ ℬ with card(C) < κ. By Lemma 7.16, ℳ realizes every n-type over C for every such C. That is, ℳ is a κ-saturated model of APA and it has density κ. Using the standard back-and-forth argument from model theory, any two such models are isomorphic.

198 � A. Berenstein and C. W. Henson Finally, we have Maharam’s theorem, which characterizes the structure of all probability algebras up to isomorphism. Theorem 7.18. Every model ℳ of Pr is determined up to isomorphism by its invariants Φℳ for the atomic part and its Maharam invariants (𝒦ℳ , Ψℳ ) for the atomless part. Proof. The definition of the Maharam invariants for general probability algebras is in the first paragraph of this section; the definition of Φℳ is in Section 4. The proof of the theorem is given above, with the key result being Corollary 7.17, which handles the maximal homogeneous components of the atomless part of ℳ. Note 7.11 indicates how the structure of ℳ is determined by what these invariants say about its component parts. Note that for each infinite cardinal κ, we identified the κ-saturated model of APA of density character κ as the Maharam homogenous model of density κ. More information on κ-saturated and κ-homogeneous models of APA can be found in [26]. The following characterization of the “atomless over” property is often useful:

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Proposition 7.19. Let ℳ = (ℬ, μ, d) be a model of APA and let 𝒜 be a σ-subalgebra of ℬ. The following are equivalent: (a) ℬ is atomless over 𝒜. (b) For an infinite set of positive integers n, there is in ℬ a partition of 1, say u = (u1 , . . . , un ), such that μ(a ∩ ui ) = n1 μ(a) for all a ∈ 𝒜 and i = 1, . . . , n. (In other words, each ui satisfies ui 𝒜 and has measure n1 .) (c) There is an atomless σ-subalgebra 𝒞 of ℬ such that 𝒜 𝒞 .

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Proof. (a) ⇒ (c) We build inductively a sequence {𝒞n }n≥1 of finite subalgebras of ℬ such that for all n ≥ 2 we have 𝒞n (𝒜 ∪ (⋃i μ(ai )μ(bj ) = μ(Ii )μ(Jj ). Therefore we have dependent realizations of the types tp(a), tp(b). This shows that no pair of nonalgebraic types over 0 is almost orthogonal. Once we allow parameters, elements may be supported over disjoint sets and we obtain more freedom: Proposition 8.13. For any e ∈ U with e ∉ {0, 1}, there are nonalgebraic p, q ∈ S1 ({e}) which are orthogonal. Proof. Let e ∈ U have 0 < μ(e) < 1 and let E = {0, 1, e, ec }. Let a, b ∈ U satisfy a ≤ e, μ(a) = μ(e)/2, b ≤ ec , and μ(b) = μ(ec )/2. Consider p = tp(a/E) and q = tp(b/E). Claim 1. p ⊥a q. We work in the measure algebra (ℬ̂, μ, d) associated to the standard Lebesgue space ([0, 1], ℬ, d), which is ℵ0 -saturated, and may assume that (ℬ̂, μ, d) is an elementary substructure of 𝒰 . Let r = μ(e). Since the type of an element is determined by its measure, we take e to be (the equivalence class of) [0, r) and ec to be (the equivalence class of) [r, 1]. Let a ≤ e have μ(a) = r/2, so a 󳀀󳨐 p; similarly let b ≤ ec have μ(b) = (1 − r)/2, so b 󳀀󳨐 q. Note that ℙ(a|E) = 21 χe and ℙ(ac |E) = 21 χe + χec . Since b ≤ ec , we have b ∩ a = 0, c b ∩ a = e ∩ a and ℙ(a|(Eb)# ) = 21 χe . Similarly, ℙ(ac |(Eb)# ) = 21 χe + χec . By Theorem 8.1, we have a ⌣ | E b, and thus Claim 1 is proved. Claim 2. p ⊥ q. Now let a small closed subalgebra F ⊆ U have E ⊆ F. Choose aF , bF ∈ U with aF 󳀀󳨐 p ↿F, bF 󳀀󳨐 q ↿F; then as before aF ≤ e and by Theorem 8.1, we have ℙ(aF |F) = 21 χe ,

208 � A. Berenstein and C. W. Henson ℙ(aFc |F) = 21 χe + χec . Likewise bF ≤ ec and we get ℙ(aF |(FbF )# ) = 21 χe = ℙ(aF |F), ℙ(aFc |(FbF )# ) = 21 χe + χec = ℙ(aFc |F). Using Theorem 8.1, we conclude aF ⌣ | F bF , as desired. From the previous proposition we get: Corollary 8.14. The theory APA is not unidimensional. Maharam’s theorem (Theorem 7.18) provides a countable set of cardinals 𝒦ℳ that helps classify a given model of APA up to isomorphism. In the classical first order setting, the existence of two or more distinct classifying cardinals is related to the existence of orthogonal types. In the example that follows, we illustrate this phenomenon in the setting of continuous model theory, for APA. Example 8.15. Let (ℬ, μ, d) 󳀀󳨐 APA and assume b1 , b2 ∈ ℬ are homogeneous elements of different density. Let C be the algebra generated by {b1 , b2 }. Let a1 , a2 ∈ ℬ satisfy 0 < ai < bi for i = 1, 2. Then tp(a1 /C) ⊥ tp(a2 /C). More generally, assume that b1 , b2 ∈ ℬ are disjoint elements and C is an algebra containing these elements. Also assume we are given a1 , a2 ∈ ℬ nonalgebraic over C with ai < bi for i = 1, 2. Then tp(a1 /C) ⊥ tp(a2 /C). Proof. We start with the first statement. Since b1 , b2 are homogeneous elements of different density we have b1 ∩ b2 = 0 and thus C = {0, 1, b1 , b2 , (b1 ∪ b2 )c }. We first show a1 ⌣ | C a2 . We see ℙ(a1 |C) =

μ(a1 ) χ μ(b1 ) b1

and ℙ(a1c |C) =

we have a2 ≤ b2 and b2 is disjoint from b1 , so

μ(b1 )−μ(a1 ) χb1 + χbc1 . On the other hand, μ(b1 ) 1) ℙ(a1 |(Ca2 )# ) = μ(a χ = ℙ(a1 |Ca2 ) and μ(b1 ) b1

1 )−μ(a1 ) χb1 + χbc1 = ℙ(a1c |C). Therefore we have a1 ⌣ | C a2 . ℙ(a1c |(Cb2 )# ) = μ(bμ(b 1) Claim. tp(a1 /C) ⊥ tp(a2 /C). Let F ⊆ U be a small closed subalgebra with C ⊆ F and let p ↿F and q ↿F be the nonforking extensions to F of p = tp(a1 /C) and q = tp(a2 /C), respectively. Choose 1) a1F , a2F ∈ U with a1F 󳀀󳨐 p ↿F, a2F 󳀀󳨐 q ↿F. By Theorem 8.1, we have ℙ(a1F |F) = μ(a χ μ(b ) b1

c and ℙ(a1F |C) =

1

μ(b1 )−μ(a1 ) χb1 + χbc1 . Since a2F ≤ b2 and b1 and b2 are disjoint, we get μ(b1 ) μ(a1 ) c c 1 )−μ(a1 ) χ = ℙ(a1F |F) and ℙ(a1F |(Fa2F )# ) = μ(bμ(b χb1 + χbc1 = ℙ(a1F |F). μ(b1 ) b1 1)

ℙ(a1F |(Fa2F )# ) = Using Theorem 8.1, we conclude a1F ⌣ | F a2F , as desired. The more general statement has a similar proof, and we leave the details to the reader. We need the following easy result. Observation 8.16. Let 0 < δ
0 such that δ ≤ ℙ(b1 |D)(x) ≤ 1 − δ

for μ-almost every x ∈ u.

(*)

We may assume 𝒰 is the probability algebra associated to a probability space (X, ℬ, μ). Let ([0, 1], 𝒞 , λ) be a standard atomless Lebesgue space and work in the probability algebra of the space (X ×[0, 1], ℬ ⊗ 𝒞 , μ⊗λ), identifying each v ∈ D with v′ = v×[0, 1] as done in the proof of Lemma 2.13. Let b′1 = {(x, y) ∈ X × [0, 1] : 0 ≤ y ≤ ℙ(b1 |D)(x)}. Just as in the proof of Lemma 2.13, we have that b′1 󳀀󳨐 p1 . Also let a′ = X × [0, 1/2] = {(x, y) ∈ X × [0, 1] : 0 ≤ y ≤ 21 }, which is a realization of q ↿D since it has measure 1/2 and is independent from all elements of U. We will prove b′1 ⌣ /| D a′ using Theorem 8.1 and Definition 2.8. Note that ℙ(b′1 |D)ℙ(a′ |D) = 21 ℙ(b′1 |D). On the other hand,

1 b′1 ∩ a′ = {(x, y) ∈ X × [0, 1] : 0 ≤ y ≤ min(ℙ(b1 |D)(x), )}. 2 For almost every x ∈ u, we get 1 δ ℙ(b′1 ∩ a′ |D)(x) − ℙ(b′1 |D)(x) ≥ 2 2 using (*) and Observation 8.16. Since u has positive measure, we get 󵄨 󵄨 ∫󵄨󵄨󵄨ℙ(b′1 ∩ a′ |D) − ℙ(a′ |D)ℙ(b′1 |D)󵄨󵄨󵄨 d(μ ⊗ λ) ≥ μ(u)δ/2 > 0 and thus b′1 ⌣ /| D a′ as desired.

9 Ranks obtained from entropy In this section we discuss the definition and main properties of entropy, following [27, Chapter 4] and [10, Chapter 4], to bring out the connection with model theoretic aspects of APA. The results given in Fact 9.4 and Corollary 9.5 show how entropy provides a rank that is closely connected to model-theoretic forking. Let (X, ℬ, μ) be an atomless probability space.

210 � A. Berenstein and C. W. Henson Definition 9.1. Let 𝒜 be a finite subalgebra of ℬ with atoms {A1 , . . . , Ak }. Let 𝒞 be a σ-subalgebra of ℬ. Then the entropy of 𝒜 given 𝒞 is H(𝒜/𝒞 ) = − ∫ ∑ ℙ(Ai |𝒞 ) ln(ℙ(Ai |𝒞 ))dμ. 1≤i≤k

We write H(𝒜) for H(𝒜/{0, X}). If 𝒜 and 𝒞 are σ-algebras, we denote by 𝒜 ∨ 𝒞 the σ-algebra generated by 𝒜 and 𝒞 . Definition 9.2. A continuous real-valued function F with domain [a, b] is convex if F(tx1 + (1 − t)x2 ) ≤ tF(x1 ) + (1 − t)F(x2 ) for all choices of x1 , x2 in [a, b] and all t ∈ [0, 1]. Note that if F: [a, b] → ℝ is continuous and is twice differentiable on (a, b), and if F ′′ (x) > 0 for all x in (a, b), then F is convex. Fact 9.3 ([10, Proposition 4.4]). Let ℰ be a σ-subalgebra of ℬ. Let F: [a, b] → ℝ be a continuous convex function, where 0 ≤ a ≤ b < ∞. Then F(𝔼ℰ (f )) ≤ 𝔼ℰ (F(f )) for each f ∈ L1 (X, ℬ, μ) with f (X) ⊆ [a, b]. Fact 9.4. Let 𝒜, 𝒞 be finite subalgebras of ℬ and let 𝒟, ℰ be σ-subalgebras of ℬ such that ℰ ⊆ 𝒟. Let {a1 , a2 , . . . , an } be the atoms in 𝒜. Then: (1) H(𝒜 ∨ 𝒞 /ℰ ) = H(𝒜/ℰ ) + H(𝒞 /𝒜 ∨ ℰ ). (2) 𝒜 ⊆ 𝒞 󳨐⇒ H(𝒜/ℰ ) ≤ H(𝒞 /ℰ ). (3) H(𝒜/ℰ ) ≥ H(𝒜/𝒟). (4) If τ is an automorphism of (ℬ, μ.d), then H(τ(𝒜)/τ(ℰ )) = H(𝒜/ℰ ). (5) H(𝒜/ℰ ) − H(𝒜/𝒟) ≥ 21 ∑nj=1 (‖𝔼𝒟 (χaj )‖22 − ‖𝔼ℰ (χaj )‖22 ) ≥ 0. Moreover, H(𝒜/ℰ ) = H(𝒜/𝒟) iff 𝒜 is independent from 𝒟 over ℰ . Proof. The first four properties are proved in [27, Section 4.3]. Note that (5) implies (3), and (4) is obvious from the definition. Throughout the argument, we let a be any element of U; we will apply the results for a ranging over the set of atoms {a1 , . . . , an }. Fact 2.6 shows that 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩𝔼𝒟 (χa )󵄩󵄩󵄩2 − 󵄩󵄩󵄩𝔼ℰ (χa )󵄩󵄩󵄩2 = 󵄩󵄩󵄩𝔼𝒟 (χa ) − 𝔼ℰ (χa )󵄩󵄩󵄩2 ≥ 0.

(A)

Applying this for a ∈ {a1 , . . . , an } proves the second inequality in (5). Next we prove the first inequality in (5). Consider F(x) = 2x ln(x) − x 2 restricted to [0, 1]. We have F ′′ (x) = 2/x − 2 > 0 for x ∈ (0, 1). Applying Fact 9.3 for this F and f = 𝔼𝒟 (χa ), we get

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2𝔼ℰ (χa ) ln(𝔼ℰ (χa )) − 𝔼ℰ (χa )2 ≤ 2𝔼ℰ (𝔼𝒟 (χa ) ln(𝔼𝒟 (χa ))) − 𝔼ℰ (𝔼𝒟 (χa )2 ). Integrating and moving the terms in the preceding inequality yields 1 󵄩 󵄩2 󵄩2 󵄩 (− ∫ ℙ(a|ℰ ) ln(ℙ(a|ℰ ))dμ) − (− ∫ ℙ(a|𝒟) ln(ℙ(a|𝒟))dμ) ≥ (󵄩󵄩󵄩𝔼𝒟 (χa )󵄩󵄩󵄩2 − 󵄩󵄩󵄩𝔼ℰ (χa )󵄩󵄩󵄩2 ). 2 (B) Summing the terms in (B) over a ∈ {a1 , . . . , an } yields the first inequality in (5). Finally, we prove the “Moreover” statement. We know 𝒜 is independent from 𝒟 over ℰ iff 𝔼𝒟 (χaj ) = 𝔼ℰ (χaj ) for all j = 1 . . . , n (by Theorem 8.1), and the latter implies H(𝒜/ℰ ) = H(𝒜/𝒟), by definition of entropy. On the other hand, by the inequalities in (5), we see H(𝒜/ℰ ) = H(𝒜/𝒟) implies ‖𝔼𝒟 (χaj )‖22 = ‖𝔼ℰ (χaj )‖22 for all j = 1, . . . , n, which in turn implies 𝔼𝒟 (χaj ) = 𝔼ℰ (χaj ) for all j = 1 . . . , n by statement (A) applied to a = a1 , . . . , an . Fact 9.4(5) provides a connection between forking and change of entropy. It has a quantitative aspect that we record here. Recall that for a = (a1 , . . . , an ) from ℬ̂ and D ⊆ C ⊆ ℬ̂, we say tp(a/C) ϵ-forks over D if d(tp(a/C), tp(a/D) ↿C) > ϵ where tp(a/D) ↿C is the unique nonforking extension of tp(a/D) to C. (See Remark 8.2.) Corollary 9.5. Let 𝒜 be a finite subalgebra of ℬ and let ℰ , 𝒟 be σ-subalgebras of ℬ such that ℰ ⊆ 𝒟. Let {a1 , a2 , . . . , an } be the events corresponding to the atoms in 𝒜, D the set of events associated to 𝒟 and E the set of events associated to ℰ . If tp((a1 , . . . , an )/D) ϵ-forks over E, then H(𝒜/ℰ ) > H(𝒜/𝒟) + ϵ2 /2. Proof. Assume that tp(a1 , . . . , an /D) ϵ-forks over E. Then by Theorems 6.11 and 8.1, for some j = 1, . . . , n we have ‖𝔼𝒟 (aj ) − 𝔼ℰ (aj )‖1 > ϵ. By Fact 2.6, this implies ‖𝔼𝒟 (aj ) − 𝔼ℰ (aj )‖2 > ϵ and thus ‖𝔼𝒟 (aj ) − 𝔼ℰ (aj )‖22 > ϵ2 . Then we get H(𝒜/ℰ ) > H(𝒜/𝒟) + ϵ2 /2 by the inequality in Fact 9.4(5).

10 Some problems In this final section we briefly indicate a few problems that seem interesting and worth investigation. (P1) Give an explicit formula for the induced distance between types in Sn (C) for APA. d(p, q) = inf{max d(ai , bi ) : (a1 , . . . , an ) 󳀀󳨐 p, (b1 , . . . , bn ) 󳀀󳨐 q}. 1≤i≤n

(P2) Provide a thorough analysis of the imaginary sorts for APA. (P3) Complete the model-theoretic background behind a generalization to continuous model theory of Shelah’s classification theory for superstable theories. (Some first

212 � A. Berenstein and C. W. Henson steps for this as applied to APA were discussed at the end of Section 8.). In particular, study appropriate versions of properties such as DOP (dimensional order property) and OTOP (omitting types order property) in the continuous setting and prove a dichotomy theorem relating a small bound for I(λ, T) to when T is superstable and has neither DOP nor OTOP, along the lines of [24, Theorem 2.3]. There is also the possibility of proving the equivalence, for continuous theories, between uncountable categoricity and being both ω-stable and unidimensional, as is true for classical first-order theories. (P4) Consider two existentially closed actions of the free group Fk on the unique separable model ℳ of APA, where 2 ≤ k ∈ ℕ∪{ω}. Are they approximately isomorphic? In the joint paper [9] of the authors with Ibarlucía, the class of existentially closed actions by a family of automorphisms of ℳ is axiomatized, and some concrete examples of such actions are given that are approximately isomorphic but not isomorphic. The answer is known to be positive when k = 1. (See [7, Remark 18.9].)

Bibliography [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13] [14] [15] [16]

I. Ben Yaacov, Schroedinger’s cat, Isr. J. Math. 153 (2006), 157–191. I. Ben Yaacov, On uniform canonical bases in Lp lattices and other metric structures, J. Log. Anal. 4 (2012) paper 12, 30 pp. I. Ben Yaacov, On theories of random variables, Isr. J. Math. 194 (2013), 957–1012. I. Ben Yaacov, On a Roelcke-precompact Polish group that cannot act transitively on a complete metric space, Isr. J. Math. 224 (2018), 105–132. I. Ben Yaacov, Star sorts, Lelek fans, and the reconstruction of non-ℵ0 -categorical theories in continuous logic, Model Theory, in press, 23 pages. arXiv:2203.02184v3. I. Ben Yaacov, A. Berenstein and C. Ward Henson, Almost indiscernible sequences and convergence of canonical bases, J. Symb. Log. 79 (2014), 460–484. I. Ben Yaacov, A. Berenstein, C. Ward Henson and A. Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis, Vol. II, ed. by Z. Chatzidakis, D. Macpherson, A. Pillay and A. Wilkie, Lecture Notes Series of the London Mathematical Society, vol. 350, Cambridge University Press, 2008, pp. 315–427. I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability, Trans. Am. Math. Soc. 362 (2010), 5213–5259. A. Berenstein, C. Ward Henson and T. Ibarlucía, Existentially closed measure-preserving actions of free groups, submitted. arXiv:2203.10178. J. R. Brown, Ergodic theory and topological dynamics, Academic Press, New York, 1976. S. Buechler, Essential stability theory, Springer, Berlin, Heidelberg, 1996. I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati and W. Winter, Model theory of C∗ -algebras, Mem. Am. Math. Soc. 1324 (2021). G. B. Folland, Real analysis, John Wiley and Sons, 1984. D. H. Fremlin, Measure algebras, Handbook of boolean algebras, vol. 3, North-Holland, 1989, pp. 877–980. D. H. Fremlin, Measure algebras, Measure theory, Vol. 3, Torres Fremlin, 2003–04. For information see http://www.essex.ac.uk/maths/staff/fremlin/mt.htm. P. A. Halmos, Measure theory, Van Nostrand, 1950.

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[17] W. Hodges, Model theory, Cambridge University Press, 1993. [18] O. Kallenberg, Foundations of modern probability, 3rd ed., Springer, 2021. [19] T. Lindstrøm, An invitation to nonstandard analysis, Nonstandard analysis and its applications, ed. by N. Cutland, London Mathematical Society, Student Texts, vol. 10, Cambridge University Press, 1988, pp. 1–105. [20] P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Am. Math. Soc. 211 (1975), 113–122. [21] D. Maharam, On homogeneous measure algebras, Proc. Natl. Acad. Sci. USA 108 (1942), 108–111. [22] D. Ross, Loeb measure and probability, Nonstandard analysis: theory and practice, ed. by L. O. Arkeryd, C. Ward Henson and N. J. Cutland, NATO Advanced Study Institutes Series C, vol. 493, Kluwer Academic Publishers, 1997, pp. 91–120. [23] H. L. Royden, Real analysis, 3rd ed., Prentice Hall, 1988. [24] S. Shelah, Classification of first order structures which have a structure theorem, Trans. Am. Math. Soc. 12 (1985), 227–232. [25] P. Shields, The theory of Bernoulli shifts, Chicago Lectures in Mathematics, The University of Chicago Press, 1973. [26] S. Song, Saturated structures from probability theory, J. Korean Math. Soc. 53 (2016), 315–329. [27] P. Walters, An introduction to ergodic theory, Springer, 1982.

David Jekel

Free probability and model theory of tracial W∗-algebras Abstract: The notion of a ∗-law or ∗-distribution in free probability is also known as the quantifier-free type in Farah, Hart, and Sherman’s model-theoretic framework for tracial von Neumann algebras. However, the full type can also be considered an analog of a classical probability distribution (indeed, Ben Yaacov showed that in the classical setting, atomless probability spaces admit quantifier elimination and hence there is no difference between the full type and the quantifier-free type). We therefore develop a notion of Voiculescu’s free microstates entropy for a full type, and we show that if X is a d-tuple in ℳ with χ 𝒰 (X : ℳ) > −∞ for a given ultrafilter 𝒰 , then there exists an embedding ι of ℳ into 𝒬 = ∏n→𝒰 Mn (ℂ) with χ(ι(X) : 𝒬) = χ(X : ℳ); in particular, such an embedding will satisfy ι(X)′ ∩ 𝒬 = ℂ by the results of Voiculescu. Furthermore, we sketch some open problems and challenges for developing model-theoretic versions of free independence and free Gibbs laws. Keywords: Free probability, free independence, free entropy, model theory, type, quantifier elimination, von Neumann algebra MSC 2020: 03C10, 46L54, 03C66, 94A17

1 Introduction This article aims to show the actual and possible applications of a model-theoretic framework to problems in free probability. In particular, we discuss an analog of Voiculescu’s free entropy for types rather than simply noncommutative ∗-laws. We use this to deduce that, for instance, a free group W∗ -algebra admits an embedding to any given matrix ultraproduct 𝒬 = ∏n→𝒰 Mn (ℂ) which has trivial relative commutant (also known as an irreducible embedding). This corollary is closely related to the work of von Neumann Acknowledgement: I thank Isaac Goldbring for the invitation to contribute to this book. Adrian Ioana brought to my attention the problem of using random matrix theory to construct embeddings into matrix ultraproducts that have trivial relative commutant, as well as von Neumann’s work on almost commuting matrices. This question was further motivated by various discussions by Sorin Popa about the factorial relative commutant embedding problem. I also thank Srivatsav Kunnawalkam Elayavalli for many discussions about model theory and 1-bounded entropy. I thank Ilijas Farah for helpful comments on a draft of this article, including pointing out the connections and references around Question 5.3. This work was supported by the National Science Foundation postdoctoral fellowship DMS-2002826. David Jekel, Department of Mathematics, University of California San Diego, 9500 Gilman Drive #0112, La Jolla, CA 92093, USA, e-mail: [email protected]; URL: https://www.davidjekel.com https://doi.org/10.1515/9783110768282-006

216 � D. Jekel in [62] and Voiculescu in [56], but we in fact prove the stronger result that there exists an embedding such that the entropy of the canonical generators in the presence of 𝒬 is finite. The author has also studied the model-theoretic version of 1-bounded entropy in [35]. We also discuss some open questions about defining a model-theoretic notion of (free) independence in noncommutative probability theory, which is intimately connected to questions about free entropy and random matrix theory. Furthermore, we explain open problems pertaining to optimization problems in free probability for which model theory may provide a way forward.

1.1 Noncommutative probability spaces and laws; free independence For this article, a tracial W∗ -algebra is a von Neumann algebra M together with a faithful normal tracial state τ : M → ℂ. We refer to Adrian Ioana’s article in this book for background. We will often denote the pair (M, τ) by a single calligraphic letter ℳ. This fits conveniently with the model-theoretic convention of using ℳ for a structure or for a model of some theory. Free probability theory, developed in large part by Voiculescu (see, e. g., [53, 61]), studies free products of operator algebras from a probabilistic viewpoint. Tracial W∗ -algebras have long been seen as an analog of probability spaces since every commutative tracial W∗ -algebra is isomorphic to an L∞ space with the trace given by the expectation (see Section 2.4 in Ioana’s article in this volume). The elements of a tracial W∗ -algebra are thus noncommutative analogs of bounded complex random variables. Now Cartesian products of groups lead to tensor products of tracial W∗ -algebras, and tensor products of commutative tracial W∗ -algebras produce independent random variables. In the same way, free products of groups lead to free products of tracial W∗ -algebras, which produces freely independent random variables. For background on free products, see Section 5.6 of Adrian Ioana’s article in this volume. The parallel between classical and free independence (and in general between probability spaces and tracial W∗ -algebras) motivated numerous constructions in free probability. First, noncommutative ∗-laws are an analog of a (compactly supported) multivariable probability distribution in the noncommutative setting. The law or distribution of a d-tuple (X1 , . . . , Xd ) of complex random variables is the unique measure satisfying ∫ f dμ = E[f (X1 , . . . , Xd )] ℂd

for all appropriate test functions f ; in the compactly supported case, we can take f to be polynomials in x1 , . . . , xd and x 1 , . . . , x d since a compactly supported probability measure μ on ℂd is uniquely determined by its ∗-moments

Free probability and model theory of tracial W∗ -algebras �

k

k

217

∫ x1 1 ⋅ ⋅ ⋅ xd d x 11 ⋅ ⋅ ⋅ x dd dμ(x1 , . . . , xd ) ℓ

ℓ

ℂd

for k1 , . . . , kd and ℓ1 , . . . , ℓd ∈ ℕ0 . In the noncommutative case, we define a noncommutative ∗-law through its noncommutative ∗-moments. Instead of the space of classical polynomials ℂ[t1 , . . . , td , t 1 , . . . , t d ], we use the space of noncommutative ∗-polynomials ℂ⟨t1 , . . . , td , t1∗ , . . . , td∗ ⟩. Definition 1.1. The noncommutative ∗-law of a d-tuple X is the map μX : ℂ⟨t1 , . . . , td , t1∗ , . . . , td∗ ⟩ → ℂ given by μX : p 󳨃→ τ(p(X)). Definition 1.2. For r > 0, we denote by Λd,r the space of noncommutative of ∗-laws of tuples X such that each ‖Xj ‖ ≤ r. From a C∗ -algebraic viewpoint, Λd,r is the space of tracial states on the C∗ -universal free product of d copies of the universal C∗ -algebra generated by an operator of norm ≤ r (for C∗ -algebraic free product, see Example 5.15 in Szabo’s article in this volume). We equip Λd,r with the weak-∗ topology obtained by viewing it as a subset of the dual of ℂ⟨t1 , . . . , td , t1∗ , . . . , td∗ ⟩ (that is, the topology of pointwise convergence on ℂ⟨t1 , . . . , td , t1∗ , . . . , td∗ ⟩); in this topology it is compact and metrizable. One of the central notions of free probability is free independence. Given a tracial W∗ -algebra ℳ, the free independence of two W∗ -subalgebras 𝒜 and ℬ is a condition on the joint moments τ(a1 b1 ⋅ ⋅ ⋅ an bn ) for aj ∈ 𝒜 and bj ∈ ℬ, which allows them to be expressed in terms the traces of elements from 𝒜 and from ℬ individually (see, e. g., [40, Theorem 19]). For our purposes, it is sufficient to use the following characterization of free independence. Proposition 1.3 (See, e. g., [3, Section 5] and [61]). Let 𝒜 and ℬ be tracial W∗ -algebras and let 𝒜 ∗ ℬ be their (tracial) free product (see Section 5.6 of Adrian’s article in this volume). If ι : 𝒜 ∗ ℬ 󳨅→ ℳ is a trace-preserving ∗-homomorphism, then ι(𝒜) and ι(ℬ) are freely independent in ℳ. One of the most important laws in classical probability theory is the Gaussian or normal distribution. The analog in free probability is the semicircular distribution (see, for instance, [61]). A semicircular random variable is a self-adjoint X from some tracial W∗ -algebra ℳ satisfying 2

1 τℳ [f (X)] = ∫ f (x)√4 − x 2 dx 2π

for f ∈ C(ℝ).

−2

The multivariate analog of a semicircular random variable is a free semicircular family, that is, a d-tuple (X1 , . . . , Xd ) such that each Xj is semicircular, and W∗ (X1 ), . . . , W∗ (Xd ) are freely independent. Similarly, the analog of a complex Gaussian is a circular ran-

218 � D. Jekel dom variable, that is, a variable of the form Z = (X + iY )/√2 where X and Y are freely independent semicircular variables; and a free circular family is a d-tuple (Z1 , . . . , Zd ) of circular random variables generating freely independent W∗ -algebras.

1.2 Free entropy If there are noncommutative laws or probability distributions, there should also be a free analog of the entropy of a probability distribution. Recall that the differential entropy of a probability measure μ on ℂd is h(μ) = − ∫ ρ log ρ dx whenever μ has a density ρ with respect to Lebesgue measure (and if there is no density, h(μ) is defined to be −∞). Voiculescu defined several analogs of h in free probability. We will primarily be concerned with the “free microstate entropy” χ(μ) for a noncommutative ∗-law of a d-tuple, defined in [55]. Unlike h, the free entropy χ cannot be defined directly by an integral formula (except in the case of a single self-adjoint random variable). Instead Voiculescu took an approach based on statistical mechanics and Boltzmann’s formulation of entropy in terms of microstates. The microstate entropy, roughly speaking, quantifies the amount of matrix tuples that have noncommutative laws close to μ. More precisely, the free microstate entropy is defined as follows. Let 𝒪 be an open set in Λd,r . Then we define the microstate space d Γ(n) r (𝒪 ) := {Y ∈ Mn (ℂ) : μY ∈ 𝒪 }.

The free entropy is defined using the exponential growth rate of the Lebesgue measure of these microstate spaces. Note that Mn (ℂ)d is an dn2 -dimensional complex innerproduct space with the inner product d

⟨X, Y⟩ = ∑ trn (Xj∗ Yj ). j=1

Thus, there is a canonical Lebesgue measure on Mn (ℂ)dsa defined by identifying it with 2

ℂdn by an isometry. Let vol Γ(n) r (𝒪 ) be its Lebesgue measure. Then χr (μ) = inf (lim sup( 𝒪∋μ

n→∞

1 log vol Γ(n) r (𝒪 ) + 2d log n)), n2

where 𝒪 ranges over all open neighborhoods in Λd,r of μ. The normalization of dividing by n2 and adding 2d log n can be motivated by the fact that this normalization will yield 2 a finite limit if we applied it to the log-volume of an r-ball in ℂdn ; similarly, one can check that this normalization works for variables whose real and imaginary parts are freely independent semicirculars [55] (for further discussion of the normalization, see [48, Appendix]). We also write χr (X) = χr (μX ) for a d-tuple X.

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Here we give the definition of χ for arbitrary (non-self-adjoint) d-tuples rather than the self-adjoint version from [55, 56]. This is because we have chosen to work with nonself-adjoint tuples in this paper because it seems to fit more naturally with the definition of types in the model theory of tracial W∗ -algebras. The self-adjoint version is defined in the same way as described above except that we restrict to self-adjoint tuples X and correspondingly consider microstate spaces of self-adjoint matrix d-tuples rather than arbitrary matrix d-tuples. Because every matrix Z can be uniquely written as X + iY where the “real and imaginary parts” X and Y are self-adjoint, one can show that the free entropy for d non-self-adjoint variables (Z1 , . . . , Zd ) agrees with the free entropy for the 2d self-adjoint variables obtained by taking the real and imaginary parts of the Zj . In the definition of χ, we do not know whether using a lim sup or lim inf will give the same answer (see [60]). Thus, it is also convenient to use an ultrafilter variant as in [57]: For a free ultrafilter 𝒰 on ℕ, set χR𝒰 (μ) = inf lim ( 𝒪∋μ n→𝒰

1 log vol(Γ(n) R (𝒪 )) + 2d log n). n2

Intuitively, if ℳ is a tracial W∗ -algebra generated by X = (X1 , . . . , Xd ), then χR𝒰 (μX ) measures the amount of ways to embed ℳ into the ultraproduct 𝒬 = ∏n→𝒰 Mn (ℂ). Indeed, an embedding of ℳ into 𝒬 is uniquely determined by where it sends the generators X, and thus corresponds to an equivalence class of sequences [Y(n) ]n∈ℕ such that Y(n) ∈ Mn (ℂ)d and limn→𝒰 λY(n) = λX . Moreover, limn→𝒰 μY(n) = μX means precisely that for every neighborhood 𝒪 of μX , we have that {n : μY(n) ∈ 𝒪} or {n : Y(n) ∈ Γ(n) R (𝒪 )} is in the ultrafilter 𝒰 . Voiculescu also defined in [56] the free entropy χ(X : Y), called the free entropy of X in the presence of Y, where X = (X1 , . . . , Xd ) and Y = (Y1 , . . . , Ym ) are tuples in a tracial W∗ -algebra ℳ. This version is defined by replacing the space of matrix microstates for X alone with the space of microstates for X which extend to microstates for (X, Y). See Section 4.3 for details. Entropy in the presence has several operator-algebraic applications. The first one which is of interest for this paper is its relationship with relative commutants. Voiculescu showed in [56, Corollary 4.3] that if χ(X : Y) > −∞, then W∗ (X)′ ∩ W∗ (X, Y) = ℂ (and in fact there are no nontrivial sequences in W∗ (X, Y) that asymptotically commute with W∗ (X)). Later, for ℳ = (M, τ) a von Neumann algebra with distinguished trace, Voiculescu defined χ(X : ℳ) = inf infm χ(X : Y). m∈ℕ Y∈M

Note that if χ(X : ℳ) > −∞, then W∗ (X)′ ∩ ℳ = ℂ since we can apply [56, Corollary 4.3] to any tuple Y from W∗ (X)′ ∩ ℳ. In this paper, we will prove the following result about entropy in the presence.

220 � D. Jekel Theorem 1.4. Let ℳ be a tracial noncommutative probability space and X ∈ ℳd . If χ 𝒰 (X : ℳ) > −∞, then there exists an embedding of ι : ℳ → 𝒬 = ∏n→𝒰 Mn (ℂ) such that χ 𝒰 (ι(X) : 𝒬) = χ 𝒰 (X, ℳ). The work of Voiculescu [56, Corollary 4.3] shows that any such embedding must satisfy ι(X)′ ∩ 𝒬 = ℂ, and hence we obtain the following corollary. Corollary 1.5. If χ 𝒰 (X : ℳ) > −∞, then there exists an embedding ι : ℳ → 𝒬 such that ι(X)′ ∩ 𝒬 = ℂ, hence also ι(ℳ)′ ∩ 𝒬 = ℂ. Similarly, one can argue as in [56, Section 5], that if χ 𝒰 (ι(X) : 𝒬) > −∞ there does not exist a diffuse abelian subalgebra of 𝒬 such that the W∗ -algebra of its normalizer contains X. For a survey of other applications of free entropy theory to operator algebras, see, for instance, the introduction and Section 2 of [32]. The hypotheses of Theorem 1.4 hold when X is a free circular family and ℳ = W∗ (X) (or more generally, the free product of W∗ (X) with some other Connes-embeddable tracial W∗ -algebra); see [56]. Thus, in particular, for d ≥ 1 there exists an embedding of a free group W∗ -algebra L(𝔽2d ) into 𝒬 with trivial relative commutant since L(𝔽2d ) is isomorphic to the tracial W∗ -algebra generated by a free circular d-tuple. (One can show the case for L(𝔽m ) with m ≥ 3 odd either by considering the self-adjoint version of entropy or observing that L(𝔽2d+1 ) = L(𝔽2d ) ∗ L(ℤ).) We remark that Corollary 1.5 could be proved directly from the same arguments as [56, Section 3]. Roughly speaking, the argument is that for δ ∈ (0, 1/2), the set of matrix tuples for which there exists some projection P of trace between δ and 1 − δ almost commuting with X has very small volume compared to the microstate space for X in the presence of ℳ. Consequently, a randomly chosen embedding for W∗ (X) extending to an embedding of ℳ could not have any nontrivial projection that commutes with ι(W∗ (X)). This probabilistic argument is similar in spirit to von Neumann’s earlier work [62] which showed that for large n most matrices A do not approximately commute with any matrices B other than those which are close to scalar multiples of identity. However, the free probabilistic statement of Theorem 1.4 is much stronger than Corollary 1.5. In general, for tracial W∗ -algebras ℳ ⊆ 𝒩 and X ∈ ℳd , we have χ 𝒰 (X : 𝒩 ) ≤ χ 𝒰 (X : ℳ), because the existence of microstates approximating the generators of 𝒩 (in a way compatible with X) is a stronger statement than the existence of microstates modeling the generators of ℳ. In the setting of our theorem, 𝒬 is a much larger tracial W∗ -algebra than ι(ℳ), and thus for a general embedding, χ 𝒰 (ι(X) : 𝒬) could be much smaller than χ 𝒰 (X : ℳ). To show that equality is achieved for some embedding ι, we rely on a compactness argument (see proof of Proposition 3.7), which distills aspects of the probabilis-

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tic argument sketched in the previous paragraph. This compactness argument requires working over a space of types, objects which record more information about a tuple than merely their noncommutative laws (which are equivalent to quantifier-free types). We do not know of any way to prove the first statement of Theorem 1.4, or the analogous statement for 1-bounded entropy in [35, Theorem 1.1], without using some version of entropy for full types or at least for one-quantifier types. As further motivation for developing free entropy for types, we point out the recent work of Chifan, Ioana, and Kunnawalkam Elayavalli [12], which has combined ideas from free entropy theory, deformation-rigidity theory, and model theory to construct a II1 -factor M without property Gamma which satisfies h(M 𝒰 ) = 0 and M is not elementarily equivalent to a free group von Neumann algebra. The next section will review the basic model-theoretic setup, and discuss another motivation for defining a version of χ for full types rather than only noncommutative laws, namely the lack of quantifier elimination in noncommutative probability spaces.

1.3 Model theory and types The work of [19, 20, 21] put tracial W∗ -algebras into the framework of continuous model theory of [5, 6]. The authors constructed a language ℒtr for tracial W∗ -algebras as well as a set of axioms Ttr that characterize when an ℒtr -structure comes from a tracial W∗ algebra. For further background, see Goldbring and Hart’s article in this volume. We will follow the treatment in [20] which introduces “domains of quantification” to cut down on the number of “sorts” needed. A language can have many sorts, and each sort can have many domains of quantification. The function and relation symbols come with a given modulus of uniform continuity for each product of domains of quantification (we will typically use S for a sort and D for a domain of quantification). For tracial W∗ -algebras, we have a single sort and the domains of quantification are operator-norm balls Dr for r ∈ (0, ∞). (Although the original paper used only integer values of r, we choose here to work with arbitrary positive values, which changes the setup of the languages but does not affect the proofs.) Model-theoretic notions such as terms and formulas correspond to familiar objects in von Neumann algebras and free probability: – Terms in ℒtr are expressions obtained from iterating scalar multiplication, addition, multiplication, and the ∗-operation on variables and the unit symbol 1. If (M, τ) is a tracial W∗ -algebra, then the interpretation of the term in ℳ is a function represented by a ∗-polynomial. – In ℒtr , a basic formula can take the form Re tr(f ) or Im tr(f ) where f is an expression obtained by iterating the ∗-algebraic operations. Thus, when evaluated in a tracial W∗ -algebra, it corresponds to the real or imaginary part of the trace of a noncommutative ∗-polynomial.

222 � D. Jekel –

Quantifier-free formulas can be obtained by applying continuous functions (“connectives”) to basic formulas. When the connectives are polynomial, this relates to an object in the free probability and random matrix literature called a trace polynomial.

General formulas are obtained from basic formulas recursively by applying such connectives and also taking the supremum or infimum in some variable over some domain of quantification. Although general formulas have not been studied much in free probability, our goal is to show that such a study is natural and worthwhile. For convenience, we will assume that our formulas do not have two copies of the same variable (i. e., if a variable is bound to a quantifier, there is no other variable of the same name that is free or bound to a different quantifier). For instance, in the formula Im tr(x1 ) sup Re tr(x1 x2 + x3 x1∗ ), x1 ∈D1

the first occurrence of x1 is free while the latter two occurrences are bound to the quantifier supx1 ∈D1 , but we can rewrite this formula equivalently as Im tr(x1 ) sup Re tr(y1 x2 + x3 y∗1 ). y1 ∈D1

We will typically denote the free variables by (xi )i∈ℕ and the bound variables by (yi )i∈ℕ . Lowercase letters will be used for formal variables while uppercase letters will be used for individual operators in operator algebras. For the most part, we will use formulas in finite or countably many free variables. Thus, we will consider an index set I (often finite or countable), an I-tuple S of sorts, and an I-tuple x = (xi )i∈I of variables in those sorts. We denote by ℱS the set of formulas with free variables (xi )i∈I where xi is from Si . Since tracial W∗ -algebras have only one sort, we will use the notation ℱI in this case. Given an ℒ-structure ℳ and an I-tuple X ∈ ∏i∈I Siℳ , each formula ϕ has an interpretation ϕℳ which can then be evaluated at X to yield ϕℳ (X). The type of X in ℳ is the map tpℳ (X) : ℱS → ℝ,

ϕ 󳨃→ ϕℳ (X).

In the case of tracial W∗ -algebras, this is very similar to how a noncommutative law of X is defined through the evaluation of the trace of a non-commutative polynomial on X. Indeed, the noncommutative law of X is obtained by restricting the linear functional tpℳ (X) to the set of basic formulas inside of ℱI . Since the value of basic formulas at X uniquely determines the value of all quantifier-free formulas, the noncommutative

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law contains the same information as the evaluation of quantifier-free formulas on X, namely the quantifier-free type tpℳ qf (X). Therefore, from a probabilistic viewpoint, the type of a tuple is an enrichment of the noncommutative ∗-law, which in turn is an analog of a classical probability distribution. One could also think of the type of a d-tuple in tracial W∗ -algebras as an analog of the type of classical random variables described in [4]. However, it turns out that the theory of atomless classical probability spaces admits quantifier elimination (see [6, Example 4.3] and [4, Fact 2.10], as well as Berenstein and Henson’s article in this volume), and so there is no practical distinction between the type and the quantifier-free type. We will state and prove this result about quantifier elimination in the framework of commutative tracial W∗ -algebras (see Theorem 2.13), showing that the ℒtr -structure L∞ [0, 1], with the trace given by Lebesgue measure, admits quantifier elimination. Hence the type of a d-tuple in L∞ [0, 1] (or any model of the same theory) is uniquely determined by its quantifier-free type. Therefore, it is worth asking whether in the noncommutative setting the full type might in some sense be a better analog for a probability distribution than the ∗-law itself. We discuss this idea further in Section 6. But if the type is to be viewed as an analog of the classical probability distribution, then we would want corresponding notions of independence and of entropy. This idea provides additional motivation for our development of the analog of Voiculescu’s entropy χ 𝒰 for types. However, we will not resolve the problem of finding a corresponding notion of independence; we give some suggestions and challenges in Section 5.

1.4 Outline 𝒰 Section 3 defines free entropy χfull (μ) for a full (or complete) type μ. Then in Section 4, we explain how the analogous construction for quantifier-free types agrees with Voiculescu’s original χ 𝒰 , and furthermore how the analogous quantity χ∃𝒰 for existential types agrees with Voiculescu’s entropy in the presence. The section concludes with the proof of Theorem 1.4. Sections 3 and 4 closely parallel [35], which handles the full-type analog of the 1-bounded entropy of Jung [39] and Hayes [31]. In Section 5, we consider the problem of defining an analog of free independence for types rather than for noncommutative laws. We sketch several approaches to independence (free products, random matrix theory, and model theory), and state many open questions based on these approaches. In Section 6, we discuss several open problems in free probability involving optimization (i. e., sup’s and inf’s), motivated by attempts to adapt the classical theory about optimal transport, entropy, and Gibbs ∗-laws to the noncommutative setting. We suggest ways in which the model-theoretic framework could bring further insight into these issues.

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2 Types, definable functions, and quantifier elimination 2.1 Types and definable predicates We use the following notation for concepts in continuous model theory (see Hart’s article in this volume for an introduction to the topic): – ℒ denotes a language. – 𝒮 denotes the collection of sorts in ℒ. – For each sort S, 𝒟S denotes the set of domains of quantification in S. – ℳ will be an ℒ-structure. – We denote the interpretation of sorts, domains, relation symbols, function symbols, etc., in ℳ using a superscript ℳ. – We usually use T for theories. – ℒtr will be the language of tracial W∗ -algebras and Ttr will be the theory of tracial W∗ -algebras given by [20]. – A tracial W∗ -algebra is a pair ℳ = (M, τ) where M is a W∗ -algebra and τ is a faithful normal tracial state on M. – ℒtr has only one sort; we do not explicitly name this sort, but its interpretation in ℳ will simply be denoted ℳ. Definition 2.1. Let I be an index set, and let S = (Si )i∈I be an I-tuple of sorts in a language ℒ. Let ℱS be the space of ℒ-formulas with free variables (xi )i∈I with xi from the sort Si . If ℳ is an ℒ-structure and X ∈ ∏i∈I Siℳ , then the type of X is the map tpℳ (X) : ℱS → ℝ given by ϕ 󳨃→ ϕℳ (X). Remark 2.2. In model theory, one often studies the type of a tuple over 𝒜 ⊆ ℳ, which records the values of formulas with coefficients in 𝒜 (see [5, Section 8]). For most of this paper, we take 𝒜 = ⌀. Definition 2.3. Let S = (Si )i∈I be an I-tuple of sorts in ℒ, and let T be an ℒ-theory. We denote by 𝕊S (T) the set of types tpℳ (X) for all X ∈ ∏i∈I Siℳ for all ℳ 󳀀󳨐 T. Similarly, if D ∈ ∏i∈I 𝒟Si , then we denote by 𝕊D (T) the set of types tpℳ (X) of all X ∈ ∏i∈I Dℳ i for all ℳ 󳀀󳨐 T. The space of types 𝕊D (T) is equipped with a weak-∗ topology as follows. If S is an I-tuple of ℒ-sorts, the set ℱS of formulas defines a real vector space. For each ℒ-structure ℳ and X ∈ ∏i∈I Sjℳ , the type tpℳ (X) is a linear map ℱS → ℝ. Thus, for each ℒ-theory

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T and D ∈ ∏i∈I 𝒟Si , the space 𝕊D (T) is a subset of the real dual ℱS† . We equip 𝕊D (T) with the weak-∗ topology (also known as the logic topology). The space 𝕊D (T) is compact in this topology [5, Proposition 8.6]. Moreover, if I is at most countable and the language is separable, then 𝕊D (T) is metrizable. In particular, this applies to ℒtr and Ttr (see [35, Observations 3.13 and 3.14] for further explanation). See Section 7 of Hart’s article in this volume for more background and explanation of types. In the setting with domains of quantification, it is also convenient to have a topology on 𝕊S (T). We say that 𝒪 ⊆ 𝕊S (T) is open if 𝒪 ∩ 𝕊D (T) is open for every D ∈ ∏j∈ℕ 𝒟Sj ; this defines a topology on 𝕊S (T), which we will also call the logic topology. One can then show that the inclusion map 𝕊D (T) → 𝕊S (T) is a topological embedding [35, Observation 3.6]. Every formula defines a continuous function on 𝕊S (T) for each theory T, but the converse is not true. The objects that correspond to continuous functions on 𝕊S (T) are a certain completion of the set of formulas, called definable predicates. Our approach to the definition will be semantic rather than syntactic, defining these objects immediately in terms of their interpretations. Definition 2.4. Let ℒ be a language and T an ℒ-theory. A definable predicate relative to T is a collection of functions ϕℳ : ∏i∈I Siℳ → ℝ (for each ℳ 󳀀󳨐 T) such that for every collection of domains D = (Dj )j∈ℕ and every ϵ > 0, there exists a finite F ⊆ I and an ℒ-formula ψ(xi : i ∈ F) such that whenever ℳ 󳀀󳨐 T and X ∈ ∏j∈ℕ Dℳ j , we have 󵄨󵄨 ℳ 󵄨 ℳ 󵄨󵄨ϕ (X) − ψ (Xi : i ∈ F)󵄨󵄨󵄨 < ϵ. Here we describe definable predicates relative to an arbitrary theory T because we want to study definable predicates that make sense uniformly for all models of some theory T (for instance, Ttr in ℒtr ) rather than merely for a single ℒ-structure or for all ℒ-structures. In our definition, two distinct formulas may result in the same definable predicate. As explained in Section 5.2 of Hart’s article in this volume, definable predicates are a natural generalization of formulas, and they can also be characterized as the separationcompletion of the space of formulas with respect to certain seminorms. In [5, Theorem 9.9], the authors show that definable predicates correspond to continuous functions on the space of types. This result adapts to the setting with multiple sorts and domains of quantification as well (in fact, our definition of the logic topology on 𝕊S (T) was chosen so that this would work!). For a proof, see [35, Proposition 3.9] (although this proposition assumes countable index sets, removing this restriction does not affect the argument). Proposition 2.5 ([35, Observation 3.11]). Let ℒ be a language and T an ℒ-theory. Let ϕ be a collection of functions ϕℳ : ∏i∈I Siℳ → ℝ for each ℳ 󳀀󳨐 T. The following are equivalent: (1) ϕ is a definable predicate relative to T.

226 � D. Jekel (2) There exists a continuous γ : 𝕊S (T) → ℝ such that ϕℳ (X) = γ(tpℳ (X)) for all ℳ 󳀀󳨐 T and X ∈ ∏i∈I Siℳ . Just like formulas, definable predicates satisfy a certain uniform continuity property with respect to the metric d ℳ . Observation 2.6. If ϕ = (ϕℳ ) is a definable predicate in sorts S over ℒ relative to T, then ϕ satisfies the following uniform continuity property: For every D ∈ ∏i∈I 𝒟Si and ϵ > 0, there exists a finite F ⊆ I and δ > 0 such that, whenever ℳ 󳀀󳨐 T and X, Y ∈ ∏j∈ℕ Dℳ j , d ℳ (Xi , Yi ) < δ

for all i ∈ F

󳨐⇒

󵄨󵄨 ℳ 󵄨 ℳ 󵄨󵄨ϕ (X) − ϕ (Y)󵄨󵄨󵄨 < ϵ.

Moreover, for every D ∈ ∏i∈I 𝒟Si , there exists a constant C such that |ϕℳ | ≤ C for all ℳ 󳀀󳨐 T. Corollary 2.7. Let ℒ be a language, ℳ a model of a theory T, let S be an I-tuple of sorts, and let D ∈ ∏i∈I 𝒟Si . Let J be a directed set, let (X(j) )j∈J be a net of I-tuples from ∏i∈I Dℳ i , ℳ (j) ℳ and let X ∈ ∏i∈I Dℳ i . If limj∈J Xi = Xi for each i ∈ I, then limj∈J tp (X ) = tp (X). (j)

Finally, we recall that there are quantifier-free versions of all the notions in this section, which are defined analogously except with quantifier-free formulas instead of all formulas; in particular: – Let ℱqf,S denote the set of quantifier-free formulas (i. e., those defined without using any sup or inf) in variables from an I-tuple S of sorts. Given X ∈ ∏i∈I Siℳ , the ℳ quantifier-free type tpℳ qf (X) is the map ℱqf,S → ℝ given by ϕ 󳨃→ ϕ (X). – Given D ∈ ∏i∈I 𝒟Si , we denote by 𝕊qf,D (T) the space of quantifier-free types of elements from ∏i∈I Dℳ for any ℳ 󳀀󳨐 T. We equip it with the weak-∗ topology as a i subset of the dual of ℱqf,S . – We denote the set of all quantifier-free types of tuples from S by 𝕊qf,S (T). As in the case of 𝕊S (T), we equip 𝕊qf,S (T) with the “union topology.” – A quantifier-free definable predicate relative to an ℒ-theory T is defined analogously to Definition 2.4 except that now the formula ψ is required to be a quantifier-free formula. – Analogous to Proposition 2.5, quantifier-free definable predicates correspond to continuous functions on 𝕊qf,S (T). The relationship between types and quantifier-free types is as follows (we will need this to relate our model-theoretic version of χ with Voiculescu’s). The quantifier-free type ℳ tpℳ qf (X) is the restriction of tp (X) from ℱS to ℱqf,S . The restriction operation defines a map πqf : 𝕊S (T) → 𝕊qf,S (T). It is immediate that for each I-tuple D of domains, the map πqf is weak-∗ continuous and surjective 𝕊D (T) → 𝕊S (T), and since the domain and codomain are Hausdorff it is therefore a topological quotient map. By definition of the

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topology on 𝕊S (T), it follows that πqf is continuous 𝕊S (T) → 𝕊qf,S (T), and furthermore it is a topological quotient map.

2.2 Definable functions Here we summarize a few definitions and facts about definable functions needed to establish the properties of entropy in Section 4. In particular, we recall the result from [35] that for an I-tuple X in a tracial W∗ -algebra, every element Y ∈ W∗ (X) can be expressed as Y = f (X) for some quantifier-free definable function f . Definition 2.8. Let ℒ be a language, T an ℒ-theory, I an index set, S an I-tuple of sorts, and S ′ another sort. A definable function f : ∏i∈I Si → S ′ relative to T is a collection of functions f ℳ : ∏i∈I Siℳ → (S ′ )ℳ for each ℳ 󳀀󳨐 T satisfying the following conditions: (1) For every D ∈ ∏i∈I 𝒟Si , there exists D′ ∈ 𝒟S′ such that each f ℳ maps ∏i∈I 𝒟Sℳ i into Dℳ S′ . (2) The function ϕℳ (X, Y ) = d ℳ (f ℳ (X), Y) for ℳ 󳀀󳨐 T is a definable predicate in variables from the sorts (S, S ′ ).

It is convenient to consider also definable functions where the output is a tuple. A definable function f : ∏i∈I Si → ∏j∈I ′ Sj′ relative to T is simply a tuple where each fj is definable. We also have the following characterization (or alternative definition if you will) of definable functions in terms of how they interact with definable predicates. For a proof of this equivalence, see [35, Proposition 3.17] (although the statement there was only given for countable index sets, it generalizes to arbitrary index sets without any essential changes). Proposition 2.9. Let S and S′ be I and I ′ -tuples of sorts in the language ℒ. Consider a collection of maps fℳ : ∏j∈ℕ Sjℳ → ∏j∈ℕ (Sj′ )ℳ for ℳ 󳀀󳨐 T. Then f is a definable function ∏i∈I Si → ∏i∈I ′ Si′ relative to the ℒ-theory T if and only if the following conditions hold: (1) For each D ∈ ∏j∈ℕ 𝒟Sj , there exists D′ ∈ ∏j∈ℕ 𝒟S′ such that for every ℳ 󳀀󳨐 T, fℳ j

′ ℳ maps ∏j∈ℕ Dℳ j into ∏j∈ℕ (Dj ) . ̃ (2) Whenever S̃ is an I-tuple of sorts and ϕ is a definable predicate relative to T in the free ′ ′ variables xj ∈ Sj for j ∈ I ′ and x̃j ∈ S̃j for j ∈ I,̃ then ϕ(f(x), x)̃ is a definable predicate in the variables x = (xj )j∈ℕ and x̃ = (x̃j )j∈ℕ .

As a consequence, if f is a definable function ∏i∈I Si → ∏i∈I ′ Si′ and ϕ is a definable predicate in the variables xj′ ∈ Sj′ , then ϕ ∘ f is a definable predicate (see [35, Lemma 3.20(1)]). We want to define quantifier-free definable functions analogously to Definition 2.8 except using quantifier-free definable predicates rather than definable predicates. Unfortunately, we do not know in general whether the analog of Proposition 2.9 holds for the quantifier-free setting for an arbitrary choice of theory T. Thus, we prefer to de-

228 � D. Jekel fine quantifier-free definable functions by the analog of the properties listed in Proposition 2.9 instead. Definition 2.10. Let S and S′ be I and I ′ -tuples of sorts in the language ℒ. Consider a collection of maps fℳ : ∏j∈ℕ Sjℳ → ∏j∈ℕ (Sj′ )ℳ for ℳ 󳀀󳨐 T. Then f is a quantifier-free definable function ∏i∈I Si → ∏i∈I ′ Si′ relative to the ℒ-theory T if and only if the following conditions hold: (1) For each D ∈ ∏j∈ℕ 𝒟Sj , there exists D′ ∈ ∏j∈ℕ 𝒟S′ such that for every ℳ 󳀀󳨐 T, fℳ j

′ ℳ maps ∏j∈ℕ Dℳ j into ∏j∈ℕ (Dj ) . ̃ (2) Whenever S̃ is an I-tuple of sorts and ϕ is a quantifier-free definable predicate relative to T in the free variables xj′ ∈ Sj′ for j ∈ I ′ and x̃j ∈ S̃j for j ∈ I,̃ then ϕ(f(x), x)̃ is a quantifier-free definable predicate in the variables x = (xj )j∈ℕ and x̃ = (x̃j )j∈ℕ .

An important consequence of this definition is that if ϕ is a quantifier-free definable predicate and f is a quantifier-free definable function, and if the domain of ϕ is the same as the codomain of f, then ϕ ∘ f is a quantifier-free definable predicate. But in any case, [35, Theorem 3.30] shows that the two possible definitions of quantifier-free definable functions proposed above are equivalent in the case of ℒtr and Ttr . Furthermore, it turns out that every element in a W∗ -algebra can be expressed as a quantifier-free definable function of the generators. Proposition 2.11 ([35, Proposition 3.32]). If ℳ = (M, τ) is a tracial W∗ -algebra and X ∈ ∏i∈I Dri generates M and Y ∈ ∏i∈I ′ Dℳ r ′ , then there exists a quantifier-free definable funci

tion f in ℒtr relative to Ttr such that Y = f(X). In fact, f can be chosen so that fkℳ maps M I into ∏i∈I ′ Dℳ r ′ for all ℳ 󳀀󳨐 T. i

A special case (and one of the key steps in the proof) of this result is that there exists a one-variable quantifier-free definable function f such that f ℳ maps M into the unit ball and equals the identity on the unit ball (this is more subtle than in the C∗ -algebraic setting because the operator norm is not a definable predicate in ℒtr ). By composing with this f , one can extend a definable predicate defined only on a product of operator-norm balls to a globally defined definable predicate. See [35, Remark 3.33] for details. Proposition 2.12. Let I be an index set and r ∈ (0, ∞)I . Let 𝕊r (Ttr ) be the set of types for I-tuples in ∏i∈I Dℳ ri for ℳ 󳀀󳨐 Ttr . Then for every continuous ϕ : 𝕊r → ℝ, there exists a continuous extension to 𝕊I (Ttr ), that is, there us a definable predicate ψ such that ψ|𝕊r (Ttr ) = ϕ.

2.3 Quantifier elimination in classical probability A theory T is said to admit quantifier elimination if for every d, every d-variable definable predicate with respect to T is a quantifier-free definable predicate. An ℒ-structure ℳ

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is said to admit quantifier elimination if its theory admits quantifier elimination. For further background and equivalent characterizations of quantifier elimination, see [5, Chapter 13] or Section 9 of Hart’s article in this volume. In this section, we show that a diffuse classical probability space admits quantifier elimination (in contrast to general tracial W∗ -algebras), which gives a model-theoretic heuristic for why noncommutative probability theory is more difficult than classical probability theory. Theorem 2.13. The ℒtr -structure L∞ [0, 1], with the trace E : L∞ [0, 1] → ℂ given by integration, admits quantifier elimination. This result is closely related to [6, Example 4.3] and [4, Fact 2.10] which showed quantifier elimination for the theory of diffuse classical probability spaces. Moreover, Berenstein and Henson’s article in this volume gives a more thorough treatment of the model theory of classical probability spaces. We will give an argument for quantifier elimination of L∞ [0, 1] directly in the W∗ -algebraic framework. We need several lemmas, the first of which is the characterization of quantifier elimination in terms of types. Lemma 2.14. Let T be a theory in a language ℒ. Then T admits quantifier elimination if and only if for every d ∈ ℕ and S = (S1 , . . . , Sd ) of sorts, the restriction map πqf : 𝕊S (T) → 𝕊qf,S (T) is injective (or in other words, an d-tuple’s type is uniquely determined by its quantifier-free type). Proof. It suffices to show that for every d ∈ ℕ and d-tuple S of sorts, the map 𝕊D (T) → 𝕊D,qf (T) is injective if and only if every definable predicate in variables x1 , . . . , xd from S1 , . . . , Sd is a quantifier-free definable predicate. Fix d and let S be a d-tuple of sorts. If the map 𝕊S (T) → 𝕊qf,S (T) is injective, then it is bijective. This implies that the restricted map 𝕊D (T) → 𝕊qf,D (T) is bijective as well, and since 𝕊D (T) is compact and 𝕊qf,D (T) is Hausdorff, the map 𝕊D (T) → 𝕊qf,D (T) is a homeomorphism. But this implies that 𝕊S (T) → 𝕊qf,S (T) is a homeomorphism. Definable predicates are equivalent to continuous functions on 𝕊S (T) while quantifier-free definable predicates are equivalent to continuous functions on 𝕊qf,S (T). The restriction map 𝕊S (T) → 𝕊qf,S (T) induces the inclusion map C(𝕊qf,S (T)) → C(𝕊S (T)) from quantifier-free definable predicates to all definable predicates. Since this map is bijective, then every definable predicate is quantifier-free definable. Conversely, suppose that every definable predicate is quantifier-free definable. If μ and ν are two distinct types, then there exists a formula ϕ such that μ(ϕ) ≠ ν(ϕ). By assumption, ϕ is a quantifier-free definable predicate, and hence μ(ϕ) ≠ ν(ϕ) implies that πqf (μ) ≠ πqf (ν). Thus, πqf is injective as desired. The following fact was observed from the viewpoint of probability algebras in [4, Fact 2.10]. Again, see Berenstein and Henson’s article for further discussion. Lemma 2.15. Every separable model of Th(L∞ [0, 1], E) is isomorphic to (L∞ [0, 1], E). In other words, Th(L∞ [0, 1], E) is ℵ0 -categorical.

230 � D. Jekel Proof. Let ℳ be a separable model of Th(L∞ [0, 1], E). Then ℳ is a tracial W∗ -algebra. Moreover, it is commutative since sup d ℳ (XY , YX) =

X,Y ∈Dℳ 1

sup

∞ [0,1]

X,Y ∈DL1

‖XY − YX‖L2 [0,1] = 0.

To show that ℳ is diffuse, consider for each n ∈ ℕ the sentence n n n 󵄨󵄨 1 󵄨󵄨󵄨 󵄨 (d(x1 + ⋅ ⋅ ⋅ + xn , 1) + ∑ d(xj2 , xj ) + ∑ d(xj , xj∗ ) + ∑󵄨󵄨󵄨τ(xj ) − 󵄨󵄨󵄨). ℳ 󵄨 n 󵄨󵄨 x1 ,...,xn ∈D1 j=1 j=1 j=1 󵄨

inf

The value of these sentences in L∞ [0, 1] is 0 since we can take xj = 1[(j−1)/n,j/n) . Therefore, the value of these sentences in ℳ is also zero. However, this could not happen if ℳ had atoms. Now we apply the fact that every diffuse separable commutative tracial W∗ -algebra is isomorphic to (L∞ [0, 1], E). (See Section 2.4 in Ioana’s article in this volume.) Lemma 2.16. Let X, Y ∈ L∞ [0, 1]d . If X and Y have the same quantifier-free type, then for every ϵ > 0, there exists an automorphism α of (L∞ [0, 1], E) such that ‖α(X)−Y‖L2 [0,1]d ≤ ϵ. Proof. Fix X and Y. Let P denote the Lebesgue (probability) measure on [0, 1]. Fix r > 0 such that ‖Xj ‖L∞ [0,1] ≤ r and ‖Yj ‖L∞ [0,1] ≤ r. Fix ϵ > 0. Let S1 , . . . , Sk be a partition of {z ∈ ℂ : |z| ≤ r}d into measurable sets each of which has diameter at most ϵ. Since X and Y have the same quantifier-free type (or probability distribution), we have P(X ∈ Sj ) = P(Y ∈ Sj ). Thus, X−1 (Sj ) and Y−1 (Sj ) are two Lebesgue-measurable subsets of [0, 1] with the same measure. This implies (by a standard exercise in measure theory; see Theorem 2.18 in Ioana’s article in this volume) that there is a measurable isomorphism fj : X−1 (Sj ) → Y−1 (Sj ). Let f : [0, 1] → [0, 1] be the measurable map given by f |X−1 (Sj ) = fj . Let α be the automorphism of L∞ [0, 1] given by Z 󳨃→ Z ∘ f −1 . Then α(X) = X ∘ f −1 maps Y−1 (Sj ) into Sj , and therefore, for ω ∈ Y−1 (Sj ), both α(X)(ω) and Y(ω) are in Sj , which implies that ‖α(X)(ω) − Y(ω)‖ℂd ≤ diam(Sj ) ≤ ϵ. Therefore, ‖α(X)(ω) − Y(ω)‖ℂd ≤ ϵ for all ω ∈ [0, 1] and in particular ‖α(X) − Y‖L2 [0,1]d ≤ ϵ.

Proof of Theorem 2.13. Fix d and let 𝕊d (Th(L∞ [0, 1], E)) denote the space of types of d-tuples in models of the theory of L∞ ([0, 1], E). Consider two types μ and ν in 𝕊d (Th(L∞ [0, 1], E)) with πqf (μ) = πqf (ν). Let X be a tuple from some model ℳ with type μ and let Y be a tuple from 𝒩 with type ν. By passing to separable elementary submodels through the Löwenheim–Skolem theorem, we may assume without loss of generality that ℳ and 𝒩 are separable. Then by Lemma 2.15, we may assume that ℳ = 𝒩 = (L∞ [0, 1], E). Because X and Y have the same quantifier-free type, Lemma 2.16 implies that for every n ∈ ℕ, there exists an automorphism αn of (L∞ [0, 1], E) such that ‖αn (X)−Y‖L2 [0,1]d < 1/n. Therefore, tpL

∞

[0,1],E

(Y) = lim tpL

∞

n→∞

[0,1],E

(αn (X)) = tpL

∞

[0,1],E

(X).

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This implies that the map πqf : 𝕊d (Th(L∞ [0, 1], E)) → 𝕊qf,d (Th(L∞ [0, 1], E)) is injective for every m, and therefore, (L∞ [0, 1], E) admits quantifier elimination by Lemma 2.14. We also remark that the matrix algebra Mn (ℂ) admits quantifier elimination as a tracial W∗ -algebra, as was observed in [15, end of Section 2]. This fact can be proved by combining Lemma 2.14 with the following result. Lemma 2.17. Let X, Y ∈ Mn (ℂ)d . The following are equivalent: (1) X and Y have the same type in (Mn (ℂ), trn ). (2) X and Y have the same quantifier-free type (Mn (ℂ), trn ). (3) There exists a unitary U such that UXU ∗ = Y. (1) 󳨐⇒ (2) and (3) 󳨐⇒ (1) are immediate. The claim (2) 󳨐⇒ (3) follows from the multivariate version of Specht’s theorem [50] observed by Wiegmann [63] and verified in [38, Theorem 2.2]. See [47] for a survey of related results. However, there are many tracial W∗ -algebras that do not admit quantifier elimination; Goldbring, Hart, and Sinclair showed in [27, Theorem 2.2] that a tracial W∗ -algebra that is locally universal and McDuff cannot have quantifier elimination (and in fact this applies to all McDuff tracial W∗ -algebras as explained in Goldbring and Hart’s article in this volume). The complete classification of C∗ -algebras with quantifier elimination was handled in [15], so the general investigation of quantifier elimination for tracial W∗ -algebras seems like an achievable and worthwhile research project. Question 2.18. Determine which separable tracial W∗ -algebras admit quantifier elimination.

2.4 Existential types We conclude our discussion of continuous model theory for tracial W∗ -algebras with a description of existential types. This subsection serves as background for the study of entropy for existential types in Section 4.2, which we will show agrees with Voiculescu’s notion of entropy in the presence. Definition 2.19. Let I be an index set, and consider variables x = (xi )i∈I from sorts (Si )i∈I . An existential formula in the language ℒ is a formula of the form ϕ(x) =

inf

y1 ∈D1 ,...,yk ∈Dk

ψ(x, y1 , . . . , yk ),

where ψ is a quantifier-free formula and D1 , . . . , Dk are domains of quantification in the appropriate sorts. Similarly, we say that ϕ is an existential definable predicate relative to T if

232 � D. Jekel ϕℳ (X) =

inf

Y∈∏j∈ℕ Dℳ j

ψℳ (X, Y)

for ℳ 󳀀󳨐 T, where ψ is a quantifier-free definable predicate. Since a quantifier-free definable predicate can be approximated on each product of domains by a quantifier-free formula in finitely many variables, one can argue that an existential definable predicate can be approximated uniformly on each product of domains of quantification by an existential formula (and this is why it is a definable predicate to begin with). Definition 2.20. Let ℳ be an ℒ-structure, S = (S1 , . . . , Sd ) a d-tuple of sorts, and X ∈ ∏dj=1 Sjℳ . Let ℱ∃,S denote the space of existential formulas. The existential type tpℳ ∃ (X) is the map tpℳ ∃ (X) : ℱ∃,S → ℝ,

ϕ 󳨃→ ϕℳ (X).

If T is an ℒ-theory, we denote the set of existential types that arise in models of T by 𝕊∃,S (T). The topology for existential types, however, is not simply the weak-∗ topology on 𝕊∃,D (T) for each tuple of domains. Rather, we define neighborhoods of a type μ = tpℳ (X) using sets of the form {ν : ν(ϕ) < μ(ϕ) + ϵ} for existential formulas ϕ. We remark that if we applied the same construction to the full type or the quantifier-free type because formulas and quantifier-free formulas are both closed under multiplication by −1 (however, existential formulas are not). The idea is that if ϕℳ (X) = infY∈∏j∈ℕ Dℳ ψℳ (X, Y) for some quantifier-free definj

able predicate ϕ, then μ(ϕ) ≤ c means that there exists Y such that ψℳ (X, Y) < c + δ for any δ > 0. Thus, a neighborhood corresponds to types ν where there exists Y that gets within ϵ of the infimum achieved for μ.

Definition 2.21. Let T be an ℒ-theory, S an d-tuple of sorts, and D ∈ ∏dj=1 𝒟Sj . We say that 𝒪 ⊆ 𝕊∃,D (T) is open if for every μ ∈ 𝒪, there exist existential formulas ϕ1 , . . . , ϕk and ϵ1 , . . . , ϵk > 0 such that {ν ∈ 𝕊∃,D (T) : ν(ϕj ) < μ(ϕj ) + ϵj for j = 1, . . . , k} ⊆ 𝒪. Moreover, we say that 𝒪 ⊆ 𝕊∃,S (T) is open if 𝒪 ∩ 𝕊∃,D (T) is open in 𝕊∃,D (T) for all D ∈ ∏j∈ℕ 𝒟Sj . Observation 2.22. – Any set of the form {ν : ν(ϕ1 ) < c1 , . . . , ν(ϕk ) < ck }, where ϕ1 , . . . , ϕk are existential formulas, is open in 𝕊∃,S (T). – The same holds if ϕj is an existential definable predicate rather than existential formula, since it can be uniformly approximated by existential formulas on each product

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–

233

of domains of quantification. Hence existential definable predicates may be used in Definition 2.21 without changing the definition. The inclusion 𝕊∃,D (T) → 𝕊∃,S (T) is a topological embedding since each of the basic open sets in 𝕊∃,D (T) given by ν(ϕj ) < μ(ϕj ) + ϵj for j = 1, . . . , k extends to an open set in 𝕊∃,S (T). The restriction map π∃ : 𝕊S (T) → 𝕊∃,S (T) is continuous.

Remark 2.23. Like the Zariski topology on the space of ideals in a commutative ring, the topology on 𝕊∃,S (T) is often non-Hausdorff. Indeed, one can check that the intersection of all neighborhoods of an existential type μ is 𝒦μ = {ν : ν(ϕ) ≤ μ(ϕ) for all ϕ ∈ ℱ∃,S }.

(2.1)

We say that ν extends μ if ν(ϕ) ≤ μ(ϕ) for all existential formulas ϕ, which is equivalent to saying that μ(ϕ) = 0 implies that ν(ϕ) = 0 for all ϕ ∈ ℱ∃,S (since max(ϕ − c, 0) is an existential formula if ϕ is). Then {μ} is closed if and only if it does not have any proper extension, or it is maximal. These closed points correspond to existential types from existentially closed models (see [26, Section 6.2]), and such maximal existential types in 𝕊∃,D (T) form a compact Hausdorff space. However, our present goal is to work with general tracial W∗ -algebras, not only those that are existentially closed.

3 Free entropy for types We define a version of Voiculescu’s free entropy for full types rather than only quantifier-free types. We will see in Section 4 that Voiculescu’s free entropy of a d-tuple X in the presence of ℳ can be realized as a special case of free entropy for a closed subset of the type space.

3.1 Definition Because we only have one sort in ℒtr , it will be convenient to use slightly different notation for the spaces of types. Specifically, 𝕊d (Ttr ) will denote the space of types for d-tuples, or in other words 𝕊S (Ttr ) for S = (M, . . . , M). Moreover, 𝕊d,r (Ttr ) will denote the space of d-tuples with operator norm bounded by r, or in other words, 𝕊D (Ttr ) where D = (Dr , . . . , Dr ). To simplify notation, we use the same r for all d-variables rather than choosing radii r1 , . . . , rd . Definition 3.1 (Microstate spaces). If 𝒦 ⊆ 𝕊d (Ttr ) and r ∈ (0, ∞), we define d

Mn (ℂ) Γ(n) : tpMn (ℂ) (Y) ∈ 𝒦}. r (𝒦) = {Y ∈ ∏ Dr j=1

234 � D. Jekel We view this as a microstate space as in Voiculescu’s free entropy theory. The entropy will be defined in terms of Lebesgue measure of the microstate spaces, with the normalization of Lebesgue measure corresponding to the inner product from the normalized trace, as described in Section 1.2. By transporting the canonical Lebesgue mea2 sure on ℂn d through such an isometry, we obtain a canonical Lebesgue measure on Mn (ℂ)d . Note that this convention differs from that of Voiculescu since we use the normalized trace rather than the unnormalized trace to define the inner product. Moreover, here we consider all matrices rather than only self-adjoint matrices like Voiculescu, because the notion of type is more natural without restricting to the selfadjoint setting. However, most of the results here would also work with the same proof for self-adjoint tuples, provided that we restrict to microstate spaces of self-adjoint matrices. Definition 3.2 (Free entropy for types). Let 𝒦 ⊆ 𝕊r (Ttr ). Then define 𝒰 χfull,r (𝒦) :=

inf

lim (

open 𝒪⊇𝒦 n→𝒰

1 log vol Γ(n) r (𝒪 ) + 2d log n), n2

where 𝒪 ranges over all neighborhoods of 𝒦 in 𝕊d,r (Ttr ). Moreover, we define 𝒰 𝒰 χfull (𝒦) := sup χfull,r (𝒦). r>0

An important special case is when 𝒦 consists of a single point μ ∈ 𝕊d (Ttr ). We will write 𝒰 𝒰 χfull,r (μ) = χfull,r ({μ}),

𝒰 𝒰 χfull (μ) = χfull ({μ}).

Looking at the free entropy of a single type μ is the analog of what Voiculescu originally did for noncommutative ∗-laws (quantifier-free types), since he defined the entropy for the noncommutative ∗-law of a tuple rather than more generally for a set of ∗-laws. However, the variational principle (Proposition 3.7) in the next section will allow us to express the entropy of a closed set in terms of the entropy of individual types. We begin with a few immediate observations that will be useful in the sequel. Observation 3.3. Since the map 𝕊d,r (Ttr ) → 𝕊d (Ttr ) is a topological embedding, every open subset 𝒪 of the former is the restriction of an open set 𝒪′ in 𝕊d (Ttr ). In fact, because 𝕊d,r (Ttr ) is closed in 𝕊d (Ttr ), we can take 𝒪 = 𝒪 ∪ [𝕊d (Ttr ) \ 𝕊d,r (Ttr )]. ′

𝒰 Thus, in the definition of χfull,r (𝒦) in Definition 3.2, we may take 𝒪 to range over all open neighborhoods of 𝒦 in 𝕊d (Ttr ) or all open neighborhoods of 𝒦 ∩ 𝕊d,r (Ttr ) in 𝕊d,r (Ttr ), and the resulting quantity will be the same.

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(n) ′ Observation 3.4 (Monotonicity). Note that if 𝒦 ⊆ 𝒦′ ⊆ 𝕊d (Ttr ), then Γ(n) r (𝒦) ⊆ Γr (𝒦 ). It follows from this that if 𝒪 ⊆ 𝕊d,r (Ttr ) is open, then 𝒰 χfull,r (𝒪) = lim ( n→𝒰

1 log vol Γ(n) r (𝒪 ) + 2d log n), n2

that is, the infimum over neighborhoods of 𝒪 is achieved by 𝒪 itself. Similarly, it follows 𝒰 𝒰 𝒰 𝒰 that if 𝒦 ⊆ 𝒦′ , then χfull,r (𝒦) ≤ χfull,r (𝒦′ ) and hence χfull (𝒦) ≤ χfull (𝒦′ ). Remark 3.5. A similar argument as in [55, Proposition 2.4] shows that if 𝒦 ⊆ 𝕊d,r (Ttr ), and if r < r1 < r2 , then 𝒰 𝒰 𝒰 χfull,r (𝒦) = χfull,r (𝒦) = χfull,r (𝒦). 1 2

3.2 Variational principle In this section, we show that the free entropy defines an upper semicontinuous function on the type space, and then deduce a variational principle for the entropy of a closed set, in the spirit of various results in the theory of entropy and large deviations. Lemma 3.6 (Upper semicontinuity). For each r ∈ (0, ∞) and d ∈ ℕ, the function μ 󳨃→ 𝒰 χfull,r (μ) is upper semicontinuous on 𝕊d (Ttr ). Proof. Fix r. For each open 𝒪 ⊆ 𝕊d (Ttr ), let χ 𝒰 (𝒪), f𝒪 (μ) = { full,r ∞,

μ ∈ 𝒪, otherwise.

𝒰 Since 𝒪 is open, f𝒪 is upper semicontinuous. Moreover, χfull,r (μ) is the infimum of f𝒪 (μ) over all open 𝒪 ⊆ 𝕊(Ttr ), and the infimum of a family of upper semicontinuous functions is upper semi-continuous.

Proposition 3.7 (Variational principle). Let 𝒦 ⊆ 𝕊d (Ttr ) and let r > 0. Then 𝒰 𝒰 𝒰 sup χfull,r (μ) ≤ χfull,r (𝒦) ≤ sup χfull,r (μ).

(3.1)

𝒰 𝒰 𝒰 sup χfull (μ) ≤ χfull (𝒦) ≤ sup χfull (μ).

(3.2)

μ∈𝒦

μ∈cl(𝒦)

Hence, μ∈𝒦

μ∈cl(𝒦)

In particular, if 𝒦 is closed, then these inequalities are equalities. 𝒰 𝒰 Proof. If μ ∈ 𝒦, then by monotonicity χfull,r ({μ}) ≤ χfull,r (𝒦). Taking the supremum over μ ∈ 𝒦, we obtain the first inequality of (3.1).

236 � D. Jekel 𝒰 For the second inequality of (3.1), let C = supμ∈cl(𝒦) χfull,r (μ). If C = ∞, there is ′ nothing to prove. Otherwise, let C > C. For each μ ∈ cl(𝒦) ∩ 𝕊d,r (Ttr ), there exists some 𝒰 open neighborhood 𝒪μ of μ in 𝕊d,r (Ttr ) such that χfull,r (𝒪μ ) < C ′ . Since {𝒪μ }μ∈cl(𝒦)∩𝕊d,r (Ttr ) is an open cover of the compact set cl(𝒦)∩𝕊d,r (Ttr ), there exist μ1 , . . . , μk ∈ cl(𝒦)∩𝕊d,r (Ttr ) such that k

cl(𝒦) ∩ 𝕊r (Ttr ) ⊆ ⋃ 𝒪μj . j=1

Let 𝒪 = ⋃kj=1 𝒪μj . Then k

(n) (n) vol Γ(n) r (𝒪 ) ≤ ∑ vol Γr (𝒪μk ) ≤ k max vol Γr (𝒪μj ). j

j=1

Thus, 1 1 1 (n) log vol Γ(n) r (𝒪 ) + 2d log n ≤ 2 log k + max 2 log vol Γr (𝒪μj ) + 2d log n. j n2 n n Taking the limit as n → 𝒰 , 𝒰 𝒰 χfull,r (cl(𝒦)) ≤ χr𝒰 (𝒪) ≤ max χfull,r (𝒪μj ) ≤ C ′ . j

Since C ′ > C was arbitrary, 𝒰 𝒰 𝒰 χfull,r (𝒦) ≤ χfull,r (cl(𝒦)) ≤ C = sup χfull,r (μ), μ∈cl(𝒦)

completing the proof of (3.1). Taking the supremum over r in (3.1), we obtain (3.2). Remark 3.8. If we assume that 𝒦 is a closed subset of 𝕊d,r (Ttr ), then χ 𝒰 (μ) = χr𝒰 (μ) for all μ ∈ 𝒦. Hence, by upper semicontinuity, the supremum in (3.2) is a maximum.

3.3 Entropy and ultraproduct embeddings Lemma 3.9 (Ultraproduct realization of types). Let 𝒬 = ∏n→𝒰 Mn (ℂ). Let μ ∈ 𝕊d (Ttr ). If 𝒰 χfull (μ) > −∞, then there exists X ∈ 𝒬d such that tp𝒬 (X) = μ. 𝒰 𝒰 Proof. If χfull (μ) > −∞, then χfull,r (μ) > −∞ for some r ∈ (0, ∞). Since 𝕊d,r (Ttr ) is metrizable, there is a sequence (𝒪k )k∈ℕ of neighborhoods of μ in 𝕊(T) such that 𝒪k+1 ⊆ 𝒪k and ⋂k∈ℕ 𝒪k = {μ}. For k ∈ ℕ, let

Ek = {n ∈ ℕ : n ≥ k, Γ(n) r (𝒪k ) ≠ ⌀}.

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Now choose X(n) ∈ Mn (ℂ)d as follows. For each n ∈ ̸ E1 , set X(n) = 0. For each n ∈ Ek \Ek+1 , let X(n) be an element of Γ(n) r (𝒪k ). Since 𝒰 is an ultrafilter, either Ek ∈ 𝒰 or Ekc ∈ 𝒰 . If we had Ekc ∈ 𝒰 , then lim (1/n2 ) log vol(Γ(n) (𝒪k )) + 2d log n

n→𝒰

would be −∞ since Γ(n) (𝒪k ) would be empty for n ∈ Ekc . Hence, Ek ∈ 𝒰 . For n ∈ Ek , we have tpMn (ℂ) (X(n) ) ∈ 𝒪k . Therefore, limn→𝒰 tpMn (ℂ) (X(n) ) ∈ 𝒪k . Since this holds for all k, limn→𝒰 tpMn (ℂ) (X(n) ) = μ. Let X = [X(n) ]n∈ℕ ∈ 𝒬d . Then tp𝒬 (X) = lim tpMn (ℂ) (X(n) ) = μ. n→𝒰

Then because 𝒬 is countably saturated, the same reasoning as [20, Lemma 4.12] implies the following result, which will later play a role in the proof of Theorem 1.4. Corollary 3.10. Suppose that ℳ is a separable tracial W∗ -algebra and X ∈ ℳd . If 𝒰 χfull (tpℳ (X)) > −∞, then there exists an elementary embedding ι : ℳ → 𝒬 (thus ℳ must be elementarily equivalent to 𝒬). Since the embedding ι : ℳ → 𝒬 is elementary, in 𝒰 𝒰 particular tp𝒬 (ι(X)) = tpℳ (X), and hence χfull (tp𝒬 (ι(X))) = χfull (tpℳ (X)).

4 Entropy for quantifier-free and existential types 𝒰 In this section, we relate the free entropy for types χfull back to Voiculescu’s original free entropy from [55], as well as free entropy in the presence from [56]. In particular, we show that Voiculescu’s free entropy χ 𝒰 (X) is the version for quantifier-free types and the entropy in the presence χ 𝒰 (X : ℳ) is the version for existential types.

4.1 Entropy for quantifier-free types Voiculescu’s definition of entropy for quantifier-free types is exactly analogous to Definitions 3.1 and 3.2. Definition 4.1 (Entropy for quantifier-free types). For 𝒦 ⊆ 𝕊qf,d (Ttr ) and r ∈ (0, ∞), we define d

M (ℂ)

n (ℂ) Γ(n) (𝒦) := {X ∈ (DM ) : tpqfn r qf,r

(X) ∈ 𝒦}.

Then we define for r ∈ (0, ∞), 𝒰 χqf,r (𝒦) :=

inf

lim (

𝒪⊇𝒦 open in 𝕊qf,d,r (Ttr ) n→𝒰

1 log vol Γ(n) (𝒪) + 2d log n), qf,r n2

238 � D. Jekel and we set 𝒰 𝒰 χqf (𝒦) := sup χqf,r (𝒦). r∈(0,∞)

𝒰 𝒰 For μ ∈ 𝕊qf (Ttr ), let χqf (μ) := χqf ({μ}).

Note that this microstate space is the same one that appeared in the initial definition of Section 1.2 without the qf subscript; however, at this point it is convenient to introduce the subscript to distinguish it from the microstate spaces for the full discussed earlier. Voiculescu [55, Definition 2.1] originally defined χqf in terms of particular open sets defined by looking at moments of order up to m being within some distance ϵ of the moments of μ; in other words, he used a neighborhood basis μ in 𝕊qf,d,r (Ttr ) rather than all open sets. The following lemma shows that our definition is equivalent. Lemma 4.2. Let r ∈ (0, ∞). Let 𝒦 ⊆ 𝕊qf,d,r (Ttr ). Let (𝒪k )k∈ℕ be a sequence of open subsets of 𝕊(Ttr ) such that 𝒪k+1 ⊆ 𝒪k and ⋂∞ k=1 𝒪k = 𝒦. Then 𝒰 𝒰 𝒰 χqf,r (𝒦) = lim χqf,r (𝒪k ) = inf χqf,r (𝒪k ). k→∞

k∈ℕ

𝒰 𝒰 The same holds with 𝕊qf,d,r (Ttr ) replaced by 𝕊d,r (Ttr ) and χqf,r replaced by χfull,r .

Proof. The argument for the case of quantifier-free types and the case of full types is the same, and we will write the argument for full types. By Observation 3.4, 𝒰 𝒰 𝒰 χfull,r (𝒦) ≤ χfull,r (𝒪k+1 ) ≤ χfull,r (𝒪k ),

so that 𝒰 𝒰 𝒰 χfull,r (𝒦) ≤ inf χfull,r (𝒪k ) = lim χfull,r (𝒪k ). k∈ℕ

k→∞

For the inequality in the other direction, fix 𝒪 ⊇ 𝒦 open. Then 𝕊d,r (Ttr ) \ 𝒪 is closed and c contained in 𝒦c = ⋃k∈ℕ 𝒪kc = ⋃k∈ℕ 𝒪k . By compactness, there is a finite subcollection c of 𝒪k ’s that covers 𝕊d,r (Ttr ) \ 𝒪. The 𝒪k ’s are nested, so there exists some k such that c 𝕊d,r (Ttr ) \ 𝒪 ⊆ 𝒪k , hence 𝒪k ∩ 𝕊d,r (Ttr ) ⊆ 𝒪. Therefore, 𝒰 𝒰 inf χfull,r (𝒪k ) ≤ χfull,r (𝒪).

k∈ℕ

𝒰 Since 𝒪 was an arbitrary neighborhood of 𝒦, we conclude that infk∈ℕ χfull,r (𝒪k ) ≤ 𝒰 χfull,r (𝒦) as desired.

This lemma also allows us to relate the entropy for quantifier-free types directly to the entropy for types.

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239

Lemma 4.3. Let π : 𝕊d (Ttr ) → 𝕊qf,d (Ttr ) be the canonical restriction map. Let 𝒦 ⊆ 𝕊qf,d (Ttr ) be closed. Then 𝒰 𝒰 χqf (𝒦) = χfull (π −1 (𝒦)).

Proof. Fix r ∈ (0, ∞)𝒩 , and let 𝒦r = 𝒦 ∩ 𝕊qf,d,r (Ttr ). Since 𝕊qf,d,r (Ttr ) is metrizable and 𝒦 ⊆ 𝕊qf,d,r is closed, there exists a sequence of open sets 𝒪k in 𝕊qf,d,r (Ttr ) such that 𝒪k+1 ⊆ 𝒪k and ⋂k∈ℕ 𝒪k = 𝒦r (and these can be extended to open sets in 𝕊qf,d (Ttr ) since the inclusion of 𝕊qf,d,r (Ttr ) is a topological embedding). Now π −1 (𝒪k ) is open in

𝕊d,r (Ttr ) and π −1 (𝒪k+1 ) ⊆ π −1 (𝒪k+1 ) ⊆ π −1 (𝒪k ) and ⋂k∈ℕ π −1 (𝒪k ) = π −1 (𝒦r ). Note that −1 Γ(n) (𝒪k ) = Γ(n) r (π (𝒪k )). Thus, using the previous lemma, qf,r 𝒰 𝒰 χqf,r (𝒦) = χqf,r (𝒦r ) 𝒰 = inf χqf,r (𝒪k ) k∈ℕ

𝒰 = inf χfull,r (π −1 (𝒪k ))

= =

k∈ℕ 𝒰 χfull,r (π −1 (𝒦r )) 𝒰 χfull,r (π −1 (𝒦)).

Taking the supremum over r completes the argument. In particular, by combining this with the variational principle (Proposition 3.7), we obtain the following corollary. Corollary 4.4. Let π : 𝕊d (Ttr ) → 𝕊qf,d (Ttr ) be the restriction map. If μ ∈ 𝕊qf,d (Ttr ), then 𝒰 𝒰 χqf (μ) = sup χfull (ν). ν∈π −1 (μ)

Remark 4.5. Furthermore, if r is large enough that μ ∈ 𝕊qf,d,r , then π −1 (μ) ⊆ 𝕊d,r . Hence, 𝒰 𝒰 for ν ∈ π −1 (μ), we have χfull,r+1 (ν) = χfull (μ) by Remark 3.5. Finally, by Lemma 3.6, χ 𝒰 −1 achieves a maximum on π (𝒦). It follows that 𝒰 𝒰 χqf (μ) = max χfull (ν). ν∈π −1 (μ)

4.2 Entropy for existential types Here we define the entropy for existential types; see Section 2.4 for background. In the next subsection, we will show that this corresponds to Voiculescu’s χ(X : ℳ). Here 𝕊∃,d (Ttr ) will denote the spaces of existential types for d-tuples from tracial W∗ -algebras and 𝕊∃,d,r (Ttr ) will denote those arising from operators bounded in operator norm by r.

240 � D. Jekel Mn (ℂ)

Definition 4.6. For 𝒦 ⊆ 𝕊∃,d (Ttr ) and r ∈ (0, ∞), let Γ(n) ∃,r (𝒦) = {X ∈ ∏j∈ℕ Dr

M (ℂ) tp∃ n (X)

:

∈ 𝒦}, and define 𝒰 χ∃,r (𝒦) =

inf

lim (

𝒪⊇𝒦 open n→𝒰

1 log vol Γ(n) ∃,r (𝒪 ) + 2d log n). n2

Then let 𝒰 χ∃𝒰 (𝒦) = sup χ∃,r (𝒦). r∈(0,∞)

Because of the non-Hausdorff nature of 𝕊∃ (Ttr ), we will restrict our attention to an individual existential type rather than to a closed set of existential types. Lemma 4.7. Let μ ∈ 𝕊∃,d (Ttr ). Let π∃ : 𝕊(Ttr ) → 𝕊∃,d (Ttr ) be the canonical restriction map. Then 𝒰 χ∃𝒰 (μ) = χfull (π −1 (𝒦μ )) =

𝒰 max χfull (ν),

ν∈π −1 (𝒦μ )

where 𝒦μ is given by (2.1). Proof. Fix r ∈ (0, ∞). If 𝒪 is a neighborhood of μ in 𝕊∃,d (Ttr ), then it contains 𝒦μ , and

(n) −1 hence π∃−1 (𝒪) is a neighborhood of π −1 (𝒦μ ) in 𝕊(Ttr ). Moreover, Γ(n) ∃,r (𝒪 ) = Γr (π∃ (𝒪 )), hence 𝒰 𝒰 χfull,r (π∃−1 (𝒦μ )) ≤ χ∃,r (μ).

It remains to show the reverse inequality. Since the space of definable predicates on Ddr relative to Ttr is separable with respect to the uniform metric, so is the space of existential definable predicates. Let (ϕj )j∈ℕ be a sequence of existential definable predicates that are dense in this space. Let 𝒪k = {ν ∈ 𝕊∃,d,r (Ttr ) : ν(ϕj ) < μ(ϕj ) +

1 for j ≤ k}. k

Note that ⋂ 𝒪k = {ν ∈ 𝕊∃,d,r (Ttr ) : ν(ϕk ) ≤ μ(ϕk ) for k ∈ ℕ} = 𝒦μ .

k∈ℕ

Moreover, π∃−1 (𝒪k+1 ) ⊆ {ν ∈ 𝕊d (Ttr ) : ν(ϕj ) ≤ μ(ϕj ) +

1 for j ≤ k + 1} ⊆ π∃−1 (𝒪k ). k+1

Therefore, by Lemma 4.2 applied to π∃−1 (𝒪k ), we have

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� 241

𝒰 𝒰 𝒰 χ∃,r (μ) ≤ inf χ∃,r (𝒪k ) = inf χfull,r (π∃−1 (𝒪k )) = χr𝒰 (π∃−1 (𝒦)), k∈ℕ

k∈ℕ

𝒰 where the last equality follows from the density of {ϕk : k ∈ ℕ}. Thus, χ∃,r (μ) = 𝒰 −1 χfull,r (π∃ (𝒦μ )). Taking the supremum over r yields the first asserted equality χ∃𝒰 (μ) = 𝒰 χfull (π∃−1 (𝒦μ )). By applying the variational principle (Proposition 3.7) to the closed set 𝒰 π∃−1 (𝒦μ ), this is equal to supν∈π −1 (𝒦μ ) χfull (ν). Then the same argument as in Remark 4.5 ∃ shows that the supremum is in fact a maximum.

4.3 Existential entropy and entropy in the presence Let us finally explain why the existential entropy defined here agrees with (the ultrafilter version of) Voiculescu’s χ 𝒰 (X, ℳ). The definition is given in terms of Voiculescu’s microstate spaces for some d-tuple X in the presence of some m-tuple Y. Definition 4.8 (Voiculescu [56, Definition 1.1]). Let ℳ be a tracial W∗ -algebra. Let d, m ∈ ℕ and let X ∈ ℳd and Y ∈ ℳm . Let r ∈ (0, ∞). Let 𝕊qf,d+m,r (Ttr ) be the set of quantifierfree types of tuples from ∏i∈I Dr × ∏j∈J Dr equipped with the weak-∗ topology. Let p : Mn (ℂ)d+m → Mn (ℂ)d be the canonical projection onto the first d coordinates. Then we define χr𝒰 (X : Y) :=

inf

lim (

𝒪∋tpℳ (X,Y) n→𝒰 qf

1 log vol p[Γ(n) (𝒪)] + 2d log n), qf,r n2

where 𝒪 ranges over all neighborhoods of tpℳ qf (X, Y) in 𝕊qf,d+m (Ttr ). Then set χ 𝒰 (X : Y) := sup χr𝒰 (X : Y). r>0

Finally, we define χ ℳ (X : ℳ) = inf infm χ 𝒰 (X : Y). m∈ℕ Y∈M

Remark 4.9. It follows from [56, Proposition 1.6] that for finite tuples X, Y, and Z, χ 𝒰 (X : Y, Z) ≤ χ 𝒰 (X : Y) with equality when Z comes from W∗ (X, Y). As a consequence, χ 𝒰 (X : W∗ (X, Y)) = χ 𝒰 (X : Y). In fact, by a similar slightly more technical argument one can show that if ℳ is generated by X and Y1 , Y2 , . . . , then χ 𝒰 (X : ℳ) = lim χ 𝒰 (X : Y1 , . . . , Yk ). k→∞

242 � D. Jekel The idea behind why χ 𝒰 (X : ℳ) corresponds to the entropy of tpℳ ∃ (X) is that a matrix tuple X′ is in the projection p[Γ(n) ( 𝒪 )] if and only if there exists some Y′ such that qf,r M (ℂ)

tpqfn

(X′ , Y′ ) ∈ 𝒪. If X′ , Y′ being in Γ(n) (𝒪) can be detected by a quantifier-free formula qf,r

being less than some c, then X′ being in p[Γ(n) (𝒪)] can be detected by an existential qf,r formula. Proposition 4.10. In the setup of Definition 4.8, we have χ 𝒰 (X : ℳ) = χ∃𝒰 (tpℳ ∃ (X)).

Proof. First, let us show that χ 𝒰 (X : ℳ) ≤ χ∃𝒰 (tpℳ ∃ (X)). Fix r ∈ (0, ∞) with ‖Xj ‖ < r. Let 𝒪 be a neighborhood of tpℳ (X) in 𝕊 (T ). Then there exist existential definable ∃,d,r tr ∃ predicates ϕ1 , . . . , ϕk and ϵ1 , . . . , ϵk > 0 such that {ν ∈ 𝕊∃,d,r (Ttr ) : ν(ϕj ) ≤ μ(ϕj ) + ϵj for j = 1, . . . , k} ⊆ 𝒪. Since existential formulas are dense in the space of existential definable predicates, we can assume without loss of generality that the ϕj ’s are existential formulas, so there exist quantifier-free formulas ψ1 , . . . , ψk such that ′ ϕ𝒩 j (X ) =

inf

mj Y′ ∈∏i=1

D𝒩 ti,j

′ ′ ′ ℕ ψ𝒩 j (X , Y ) for all 𝒩 󳀀󳨐 Ttr and X ∈ 𝒩 .

By rescaling the variables Y′ in these formulas, we can assume without loss of generality m that ti,j < r. Moreover, for our particular ℳ and X, there exists Yj ∈ ∏i=1j Dℳ ti,j such that ψℳ j (X, Yj ) < μ(ϕj ) + ϵj . Then 𝒪 := {tpqf (X , Y1 , . . . , Yk ) ∈ 𝕊qf,d+m1 +⋅⋅⋅+mk ,r′ (Ttr ) : ψj (X , Yj ) < μ(ϕj ) + ϵj for j = 1, . . . , k} ′

𝒩

′

′

′

′

′

is a neighborhood of tpℳ qf (X, Y1 , . . . , Yk ) in 𝕊qf,d+m1 +⋅⋅⋅+mk ,r (Ttr ) such that p[Γ(n) (𝒪′ )] ⊆ Γ(n) ∃,r (𝒪 ). qf,r Therefore, vol p[Γ(n) (𝒪′ )] ≤ vol Γ(n) ∃,r (𝒪 ), qf,r which implies χr𝒰 (X : Y1 , . . . , Yk ) ≤ lim ( n→𝒰

1 𝒰 log vol p[Γ(n) (𝒪)] + 2d log n) ≤ χ∃,r (𝒪). qf,r n2

Free probability and model theory of tracial W∗ -algebras �

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By the same reasoning as [55, Proposition 2.4], since r > ‖Xj ‖∞ and r > ‖Yi,j ‖∞ , we have 𝒰 χ 𝒰 (X : ℳ) ≤ χ 𝒰 (X : Y1 , . . . , Yk ) = χr𝒰 (X : Y1 , . . . , Yk ) ≤ χ∃,r (𝒪).

Since 𝒪 was an arbitrary neighborhood of tpℳ ∃ (X), we have 𝒰 𝒰 ℳ χ 𝒰 (X : ℳ) ≤ χ∃,r (tpℳ ∃ (X)) ≤ χ∃ (tp∃ (X)).

To prove the second inequality, fix m and an m-tuple Y ∈ M m . Let 𝒪′ be a neighborhood of tpℳ qf (X, Y). By definition of the weak-∗ topology on Σqf,d+m,r , there exist quantifier-free formulas ψ1 , . . . , ψk and ϵ1 , . . . , ϵk > 0 such that 󵄨 󵄨󵄨 ′ {σ ∈ 𝕊qf,d+m,r : 󵄨󵄨󵄨σ(ψj ) − ψℳ j (X, Y)󵄨󵄨 < ϵj for j = 1, . . . , k} ⊆ 𝒪 . Now consider the existential formula ϕ𝒩 (X′ ) =

inf

max (

′ ′ ℳ |ψ𝒩 j (X , Y ) − ψj (X, Y)|

m j=1,...,k Y′ ∈(D𝒩 r )

ϵj

)

for every tracial W∗ -algebra 𝒩 . Note that ϕ𝒩 (X′ ) < 1 implies that there exists some tuple m 𝒩 ′ ′ ℳ 𝒩 ′ ′ Y′ from (D𝒩 r ) with |ψj (X , Y ) − ψj (X, Y)| < ϵj for j = 1, . . . , k and hence tpqf (X , Y ) ∈ ′ 𝒪 . In particular, the set 𝒪 := {μ ∈ 𝕊∃,d,r (Ttr ) : μ(ϕ) < 1}

is an open subset of 𝕊∃,d,r (Ttr ) such that (n) ′ Γ(n) ∃,r (𝒪 ) ⊆ p[Γqf,r (𝒪 )]. 𝒰 𝒰 Because for every 𝒪′ there exists such an 𝒪, we obtain χ∃,r (tpℳ ∃ (X)) ≤ χr (X : Y). Then we take the supremum over r and the infimum over m and Y to complete the argument.

4.4 Proof of Theorem 1.4 Now we complete the proof of Theorem 1.4. We state the theorem here with slight additional claims that we could not state earlier because we had not yet defined entropy for types. Theorem 4.11. Let ℳ be a separable tracial W∗ -algebra and X ∈ M d . If χ 𝒰 (X : ℳ) > −∞, then there exists an embedding of ι : ℳ → 𝒬 = ∏n→𝒰 Mn (ℂ) such that 𝒰 χ 𝒰 (ι(X) : 𝒬) = χfull (tp𝒬 (ι(X))) = χ 𝒰 (X : ℳ).

244 � D. Jekel Proof. Let π∃ be the projection from full types to existential types, and let μ = tpℳ ∃ (X). By Lemma 4.7, there exists ν ∈ π∃−1 (𝒦μ ) such that χ 𝒰 (ν) = χ∃𝒰 (μ). By Lemma 3.9, there exists X′ ∈ 𝒬d with tp𝒬 (X′ ) = ν. Let Y ∈ ∏j∈ℕ Dℳ R be an ℕ-tuple that generates ℳ, and write (X, Y) = (X1 , . . . , Xd , Y1 , Y2 , . . . ). Since the space of quantifier-free types 𝕊qf,ℕ,R (Ttr ) for ℕ-tuples is compact and metrizable, there exists a quantifier-free definable predicate ϕ ≥ 0 such that for tracial 𝒩 𝒩 W∗ -algebras 𝒩 and for Z ∈ ∏dj=1 D𝒩 R and W ∈ ∏j∈ℕ DR , we have ϕ (Z, W) = 0 if and ℳ only if tp𝒩 qf (Z, W) = tpqf (X, Y). Note that

ψ𝒩 (Z) =

inf

W∈∏j∈ℕ D𝒩 R

ϕ(Z, W)

is an existential definable predicate over Ttr in the variables Z1 , . . . , Zd , and 0 ≤ ψ𝒬 (X′ ) ≤ ψℳ (X) ≤ ϕℳ (X, Y) = 0, where we have applied the fact that tp∃ (X′ ) = π∃ (ν) ∈ 𝒦μ .

M (ℂ)

Now we may write X′ = [X(n) ]n∈ℕ where X(n) ∈ ∏j∈ℕ Dr n (this is well known, and it follows for instance from the construction of X′ through Lemma 3.9). Then lim ψMn (ℂ) (X(n) ) = ψ𝒬 (X′ ) = 0,

n→𝒰

M (ℂ)

hence there exists Y(n) ∈ ∏j∈ℕ Dr′ n

such that

lim ϕMn (ℂ) (X(n) , Y(n) ) = 0.

n→𝒰

Let Y′ = [Y(n) ]n∈ℕ ∈ ∏j∈ℕ D𝒬 . Then ϕ𝒬 (X′ , Y′ ) = 0, and therefore, tp𝒬 (X′ , Y′ ) = qf r′ j

′ ′ ′ ′ tpℳ qf (X , Y ). Hence, there exists an embedding ι : ℳ → 𝒬 with ι(X, Y) = (X , Y ). By Lemma 4.7 and Proposition 4.10, we have 𝒰 𝒰 ′ 𝒰 ′ 𝒰 χfull (ν) = χfull (tp𝒬 (X′ )) ≤ χ∃𝒰 (tp𝒬 ∃ (X )) = χ (X : 𝒬) = χ∃ (π∃ (ν)).

Meanwhile, since Kπ∃ (ν) ⊆ 𝒦μ , χ∃𝒰 (π∃ (ν)) ≤ χ∃𝒰 (μ) = χ 𝒰 (X : ℳ). 𝒰 By our choice of ν, χfull (ν) = χ∃𝒰 (μ), so all these inequalities are equalities.

Remark 4.12. We mentioned in Section 1.2 that one can prove the existence of an embedding into a matrix ultraproduct that has trivial relative commutant by showing that a randomly chosen embedding has this property. This is an example of a probabilistic argument showing that there exists some x in a given set satisfying some conditions by establishing that a random x satisfies these conditions with high (or at least posi-

Free probability and model theory of tracial W∗ -algebras �

245

tive) probability. Our proof of Theorem 1.4 is in spirit also a probabilistic argument. The variational principle shows that there must be some μ ∈ π∃−1 (𝒦μ ) with entropy equal to χ∃𝒰 (μ). Unwinding the proof of Proposition 3.7, our intuition is that an existential microstate space for μ can be covered by microstate spaces for finitely many ν ∈ π∃−1 (μ) and so there must be some ν which carries part of the mass of the microstate space for μ that is not exponentially small compared to the whole.

5 Toward a notion of independence In this section, we explore possible ways of adapting free independence to the modeltheoretic framework for tracial W∗ -algebras. We state many related open questions at the intersection of random matrix theory and model theory.

5.1 Noncommutative probability viewpoint The notion of free independence mentioned in the introduction leads to the notion of free convolution analogous to the convolution of two probability measures on ℝd in classical probability. Suppose we have two ∗-laws μ, ν ∈ 𝕊qf,d (Ttr ). If X and Y are d-tuples ℬ 𝒜∗ℬ from 𝒜 and ℬ with tp𝒜 (X + Y) is qf (X) = μ and tpqf (Y) = ν, then it turns out that tpqf uniquely determined by μ and ν. This quantifier-free type tp𝒜∗ℬ (X + Y) is called the qf (additive) free convolution of μ and ν and it is denoted μ ⊞ ν.

Question 5.1. Is there a model-theoretic analog of free independence? Is there a notion of free convolution for full types? A naïve approach to constructing free convolution for types would be as follows. Consider types μ and ν. Let X and Y be d-tuples from 𝒜 and ℬ respectively with tp𝒜 (X) = μ and tpℬ (Y) = ν. Then we may try to define tp𝒜 (X) + tpℬ (Y) as tp𝒜∗ℬ (X + Y); however, it is not clear whether this is well defined. Question 5.2. Suppose that tp𝒜1 (X) = tp𝒜2 (X′ ) and tpℬ1 (Y) = tpℬ2 (Y′ ). Then does tp𝒜1 ∗ℬ1 (X + Y) = tp𝒜2 ∗ℬ2 (X′ + Y′ )? This in turn provokes an even more basic question about the relationship between free products and model theory. Observe that tp𝒜1 (X) = tp𝒜2 (X′ ) implies that 𝒜1 ≡ 𝒜2 because every sentence can be considered as a formula in the free variables x1 , . . . , xd (which happens to be independent of those variables). Question 5.3. Do free products preserve elementary equivalence? In other words, if 𝒜1 ≡ 𝒜2 and ℬ1 ≡ ℬ2 , then does it necessarily follow that 𝒜1 ∗ 𝒜2 ≡ ℬ1 ∗ ℬ2 ? The analogous question for free products of groups was highly nontrivial, and was answered in the affirmative by Sela [46, Theorem 7.1]. An affirmative answer for the

246 � D. Jekel groupoid case was given much earlier by Olin [42]. One can also ask the analogous question but with tensor products rather than free products.1 For modules, tensor products do not preserve elementary equivalence [41]. For C∗ -algebras, tensor products do not preserve elementary equivalence in general, but tensoring with continuous functions on the Cantor set does [17, Section 6]. This question in the von Neumann algebra case has been partially answered by Farah and Ghasemi [18].

5.2 Random matrix viewpoint However, it is unclear whether the free product is even the “right” construction of independence for types from a random matrix viewpoint. In the quantifier-free setting, free independence can be characterized through random matrix theory as follows. Recall that if Xn is a random variable on a probability space (Ωn , ℱn , Pn ) with values in a topological space 𝒳 and if x ∈ 𝒳 , we say that Xn converges to x in probability if for every neighborhood 𝒪 of x, we have Pn (Xn ∈ 𝒪) = Pn (Xn−1 (𝒪)) → 1. Similarly, if Xn is a real random variable and x ∈ ℝ, we say that lim supn→∞ Xn ≤ x in probability if for every ϵ > 0, we have limn→∞ Pn (Xn < x + ϵ) = 1. Theorem 5.4 (Voiculescu [54, 58]). Let X and Y be d-tuples in a tracial W∗ -algebra ℳ that are freely independent. Let X(n) and Y(n) be random variables in Mn (ℂ)d such that M (ℂ) Mn (ℂ) (n) (1) tpqfn (X(n) ) → tpℳ (Y ) → tpℳ qf (X) and tpqf qf (Y) in probability (as random elements of the space 𝕊qf,d (Ttr )).

(2) For some constant C, we have lim supn→∞ ‖Xj(n) ‖ ≤ C and lim supn→∞ ‖Yj(n) ‖ ≤ C in probability. (3) For each n ∈ ℕ, X(n) and Y(n) are classically independent as Mn (ℂ)d -valued random variables, or in other words, the family (Xj(n) )k,ℓ for j = 1, . . . , d and k, ℓ = 1, . . . , N and the family (Yj(n) )k,ℓ for j = 1, . . . , d and k, ℓ = 1, . . . , N are classically independent.

(4) The probability distributions of X(n) and of Y(n) are invariant under conjugation by each n × n unitary matrix U. M (ℂ)

Then tpqfn

M (ℂ)

n (X(n) , Y(n) ) → tpℳ qf (X, Y) in probability. Thus, in particular, tpqf

(X(n) +

ℳ Y(n) ) → tpℳ qf (X) ⊞ tpqf (Y).

Question 5.5. Is there some analog of Theorem 5.4 for full types rather than only quantifier-free types? There are several issues in addressing this broad question. The first is whether we want to use a limit as n → ∞ or an ultralimit as n → 𝒰 . In order for tpMn (ℂ) (X(n) )

1 These questions are also inspired by the Feferman–Vaught theorem which asserts that direct products preserve elementary equivalence [22]. The continuous version was given in [25].

Free probability and model theory of tracial W∗ -algebras

� 247

to converge as n → ∞ to tpℳ (X), it would be necessary for Th(Mn (ℂ)) → Th(ℳ) as n → ∞. Indeed, every sentence is a formula with no free variables which can also be viewed as a formula in variables x1 , . . . , xd . Thus, the type of a d-tuple includes the information of the theory of the tracial W∗ -algebra that they came from. This question was asked by Popa and also by Farah, Hart, and Sherman in [20]. Question 5.6. Does limn→∞ Th(Mn (ℂ)) exist, or in other words, for each formula ϕ with no free variables, does limn→∞ ϕMn (ℂ) exist? Equivalently, are ∏n→𝒰 Mn (ℂ) and ∏n→𝒱 Mn (ℂ) elementarily equivalent for all free ultrafilters 𝒰 and 𝒱 on ℕ? This question is still wide open. Many of the tools used in asymptotic random matrix theory to prove results like Theorem 5.4 work for every large value of n. As such, they are not suited to distinguish between Th(Mn (ℂ)) and Th(Mm (ℂ)) for large values of n and m. However, perhaps group representations or quantum games could be used if one wants to show that matrix ultraproducts are not elementarily equivalent. To circumvent this question for the moment, let us consider limits with respect to a fixed ultrafilter 𝒰 . The second issue in Question 5.5 is that hypothesis (4) of unitary invariance is not strong enough to make an analog of Theorem 5.4 for full types. We illustrate this with Example 5.8 below. We will need the following lemma, which is an adaptation of the result that if all embeddings of 𝒜 into 𝒬 are unitarily conjugate, then the 1-bounded entropy h(𝒜) is less than or equal to zero, which implies that χ 𝒰 (X : 𝒜) = −∞ for every tuple X from 𝒜 (see [31]); the argument is similar to [56] and [31]. Lemma 5.7. Let μ ∈ 𝕊d (Ttr ) be a type realized in 𝒬. Let 𝒬 = ∏n→𝒰 Mn (ℂ). Suppose that for every X and Y ∈ 𝒬d realizing μ, there exists a unitary U ∈ 𝒬 such that UXU ∗ = Y, that is, UXj U ∗ = Yj for each j. Then χ 𝒰 (μ) = −∞. Proof. Fix a large r > 0. Let 𝒬 d

𝒦 = {tp (X, Y) : X, Y ∈ (Dr ) , tp (X) = tp (Y) = μ}. 𝒬

𝒬

𝒬

Since 𝒬 is countably saturated, 𝒦 is closed. Consider the definable predicate ψ

ℳ

d

(X, Y) = inf (∑ d Z∈D1

=

ℳ

j=1

inf

unitaries U∈ℳ

(e

πi(Z+Z ∗ )

Xj e

−πi(Z+Z ∗ )

2

1/2

, Yj ) )

󵄩󵄩 󵄩 ∗ 󵄩󵄩UXU − Y󵄩󵄩󵄩2 .

One can check that ψℳ is a definable predicate by expressing eπi(Z+Z ) as a power series and hence approximating it by noncommutative ∗-polynomials uniformly on each operator norm ball. Moreover, since every unitary U in a tracial W∗ -algebra can be ex∗ pressed as e2πiZ for some self-adjoint Z, we see that eπi(Z+Z ) for Z ∈ D1 will produce all the unitaries in ℳ. ∗

248 � D. Jekel Now by our assumption, ψ𝒬 (X, Y) = 0 whenever tp𝒬 (X, Y) ∈ 𝒦. Let 𝒪k be a sequence of neighborhoods of μ in 𝕊d,r (Ttr ) such that 𝒪k+1 ⊆ 𝒪k and ⋂k∈ℕ 𝒪k = {μ}. Then let ℳ

𝒪k = {tp ′

(X, Y) : ℳ 󳀀󳨐 Ttr , tpℳ (X) ∈ 𝒪k , tpℳ (Y) ∈ 𝒪k }.

′ ⊆ 𝒪k′ and ⋂k∈ℕ 𝒪k′ = 𝒦. Fix ϵ > 0. Since 𝒦 ⊆ {γ ∈ 𝕊2d,r : γ(ψ) = 0}, we see Then 𝒪k+1

that (𝕊2d,r (Ttr ) \ 𝒪k )k∈ℕ is an open cover of {γ ∈ 𝕊2d,r (Ttr ) : γ(ψ) ≥ ϵ}. Since the latter set is compact and the 𝒪k ’s are nested, there exists some k such that ′

{γ ∈ 𝕊2d,r : γ(ψ) ≥ ϵ} ⊆ 𝕊2d,r (Ttr ) \ 𝒪k , ′

or in other words 𝒪k ⊆ {γ ∈ 𝕊2d,r : γ(ψ) < ϵ}. ′

(n) (n) ′ Mn (ℂ) This implies that for all (X, Y) ∈ Γ(n) (X, Y) < ϵ, r (𝒪k ) × Γr (𝒪k ) = Γr (𝒪k ), we have ψ ∗ or in other words, there exists a unitary U ∈ Mn (ℂ) such that ‖UXU − Y‖2 < ϵ. 2 By [51, Theorem 7], the unitary group U(n) can be covered by (C/ϵ)n operator-norm balls of radius ϵ, where C is a universal constant (independent of n). Let (Ui )m i=1 be unitaries such that the balls B‖⋅‖ (Ui , ϵ) cover the unitary group. Now we claim that the balls B‖⋅‖2 (Ui XUi∗ , (2√dr + 1)ϵ) cover Γ(n) r (𝒪k ). To see this, let Y be in the microstate space (n) Γr (𝒪k ). Then there exists a unitary U such that ‖UXU ∗ − Y‖2 < ϵ. Next, there exists a unitary Uj such that ‖Uj − U‖ < ϵ. Then

󵄩󵄩 󵄩 󵄩 ∗ ∗󵄩 ∗󵄩 ∗ ∗ 󵄩 󵄩󵄩Ui Xj Ui − UXj U 󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩(Ui − U)Xj Ui 󵄩󵄩󵄩2 + 󵄩󵄩󵄩UXj (Ui − U )󵄩󵄩󵄩2 󵄩 󵄩 󵄩 󵄩 ≤ ‖Ui − U‖‖Xj ‖2 󵄩󵄩󵄩Ui∗ 󵄩󵄩󵄩 + ‖U‖‖Xj ‖2 󵄩󵄩󵄩Ui∗ − U ∗ 󵄩󵄩󵄩 ≤ 2rϵ. Thus, d

1/2

󵄩󵄩 󵄩 ∗ ∗󵄩 ∗ ∗ 󵄩2 󵄩󵄩Ui XUi − UXU 󵄩󵄩󵄩2 = (∑󵄩󵄩󵄩Ui Xj Ui − UXj U 󵄩󵄩󵄩2 ) j=1

≤ 2√drϵ.

So by the triangle inequality, ‖Ui XUi∗ − Y‖2 < 2√drϵ + ϵ. Hence, Γ(n) r (𝒪k ) is covered by the balls B‖⋅‖2 (Ui XUi∗ , (2√dr + 1)ϵ). 2

Now Mn (ℂ)d is a complex vector space of dimension dn2 , hence isometric to ℂdn . A ball of radius (2√dr + 1)ϵ therefore has Lebesgue measure 2

πn d 2n2 d [(2√dr + 1)ϵ] . 2 (n d)!

Free probability and model theory of tracial W∗ -algebras �

249

2

The number of balls needed to cover the microstate space is (C/ϵ)n , and hence 2

2

vol Γ(n) r (𝒪k )

2 2 2 2 2 2 C n π dn ≤ 2 2 (2√dr + 1)2dn ϵ2dn = C n π dn (2√dr + 1)2dn ϵ(2d−1)n . n ϵ (n d)!

Thus, 1 1 2 √ log vol Γ(n) r (𝒪k ) − 2 log(n d)! ≤ log C + d log π + 2d log(2 dr + 1) + (2d − 1) log ϵ. n2 n Now using Stirling’s formula, log(n2 d)! = n2 d log(n2 d) − n2 d + O(log(n2 d)), so that 1 log(n2 d)! = 2d log n + d(log d − 1) + O(log(n2 d)/n2 ), n2 where the last error term goes to zero as n → ∞. Hence, replacing (1/n2 ) log(n2 d)! by 2d log n does not change the asymptotic behavior of the above equation. After this replacement, sending n → ∞ results in χr𝒰 (μ) ≤ log C + d log π − d(log d − 1) + 2d log(2√dr + 1) + (2d − 1) log ϵ. Since ϵ was arbitrary, χr𝒰 (μ) = −∞, and since r was arbitrary χ 𝒰 (μ) = −∞. Example 5.8. This example shows that unitary invariance (4) is not a strong enough hypothesis to produce an analog of Theorem 5.4 for full types. Indeed, choose some type μ with χ 𝒰 (μ) > −∞. It follows that there exist two tuples X = [X(n) ]n∈ℕ and Y = [Y(n) ]n∈ℕ in 𝒬 with type μ that are not unitarily conjugate. Letting ψ be the formula in the previous lemma, we have lim ψMn (ℂ) (X(n) , Y(n) ) = ψ𝒬 (X, Y) > 0.

n→𝒰

(If ψ𝒬 (X, Y) were equal to zero, then since 𝒬 is countably saturated there would exist a unitary U with ‖UXU ∗ − Y‖2 = 0.) Now let U (n) and V (n) be independent random variables, each chosen according to the Haar probability measure on the n × n unitary group. Consider the two pairs of matrix models: (A) U (n) X(n) (U (n) )∗ and V (n) X(n) (V (n) )∗ ; (B) U (n) X(n) (U (n) )∗ and V (n) Y(n) (V (n) )∗ . Then both pairs of random matrix models satisfy the hypotheses of Theorem 5.4: (1) The types of U (n) X(n) (U (n) )∗ and V (n) X(n) (V (n) )∗ and V (n) Y(n) (V (n) )∗ in Mn (ℂ) converge to μ almost surely since the type is invariant under unitary conjugation.

250 � D. Jekel (2) The operator norms of X(n) and Y(n) are bounded. (3) Since X and Y are deterministic and U (n) and V (n) are independent, the tuples from (A) and (B) respectively are independent. (4) The probability distributions are unitarily invariant by construction because the Haar measure on the unitary group is left-invariant. However, we have ψMn (ℂ) (U (n) X(n) (U (n) ) , V (n) X(n) (V (n) ) ) = ψMn (ℂ) (X(n) , X(n) ) → 0 ∗

∗

and ψMn (ℂ) (U (n) X(n) (U (n) ) , V (n) Y(n) (V (n) ) ) = ψMn (ℂ) (X(n) , Y(n) ) → ψ𝒬 (X, Y) > 0. ∗

∗

So, tpMn (ℂ) (U (n) X(n) (U (n) )∗ , V (n) X(n) (V (n) )∗ ) and tpMn (ℂ) (U (n) X(n) (U (n) )∗ , V (n) Y(n) (V (n) )∗ ) cannot converge in probability to the same limit. If we want some analog of Theorem 5.4 for full types, what condition could we make that is stronger than unitary invariance? Intuitively, we want something like invariance under automorphisms of 𝒬, since (under the continuum hypothesis) any two tuples with the same type are conjugate by an automorphism of 𝒬. This would translate into invariance under approximate automorphisms on the matrix level. However, it is not clear to me how to formulate this notion precisely. Rather, we could think of an automorphism-invariant measure on the microstate space as one which gives equal weight to all the different microstates for the same type, or in other words, a uniform distribution on the microstate space. This motivates the following definition, which analogous to a known property of the independent join for ∗-laws [58, Lemma 3.5]. Definition 5.9 (Independent join). Let 𝒬 = ∏n→𝒰 Mn (ℂ). Let μ1 ∈ 𝕊d1 (Th(𝒬)) and μ2 ∈ 𝕊d2 (Th(𝒬)). We say that a type μ ∈ 𝕊d1 +d2 (Th(𝒬)) is an independent join of μ1 and μ2 if for every sufficiently large r, we have lim inf

lim inf

lim

𝒪↘μ (𝒪1 ,𝒪2 )↘(μ1 ,μ2 ) n→𝒰

(n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ))

= 1,

where the lim inf as 𝒪 ↘ γ denotes the lim inf over the directed system of neighborhoods of μ in 𝕊d1 +d2 ,r (Ttr ) ordered by reverse inclusion, and similarly (𝒪1 , 𝒪2 ) ↘ (μ1 , μ2 ) denotes the directed system of pairs of neighborhoods of μ1 and μ2 in 𝕊d1 ,r (Ttr ) and 𝕊d2 ,r (Ttr ), respectively, ordered by reverse inclusion of both elements of the pair. Intuitively, this definition says that for each neighborhood 𝒪 of γ, if the neighborhoods 𝒪1 and 𝒪2 are sufficiently small, then a tuple X chosen uniformly at random from (n) (n) Γ(n) r (𝒪1 ) × Γr (𝒪2 ) has a high probability of being in Γr (𝒪 ). We remark that the same definition makes sense for a triple (μ1 , μ2 , μ3 ) or in general any finite tuple of types.

Free probability and model theory of tracial W∗ -algebras

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The main issue with Definition 5.9 is existence: Question 5.10. Given μ1 ∈ 𝕊d1 ,r (Th(𝒬)) and μ2 ∈ 𝕊d2 ,r , does an independent join exist? Resolving this question will likely require a more detailed understanding of the behavior of formulas in independent random matrix tuples, just as Theorem 5.4 required analyzing the behavior of terms and quantifier-free formulas. This endeavor may also be entangled with Question 5.6. However, Definition 5.9 at least has several desirable properties. Observation 5.11 (Uniqueness). Given μ1 ∈ 𝕊d1 ,r (Th(𝒬)) and μ2 ∈ 𝕊d2 ,r , there is at most one independent join. Proof. Suppose that μ and μ′ are both independent joins of μ1 and μ2 . Fix a large r > 0. Since 𝕊d1 +d2 ,r (Ttr ) is Hausdorff, any sufficiently small neighborhoods of 𝒪 of μ and 𝒪′ of μ′ will be disjoint. Therefore, for all neighborhoods 𝒪1 and 𝒪2 of μ1 and μ2 , we will have (n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ))

+

′ (n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ))

≤ 1.

This implies that the limits in Definition 5.9 cannot be 1 for both μ and μ′ . Observation 5.12 (Symmetry). Suppose that tp𝒬 (X, Y) is the independent join of tp𝒬 (X) and tp𝒬 (Y). Then tp𝒬 (Y, X) is the independent join of tp𝒬 (Y) and tp𝒬 (X). This follows from the symmetry of the definition and the symmetry of the notion of neighborhoods in the logic topology. We leave the details to the reader. Observation 5.13 (Amalgamation property). Let μ1 ∈ 𝕊d1 (Th(𝒬)), μ2 ∈ 𝕊d2 (Th(𝒬)), and μ3 ∈ 𝕊d3 (Th(𝒬)). Let μ be an independent join of μ1 and μ2 , and let μ′ be an independent join of μ and μ3 . Then μ′ is an independent join of μ1 , μ2 , μ3 . Proof. Per the three-type analog of Definition 5.9, we need to show that lim′ inf ′

lim inf

lim

𝒪 ↘μ (𝒪1 ,𝒪2 ,𝒪3 )↘(μ1 ,μ2 ,μ3 ) n→𝒰

′ (n) (n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 ) × Γr (𝒪3 )]) (n) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ) × Γr (𝒪3 ))

= 1.

The lim inf on the left will not change if we insert an auxiliary variable 𝒪 representing a neighborhood of μ, so we can write our goal as lim′ inf lim inf ′ 𝒪 ↘μ

lim inf

lim

𝒪↘μ (𝒪1 ,𝒪2 ,𝒪3 )↘(μ1 ,μ2 ,μ3 ) n→𝒰

′ (n) (n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 ) × Γr (𝒪3 )]) (n) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ) × Γr (𝒪3 ))

Let 𝒪̃ := {tp

ℳ

(X, Y) : ℳ 󳀀󳨐 Ttr , tpℳ (X, Y) ∈ 𝒪, tpℳ (X) ∈ 𝒪1 , tpℳ (Y) ∈ 𝒪2 }.

= 1.

252 � D. Jekel (n) (n) ̃ Since Γ(n) r (𝒪 ) ⊆ Γr (𝒪1 )×Γr (𝒪2 ), the quantity that we want to estimate can be bounded from below by (n) (n) ′ ̃ vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪 ) × Γr (𝒪3 )])

(n) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ) × Γr (𝒪3 )) ′ (n) (n) (n) ̃ ̃ vol(Γ(n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪 ) × Γr (𝒪3 )]) r (𝒪 ) × Γr (𝒪3 )) = . (5.1) (n) (n) (n) (n) vol(Γr (𝒪̃ ) × Γr (𝒪3 )) vol(Γr (𝒪1 ) × Γr (𝒪2 ) × Γ(n) r (𝒪3 ))

To estimate the first term, we observe that for a fixed 𝒪, the net (𝒪̃ , 𝒪3 ) indexed by (𝒪1 , 𝒪2 , 𝒪3 ) ↘ (μ1 , μ2 , μ3 ) is a subnet of the net (𝒪1,2 , 𝒪3 ) of neighborhoods of (μ, μ3 ) (here 𝒪1,2 is another variable distinct from 𝒪 but also representing neighborhoods of μ). We therefore have lim inf

lim

(𝒪1 ,𝒪2 ,𝒪3 )↘(μ1 ,μ2 ,μ3 ) n→𝒰

≥

lim inf

lim

(𝒪1,2 ,𝒪3 )↘(μ,μ3 ) n→𝒰

(n) (n) ′ ̃ vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪 ) × Γr (𝒪3 )]) (n) (n) vol(Γr (𝒪̃ ) × Γr (𝒪3 ))

(n) (n) ′ vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1,2 ) × Γr (𝒪3 )]) (n) vol(Γ(n) r (𝒪1,2 ) × Γr (𝒪3 ))

.

On the right-hand side, applying lim inf𝒪↘μ does nothing, and then applying lim inf𝒪′ ↘μ′ yields the limit in the definition of μ′ being an independent join of μ and μ3 , and thus lim′ inf lim inf ′ 𝒪 ↘μ

lim inf

lim

𝒪↘μ (𝒪1 ,𝒪2 ,𝒪3 )↘(μ1 ,μ2 ,μ3 ) n→𝒰

′ (n) (n) ̃ vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪 ) × Γr (𝒪3 )]) ≥ 1. (n) ̃ vol(Γ(n) r (𝒪 ) × Γr (𝒪3 ))

Now the second term on the right-hand side of (5.1) is (n) ̃ vol(Γ(n) r (𝒪 ) × Γr (𝒪3 )) (n) (n) vol(Γr (𝒪1 ) × Γr (𝒪2 ) × Γ(n) r (𝒪3 ))

= =

̃ vol(Γ(n) r (𝒪 )) (n) (n) vol(Γr (𝒪1 ) × Γr (𝒪2 )) (n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) . (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ))

Taking the lim inf as (𝒪1 , 𝒪2 , 𝒪3 ) ↘ (μ1 , μ2 , μ3 ) is equivalent to taking the lim inf as (𝒪1 , 𝒪2 ) ↘ (μ1 , μ2 ) since the expression is independent of 𝒪3 . After that, we apply lim inf𝒪↘μ , which results in the limit in the definition of μ being an independent join of μ1 and μ2 . Therefore, lim′ inf lim inf ′ 𝒪 ↘μ

lim inf

lim

𝒪↘μ (𝒪1 ,𝒪2 ,𝒪3 )↘(μ1 ,μ2 ,μ3 ) n→𝒰

(n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ))

= lim′ inf 1 = 1. ′ 𝒪 ↘μ

Thus, when we multiply out the two terms in (5.1), we obtain lim′ inf lim inf ′ 𝒪 ↘μ

lim inf

lim

𝒪↘μ (𝒪1 ,𝒪2 ,𝒪3 )↘(μ1 ,μ2 ,μ3 ) n→𝒰

′ (n) (n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 ) × Γr (𝒪3 )]) (n) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ) × Γr (𝒪3 ))

≥ 1.

Free probability and model theory of tracial W∗ -algebras �

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On the other hand, this limit is clearly less than or equal to 1 by monotonicity of Lebesgue measure, so the proof is complete. We remark that if μ is an independent join of μ1 and μ2 according to Definition 5.9, then χ 𝒰 (μ) = χ 𝒰 (μ1 ) + χ 𝒰 (μ2 ). The argument is the same as [58, Theorem 3.8]. Indeed, the inequality χ 𝒰 (μ) ≤ χ 𝒰 (μ1 ) + χ 𝒰 (μ2 ) follows because for all neighborhoods 𝒪1 and 𝒪2 of μ1 and μ2 respectively, (n) Γ(n) r (𝒪1 ) × Γr (𝒪2 ) is a microstate space for μ. On the other hand, for any neighborhood 𝒪 of μ, we have 1 log vol Γ(n) r (𝒪 ) + (d1 + d2 ) log n n2 1 (n) (n) ≥ 2 log vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) + (d1 + d2 ) log n n (n) (n) vol(Γ(n) 1 1 r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) = 2 log + 2 log vol Γ(n) r (𝒪1 ) + d1 log n (n) (n) n n vol(Γr (𝒪1 ) × Γr (𝒪2 )) 1 log vol Γ(n) r (𝒪2 ) + d2 log n n2 (n) (n) vol(Γ(n) 1 r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) ≥ log + 2 log vol Γ(n) r (𝒪1 ) + d1 log n (n) (n) n vol(Γr (𝒪1 ) × Γr (𝒪2 )) 1 + 2 log vol Γ(n) r (𝒪2 ) + d2 log n. n +

Taking the limit as n → 𝒰 , we get χr𝒰 (𝒪) ≥ lim log n→𝒰

(n) (n) vol(Γ(n) r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) (n) vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ))

+ χr𝒰 (μ1 ) + χr𝒰 (μ2 ).

Then by taking the lim inf as (𝒪1 , 𝒪2 ) ↘ (μ1 , μ2 ) and then the lim inf as 𝒪 ↘ μ, we obtain χr𝒰 (μ) ≥ χr𝒰 (μ1 ) + χr𝒰 (μ2 ). On the other hand, we do not know if χ 𝒰 (μ) = χ 𝒰 (μ1 ) + χ 𝒰 (μ2 ) implies that μ is an independent join of μ1 and μ2 according to Definition 5.9. Rather it only implies the weaker condition that lim inf

lim inf

lim

𝒪↘μ (𝒪1 ,𝒪2 )↘(μ1 ,μ2 ) n→𝒰

(n) (n) vol(Γ(n) 1 r (𝒪 ) ∩ [Γr (𝒪1 ) × Γr (𝒪2 )]) log = 0. 2 (n) n vol(Γ(n) r (𝒪1 ) × Γr (𝒪2 ))

For Voiculescu’s free entropy for noncommutative ∗-laws (with limits respect to an 𝒰 ultrafilter 𝒰 ), the freely independent join is the unique ∗-law μ satisfying χqf (μ) =

𝒰 𝒰 χqf (μ1 ) + χqf (μ2 ); the case of d1 = d2 = 1 is handled in [57, Proposition 4.3]. However, the natural proof for d1 , d2 > 1 based on the ideas of [58] would rely on unitary invari-

254 � D. Jekel ance to obtain freeness, which as explained above does not work in the setting of full types. Question 5.14. Let μ1 and μ2 be types of d1 -tuple and d2 -tuple, respectively. Under what conditions is there a unique join μ such that χ 𝒰 (μ) = χ 𝒰 (μ1 ) + χ 𝒰 (μ2 )?

5.3 Independence axioms from model theory Independence has been studied in model theory in the context of stable theories. It turns out that the theory of any II1 factor is not stable [19]. However, there are still situations where unstable theories admit a “well-behaved” independence relation. For the present discussion, we will focus on the axioms that we want an independence relation in ℒtr to satisfy. Definition 5.15 (See [1, 16]). Let 𝒬 be a “large” tracial W∗ -algebra. For “small” sets A, B, C ⊆ 𝒬, we say that A ≡C B if there is an automorphism of 𝒬 mapping A to B and fixing C pointwise (and A ≡ B if there is any automorphism mapping A to B). An | C B between “small” sets in 𝒬 (read as independence relation is a ternary relation A ⌣ “A is independent from B over C”) satisfying the following properties: | C B if and only if A′ ⌣ | C ′ B′ . (1) (Invariance) If A, B, C ≡ A′ , B′ , C ′ , then A ⌣ ′ ′ ′ ′ | C B, then A ⌣ |C B. (2) (Monotonicity) If A ⊆ A and B ⊆ B and A ⌣ | C B, then A ⌣ | D B. (3) (Base monotonicity) Suppose D ⊆ C ⊆ B. If A ⌣ | C A and C ⌣ | D A, then B ⌣ | D A. (4) (Transitivity) Suppose D ⊆ C ⊆ B. If B ⌣ | | (5) (Normality) A ⌣C B implies A ∪ C ⌣C B. | C B and B ⊆ B,̃ then there exists A′ ≡B∪C A such that A′ ⌣ | C B.̃ (6) (Extension) If A ⌣ | C B for all finite A0 ⊆ A, then A ⌣ | C B. (7) (Finite character) If A0 ⌣ (8) (Local character) For every A, there is a cardinal κ(A) such that for every B there | C B. exists C ⊆ B with |C| < κ(A) such that A ⌣ We also define the following properties that an independence relation may have | C B, then B ⌣ | C A. (9) (Symmetry) If A ⌣ | C B. (10) (Full existence) For every A, B, and C, there exists A′ ≡C A such that A′ ⌣ Goldbring, Hart, and Sinclair [27] asked whether there was some independence re| C B related to W∗ (A, C) being freely independent from W∗ (B, C) with amalgalation A ⌣ mation over W∗ (C). Free independence with amalgamation is a moment condition similar to free independence except using the conditional expectation onto W∗ (C) rather than the trace. The properties (1)–(5), (7), (9) above hold for free independence with amalgamation. However, free independence with amalgamation only depends on the quantifier-free type of (A, B, C), and ideally we would want an independence relation that depends on the full type.

Free probability and model theory of tracial W∗ -algebras �

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Question 5.16. Let 𝒬 = ∏n→𝒰 Mn (ℂ). Does there exist a nontrivial independence relation on countable subsets of 𝒬 (that is, a relation satisfying (1)–(8))? Furthermore, does | C B implies that W∗ (A, C) and there exist such an independence relation such that A ⌣ W∗ (B, C) are freely independent with amalgamation over W∗ (C)? One might naïvely hope that there could be an independence relation satisfying the above properties such that the full type of (A, B, C) is determined by the types of (A, C) | C B. However, this is too much and (B, C) and the specification of independence A ⌣ to ask because having a unique way of creating an independent join would imply that theory T is stable (see [52, Section 8] for the discrete model theory case), but [19] showed that the theory of a II1 factor is never stable. Returning to Question 5.16, the main difficulty with free independence with amalgamation in the framework of the independence axioms above concerns the extension property (6) and the related full existence property (10). By [1, Remark 1.2], if (1)–(5), (7), and (9) hold, then the extension property (6) is equivalent to the full existence property (10). To show (10), we would want to know that independent products always exist in 𝒬 in the following sense: If ℳ ⊆ ℳ1 ⊆ 𝒬 and ℳ ⊆ ℳ2 ⊆ 𝒬, then there exists a copy ℳ̃ 2 of ℳ2 and an isomorphism Φ : ℳ2 → ℳ̃ 2 that restricts to the identity on ℳ, such that ℳ1 and ℳ̃ 2 are “independent” over ℳ with respect to whatever independence relation we are studying, and such that the ℳ-type of ϕ(X) is the same as the ℳ-type of X for all tuples X in ℳ2 . We can state the same problem in terms of automorphisms of 𝒬 as follows. Assuming the continuum hypothesis, if α and β are embeddings of ℳ into 𝒬 such that α(X) and β(X) have the same type for every tuple X from ℳ, then α and β are conjugate by an automorphism of 𝒬. Thus, given ℳ ⊆ ℳ1 ⊆ 𝒬 and ℳ ⊆ ℳ2 ⊆ 𝒬, we would need there to exist an automorphism ϕ of 𝒬 that fixes ℳ pointwise and such that ℳ1 and ϕ(ℳ2 ) are independent according to whatever independence relation we are studying. Thus, the question is whether there are “enough” automorphisms fixing ℳ that we can move the ℳ2 into an “independent” position from ℳ1 . But at this point, it is not even known whether we can move ℳ2 so as to become freely independent from ℳ1 with amalgamation over ℳ (although this is true for ℳ = ℂ). The work of Popa on freely independent subalgebras inside ultraproducts [43, 44, 45] shed some light on this question when ℳ is abelian. But it is not known in general whether there even exists an embedding of the amalgamated free product ℳ1 ∗ℳ ℳ2 into 𝒬 (Brown, Dykema, and Jung [11] proved this when ℳ is amenable). This issue is also discussed in Section 6.1 of Goldbring and Hart’s article in this volume. As in Definition 5.9, we could try choosing the embeddings of ℳ1 and ℳ2 “uniformly at random” out of all embeddings that restrict to a given embedding of ℳ. In other words, fixing generators X and Y for ℳ1 and ℳ2 and generators Z for ℳ, we would study the relative microstate spaces for X conditioned on Z and Y conditioned on Z. Let us assume for the sake of the heuristic discussion that X, Y, and Z are finite tuples. Fixing a sequence of matrix approximations Z(n) such that Z = [Z(n) ]n∈𝒰 in 𝒬,

256 � D. Jekel we define for neighborhoods 𝒪1 of tp𝒬 (X, Z) the microstate space (n) Γ(n) 󴁄󴀼 Z) := {X′ ∈ Mn (ℂ)d1 : tpMn (ℂ) (X′ , Z(n) ) ∈ 𝒪1 } r (𝒪1 |Z

and the analogous microstate space for each neighborhood 𝒪2 of tp𝒬 (Y, Z). Then just as in Definition 5.9, we can ask: Question 5.17. In the setup above, does there exist some Y′ ∈ 𝒬d with tp𝒬 (Y′ , Z) = tp𝒬 (Y, Z) and lim inf

lim inf

lim

𝒪↘tp𝒬 (X,Y′ ,Z) (𝒪1 ,𝒪2 )↘(tp𝒬 (X,Z),tp𝒬 (Y,Z)) n→𝒰 (n) (n) (n) 󴁄󴀼 Z) ∩ [Γ(n) 󴁄󴀼 Z) × Γ(n) 󴁄󴀼 Z)]) vol(Γ(n) r (𝒪1 | Z r (𝒪2 | Z r (𝒪 | Z (n) 󴁄󴀼 Z) × Γ(n) (𝒪 | Z(n) 󴁄󴀼 Z)) vol(Γ(n) r (𝒪1 | Z r 2

= 1?

We also point out that even if such a Y′ exists, it is not immediate whether tp (X, Y′ , Z) only depends on tp𝒬 (X, Z) and tp𝒬 (Y, Z), independent of the particular choice of Z and Z(n) realizing those types. 𝒬

6 Model theory and noncommutative optimization problems 6.1 Wasserstein distance Biane and Voiculescu [9] studied a noncommutative analog of the Wasserstein distance. For two quantifier-free types μ, ν ∈ 𝕊qf,d (Ttr ), the Wasserstein distance is ℳ dW (μ, ν) = inf{‖X − Y‖L2 (ℳ) : ℳ 󳀀󳨐 Ttr , tpℳ qf (X) = μ, tpqf (Y) = ν}.

(6.1)

This is the noncommutative tracial version of the classical Wasserstein distance defined for probability measures on ℝd with finite second moment, which plays a major role in optimal transport theory. The Wasserstein distance is defined by an optimization problem, namely, to minimize the L2 -distance between some X and Y with quantifier-free types μ and ν, respectively. We remark that Song studied the L1 classical Wasserstein distance between types in atomless probability spaces from a model-theoretic viewpoint in [49, Sections 5.3–5.4]. The analogous distance for full types μ and ν in 𝕊d (Ttr ) is dW (μ, ν) = inf{‖X − Y‖L2 (ℳ) : ℳ 󳀀󳨐 Ttr , tpℳ (X) = μ, tpℳ (Y) = ν}.

(6.2)

Of course, since the type determines the theory of ℳ, we have dW (μ, ν) = ∞ unless the types μ and ν can be realized in the same tracial W∗ -algebra. For this reason, it is

Free probability and model theory of tracial W∗ -algebras

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natural to fix a complete theory T (for instance, the theory of the hyperfinite II1 factor ℛ or the theory of some matrix ultraproduct 𝒬) and then study the Wasserstein distance on 𝕊d (T). This Wasserstein distance for types is exactly the d-metric between types for a given complete theory described in [5, Section 8, p. 44]. For each r, the Wasserstein distance dW provides a complete metric on 𝕊d,r (T) as observed in [5, Proposition 8.8] and the topology of dW refines the logic topology [5, Proposition 8.7]. The analogous properties were also shown for the non-commutative Wasserstein distance on quantifier-free types. The topology refines the weak-∗ topology by [9, Proposition 1.4(b)(c)]. One can also argue similarly to [5, Proposition 8.8] that dW gives a complete metric on the space of quantifier-free types. In classical probability theory, for any given compactly supported probability measures μ and ν, one can always find X and Y in L∞ [0, 1] with probability distributions μ and ν that achieve the infimum in (6.1) (such a pair achieving the infimum is called an optimal coupling). By contrast, in the noncommutative setting of [9], one has very little control over which tracial von Neumann algebra is needed to achieve an optimal coupling, as explained in [24, Section 5]. For instance, given the negative solution to the Connes embedding problem proposed in [37], for some d and r, there exist μ and ν ∈ 𝕊qf,d,r (Ttr ) such that each of μ and ν is the ∗-law of a some d-tuple in a matrix algebra Mn (ℂ), and yet the infimum (6.1) can only be achieved using a non-Connes embeddable algebra ℳ [24, Corollary 5.14] (for background on the Connes embedding problem, see Goldbring’s article in this volume). This is related to the issue discussed in Section 2.3 that L∞ [0, 1] is ℵ0 -categorical and admits quantifier elimination, but very few tracial W∗ -algebras have these properties. This difficulty in the tracial W∗ -setting seems to be ameliorated by using Wasserstein distance for full types rather than quantifier-free types. Indeed, for any countably saturated model ℳ of a given theory T, then the Wasserstein distance between two types μ, ν ∈ 𝕊d (T) can always be achieved by some coupling in ℳ. Moreover, in any model of T, there are approximately optimal couplings. Thus, consistent with the discussion in Section 1.3, it is worth considering whether the space of full types relative to a complete theory T ⊇ Ttr may be a better analog for the space of probability distributions on ℝd than the space of quantifier-free types. However, even in this setting, the weak-∗ and Wasserstein topologies will usually not agree; by the continuous Ryll–Nardzewski theorem [5, Proposition 12.10] for a complete theory T, the logic topology and d-metric topology agree if and only if T is ℵ0 -categorical. More precisely, using similar reasoning to [24, Section 5.4], we can characterize the agreement of the weak-∗ and Wasserstein topologies at a particular point μ ∈ 𝕊d,r (T) through a stability or property (Corollary 6.4 below), which follows from the ultraproduct characterizations of weak-∗ convergence and of Wasserstein convergence. The characterization of weak-∗ convergence is as follows. Lemma 6.1. Fix r ∈ (0, ∞) and a free ultrafilter 𝒰 on ℕ. For n ∈ ℕ, let ℳ(n) 󳀀󳨐 Ttr and (n) d X(n) ∈ (Dℳ )d . Let ℳ be a separable model of T and X ∈ (Dℳ r r ) . Then the following are equivalent:

258 � D. Jekel (n)

(1) limn→𝒰 tpℳ (X(n) ) = tpℳ (X) in the weak-∗ topology on 𝕊d,r (T). (2) There exists an elementary embedding ι : ℳ → ∏n→𝒰 ℳ(n) such that ι(X) = [X(n) ]n∈ℕ . Proof. (2) 󳨐⇒ (1) because by the fundamental theorem of ultraproducts (n) (n) tp∏n→𝒰 ℳ ([X(n) ]n∈ℕ ) = limn→𝒰 tpℳ (X(n) ). (1) 󳨐⇒ (2) follows by similar reasoning as [20, Lemma 4.12]. To characterize Wasserstein convergence, we use the model-theoretic analog of the factorizable maps of [2]. Definition 6.2. Let ℳ and 𝒩 be models of a complete T ⊇ Ttr . An elementarily factorizable map Φ : ℳ → 𝒩 is a map of the form Φ = β∗ α where α : ℳ → 𝒫 and β : 𝒩 → 𝒫 are elementary embeddings into another 𝒫 󳀀󳨐 T, and β∗ : 𝒫 → 𝒩 is the trace-preserving conditional expectation. Lemma 6.3. Fix r ∈ (0, ∞) and a free ultrafilter 𝒰 on ℕ. For n ∈ ℕ, let ℳ(n) 󳀀󳨐 Ttr and (n) d X(n) ∈ (Drℳ )d . Let ℳ be a separable model of T and X ∈ (Dℳ r ) . Then the following are equivalent: (n) (1) limn→𝒰 tpℳ (X(n) ) = tpℳ (X) with respect to Wasserstein distance. (2) There exists an elementary embedding ι : ℳ → ∏n→𝒰 ℳ(n) such that ι(X) = [X(n) ]n∈ℕ , and there exist elementarily factorizable maps Φ(n) : ℳ → ℳ(n) such that ι(Z) = [Φ(n) (Z)]n∈𝒩

for all Z ∈ W∗ (X).

In fact, if (1) holds, then every elementary ι : ℳ → ℳ(n) with ι(X) = [X(n) ]n∈ℕ admits such a family Φ(n) of elementary factorizable maps as in (2). Proof. (2) 󳨐⇒ (1) Suppose that (2) holds. Let Φ(n) = (β(n) )∗ α(n) where α(n) : ℳ → 𝒫 (n) and β(n) : ℳ(n) → 𝒫 (n) are elementary embeddings. Then 2

(n)

dW (tpℳ (X(n) ), tpℳ (X)) 󵄩 󵄩2 ≤ 󵄩󵄩󵄩α(n) (X) − β(n) (X(n) )󵄩󵄩󵄩L2 (𝒫 (n) )d 󵄩 󵄩2 󵄩 󵄩2 = 󵄩󵄩󵄩α(n) (X)󵄩󵄩󵄩L2 (𝒫 (n) )d − 2 Re⟨α(n) (X), β(n) (X(n) )⟩L2 (𝒫 (n) )d + 󵄩󵄩󵄩β(n) (X(n) )󵄩󵄩󵄩L2 (𝒫 (n) )d 󵄩 󵄩2 = ‖X‖2L2 (ℳ)d − 2 Re⟨Φ(n) (X), X(n) ⟩L2 (ℳ(n) )d + 󵄩󵄩󵄩X(n) 󵄩󵄩󵄩L2 (ℳ(n) )d 󵄩 󵄩2 󵄩 󵄩2 = ‖X‖2L2 (ℳ)d − 󵄩󵄩󵄩Φ(n) (X)󵄩󵄩󵄩L2 (ℳ(n) )d + 󵄩󵄩󵄩Φ(n) (X) − X(n) 󵄩󵄩󵄩L2 (ℳ(n) )d . The latter goes to zero as n → 𝒰 since ι(X) = [Φ(n) (X)]n∈ℕ = [X(n) ]n∈ℕ in ∏n→𝒰 ℳ(n) . Now let us prove (1) 󳨐⇒ (2), as well as the final claim of the lemma. Assume that (1) holds. Let ι : ℳ → ∏n→𝒰 ℳ(n) be any elementary embedding with ι(X) = [X(n) ]n∈ℕ (which exists by Lemma 6.1 since Wasserstein convergence implies weak-∗ (n)

convergence). Since tpℳ (X(n) ) converges to tpℳ (X) in the Wasserstein distance, there

Free probability and model theory of tracial W∗ -algebras �

259

exists models 𝒫 (n) 󳀀󳨐 T and Y(n) , Z(n) ∈ (D𝒫 )d such that r (n)

(n)

(n)

tp𝒫 (Y(n) ) = tpℳ (X),

(n)

tp𝒫 (Z(n) ) = tpℳ (X(n) ),

󵄩 󵄩 lim 󵄩󵄩Y(n) − Z(n) 󵄩󵄩󵄩L2 (𝒫 (n) ) = 0.

n→𝒰 󵄩

By enlarging the model 𝒫 (n) if necessary, we can arrange that there are elementary embeddings α(n) : ℳ → 𝒫 (n) and β(n) : ℳ(n) → 𝒫 (n) such that α(n) (X) = Y(n) and β(n) (X(n) ) = Z(n) . Now consider the elementarily factorizable map Φ(n) = (β(n) )∗ α(n) . We claim that ι(Z) = [Φ(n) (Z)]n∈ℕ for all Z ∈ W∗ (X). Fix Z. By Proposition 2.11, there exist a quantifier-free definable function f relative to Ttr such that Z = f ℳ (X). Since the conditional expectation is contractive in L2 , (n) ∗ ∗ 󵄩󵄩 (n) 󵄩 󵄩 󵄩 ℳ(n) (X(n) )󵄩󵄩󵄩L2 (ℳ(n) )d = 󵄩󵄩󵄩(β(n) ) α(n) (f ℳ (X)) − (β(n) ) β(n) (f ℳ (X(n) ))󵄩󵄩󵄩L2 (ℳ(n) )d 󵄩󵄩Φ (Z) − f (n) 󵄩 󵄩 ≤ 󵄩󵄩󵄩α(n) (f ℳ (X)) − β(n) (f ℳ (X(n) ))󵄩󵄩󵄩L2 (𝒫 (n) )d (n) 󵄩 (n) 󵄩 = 󵄩󵄩󵄩f 𝒫 (Y(n) ) − f 𝒫 (Z(n) )󵄩󵄩󵄩L2 (𝒫 (n) )d .

The right-hand side goes to zero as n → 𝒰 because of the fact that ‖Y(n) −Z(n) ‖L2 (𝒫 (n) )d → 0 and the L2 -uniform continuity property of definable functions (see [5, Proposition 9.23] and [35, Lemma 3.19]). This implies that (n)

(n)

[Φ(n) (Z)]n∈ℕ = [f ℳ (X(n) )]n∈ℕ = f ∏n→𝒰 ℳ ([X(n) ]n∈ℕ ) = ι(f ℳ (X)) = ι(Z), where we have used the fact that quantifier-free definable functions commute with embeddings. A type μ ∈ 𝕊d,r (T) is called principal if the weak-∗ and Wasserstein topologies agree at μ, meaning that every weak-∗ neighborhood of μ in 𝕊d,r (T) contains a Wasserstein neighborhood and vice versa (see [5, Definition 12.2 and Proposition 12.4], and also Section 8.3 of Hart’s article in this volume). Since both the weak-∗ and the Wasserstein topologies are metrizable, this is equivalent to saying that for a given free ultrafilter 𝒰 on ℕ, for every sequence μ(n) in 𝕊d,r (T), we have limn→𝒰 μ(n) = μ in the weak-∗ topology if and only if limn→𝒰 μ(n) = μ in the Wasserstein distance. Combining Lemmas 6.1 and 6.3, we obtain the following result. Corollary 6.4. Let T ⊇ Ttr be a complete theory, let ℳ be a separable model of T, and let d X ∈ (Dℳ r ) . Let 𝒰 be a free ultrafilter on ℕ. Then the following are equivalent: ℳ (1) tp (X) is a principal type. (2) Given any models ℳ(n) 󳀀󳨐 T and any elementary embedding ι : ℳ → ∏n→𝒰 ℳ(n) , where ℳ(n) 󳀀󳨐 T, there exist elementarily factorizable maps Φ(n) : ℳ → ℳ(n) such that ι(Z) = [Φ(n) (Z)]n∈ℕ

for all Z ∈ W∗ (X).

260 � D. Jekel In the analogous situation for noncommutative ∗-laws, the stability or lifting property that characterized agreement of the weak-∗ and Wasserstein topologies turned out to be closely related to amenability (to be more precise, it was equivalent to amenability under the assumption of Connes-embeddability [24, Proposition 5.26]). One direction of this argument adapts readily to the model-theoretic setting. Observation 6.5. Suppose that T is the complete theory of some II1 factor. Let ℳ 󳀀󳨐 T and X ∈ M d . If W∗ (X) is amenable, then tpℳ (X) is principal. Proof. Let ι : ℳ → ∏n→𝒰 ℳ(n) be an elementary embedding, where ℳ(n) 󳀀󳨐 T. Since ℳ and ℳ(n) are elementarily equivalent, there exists some 𝒩 (n) 󳀀󳨐 T that contains ℳ and 𝒩 (n) as elementary submodels; let αn : ℳ → 𝒩 (n) and βn : ℳ(n) → 𝒩 (n) be the elementary embeddings. Consider the induced elementary embeddings α : ℳ → ∏n→𝒰 𝒩 (n) and β : ∏n→𝒰 ℳ(n) → ∏n→𝒰 𝒩 (n) . Then α and β ∘ ι are both elementary embeddings of ℳ into ∏n→𝒰 𝒩 (n) . Since 𝒜 = W∗ (X) is amenable and ∏n→𝒰 𝒩 (n) is an ℵ0 -saturated II1 -factor, the embeddings α|𝒜 and β ∘ ι|𝒜 are unitarily conjugate, so there exist unitaries Un ∈ 𝒩 (n) such that β ∘ ι(Z) = [Un αn (Z)Un∗ ]n∈ℕ

for Z ∈ 𝒜.

Then Φn = βn∗ adUn ∘αn is the desired elementarily factorizable map ℳ → ℳ(n) . (For further detail, see the proof of [24, Proposition 5.26].) This motivates the following question concerning the lifting property in Corollary 6.4 for future research. Question 6.6. Let T ⊇ Ttr be a complete theory. Let ℳ 󳀀󳨐 T and let 𝒜 be a tracial W∗ subalgebra of ℳ. Under what conditions on T, ℳ, and/or 𝒜 does it hold that, for every elementary embedding ι of ℳ into an ultraproduct ∏n→𝒰 ℳ(n) of models of T, there exist elementarily factorizable maps Φ(n) : ℳ → ℳ(n) such that ι(Z) = [Φ(n) (Z)]n∈ℕ for all Z ∈ 𝒜?

6.2 Free Gibbs ∗-laws and entropy Free Gibbs laws are an important construction in the statistical mechanics / large deviations approach to random matrix theory studied in [10]. The multivariable analog of free Gibbs laws has been studied in [8, 28, 29, 30, 14, 13, 33, 34, 36]. Here we shall study free Gibbs ∗-laws for tuples of non-self-adjoint operators. Fix a d-variable definable predicate V relative to Ttr , and assume that a + b‖X‖2L2 (ℳ)d ≤ V ℳ (X) ≤ A + B‖X‖2L2 (ℳ)d for some a, A ∈ ℝ and b, B ∈ (0, ∞). Let V (n) = V Mn (ℂ) and let μ(n) be the probability measure on Mn (ℂ)d given by

Free probability and model theory of tracial W∗ -algebras

dμ(n) (X) =

1 ∫M

n

(ℂ)d

2 (n) e−n V

2

e−n V

(n)

(X)

� 261

dX,

where dX denotes Lebesgue measure (our assumed upper and lower bounds for V guar2 (n) antee that the integral of e−n V converges). Let X(n) be a random element of Mn (ℂ)d chosen according to the measure μ(n) . Question 6.7. Under what conditions does tpMn (ℂ) (X(n) ) converge in probability as n → ∞ (or as n → 𝒰 for some given free ultrafilter 𝒰 )? As in [36, Proposition 7.11 and Corollary 7.12], one can show that if there is a unique type μ maximizing 𝒰 χfull (μ) − μ[V ],

where μ[V ] denotes the evaluation of μ viewed as an element of C(𝕊)d (Ttr ) on V , then Question 6.7 has a positive answer, and in fact, for every neighborhood 𝒪 of μ and every r > 0, we have lim

n→𝒰

1 log(1 − μ(n) (Γ(n) r (𝒪 ))) < 0, n2

meaning that a matrix chosen randomly according to μ(n) has an exponentially small probability of its type landing outside a given neighborhood of μ. When V is quantifierfree definable, a ∗-law which maximizes χqf (μ) − μ[V ] is called a free Gibbs ∗-law for V . Similarly, we can call μ a free Gibbs type for a given definable predicate V if it maximizes χfull (μ) − μ[V ]. This prompts the following related question. Question 6.8. Under what conditions does there exist a unique type μ that maximizes χ 𝒰 (μ) − μ[V ]? Does this μ depend on the choice of free ultrafilter 𝒰 ? In the case where V is quantifier-free definable and satisfies certain smoothness assumptions and estimates on the derivatives, then one can obtain a unique solution by the stochastic diffusion techniques of [8] or the Dyson–Schwinger equation techniques of [28] or a combination of the two; see, for instance, [29] or [36, Proposition 8.1]. However, it is not clear to me at the time of writing how these techniques should be adapted to the setting of full types. For the diffusion techniques, we would need to understand how the logical operations of sup and inf over operator norm balls combine with the theory of free stochastic differential equations (see below for further discussion). Similarly, to adapt the Dyson–Schwinger equation approach to the setting of full types, we would some notion of derivatives and smoothness for definable predicates and definable functions. An open problem in Voiculescu’s free entropy theory is whether, for a Connesembeddable noncommutative ∗-law μ, the microstate version χ(μ) or χ 𝒰 (μ) from [55, 56] agrees with another version of free entropy called χ ∗ (μ), which is defined by studying

262 � D. Jekel the free Fisher information Φ∗ (μ) [59]. The inequality χ(μ) ≤ χ ∗ (μ) is known thanks to [7]. Moreover, [13] and [33] established equality when μ is a free Gibbs law for a quantifier-free definable predicate V satisfying some convexity / semiconcavity assumptions. A key part of this analysis is to study the smoothness of the potential V (n) (X, t) obtained from convolving μ(n) with a Gaussian distribution. For the sake of Proposition 6.9 below, we state the problem in terms of stochastic analysis. We recall that a filtration on a probability space Ω is a collection of σ-algebras such (ℱt )t∈[0,∞) with ℱs ⊆ ℱt for s ≤ t. A stochastic process (Xt )t∈[0,∞) (such as Brownian motion) is adapted to (ℱt )t∈[0,∞) if Xt is ℱt -measurable for each t; moreover, (Xt )t∈[0,∞) is progressively measurable if for each u ∈ [0, ∞), the function (s, ω) 󳨃→ Xs (ω) on [0, u] × Ω is measurable with respect to the product of the Borel σ-algebra on [0, u] with ℱu . Fix a probability space Ω with a filtration (ℱt )t∈[0,∞) , and fix an Mn (ℂ)d -valued (n) 2 (n) Brownian motion Z(n) t adapted to ℱt , normalized so that 𝔼‖Zt ‖M (ℂ)d = 2td. Let μt n

(n) d be the probability distribution of X(n) + Z(n) t , and define V (⋅, t) : Mn (ℂ) → ℝ by

dμ(n) t (X) =

1 ∫M

d n (ℂ)

2 (n) e−n V

2

e−n V

(n)

(X,t)

dX.

The following expression for V (n) (X, t) can be obtained from stochastic optimal control theory (see for instance [23, Section VI]). Proposition 6.9. t

V

(n)

(X, t) = inf[𝔼[V A

(n)

1 󵄩 󵄩2 (Xt ) + 2 ∫󵄩󵄩󵄩A(u)󵄩󵄩󵄩L2 (M (ℂ))d du]], n 2t

(6.3)

0

where A ranges over all progressively measurable functions Ω × [0, t] → Mn (ℂ)d , and where X0 = X, dXt = dZ(n) t + A(t) dt. Roughly speaking, [33, Section 6] showed that when V (n) is quantifier-free definable and satisfies some convexity and semiconcavity hypotheses, then V (n) (X, t) asymptotically approaches some quantifier-free definable predicate as n → ∞. In other words, the formula above which has an inf in it can be reexpressed in another way that does not have any quantifiers! Heuristically speaking, this enabled the study of Vt(n) as n → ∞ to proceed along the same lines as in classical probability theory where we have quantifier elimination, even though a priori the large-n behavior would seem to fall in the realm of tracial W∗ -algebras which do not have quantifier elimination. However, the regularity coming from the convexity assumptions on V (n) was crucial

Free probability and model theory of tracial W∗ -algebras �

263

in the argument, so I expect that studying more general V will require the analysis of quantifiers. Motivated by the strategy of [33, Section 6] (closely related to the earlier strategies of [7, 13]), we want to show that this Vt(n) will asymptotically be described by some definable predicate. Question 6.10. Let V be a definable predicate, and let V (n) (X, t) as above. Does there exist a definable predicate Vt such that for every r > 0, lim

n→𝒰

sup

M (ℂ) X∈(Dr n )d

M (ℂ) 󵄩󵄩 (n) 󵄩 󵄩󵄩V (X, t) − Vt n (X)󵄩󵄩󵄩L2 (Mn (ℂ))d = 0?

The following is a heuristic attempt to address this question, which shows that although we may not be able to obtain a quantifier-free definable Vt for each quantifierfree definable V , there is some hope of obtaining a definable Vt . This is part of our motivation for exploring free probability outside the quantifier-free setting. To address Question 6.10, we want to make a more precise connection between Proposition 6.9 and the model theory of tracial W∗ -algebras. Rather than getting into the technicalities of stochastic differential equations, we will look at discrete-time approximation of (6.3) as in [33, Section 6]. Let (n) Φ(n) t V (X) =

inf

A∈Mn

(ℂ)d

[V (n) (X + A + Z(n) t )+

1 ‖A‖2L2 (M (ℂ))d ]. n 2t

We conjecture that k

V (n) (X, t) = lim (Φ(n) ) V (n) (X); t/k k→∞

the results of [33, Section 6] essentially prove this assuming that V (n) is uniformly convex and semiconcave. (n) Once again, let 𝒬 = ∏n→𝒰 Mn (ℂ). Heuristically, the limit of Φ(n) as n → 𝒰 t V 𝒬 d should be described by some function Φt V : Q → ℝ given by Φt V 𝒬 (X) = inf{V 𝒬 (X + A + t 1/2 Z) : A ∈ 𝒬d ,

Z ∈ 𝒬d free circular family free from X and A}.

This Φt V 𝒬 is not a priori a definable predicate relative to Th(𝒬), but it can be described as a (pointwise) limit of definable predicates. Indeed, one can verify that d

{tp𝒬 (X, A, Z) : X, A, Z ∈ (D𝒬 r ) , Z free circular family freely independent of X, A}. is a closed subset of 𝕊3d,r (Ttr ). Hence, by Urysohn’s lemma, there exists ψr ∈ C(𝕊3d,r (Ttr )) with ψr ≥ 0 and ϕr = 0 precisely on this set. This ψr may be viewed as a definable

264 � D. Jekel predicate on D3d r , and using Proposition 2.12, ψr may be extended globally to a definable predicate on 𝕊3d (Ttr ) satisfying ψr ≥ 0 and ψℳ r (X, A, Z) = 0 if and only if Z is a standard free circular family freely independent of X and A. Let ϕt,ϵ be the definable predicate given by ϕℳ t,ϵ (X) =

inf

A,Z∈Dℳ r

V ℳ (X + A + Z) +

1 1 ‖A‖2L2 (ℳ)d + ψℳ (X, A, Z) 2t ϵ r

for ℳ 󳀀󳨐 Ttr . Then ϕt,ϵ increases as ϵ decreases. Furthermore, one can check that for r > 2, for X ∈ D𝒬 r , 𝒬 1/2 lim ϕ𝒬 t,ϵ (X) = inf{V (X + A + t Z) +

ϵ→0+

1 d ‖A‖L2 (𝒬)d : A, Z ∈ (D𝒬 r ) , ψr (X, A, Z) = 0}. 2t

This is the same as Φt V 𝒬 except that now the A in the infimum is restricted to an operator norm ball. After taking the limit as r → ∞, we obtain Φt V 𝒬 . Question 6.11. Can one show, under certain hypotheses on V , that the limits as ϵ → 0+ and r → 0+ described above occur uniformly on each operator norm ball? Then can one show that Vt𝒬 = limk→∞ (Φt/k )k V 𝒬 exists uniformly on each operator norm ball? Finally, if so, does this Vt provide a positive answer to Question 6.10? One possible flaw in this approach is that our construction only required the free circular family Z to be freely independent of A and X, which only describes something about the quantifier-free type of (A, X, Z). If we can discover a good notion of independence that pertains to the full type rather than only the quantifier-free type as proposed in Section 5, then it may be necessary or natural to require independence of Z from (A, X) with respect to their full types, not only the quantifier-free types. This would be potentially more restrictive on the possible choices of Z and thus lead to a larger infimum. We hope to choose the right setup for the infimum that will accurately describe the large-n limit of V (n) (⋅, t) as in Question 6.10. If this can be done, then another goal would be to adapt the work of [13, 33] to show some result that the free microstate entropy χ 𝒰 of a type agrees with some version of Voiculescu’s free nonmicrostate entropy χ ∗ . Of course, a prerequisite for this endeavor would be to sort out what nonmicrostate entropy for types even is. Question 6.12. Is there an analog of free Fisher information Φ∗ (μ) and free entropy χ ∗ (μ) for a full type rather than a quantifier-free type? I do not have much to say about this question now, but I hope that future researchers will explore the many interesting questions that arise from the combination of model theory and free probability.

Free probability and model theory of tracial W∗ -algebras �

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Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

H. Adler, A geometric introduction to forking and thorn-forking, J. Math. Log. 09 (2009), no. 01, 1–20. C. Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Relat. Fields 135 (2006), 520–546. G. W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2009. I. Ben Yaacov, On theories of random variables, Isr. J. Math. 194 (2013), no. 2, 957–1012. I. Ben Yaacov, A. Berenstein, C. Ward Henson and A. Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis, Vol. II, ed. by Z. Chatzidakis et al., London Mathematical Society Lecture Notes Series, vol. 350, Cambridge University Press, 2008, pp. 315–427. I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability, Trans. Am. Math. Soc. 362 (2010), no. 10, 5213–5259. P. Biane, M. Capitaine and A. Guionnet, Large deviation bounds for matrix Brownian motion, Invent. Math. 152 (2003), 433–459. P. Biane and R. Speicher, Free diffusions, free entropy and free fisher information, Ann. Inst. Henri Poincaré Probab. Stat. 37 (2001), no. 5, 581–606. P. Biane and D.-V. Voiculescu, A free probability analogue of the Wasserstein metric on the trace-state space, Geom. Funct. Anal. 11 (2001), 1125–1138. A. Boutet de Monvel, L. Pastur and M. Shcherbina, On the statistical mechanics approach in the random matrix theory: Integrated density of states, J. Stat. Phys. 79 (1995), 585–611. N. P. Brown, K. J. Dykema and K. Jung, Free entropy dimension in amalgamated free products, Proc. Lond. Math. Soc. 97 (2008), no. 2, 339–367. I. Chifan, A. Ioana and S. Kunnawalkam Elayavalli, Two non-elementarily equivalent II1 factors without property gamma, 2022. arXiv:2209.10645. Y. Dabrowksi, A Laplace principle for Hermitian Brownian motion and free entropy I: The convex functional case, 2017. arXiv:1604.06420. Y. Dabrowski, A. Guionnet and D. Shlyakhtenko, Free transport for convex potentials, N.Z. J. Math. 52 (2021), 259–359. https://doi.org/10.53733/102. C. J. Eagle, I. Farah, E. Kirchberg and A. Vignati, Quantifier elimination in C∗ -algebras, Int. Math. Res. Not. 2017 (2016), no. 24, 7580–7606. C. Ealy and I. Goldbring, Thorn-forking in continuous logic, J. Symb. Log. 77 (2012), 63–93. I. Farah, Between reduced powers and ultrapowers, J. Eur. Math. Soc. (2022). I. Farah and S. Ghasemi, Preservation of elementarity by tensor products of tracial von Neumann algebras, 2023. arXiv:2304.08437. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras I: Stability, Bull. Lond. Math. Soc. 45 (2013), no. 4, 825–838. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras II: Model theory, Isr. J. Math. 201 (2014), no. 1, 477–505. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras III: Elementary equivalence and II1 factors, Bull. Lond. Math. Soc. 46 (2014), no. 3, 609–628. S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, J. Symb. Log. 32 (1967), no. 2. W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, Applications of Mathematics, vol. 1, Springer, Berlin, Heidelberg, New York, 1975. W. Gangbo, D. Jekel, K. Nam and D. Shlyakhtenko, Duality for optimal couplings in free probability, Commun. Math. Phys. 396 (2022), 903–981. https://doi.org/10.1007/s00220-022-04480-0. S. Ghasemi, Reduced products of metric structures: a metric Feferman–Vaught theorem, J. Symb. Log. 81 (2016), no. 3, 856–875. I. Goldbring, Enforceable operator algebras, J. Inst. Math. Jussieu 20 (2021), 31–63.

266 � D. Jekel

[27] I. Goldbring, B. Hart and T. Sinclair, The theory of tracial von Neumann algebras does not have a model companion, J. Symb. Log. 78 (2013), no. 3, 1000–1004. [28] A. Guionnet and E. Maurel-Segala, Combinatorial aspects of random matrix models, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 241–279. [29] A. Guionnet and D. Shlyakhtenko, Free diffusions and matrix models with strictly convex interaction, Geom. Funct. Anal. 18 (2009), no. 6, 1875–1916. [30] A. Guionnet and D. Shlyakhtenko, Free monotone transport, Invent. Math. 197 (2014), no. 3, 613–661. [31] B. Hayes, 1-bounded entropy and regularity problems in von Neumann algebras, Int. Math. Res. Not. 1 (2018), no. 3, 57–137. [32] B. Hayes, D. Jekel, B. Nelson and T. Sinclair, A random matrix approach to absorption theorems for free products, Int. Math. Res. Not. 2021 (2021), no. 3, 1919–1979. [33] D. Jekel, An elementary approach to free entropy theory for convex potentials, Anal. PDE 13 (2020), no. 8, 2289–2374. doi:10.2140/apde.2020.13.2289. [34] D. Jekel, Conditional expectation, entropy, and transport for convex Gibbs laws in free probability, Int. Math. Res. Not. 2022 (2022), no. 6, 4516–4619. doi:10.1093/imrn/rnaa181. [35] D. Jekel, Covering entropy for types in tracial W∗ -algebras, J. Log. Anal. (2022), in press. arXiv:2204.02582. [36] D. Jekel, W. Li and D. Shlyakhtenko, Tracial non-commutative smooth functions and the free Wasserstein manifold, Diss. Math. 580 (2022), 1–150. [37] Z. Ji, A. Natarajan, T. Vidick, J. Wright and H. Yuen, MIP* = RE, 2020. arXiv:2001.04383. [38] N. Jing, Unitary and orthogonal equivalence of sets of matrices, Linear Algebra Appl. 481 (2015), 235–242. [39] K. Jung, Strongly 1-bounded von Neumann algebras, Geom. Funct. Anal. 17 (2007), no. 4, 1180–1200. [40] J. A. Mingo and R. Speicher, Free probability and random matrices, Fields Institute Monographs, vol. 35, Springer, New York, 2017. [41] P. Olin, Direct multiples and powers of modules, Fundam. Math. 73 (1971), no. 2, 113–124. [42] P. Olin, Free products and elementary equivalence, Pac. J. Math. 52 (1974), no. 1, 175–184. [43] S. Popa, Free-independent sequences in type II1 factors and related problems, Astérisque 232 (1995), 187–202. [44] S. Popa, A II1 factor approach to the Kadison–Singer problem, Commun. Math. Phys. 332 (2014), 379–414. [45] S. Popa, Independence properties in subalgebras of ultraproduct II1 factors, J. Funct. Anal. 266 (2014), 5818–5846. [46] Z. Sela, Diophantine geometry over groups X: The elementary theory of free products of groups, 2010. arXiv:1012.0044. [47] H. Shapiro, A survey of canonical forms and invariants for unitary similarity, Linear Algebra Appl. 147 (1991), 101–167. [48] D. Shlyakhtenko and T. Tao, Fractional free convolution powers, Indiana Univ. Math. J. in press. With an appendix by David Jekel. [49] S. Song, Model theory and probability, PhD thesis, University of Illinois at Urbana-Champaign, 2011. [50] W. Specht, Zur Theorie der Matrizen, II, Jahresber. Dtsch. Math.-Ver. 50 (1940), 19–23. [51] S. Szarek, Metric entropy of homogeneous spaces, Quantum probability, Banach Center Publications, vol. 43, Polish Academy of Science, Warsaw, 1998, pp. 395–410. [52] K. Tent and M. Ziegler, A Course in model theory, Lecture Notes in Logic, Cambridge University Press, 2012. [53] D.-V. Voiculescu, Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323–346. [54] D.-V. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (Dec 1991), no. 1, 201–220.

Free probability and model theory of tracial W∗ -algebras

� 267

[55] D.-V. Voiculescu, The analogues of entropy and of Fisher’s information in free probability, II, Invent. Math. 118 (1994), 411–440. [56] D.-V. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability, III: Absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), 172–199. [57] D.-V. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, IV: Maximum entropy and freeness, Free probability theory, ed. by D.-V. Voiculescu, Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, 1997, pp. 293–302. [58] D.-V. Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy, Int. Math. Res. Not. 1998 (1998), no. 1, 41–63. [59] D.-V. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, VI: Liberation and mutual free information, Adv. Math. 146 (1999), 101–166. [60] D.-V. Voiculescu, Free entropy, Bull. Lond. Math. Soc. 34 (2002), 257–278. [61] D.-V. Voiculescu, K. J. Dykema and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. [62] J. von Neumann, Approximative properties of matrices of high finite order, Port. Math. 3 (1942), 1–62. [63] N. Wiegmann, Necessary and sufficient conditions for unitary similarity, J. Aust. Math. Soc. 2 (1961), 122–126.

Ionuţ Chifan, Daniel Drimbe, and Adrian Ioana

Tensor product indecomposability results for existentially closed factors Abstract: In the first part of the paper, we survey several results from Popa’s deformation/rigidity theory on the classification of tensor product decompositions of large natural classes of II1 factors. Using a mélange of techniques from deformation/rigidity theory, model theory, and the recent works [12, 16], we highlight an uncountable family of existentially closed II1 factors M which do not admit tensor product decompositions M = P⊗Q into diffuse factors where Q is full. In the last section we discuss several open problems regarding the structural theory of existentially closed factors. Keywords: II1 factor, full factor, McDuff factor, s-prime factor, property (T), Popa’s intertwining by bimodules techniques, finite index inclusions of von Neumann algebras, wreath-like product group, tracial wreath-like product von Neumann algebra, tensor product of von Neumann algebras, spectral gap property, existentially closed factor, embedding universality MSC 2010: 46L10, 03C66, 22D55, 46L36

1 Introduction A von Neumann algebra is an algebra of bounded linear operators on a Hilbert space which is closed under the adjoint operation and in the weak operator topology. A central theme in operator algebras is the study of tensor product decompositions of II1 factors: indecomposable infinite-dimensional von Neumann algebras which admit a trace. A II1 factor which does not admit a tensor product decomposition into diffuse factors is called Acknowledgement: I. C. was partially supported by NSF Grants DMS-2154637 and FRG-DMS-1854194; D. D. was supported by the postdoctoral fellowship fundamental research 12T5221N of the Research Foundation Flanders; A. I. was partially supported by NSF Grants DMS-2153805 and FRG-DMS-#1854074, and a Simons Fellowship. The authors are very grateful to Isaac Goldbring and to the anonymous referees for proofreading our manuscript and offering numerous comments and suggestions which greatly improved the exposition and the overall quality of the paper. Ionuţ Chifan, 14 MacLean Hall, Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA, e-mail: [email protected]; URL: https://sites.google.com/view/i-chifan-website/ Daniel Drimbe, Department of Mathematics, KU Leuven, Celestijnenlaan 200b, B-3001 Leuven, Belgium, e-mail: [email protected]; URL: https://sites.google.com/view/danieldrimbe/home?authuser=0 Adrian Ioana, Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA, e-mail: [email protected]; URL: https://mathweb.ucsd.edu/~aioana/ https://doi.org/10.1515/9783110768282-007

270 � I. Chifan et al. prime. Using the notion of ∗-orthogonal von Neumann algebras, Popa showed in [64] that the II1 factor L(𝔽S ) arising from the free group 𝔽S with uncountably many generators S is prime. Using Voiculescu’s free probability theory, Ge obtained the first examples of separable prime II1 factors by showing that the free group factors L(𝔽n ), with n ≥ 2, have this property [30]. These results have been since generalized and strengthened in several ways. Ozawa used strong C∗ -algebraic techniques to prove that for any icc hyperbolic group Γ, the II1 factor L(Γ) is solid, meaning that the relative commutant of any diffuse subalgebra is amenable [57]. By developing a new approach on closable derivations, Peterson showed primeness of L(Γ), whenever Γ is any nonamenable icc group with positive first Betti number [61]. Using the framework of his powerful deformation/rigidity theory, Popa gave a new proof of the solidity of L(𝔽n ) [71]. Subsequently, an intense activity in the study of tensor product decompositions of II1 factors led to a plethora of new examples of prime II1 factors arising from various classes of countable groups and their measure preserving actions [4, 11, 13, 14, 17, 21, 24, 27, 38, 39, 48, 58, 73, 81]. In most of these results, the groups Γ for which L(Γ) was proven to be prime either satisfy an algebraic assumption (e. g., Γ is an amalgamated free product group or a wreath product group) or have some “rank one” properties (e. g., Γ is a lattice in a rank one simple Lie group or admits a certain unbounded quasicocycle). On the other hand, the primeness problem for higher rank lattices is wide open. A general conjecture predicts that any icc irreducible lattice Γ in a product G1 × ⋅ ⋅ ⋅ × Gn of connected noncompact simple real Lie groups with finite center gives rise to a prime II1 factor [44, Problem V]. This is a von Neumann algebraic counterpart of the fact, implied by Margulis’ normal subgroup theorem (see [82, Theorem 8.1.1]), that Γ does not admit any direct product decomposition into infinite groups. Using methods from Popa’s deformation/rigidity theory, this conjecture has been confirmed when G1 , . . . , Gn have rank one [27]. On the other hand, when at least one of the groups G1 , . . . , Gn has rank greater than one, the conjecture is notoriously hard. Some positive evidence in this case has been obtained in [48]. Using results from [6, 41], it was shown in [48] that any profinite free ergodic probability measure preserving action of Γ gives rise to a prime II1 factor. The first main goal of the paper is to discuss Popa’s deformation/rigidity theory. This is a remarkable framework that, starting in the early 2000s, has led to unprecedented progress in the theory of von Neumann algebras (see the survey papers [43, 44, 72, 80]). At the heart of Popa’s theory is the innovative idea of using deformations of II1 factors to locate rigid subalgebras. We briefly explain this principle in Section 3 and, in addition, we present Popa’s proof [71] of Ozawa’s solidity result [57] of the free group factors. The second main goal of this paper is to study tensor product decompositions for certain natural families of II1 factors arising from model theory [28, 34]. To motivate our results, we recall that existentially closed groups are simple [37] and as such do not admit any nontrivial direct or semidirect product decompositions. Since existentially closed factors are von Neumann algebraic analogues of existentially closed groups, it is reasonable to expect they share similar indecomposability properties. It is known that

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̄ where R is the hyperfinite II1 existentially closed factors M are McDuff, i. e., M ≅ M ⊗R factor, [34], but not strongly McDuff,1 i. e., M does not admit a tensor product decomposī with B a full II1 factor and R the hyperfinite II1 factor, [2]. However, besides this tion B⊗R result, little is known regarding possible tensor decompositions of existentially closed factors. In this paper we make progress on this problem by obtaining in Sections 6 and 7 a series of tensor indecomposability results for existentially closed factors. First, by building upon [12, 16] (see Section 4) and using Popa’s deformation/rigidity theory we provide in Theorem 7.1 an uncountable family of existentially closed II1 factors M that do not admit tensor product decompositions M = P⊗Q into diffuse factors with Q full. In fact, we believe that this property holds for any existentially closed factor M and we provide several instances when it holds under certain additional assumptions on P or Q. For instance, we establish the property whenever Q is a group von Neumann algebra of a non-inner amenable group, see Theorem 7.6. Finally, we conclude our paper by proposing in Section 8 several open problems regarding the structure of existentially closed factors beyond tensor indecomposability.

2 Preliminaries on von Neumann algebras 2.1 Terminology A tracial von Neumann algebra is a pair (M, τ) consisting of a von Neumann algebra M and a faithful normal tracial state τ : M → ℂ. This induces a norm on M by the formula ‖x‖2 = τ(x ∗ x)1/2 , for all x ∈ M. We denote by U (M) the group of unitary elements of M, by 𝒵 (M) the center of M and by Aut(M) the group of τ-preserving automorphisms of M. For u ∈ U (M), we denote by Ad(u) ∈ Aut(M) the inner automorphism of M given by Ad(u)(x) = uxu∗ . The group of all inner automorphisms of M is denoted by Inn(M). For a set I, we denote by (M I , τ) the tensor product tracial von Neumann algebra ⨂i∈I (M, τ). For J ⊂ I, we view M J ⊂ M I , in the natural way. For i ∈ I, we write M i instead of M {i} . All inclusions P ⊂ M of von Neumann algebras are assumed unital unless otherwise stated. We denote by EP : M → P the unique τ-preserving conditional expectation from M onto P, by eP : L2 (M) → L2 (P) the orthogonal projection onto L2 (P) and by ⟨M, eP ⟩ the Jones’ basic construction of P ⊂ M. We also denote by P′ ∩ M = {x ∈ M : xy = yx, for all y ∈ P} the relative commutant of P in M and by NM (P) = {u ∈ U (M) : uPu∗ = P} the normalizer of P in M. For two von Neumann subalgebras P1 , P2 ⊂ M, we denote by P1 ∨ P2 = W ∗ (P1 ∪ P2 ) the von Neumann algebra generated by P1 and P2 . Let ω be a free ultrafilter on ℕ. Consider the C∗ -algebra ℓ∞ (ℕ, M) = {(xn )n ⊂ M : supn ‖xn ‖ < ∞} and its closed ideal ℐ = {(xn )n ∈ ℓ∞ (ℕ, M) : limn→ω ‖xn ‖2 = 0}. The ul1 This terminology was introduced by Popa in [74].

272 � I. Chifan et al. trapower of M is the tracial von Neumann algebra M ω := ℓ∞ (ℕ, M)/ℐ whose canonical trace is given by τω (x) = limn→ω τ(xn ), for any x = (xn )n ∈ M ω . Finally, for any positive integer n, we denote by 1, n the set {1, . . . , n}.

2.2 Finite index inclusions of von Neumann algebras The Jones index for an inclusion P ⊆ M of II1 factors is the dimension of L2 (M) as a left P-module [50]. Pimsner and Popa defined a probabilistic notion of index for an inclusion P ⊆ M of arbitrary von Neumann algebras with conditional expectation, which extends Jones’ index for inclusions of II1 factors [63, Theorem 2.2]. Specifically, the inclusion P ⊆ M of tracial von Neumann algebras is said to have probabilistic index [M : P] = λ−1 , where 󵄩 󵄩2 λ = inf{󵄩󵄩󵄩EP (x)󵄩󵄩󵄩2 ‖x‖−2 2 : x ∈ M+ , x ≠ 0}. Here we use the convention that 01 = ∞. We continue by recording several basic facts from the literature concerning finite index inclusions of von Neumann algebras which we will use in the proofs of our main results in Sections 5 and 7. First, recall that a von Neumann algebra M is called completely atomic if 1 is an orthogonal sum of minimal projections in M. The first three items are essentially contained in [66], and we refer the reader to [11, Proposition 2.3] for a proof. The fourth item is precisely [50, Proposition 2.1.15], while the last one follows directly from the definition. Proposition 2.1. Let N ⊂ M be an inclusion of tracial von Neumann algebras with [M : N] < ∞ and P a tracial factor. Then the following hold: (1) If p ∈ N is a nonzero projection, then [pMp : pNp] < ∞. (2) If N is a factor and r ∈ N ′ ∩ M is a nonzero projection, then [rMr : Nr] < ∞. (3) If 𝒵 (M) is completely atomic, then 𝒵 (N) is completely atomic. ̄ : N ⊗P] ̄ < ∞. (4) If M and N are factors, then [M ⊗P (5) Let R ⊂ N be a von Neumann subalgebra satisfying [N : R] < ∞. Then [M : R] = [M : N][N : R]. We end this section by recalling two more results on finite index inclusions, that will be essential to deriving our main results in Section 7. Lemma 2.2 ([67, Lemma 3.1]). Let P ⊂ Q ⊂ M be an inclusion of tracial von Neumann algebras such that P ⊆ Q is a finite index inclusion of II1 factors. Then [P′ ∩ M : Q′ ∩ M] < ∞. Lemma 2.3 ([15, Lemma 2.3]). Let M, N ⊆ P be von Neumann algebras. Let M0 be the von Neumann algebra generated by EN (M) inside N, M0 := EN (M)′′ ⊆ N. If [P : M] < ∞, then [N : M0 ] < ∞.

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2.3 Popa’s intertwining techniques Almost two decades ago, S. Popa introduced in [70, Theorem 2.1 and Corollary 2.3] a powerful analytic criterion for identifying intertwiners between arbitrary subalgebras of tracial von Neumann algebras, see Theorem 2.4 below. This technique, known as Popa’s intertwining-by-bimodules technique, has played an essential role in the classification of von Neumann algebras program via Popa’s deformation/rigidity theory. Theorem 2.4 ([70]). Let (M, τ) be a tracial von Neumann algebra and let P, Q ⊆ M be (not necessarily unital) von Neumann subalgebras. Then the following are equivalent: (1) There exist projections p ∈ P, q ∈ Q, a ∗-homomorphism θ : pPp → qQq and a partial isometry 0 ≠ v ∈ M such that v∗ v ≤ p, vv∗ ≤ q and θ(x)v = vx, for all x ∈ pPp. (2) For any group 𝒢 ⊂ U (P) such that 𝒢 ′′ = P there is no net (un )n ⊂ 𝒢 satisfying ‖EQ (xun y)‖2 → 0, for all x, y ∈ M. (3) There exist finitely many xi , yi ∈ M and C > 0 such that ∑i ‖EQ (xi uyi )‖22 ≥ C for all u ∈ U (P). (4) There exists a nonzero projection f ∈ P′ ∩ ⟨M, eQ ⟩ such that Tr(f ) < ∞. (5) There exists a P-Q-subbimodule ℋ of 1P L2 (M)1Q such that dim(ℋQ ) < +∞. If one of the three equivalent conditions from Theorem 2.4 holds, then we say that a corner of P embeds into Q inside M, and write P ≺M Q. If we, moreover, have that Pp′ ≺M Q, for any projection 0 ≠ p′ ∈ P′ ∩ 1P M1P , then we write P ≺sM Q (where the superscript “s” stands for strong). For further use, we recall several useful intertwining results for von Neumann subalgebras. Lemma 2.5. Let (M, τ) be a tracial von Neumann algebra and P ⊂ pMp, Q ⊂ qMq, R ⊂ rMr be von Neumann subalgebras, where p, q, r ∈ M are projections. Then the following hold: (1) [27, Lemma 2.4(2)] If Pz ≺M Q for any nonzero projection z ∈ NpMp (P)′ ∩ M, then P ≺sM Q. (2) [27, Lemma 2.4(3)] Assume P ≺M Q. Then there is a nonzero projection z ∈ 𝒩pMp (P)′ ∩ M such that Pz ≺sM Q. (3) [79, Lemma 3.5] If P ≺M Q, then Q′ ∩ qMq ≺M P′ ∩ pMp. (4) [79, Lemma 3.7] If P ≺M Q and Q ≺sM R, then P ≺M R. (5) If S ⊆ P is a finite index von Neumann subalgebra, then S ≺M R if and only if P ≺M R. Lemma 2.6 ([63, Lemma 2.3]). Let N ⊂ M be an inclusion of tracial von Neumann algebras satisfying [M : N] < ∞. The following hold: (1) M ≺sM N. (2) If 𝒵 (N) is completely atomic, then M ≺M Nq, for any nonzero projection q ∈ N ′ ∩ M. For proofs of (1) and (2), we refer the reader to [15, Lemma 2.4] and [25, Lemma 2.11], respectively.

274 � I. Chifan et al. We end this section with the following elementary lemma. For completeness, we also include a short proof and refer the reader to Definition 4.5 for the notion of cocycle action. Lemma 2.7. Let Γ ↷σ,α A be a cocycle action and let M = A ⋊σ,α Γ the corresponding twisted crossed product von Neumann algebra. Then M ≺M A if and only if Γ is finite. Proof. If Γ is finite then A ⊆ M has finite index and thus Lemma 2.6(1) implies that M ≺M A. To see the converse assume M ≺M A. Then part (3) in Theorem 2.4 implies the existence of finitely many x1 , . . . , xn , y1 , . . . , yn ∈ M and C > 0 such that ∑ki=1 ‖EA (xi ug yi )‖22 ≥ C, for all g ∈ Γ. Approximating xi and yi , via the Kaplansky density theorem, by elements in the ∗-algebra generated by the linear span of AΓ, this inequality further implies the existence of an integer l ≥ k, a constant C ′ > 0, and xi′ = ai ugi , y′i = bi uhi , where ai , bi ∈ A and gi , hi ∈ Γ for all i ∈ 1, l, such that l

󵄩 󵄩2 ∑󵄩󵄩󵄩EA (xi′ ug y′i )󵄩󵄩󵄩2 ≥ C ′ , i=1

for all g ∈ Γ.

(2.1)

Now notice that EA (xi′ ug y′i ) = ai EA (ugi ug uhi )σh−1 (bi ) = ai EA (α(gi , g)α(gi g, hi )ugi ghi ) × i σh−1 (bi ) = τ(ugi ghi )ai α(gi , g)α(gi g, hi )σh−1 (bi ). Since α(gi , g)α(gi g, hi ) ∈ U (A), basic esi

i

timates further imply that ‖EA (xi′ ug y′i )‖2 ≤ ‖ai ‖∞ ‖bi ‖∞ |τ(ugi ghi )| = δgi ghi ,1 ‖ai ‖∞ ‖bi ‖∞ . Combining this with relation (2.1), we get l

∑ δgi ghi ,1 ≥ i=1

C′

maxli=1 ‖ai ‖∞ ‖bi ‖∞

> 0.

This implies Γ ⊆ {gi−1 hi−1 : i ∈ 1, l}, entailing Γ is finite.

2.4 Relative amenability and weak containment of bimodules A tracial von Neumann algebra (M, τ) is amenable if there exists a net (ξn )n ∈ L2 (M) ⊗ L2 (M) such that ⟨xξn , ξn ⟩ → τ(x) and ‖xξn − ξn x‖2 → 0, for all x ∈ M. Connes’ celebrated classification of amenable factors [19] shows that M is amenable if and only if M is approximately finite dimensional. Next, we recall the notion of relative amenability introduced by Ozawa and Popa in [60]. Let (M, τ) be a tracial von Neumann algebra. Let p ∈ M be a projection and P ⊂ pMp, Q ⊂ M be von Neumann subalgebras. Following [60, Section 2.2], we say that P is amenable relative to Q inside M if there exists a net (ξn )n ∈ L2 (⟨M, eQ ⟩) such that ⟨xξn , ξn ⟩ → τ(x) for all x ∈ pMp and ‖yξn − ξn y‖2 → 0, for all y ∈ P. It is a fact that P is amenable relative to ℂ inside M if and only if P is amenable.

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Let M, N be tracial von Neumann algebras. An M–N-bimodule M ℋN is a Hilbert space ℋ equipped with two commuting normal unital ∗-homomorphisms M → 𝔹(ℋ) and N op → 𝔹(ℋ). If M = N, we say, for simplicity, that M ℋM is an M-bimodule. For two bimodules M ℋN and M 𝒦N , we say that M ℋN is weakly contained in M 𝒦N if for any ϵ > 0, finite sets F ⊂ M, G ⊂ N and ξ ∈ ℋ, there exist η1 , . . . , ηn ∈ 𝒦 such that 󵄨󵄨 󵄨󵄨 n 󵄨 󵄨󵄨 󵄨󵄨⟨xξy, ξ⟩ − ∑⟨xηi y, ηi ⟩󵄨󵄨󵄨 ≤ ϵ, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 i=1

for all x ∈ F and y ∈ G.

Examples of bimodules include the trivial M-bimodule M L2 (M)M and the coarse M–Nbimodule M L2 (M) ⊗ L2 (N)N . Finally, it is a fact that a subalgebra P ⊂ pMp is amenable relative to Q inside M if and only if P L2 (pM)M is weakly contained in P L2 (⟨M, eQ ⟩)M (see, for instance, [1, Theorem 13.4.4]).

3 Popa’s deformation/rigidity theory 3.1 Deformations In the last two decades, Popa’s deformation/rigidity theory has led to unprecedented progress in the theory of von Neumann algebras, see the survey papers [43, 44, 72, 80]. Popa’s theory is centered on the remarkable idea of using deformations of II1 factors to locate rigid subalgebras. In Theorem 3.6 we will illustrate this principle by presenting Popa’s proof [71] of Ozawa’s solidity result [57] for the free group factors. First, we make precise the notion of a deformation. Definition 3.1. A deformation of the identity of a tracial von Neumann algebra (M, τ) is a sequence of unital, trace preserving, completely positive maps ϕn : M → M satisfying ‖ϕn (x) − x‖2 → 0, for all x ∈ M. A linear map ϕ : M → M is called completely positive if the amplification ϕ(m) : 𝕄m (ℂ) → 𝕄m (ℂ) given by ϕ(m) ([xi,j ]) = [ϕ(xi,j )] is positive, for any m ≥ 1. Before presenting some examples of deformations, we recall some terminology. Let (M, τ) be a tracial separable von Neumann algebra. A map ϕ : M → M is called subunital if ϕ(1) ≤ 1 and subtracial if τ ∘ ϕ ≤ τ. For a von Neumann subalgebra P ⊂ M, the map ϕ is called P-bimodular if ϕ(axb) = aϕ(x)b, for all a, b ∈ P and x ∈ M. Let P ⊂ M be a von Neumann subalgebra. Following [69, Section 2], we say that M has the Haagerup property relative to P if there exists a sequence of normal subunital subtracial completely positive P-bimodular maps ϕn : M → M such that ‖ϕn (x) − x‖2 → 0, for any x ∈ M and such that every ϕn satisfies the following relative compactness property: if (wk )k is a sequence of unitaries in M satisfying ‖EP (awn y)‖2 → 0, for all a, b ∈ M, then ‖ϕn (wk )‖2 → 0 when k → ∞.

276 � I. Chifan et al. Next, we continue by presenting two examples of deformations. For a comprehensive list of examples, we refer the reader to [43, Section 3] (see also [44, Section 3]). The first one was used by Popa by using the Haagerup property to show that certain von Neumann algebras have trivial fundamental group [69], thereby solving a longstanding question in operator algebras. Example 3.2. Let Γ ↷ (X, μ) be a probability measure preserving action of a countable group and let φn : Γ → ℂ be a sequence of positive definite functions such that φn (g) → 1, for all g ∈ Γ. Then the sequence ϕn : L∞ (X) ⋊ Γ → L∞ (X) ⋊ Γ given by ϕn (aug ) = φn (g)aug is a deformation of the identity of L∞ (X) ⋊ Γ. If Γ has the Haagerup property [36] (which is equivalent to L(Γ) having the Haagerup property relative to ℂ), one can choose φn : Γ → ℂ to satisfy φn ∈ c0 (Γ), for any n. In addition, if Γ = SL2 (ℤ), then the sequence ϕn is a deformation of the II1 factor M = L∞ (𝕋2 ) ⋊ SL2 (ℤ) which witnesses the fact that M has the Haagerup property relative to L∞ (𝕋2 ). This fact is an essential ingredient in Popa’s proof that M has trivial fundamental group. Now, we now turn our attention to the foundational notions of malleable and s-malleable deformations introduced by Popa in [69, 70] which consists of a special path of completely positive maps. This novel concept was used with great effect in classifying large classes of factors. Definition 3.3. Let (M, τ) be a tracial von Neumannn algebra. A pair (M,̃ (αt )t∈ℝ ) is called a malleable deformation of M if the following conditions hold: ̃ ; (1) (M,̃ τ)̃ is a tracial von Neumann algebra such that M ⊂ M̃ and τ = τ|M (2) (αt )t∈ℝ ⊂ Aut(M,̃ τ)̃ is a 1-parameter group with limt→0 ‖αt (x) − x‖2 = 0, for any x ∈ M;̃ (3) αt does not converge uniformly to the identity on (M)1 as t → 0. In addition, if also the following condition holds (4) There exists β ∈ Aut(M,̃ τ)̃ that satisfies β|M = IdM , β2 = IdM̃ and βαt = α−t β, for any t ∈ ℝ. then (M,̃ (αt )t∈ℝ , β) is called an s-malleable deformation of M. There are several natural classes of von Neumann algebras that admit (s-)malleable deformations: – Free group factors, and more generally, amalgamated free product von Neumann algebras [45, 65, 71]. – HNN extensions von Neumann algebras [29]. – Wreath products von Neumann algebras [40, 68, 70]. – Von Neumann algebras that are constructed from trace preserving action of groups that admit unbounded cocycles valued into various orthogonal representations, [76]. – Von Neumann algebras that admit certain unbounded closable derivations [21].

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Our second deformation that we concretely exemplify is the malleable deformation of free groups factors which was used by Popa [71] to prove Ozawa’s solidity result [57] for free group factors. Example 3.4. Free group factors admit natural malleable deformations as follows [71]. For simplicity, we recall the deformation of the group factor M = L(𝔽n ) with 2 ≤ n ≤ ∞. Let {ak }nk=1 ∪{bk }nk=1 be the generators of 𝔽2n and denote M̃ = M ∗M. By viewing 𝔽n as the subgroup of 𝔽2n generated by {ak }nk=1 , we obtain an embedding of M inside M.̃ We denote still by {ak }nk=1 ∪ {bk }nk=1 the canonical unitaries in M.̃ We now construct a 1-parameter group of automorphisms (αt )t∈ℝ of M̃ as follows. Let {hk }nk=1 ⊂ M̃ be hermitian elements such that bk = exp(ihk ), for any k ∈ 1, n. For every t ∈ ℝ, we define a trace preserving automorphism αt of M̃ by letting αt (ak ) = exp(ithk )ak ,

αt (bk ) = bk ,

for every k ∈ 1, n.

In fact, this malleable deformation is s-malleable as witnessed by defining a trace preserving automorphism β of M̃ by letting β(ak ) = ak ,

β(bk ) = b∗k ,

for every k ∈ 1, n.

3.2 Deformation versus rigidity An important concept in deformation/rigidity theory and a main source of rigidity is the notion of property (T) for von Neumann algebras introduced by Connes and Jonnes [20] and its relative version for inclusions of von Neumann algebras defined by Popa [69]. Definition 3.5. An inclusion of tracial von Neumann algebras P ⊂ M has the relative property (T) if any deformation ϕn : M → M of the identity must converge uniformly to the identity on the unit ball of P. A tracial von Neumann algebra M has property (T) if the inclusion M ⊂ M has relative property (T). Relative property (T) should be thought of as rigidity by noticing that no property (T) von Neumann algebra can have “lots” of deformations. To exemplify this, we note that if a tracial von Neumann algebra M has property (T) and the Haagerup property relative to ℂ (which provides “nontrivial” deformations), then M is completely atomic. Indeed, using the Haagerup property we let ϕn : M → M be a sequence of normal subunital subtracial completely positive maps such that ‖ϕn (x)−x‖2 → 0, for any x ∈ M, and every ϕn satisfies ‖ϕn (wk )‖2 → 0 for any sequence of unitaries (wk )k in M that weakly goes to 0. Since property (T) can also be characterized by using completely positive maps that are only subunital and subtracial (see [69]), we obtain that ϕn must converge uniformly to the identity on the unit ball of M. Hence, there is n0 ≥ 1 such that ‖ϕn0 (x) − x‖2 ≤ 1/2 for any x ∈ 𝒰 (M). If we assume by contradiction that M is not diffuse, then there is a sequence of unitaries (wk )k in M that weakly goes to 0, and hence, ‖ϕn0 (wk )‖2 → 0.

278 � I. Chifan et al. This gives the contradiction that 1 = limk→∞ ‖ϕn0 (wk ) − wk ‖2 ≤ 1/2, and hence, M is not diffuse. Moreover, since property (T) and the relative Haagerup property pass to corners of M (see [69]), it follows that any corner of M is not diffuse, and thus, M is completely atomic. The natural correspondence between completely positive maps and bimodules provides a reformulation of relative property (T) in terms of bimodules. First, recall that if ℋ is an M-bimodule, then a vector ξ ∈ M is P-central if xξ = ξx, for all x ∈ P, and tracial if ⟨xξ, ξ⟩ = ⟨ξx, ξ⟩ = τ(x), where M is endowed with the trace τ. A net of vectors (ξn )n ⊂ ℋ is called P-almost central if ‖xξn − ξn x‖2 → 0, for any x ∈ P. Following [69], the inclusion P ⊂ M has relative property (T) if and only if any M-bimodule without central P-vectors does not admit a net (ξn )n of tracial, M-almost central vectors. Hence, relative property (T) requires that all M-bimodules satisfy a certain spectral gap condition. A remarkable discovery of Popa [71, 73] is that one can obtain rigidity by only using that certain bimodules have spectral gap. For instance, if P ⊂ M is a nonamenable subalgebra, then there does not exist a net (ξn )n ∈ L2 (M) ⊗ L2 (M) of tracial, P-almost central vectors; this follows from the fact that P is non-amenable relative to ℂ inside M, see Section 2.4. To explain the spectral gap terminology, we note that Connes’ theorem [19] gives that a II1 factor P is nonamenable if and only if there exist a finite set S ⊂ P and k > 0 such that ‖ξ‖2 ≤ k ∑ ‖yξ − ξy‖2 , y∈S

for all ξ ∈ L2 (P) ⊗ L2 (P).

This principle of using spectral gap from nonamenable subalgebras is illustrated below in the proof of Theorem 3.6.

3.3 Solidity of free group factors In this section we present Popa’s deformation/rigidity theory proof [71] of Ozawa’s solidity result for the free group factors. Theorem 3.6 ([57]). Let M = L(𝔽n ), for some 2 ≤ n ≤ ∞. Then M is solid, i. e., for every diffuse von Neumann subalgebra A ⊂ M, the relative commutant A′ ∩ M is amenable. The proof uses the free malleable deformation (M,̃ (αt )t∈ℝ ) from Example 3.4. Recall that M̃ = M ∗ M. We will use the fact that the contraction EM ∘ αt : L2 (M) → L2 (M) is a compact operator, for any t > 0. In fact, the following weaker property will suffice for our purposes: Lemma 3.7. If (un )n ⊂ 𝒰 (M) is a sequence that converges weakly to 0, then for any t > 0 we have limn→∞ ‖EM (αt (un ))‖2 = 0. Proof. Denote ρ(t) = sin(πt) and notice that 0 < ρ(t) < 1, for all t > 0. Note that exp(ihk ) πt can be seen as the canonical generating unitary of L(ℤ) for any k ∈ 1, n. Thus, by iden-

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tifying L(ℤ) with L∞ (𝕋), we derive that π

τ(exp(ithk )) =

1 ∫ exp(itθ) dθ = ρ(t), 2π

for all t > 0 and k ∈ 1, n.

−π

This implies that EM (αt (ug )) = ρ(t)|g| ug ,

for all t ∈ ℝ and g ∈ 𝔽n .

(3.1)

Here, |g| denotes the word length of g ∈ 𝔽n with respect to the generating set {ak }nk=1 . Consider the Fourier expansion un = ∑g∈𝔽n τ(un ug∗ )ug . Then by formula

(3.1), we get that EM (αt (un )) = ∑g∈𝔽n ρ(t)|g| τ(un ug∗ )ug and thus ‖EM (αt (un ))‖22 =

∑g∈𝔽n ρ(t)2|g| |τ(un ug∗ )|2 . Thus, if N ≥ 1 is an integer, then using that ∑g∈𝔽n |τ(un ug∗ )|2 = ‖un ‖22 = 1, we get that

󵄩󵄩 󵄩2 󵄨 󵄨 󵄨2 󵄨 󵄨2 ∗ 󵄨2 2N ∑ 󵄨󵄨󵄨τ(un ug∗ )󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨τ(un ug∗ )󵄨󵄨󵄨 + ρ(t)2N . 󵄩󵄩EM (αt (un ))󵄩󵄩󵄩2 ≤ ∑ 󵄨󵄨󵄨τ(un ug )󵄨󵄨󵄨 + ρ(t) |g|≤N

|g|≥N

|g|≤N

Since the set {g ∈ Fn : |g| ≤ N} is finite and limn→∞ τ(un ug∗ ) = 0, for all g ∈ 𝔽n , we

conclude that lim supn→∞ ‖EM (αt (un ))‖22 ≤ ρ(t)2N , for all N ≥ 1. Since 0 < ρ(t) < 1, the conclusion of the corollary follows.

The source of rigidity in the proof of Theorem 3.6 is given by the following spectral gap result. ̃ ⊖ L2 (M) is isomorphic to an infinite multiple Lemma 3.8. The Hilbert M-bimodule L2 (M) ̄ 2 (M))⊕∞ . of the coarse M-bimodule, (L2 (M)⊗L In particular, if B ⊂ M is a nonamenable von Neumann subalgebra, then the trivial ̃ ⊖ L2 (M). B-bimodule is not weakly contained in the B-bimodule L2 (M) Proof. Let S ⊂ 𝔽2n be the set of elements g ∈ 𝔽n whose reduced form begins and ends ̃ ⊖ L2 (M) = ⨁ sp(Mug M), with a nonzero power of bk for some k ∈ 1, n. Since L2 (M) g∈S ̄ 2 (M), as in order to prove the assertion, it suffices to show that sp(Mug M) ≅ L2 (M)⊗L Hilbert M-bimodules, for any g ∈ S. If g ∈ S, then g −1 𝔽n g ∩ 𝔽n = {e}. Hence, τ(ug∗ uh ug uk ) = δg −1 hg,k −1 = δh,e δk,e = τ(uh )τ(uk ), for all h, k ∈ 𝔽n . This implies τ(ug∗ aug b) = τ(a)τ(b), for all a, b ∈ M, and further that ⟨xug y, zug t⟩ = τ(ug∗ z∗ xug yt ∗ ) = τ(z∗ x)τ(yt ∗ ) = ⟨x ⊗ y, z ⊗ t⟩L2 (M)⊗L̄ 2 (M) , for all x, y, z, t ∈ M.

̄ 2 (M) ≅ Thus, x ⊗ y 󳨃→ xug y extends to an isomorphism of Hilbert M-bimodules L2 (M)⊗L sp(Mug M).

280 � I. Chifan et al. In order to prove the last part of the lemma, let B ⊂ M be a nonamenable von Neumann subalgebra. First, note that the B-bimodule L2 (M) ⊗ L2 (M) is weakly contained in the coarse B-bimodule L2 (B) ⊗ L2 (B). This follows from the fact that any left (respectively, right) B-module is contained in L2 (B)⊕∞ as a left (respectively, right) B-module (see, for instance, [1, Proposition 8.2.3]). Next, we derive that the B-bimodule ̄ 2 (M))⊕∞ is weakly contained in the B-bimodule (L2 (B)⊗L ̄ 2 (B))⊕∞ , and hence, (L2 (M)⊗L 2 2 ̄ (B). Finally, if we assume by contradiction that the in the coarse B-bimodule L (B)⊗L 2 ̃ trivial B-bimodule is weakly contained in the B-bimodule L2 (M)⊖L (M), we derive from the first part of the lemma that the trivial B-bimodule is weakly contained in the coarse ̄ 2 (B). This implies that B is amenable, contradiction. This proves the B-bimodule L2 (B)⊗L lemma. Proof of Theorem 3.6. Let A ⊂ M be a diffuse von Neumann subalgebra and let (uk )k ⊂ 𝒰 (A) be a sequence which converges weakly to 0. Denote B = A′ ∩ M. Put tn = 1/2n , for every n ≥ 1. By Lemma 3.7, we can find a subsequence (vn )n of (uk )k such that limn→∞ ‖EM (αtn (vn ))‖2 = 0. By letting ξn = αtn (vn ) − EM (αtn (vn )), we claim that 󵄩 󵄩 lim 󵄩󵄩[x, ξn ]󵄩󵄩󵄩2 = 0,

n→∞󵄩

for every x ∈ B,

lim ⟨xξn , ξn ⟩ = τ(x),

n→∞

and

for every x ∈ M.̃

(3.2) (3.3)

To prove (3.2), note that for any x ∈ B we have ‖[x, EM (αtn (vn ))]‖2 ≤ 2‖x‖‖EM (αtn (vn ))‖2 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩[x, αtn (vn )]󵄩󵄩󵄩2 = 󵄩󵄩󵄩[α−tn (x), vn ]󵄩󵄩󵄩2 = 󵄩󵄩󵄩[α−tn (x) − x, vn ]󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩α−tn (x) − x 󵄩󵄩󵄩2 .

To prove (3.3), note that limn→∞ ‖EM (αtn (vn ))‖2 = 0 implies that for every x ∈ M,̃ we have lim ⟨xξn , ξn ⟩ = lim ⟨xαtn (vn ), αtn (vn )⟩ = τ(x).

n→∞

n→∞

Combining (3.2) and (3.3), it follows that the trivial B-bimodule is weakly contained 2 ̃ in the B-bimodule L2 (M)⊖L (M). Hence, by Lemma 3.8, it follows that B is amenable.

4 Wreath-like products and wreath-like von Neumann algebras For further use, we recall in this section the main result of our very recent work [12]. This asserts that property (T) II1 factors form an embedding universal family, i. e., every separable tracial von Neumann algebra embeds into a property (T) II1 factor, see [12, Theorem A]. This result is proved using the so-called wreath-like product von Neumann algebras. In turn, these are built from wreath-like product groups which were

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introduced and studied in [16] through the lenses of geometric group theory. To provide some context we first recall these notions along with some of their basic properties.

4.1 Wreath-like product groups Following [16], let A and B be countable groups. Then G is a regular wreath-like product of A and B if it can be realized as a group extension ε

1 → ⨁ Ab 󳨅→ G 󴀀󴀤 B → 1

(4.1)

b∈B

which satisfies the following properties: (a) Ab ≅ A for all b ∈ B, and (b) the conjugation action of G on ⨁b∈B Ab permutes the direct summands according to the rule gAb g −1 = Aε(g)b

for all g ∈ G, b ∈ B.

The class of all such wreath-like groups is denoted by 𝒲ℛ(A, B). When the extension (4.1) splits, G is the classical wreath product of A and B, G = A ≀ B. Next we recall the concept of a cocycle semidirect product group; see [7, pages 104–105]. Definition 4.1. A cocycle action of a group B on a group A is a pair (α, v) consisting of two maps α : B → Aut(A) and v : B × B → A which satisfy the following: (1) αb αc = Ad(vb,c )αbc , for every b, c ∈ B, (2) vb,c vbc,d = αb (vc,d )vb,cd , for every b, c, d ∈ B, and (3) vb,1 = v1,b = 1, for every b ∈ B. Definition 4.2. Let (α, v) be a cocycle action of a group B on group A. Then the set A × B endowed with the unit 1 = (1, 1) and the multiplication operation (x, b) ⋅ (y, c) = (xαb (y)vb,c , bc) is a group, denoted A ⋊α,v B, and called the cocycle semidirect product i

γ

group. Moreover, we have a short exact sequence 1 → A → 󳨀 A ⋊α,v B → 󳨀 B → 1, where i(a) = (a, 1) and γ(a, b) = b. Remark 4.3. It is not hard to prove that a group G belongs to 𝒲ℛ(A, B) if and only if it is isomorphic to A(B) ⋊α,v B, for a cocycle action (α, v) on B on A(B) such that αb (Ac ) = Abc , for every b, c ∈ B; see [12, Corollary 2.11]. Wreath-like product groups admit a special cocycle semidirect product decomposition. Let σ be the shift action of B on AB = ∏b∈B A given by σb (x) = (xb−1 c )c∈B , for every x = (xc )c∈B ∈ AB and b ∈ B. Note that σ leaves invariant the normal subgroup A(B) = ⨁b∈B A of AB . The following result has been established in [16]; see [12, Lemmas 2.12 and 2.13] for proofs.

282 � I. Chifan et al. Proposition 4.4 ([12, 16]). Let A, B be groups. Then G ∈ 𝒲ℛ(A, B) if and only if there is a (B) function ρ : B → AB such that setting vb,c := ρb σb (ρc )ρ−1 bc ∈ A , for every b, c ∈ B, and (B) ρ1 = 1, and letting αb := Ad(ρb )σb ∈ Aut(A ), for every b ∈ B, we have that G ≅ A(B) ⋊α,v B.

4.2 Tracial wreath-like product von Neumann algebras By analogy with wreath-like product groups there is a notion of wreath-like product for tracial von Neumann algebras. First, we need to briefly recall the notion of tracial cocycle crossed product von Neumann algebra. Definition 4.5. A cocycle action of a group B on a tracial von Neumann (M, τ) is a pair (β, w) consisting of two maps β : B → Aut(M) and w : B × B → U (M) which satisfy the following: (1) βb βc = Ad(wb,c )βbc , for every b, c ∈ B, (2) wb,c wbc,d = βb (wc,d )wb,cd , for every b, c, d ∈ B, and (3) wb,1 = w1,b = 1, for every b ∈ B. Definition 4.6. Let (β, w) be a cocycle action of a group B on a tracial von Neumann algebra (M, τ). The cocycle crossed product von Neumann algebra M ⋊β,w B is a tracial von Neumann algebra which is generated by a copy of M and unitary elements {ub }b∈B such that ub xub∗ = βb (x), ub uc = wb,c ubc and τ(xub ) = τ(x)δb,1 , for every b, c ∈ B and x ∈ M. Definition 4.7. Let (M, τ) be a tracial von Neumann algebra and B be a group. A tracial von Neumann algebra N is said to be a wreath-like product of M and B if it is isomorphic to M B ⋊β,w B, where (β, w) is a cocycle action of B on M B such that βb (M c ) = M bc , for every b, c ∈ B. We denote by 𝒲ℛ(M, B) the class of all wreath-like products of M and B. Example 4.8. If G ∈ 𝒲ℛ(A, B), then L(G) ∈ 𝒲ℛ(L(A), B); see [12, Example 3.2]. Notation 4.9. Let (M, τ) be a tracial von Neumann algebra and B be a group. We denote by (1) γ : U (M)(B) → U (M B ) the homomorphism given by γ((xb )b∈B ) = ⨂b∈B xb . (2) η : U (M)B → Aut(M B ) the homomorphism given by η((yb )b∈B ) = ⨂b∈B Ad(yb ). (3) B ↷σ U (M)B the shift action of B (which preserves the subgroup U (M)(B) < U (M)B ). (4) B ↷λ M B the Bernoulli shift action given by λb (x) = ⨂c∈B xb−1 c , for x = ⨂c∈B xc ∈ M B. With this notation, we have: Lemma 4.10. Let (M, τ) be a tracial von Neumann algebra and B a group. Let ξ : B → −1 U (M)B be a map such that ξb σb (ξc )ξbc ∈ U (M)(B) , for every b, c ∈ B. Define βb = η(ξb )λb ∈

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−1 Aut(M B ) and wb,c = γ(ξb σb (ξc )ξbc ) ∈ U (M B ), for every b, c ∈ B. Then (β, w) is a cocycle B B action of B on M and M ⋊β,w B ∈ 𝒲ℛ(M, B).

Using this lemma in combination with Proposition 4.4, it was shown in [12] that given G ∈ 𝒲ℛ(A, B), any homomorphism π : A → U (M), where (M, τ) is a tracial von Neumann algebra, extends to a homomorphism π̃ : G → U (N), for some N ∈ 𝒲ℛ(M, B). More precisely, using Proposition 4.4, we write G = A(B) ⋊α,v B, where (α, v) is a cocycle action of B on A(B) given by αb = Ad(ρb )σb and vb,c = ρb σb (ρc )ρ−1 bc , for some map B ρ : B → A . Then we have the following: Proposition 4.11. Let π : A → U (M) be a homomorphism, where (M, τ) is a tracial von −1 Neumann algebra. Define ξ := π B (ρb ) ∈ U (M)B , for every b ∈ B. Then ξb σb (ξc )ξbc ∈ (B) B −1 U (M) , for every b, c ∈ M. Define βb = η(ξb )λb ∈ Aut(M ) and wb,c = γ(ξb σb (ξc )ξbc ) ∈ U (M B ), for every b, c ∈ B. Then (β, w) is a cocycle action of B on M B , N := M B ⋊β,w B ∈ ̃ : G → U (N) given by π̃ (x) = γ(π B (x)) = 𝒲ℛ(M, B) and there is a homomorphism π ⨂b∈B π(xb ) and π̃ (e, c) = uc , for every x = (xb )b∈B ∈ A(B) and c ∈ B. Finally, using Propositions 4.4 and 4.11 and the fact that any acylindrically hyperbolic group has plenty of wreath-like quotients (see [16, Theorems 4.20]) and that there exist plenty of property (T) wreath-like product groups (see [16, Theorems 6.9]), the following result was shown in [12, Theorem A]. Theorem 4.12. Let (M, τ) be any separable von Neumann algebra. Then the following hold: (1) For every acylindrically hyperbolic group H, M embeds into a II1 factor N which is generated by a representation π : H → U (N). Thus, if H has property (T), then N has property (T). (2) M embeds into a property (T), wreath-like product II1 factor P ∈ 𝒲ℛ(Q, B) where B is a hyperbolic property (T) group, Out(P) = 1, and ℱ (P) = {1}. More precisely, Q can be taken to be Q = (M ∗ L(𝔽2 ))⊗R, where R is the hyperfinite II1 factor. We now explain the terminology used in the theorem. The class of acylindrically hyperbolic groups includes all nonelementary hyperbolic and relatively hyperbolic groups, mapping class groups of closed surfaces of nonzero genus and Out(𝔽n ), for n ≥ 2. We refer the reader to [16, Section 3.2] and to the survey [Osi18] for the precise definition of acylindrically hyperbolic groups and for more details. For a II1 factor P, we denote by Out(P) = Aut(P)/Inn(P) the outer automorphism group of P and by ℱ (P) = {τ(e)/τ(f ) | e, f ∈ P projections, ePe ≅ fPf } the fundamental group of P [54]. To this end, we make several remarks regarding the previous theorem. Part (1) in Theorem 4.12 should be viewed as a von Neumann algebraic analogue of the SQ-universality property for (acylindrically) hyperbolic groups established be Olshanskii, [55], Delaznt [23], and Dahmani–Guirardel–Osin [22]. Recall that a countable group

284 � I. Chifan et al. H is called SQ-universal if every countable group embeds into a quotient of H. Since property (T) passes to quotients, by taking H to be a hyperbolic group with property (T) it follows that every countable group embeds into a countable group with property (T). Therefore, Theorem 4.12 provides an analogue of this fact for II1 factors. Part (1) applies in particular to icc cocompact lattices H in any rank one simple real Lie group with finite center (e. g., Sp(n, 1), n ≥ 2), as such H are hyperbolic. Hence, Theorem 4.12(1) implies that the family of II1 factors generated by representations of H is embedding universal. This is in sharp contrast with the higher rank case since the work of Peterson [62] (see also [3, 5]) shows that if G is any icc lattice in higher rank simple real Lie group with finite center (e. g., SLm (ℝ), m ≥ 3), then L(G) is the only II1 factor generated by a representation of G. To put Theorem 4.12(2) into a better perspective, note that the first examples of II1 factors P with ℱ (P) = {1} have been obtained by Popa in [69] and the first examples of II1 factors P with Out(P) = {e} and ℱ (P) = {1} were obtained in [45]. Note that none of these II1 factors have property (T), although Popa’s strengthening of Connes’ rigidity conjecture (see [72, Section 3]) predicts that Out(L(G)) = {e} and ℱ (L(G)) = {1}, whenever G is an icc property (T) group with Out(G) = {e} and no characters. This conjecture has been confirmed for an uncountable class of groups in [16, Corollary 2.7] and thus showing that the class 𝒯 of all II1 factors P with property (T) which satisfy Out(P) = {e} and ℱ (P) = {1} is uncountable. Theorem 4.12(2) shows that 𝒯 is in fact embedding universal.

5 Primeness results for wreath-like product von Neumann algebras In this section we show that many of the wreath-like product von Neumann algebras introduced in the prior section are (virtually) prime; see Definition 5.5 and Theorem 5.6. We start with two preliminary results on intertwining von Neumann subalgebras in cocycle crossed product von Neumann algebras. These are modest extensions of prior results from [15, 16], but we include detailed proofs for the readers’ convenience. Notation 5.1. Let G ↷σ,α (Q, τ) be a trace preserving cocycle action on a tracial von Neumann algebra (Q, τ). Let π : G → H be a group epimorphism. Let M = Q ⋊σ,α G. Following [15, Section 2], we consider the ∗-embedding Δπ : M 󳨅→ M⊗L(H) =: M̃ given by Δπ (aug ) = aug ⊗ vπ(g) for every a ∈ M and g ∈ G. Here we have denoted by (vh )h∈H ⊂ L(H) the canonical group unitaries. When G = H and π = id, we denote Δπ simply by Δ. Our first result is a straightforward extension of [15, Proposition 3.4].

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Proposition 5.2 ([15]). Assume the Notation 5.1 and let p ∈ M be a projection. Then for any subgroup K < H and any von Neumann subalgebra P ⊆ pMp satisfying Δπ (P) ≺M̃ M⊗L(K), we have that P ≺M Q ⋊σ,α π −1 (K). Proof. Since Δπ (P) ≺M̃ M⊗L(K), using Theorem 2.4 and ‖ ⋅ ‖2 -approximations one can find group elements g1 , . . . , gn , h1 , . . . , hn ∈ H and a scalar C > 0 such that for all x ∈ U (P) we have n

󵄩 󵄩 ∑󵄩󵄩󵄩EM⊗L(K) ((1 ⊗ vgi )Δπ (x)(1 ⊗ vhi ))󵄩󵄩󵄩2 ≥ C.

(5.1)

i=1

Now pick elements ki , li ∈ G satisfying π(ki ) = gi and π(li ) = hi , for all i ∈ 1, n. Using this, together with the relation EM⊗L(K) ∘ Δπ = Δπ ∘ EQ⋊σ,α π −1 (K) , we can see that EM⊗L(K) ((1 ⊗ vgi )Δπ (x)(1 ⊗ vhi )) = EM⊗L(K) ((uk −1 ⊗ 1)Δπ (uki xuli )(ul−1 ⊗ 1)) i

i

= (uk −1 ⊗ 1)EM⊗L(K) (Δπ (uki xuli ))(ul−1 ⊗ 1) i

π

i

= (uk −1 ⊗ 1)Δ (EQ⋊σ,α π −1 (K) (uki xuli ))(ul−1 ⊗ 1). i

i

In particular, this shows that ‖EM⊗L(K) ((1 ⊗ vgi )Δπ (x)(1 ⊗ vhi ))‖2 = ‖EQ⋊σ,α π −1 (K) (uki xuli )‖2 and in combination with (5.1) we obtain ∑ni=1 ‖EQ⋊σ,α π −1 (K) (uki xuli )‖2 ≥ C, for all x ∈ U (P). Then Theorem 2.4 yields P ≺M Q ⋊σ,α π −1 (K), as desired.

In preparation for our second preliminary result we recall a deep result of Popa– Vaes from [75] regarding the classification of normalizers of amenable subalgebras in various crossed products von Neumann algebras. Since we need this theorem only for tensor products we will state it in this form. Theorem 5.3 ([75, Theorem 1.4]). Let Γ be a group that is biexact [8] and weakly amenable [59]. Let P be a tracial von Neumann algebra and denote by M = P⊗L(Γ). Let q ∈ M be a projection and let Q ⊂ qMq be a von Neumann subalgebra that is amenable relative to P inside M. Then one of the following must hold: (1) Q ≺M P; (2) NqMq (Q)′′ is amenable relative to P inside M. We note that when Γ is amenable then for any von Neumann subalgebras Q ⊆ qMq we automatically have that both Q and NqMq (Q)′′ are amenable relative to P inside M, [60, Proposition 2.4]. Thus in this case the result does not provide any meaningful information, and thus when using this result we are interested only in the case when Γ is nonamenable. Theorem 5.3 covers a variety of groups Γ which are very important to the structural study of von Neumann algebras. Next we highlight only one such class that is relevant for our results. Ozawa established that all hyperbolic groups are biexact [8, 57] and weakly amenable [59]. Since both these properties are hereditary, they are satisfied by all subgroups of hyperbolic groups. Using the strong Tits alternative for hyperbolic

286 � I. Chifan et al. groups [35] it follows that every amenable subgroup of a hyperbolic group is elementary;2 thus, Theorem 5.3 is meaningful and applies to all nonelementary subgroups of a given hyperbolic group. Our second preliminary result is a straightforward extension of [16, Theorem 6.15]. The proof is almost identical with the one presented there. Theorem 5.4 ([16]). Let G be a nonelementary subgroup of a hyperbolic group, and let G ↷σ,α (Q, τ) be a trace preserving cocycle action on a tracial von Neumann algebra (Q, τ). Let M = Q ⋊σ,α G and let p ∈ M be a projection. Then the following hold: 1. Let P ⊂ pMp be a von Neumann subalgebra which is amenable relative to Q inside M, and let N = NpMp (P)′′ . If there is a von Neumann subalgebra S ⊆ N with the relative property (T) such that S ⊀M Q, then P ≺sM Q. 2. Let A, B ⊂ pMp be commuting von Neumann subalgebras and let N = NpMp (A ∨ B)′′ . If there is a von Neumann algebra S ⊆ N with the relative property (T) and such that S ⊀M Q, then A ≺M Q or B ≺M Q. Proof. 1. Throughout the proof, we use freely the Notation 5.1. Assuming the conclusion is false, by Lemma 2.5(1) we find a nonzero projection z ∈ N ′ ∩ pMp such that Pz ⊀M Q. Since P is amenable relative to Q inside M, we get that Δ(P) is amenable relative to M⊗1 inside M.̃ This follows from Δ(Q) = Q ⊗ 1 ⊂ M ⊗ 1 and Δ(P) being amenable relative to Δ(Q) inside M̃ (see also [60, Proposition 2.4(3)]). Since G is a nonelementary subgroup of a hyperbolic group, applying Theorem 5.3 in the special case of tensor product to Δ(Pz) ⊂ M⊗L(G) gives that either (a) Δ(Pz) ≺M⊗L(G) M⊗1 or (b) Δ(Nz) is amenable relative to M⊗1 inside M⊗L(G). If (a) holds, then by Proposition 5.2 we have that Pz ≺M Q, which is a contradiction. In what follows we will use the L2 and L1 spaces of a semifinite von Neumann algebras (specifically the basic construction) introduced in Remark 4.25 in Ioana’s article. If (b) holds, then there is a sequence ηn ∈ L2 (Δ(z)⟨M⊗L(G), M⊗1⟩Δ(z))⨁ ∞ such that ‖⟨⋅ηn , ηn ⟩ − τ(⋅)‖ → 0, ‖⟨ηn ⋅, ηn ⟩ − τ(⋅)‖ → 0, and ‖yηn − ηn y‖2 → 0, for every y ∈ Δ(Nz) (see [16, Remark 7.1]). Since S ⊂ N has the relative property (T), by [69, Proposition 4.7], so does Δ(Sz) ⊂ Δ(Nz). Hence, there is a nonzero η ∈ L2 (Δ(z)⟨M⊗L(G), M⊗1⟩Δ(z)) such that yη = ηy, for every y ∈ Δ(Sz). Then ζ = η∗ η ∈ L1 (Δ(z)⟨M⊗L(G), M⊗1⟩Δ(z)) is nonzero and satisfies ζ ≥ 0 and yζ = ζy, for every y ∈ Δ(Sz). Let t > 0 such the spectral projection a = 1[t,∞) (ζ ) of ζ is nonzero. Then a ∈ Δ(Sz)′ ∩ Δ(z)⟨M⊗L(G), M⊗1⟩Δ(z). As ta ≤ ζ , we get that Tr(a) ≤ Tr(ζ )/t < ∞. Theorem 2.4 implies that Δ(Sz) ≺M⊗L(G) M⊗1. Applying Proposition 5.2 again, we get that Sz ≺M Q and hence S ≺M Q, a contradiction. 2. Let X ⊂ Δ(A) be an arbitrary amenable von Neumann subalgebra. Since X and Δ(B) commute, by Theorem 5.3 we have that either (a) X ≺M⊗L(G) M⊗1 or (b) Δ(B) is amenable relative to M⊗1 inside M⊗L(G). If (a) holds for all such X, then [8, Corollary F.14] implies that (c) Δ(A) ≺M⊗L(G) M⊗1. If (b) holds, by applying Theorem 5.3, we 2 A group is called elementary if it contains a cyclic subgroup of finite index.

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get that either (d) Δ(B) ≺M⊗L(G) M⊗1 or (e) Δ(A ∨ B) is amenable relative to M⊗1 inside M⊗L(G). If (e) holds then applying Theorem 5.3 again we further obtain either (f) Δ(A∨B) ≺M⊗L(G) M⊗1 or (g) Δ(N) is amenable relative to M⊗1 inside M⊗L(G). By Proposition 5.2, (d) implies that B ≺M Q while (c) and (f) imply A ≺M Q. So it remains to treat (g). If (g) holds, then by arguing as in part 1., we get that S ≺M Q, which is a contradiction. For our results we need a strengthening of the notion of primeness for von Neumann algebras. Definition 5.5. A von Neumann algebra M is called s-prime if it satisfies the following property: given any nonzero projection p ∈ M, if A, B ⊆ pMp are commuting von Neumann subalgebras such that A ∨ B ⊆ pMp has finite index, then either A or B is finite dimensional. From the definition, it is clear that s-primeness implies primeness. The converse, however, does not hold true in general. Constructing such examples is typically involved technically and exceeds the purpose of this paper. However, to give the reader a flavor on the presentation of such factors, we highlight an example below that arises from [27] (see also [10]). Consider any action by outer automorphisms K ↷ρ 𝔽2 of a finite group K on the free group with two generators, 𝔽2 . Now let K ↷ρ̃ 𝔽2 × 𝔽2 be the canonical diagonal action and let K = (𝔽2 × 𝔽2 ) ⋊ρ̃ K be the corresponding semidirect product group. Then [27, Theorem C] yields that L(K) is a prime factor. However, one can easily see that L(K) is not s-prime as it contains the finite index nonprime subfactor, L(𝔽2 × 𝔽2 ) = L(𝔽2 )⊗L(𝔽2 ). Additional examples of prime factors associated with fibered products and other groups have been constructed in [10, 26, 27]. Using our preliminary results together with standard technology on wreath product von Neumann algebras from [40, 46, 70], we now derive the main primeness result of the section. Theorem 5.6. Let Q be any nontrivial tracial von Neumann algebra and let Γ be any icc subgroup of a hyperbolic group. Then any property (T) wreath-like product factor M ∈ 𝒲ℛ(Q, Γ) is s-prime. Proof. Fix a projection 0 ≠ p ∈ M and two commuting von Neumann subalgebras A, B ⊆ pMp such that [pMp : A ∨ B] < ∞. Since M is a factor, we can apply Lemma 2.1(3) to obtain that 𝒵 (A ∨ B) is completely atomic. Hence, we may assume that A ∨ B is a factor, up to replacing p by a smaller projection. This implies that both A and B are factors, and thus A ∨ B = A⊗B. Since M has property (T), so does pMp. Since [pMp : A ∨ B] < ∞, then using [69, Proposition 5.7.1 and Proposition 4.6.1], we get that both A and B have property (T). Now assume by contradiction that neither A nor B is finite dimensional. This implies that both A and B are diffuse von Neumann algebras. Denote P = ⊗Γ Q ⊂ M. Notice that since [pMp : A ∨ B] < ∞ and Γ is infinite then, using Lemma 2.7 and part (5) in Lemma 2.5, we have that A ∨ B ⊀M P. Thus using part

288 � I. Chifan et al. (2) in Theorem 5.4 (for R = A ∨ B), we have either A ≺M P or B ≺M P. Due to symmetry, we may assume that A ≺M P. Next we argue there exists a finite subset F ⊂ Γ such that A ≺M ⊗F Q.

(5.2)

As A ≺M P one can find nonzero projections a0 ∈ A, b0 ∈ P, a partial isometry w ∈ b0 Ma0 , and a ∗-isomorphism onto its image ψ : a0 Aa0 → ψ(a0 Aa0 ) =: P0 ⊆ b0 Pb0 such that ψ(x)w = wx

for all x ∈ a0 Aa0 .

(5.3)

Next we show that one can find a finite subset F ⊂ Γ such that P0 ≺P ⊗F Q.

(5.4)

Towards this, note that since A has property (T) then so does a0 Aa0 . Therefore P0 has property (T) as well. Now fix an exhaustion Fn ↗ Γ by a sequence of increasing finite subsets, where n ≥ 1. Thus, if Pn := ⊗Fn Q we notice that (Pn )n ⊂ P forms an increaswot

ing sequence of von Neumann algebras such that ⋃n Pn = P. Hence, letting EPn be the conditional expectation from P onto Pn we have that EPn → Id, ‖ ⋅ ‖2 -pointwise on P. As P0 has property (T) it follows that for every ε > 0 there is nε ∈ ℕ such that ‖EPn (x) − x‖2 < ε for all x ∈ U (P0 ) and n ≥ nε . Picking ε ≤ 2−1 τ(b0 )1/2 and using the triangle inequality above, we have ‖EPn (x)‖2 ≥ ‖x‖2 − ‖x − EPn (x)‖2 ≥ ‖x‖2 − ε ≥ 2−1 τ(b0 )1/2 > 0, for all x ∈ U (P0 ). Then using Theorem 2.4, we get the intertwining (5.4), as desired. Finally, the intertwinings (5.4) and (5.3), together with the transitivity property from [45, Lemma 1.4.5] (see also [79, Remark 3.8]), yield the intertwining (5.2). Consequently, letting S = ⊗F Q, one can find nonzero projections a ∈ A, r ∈ S, a nonzero partial isometry v ∈ rMa, and a ∗-isomorphism onto its image ϕ : aAa → ϕ(aAa) =: D ⊆ rSr such that ϕ(x)v = vx

for all x ∈ aAa.

(5.5)

The intertwining relation also implies that vv∗ ∈ D′ ∩ rMr and v∗ v ∈ (A′ ∩ pMp)a. Next we prove the following: Claim 5.7. There exists a finite set K ⊂ Γ such that D′ ∩ rMr ⊆ PK. Proof of the claim. Fix y ∈ D′ ∩ rMr with ‖y‖∞ ≤ 1. Let y = ∑g yg ug be its Fourier expansion, where yg ∈ rP for all g ∈ Γ. Since xy = yx for all x ∈ D ⊂ P, we get that xyg = yg σg (x)

for all x ∈ D, g ∈ Γ.

(5.6)

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Let K = F −1 F and notice that for every g ∈ Γ \ K we have that gF ∩ F = 0. Next, we prove that yg = 0 for all g ∈ Γ \ K which will conclude the proof of the claim. Fix ε > 0. Since the Fourier coefficients satisfy ‖yg ‖∞ ≤ ‖y‖∞ ≤ 1 one can pick a1 , . . . , ak ∈ rS and b1 , . . . , bk ∈ ⊗Γ\F Q such that 󵄩󵄩 k 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩∑ ai ⊗ bi 󵄩󵄩󵄩 ≤ 1, 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩∞ 󵄩i=1 󵄩󵄩 󵄩󵄩 k 󵄩󵄩 󵄩 󵄩󵄩yg − ∑ ai ⊗ bi 󵄩󵄩󵄩 ≤ ε. 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩2 i=1

and

(5.7) (5.8)

Using the intertwining relation (5.6) then inequalities (5.8), (5.7), and (5.8) again, basic calculations show that for every unitary x ∈ D we have 󵄩󵄩 󵄩 󵄩 ∗ 󵄩 ∗ 󵄩 ∗ 󵄩 󵄩󵄩ES (yg yg )󵄩󵄩󵄩2 = 󵄩󵄩󵄩xES (yg yg )󵄩󵄩󵄩2 = 󵄩󵄩󵄩ES (yg σg (x)yg )󵄩󵄩󵄩2 󵄩󵄩 󵄩󵄩 k 󵄩󵄩 󵄩 ∗ 󵄩 󵄩 ≤ ε + 󵄩󵄩ES ((∑ ai ⊗ bi )σg (x)yg )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 󵄩 󵄩2 󵄩󵄩 󵄩󵄩 k k 󵄩󵄩 󵄩󵄩 ≤ 2ε + 󵄩󵄩󵄩ES ((∑ ai ⊗ bi )σg (x)(∑ aj∗ ⊗ b∗j ))󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 j=1 󵄩 󵄩2 󵄩󵄩 󵄩󵄩 k 󵄩󵄩 󵄩󵄩 = 2ε + 󵄩󵄩󵄩ES ( ∑ (ai aj∗ ) ⊗ (bi σg (x)b∗j ))󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩 i,j=1 󵄩 󵄩2

(5.9)

Using that ai aj∗ ∈ S, the S-bimodularity of the expectation, ES (x) = τ(x)1 for all x ∈ ⊗Γ\K Q, and the triangle inequality we can also see that 󵄩󵄩 󵄩󵄩 󵄩󵄩 k 󵄩󵄩 k 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ES ( ∑ (ai a∗ ) ⊗ (bi σg (x)b∗ ))󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ (ai a∗ ) ⊗ ES (bi σg (x)b∗ )󵄩󵄩󵄩 j j j j 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩2 󵄩󵄩i,j=1 i,j=1 󵄩2 󵄩󵄩 k 󵄩󵄩 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∑ τ(σg (x)b∗j bi )ai aj∗ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩i,j=1 󵄩2 k

󵄨 󵄨󵄩 󵄩 ≤ ∑ 󵄨󵄨󵄨τ(xσg −1 (b∗j bi ))󵄨󵄨󵄨󵄩󵄩󵄩ai aj∗ 󵄩󵄩󵄩2 . i,j=1

(5.10)

Since A is diffuse, so is D, and thus one can find a sequence (xn )n ⊂ U (D) such that xn → 0 weakly, as n → ∞. In particular, we have that limn→∞ |τ(xn σg −1 (b∗j bi ))| = 0, for all i, j ∈ 1, k. Therefore, applying inequalities (5.9) and (5.10) to x = xn and taking the limit as n → ∞ we get that ‖ES (yg y∗g )‖2 ≤ 2ε. Since ε > 0 was arbitrary we get ES (yg y∗g ) = 0 and hence yg = 0, as desired. Now, fix z ∈ A′ ∩ pMp. Using relation (5.5) twice, we can see that for all x ∈ aAa we have ϕ(x)vzv∗ = vxzv∗ = vzxv∗ = vzv∗ ϕ(x) and therefore vzv∗ ∈ D′ ∩ rMr. Thus

290 � I. Chifan et al. v(A′ ∩pMp)v∗ ⊆ D′ ∩rMr. From (5.5) we also have that vAv∗ = Dvv∗ where vv∗ ∈ D′ ∩rMr. Combining these with the Claim 5.7, we can see that v((A)(A′ ∩ pMp))v∗ = vAv∗ v(A′ ∩ pMp)v∗ ⊂ D(D′ ∩ rMr) ⊆ PK. This implies that v(A ∨ (A′ ∩ pMp))v∗ ⊆ PK.

(5.11)

Let 𝒫PK be the orthogonal projection from L2 (M) onto the ‖ ⋅ ‖2 -closure of span{xug : x ∈ P, g ∈ K}. One can check for every x ∈ M we have 𝒫PK (x) = ∑ EP (xug −1 )ug . g∈K

(5.12)

By (5.11), for all x ∈ v(A ∨ (A′ ∩ pMp))v∗ we have that 𝒫PK (x) = x. Using this together with formula (5.12) and basic calculations, for all unitaries u ∈ v∗ v(A ∨ (A′ ∩ pMp))v∗ v we have 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 0 < τ(v∗ v) = ‖u‖22 = 󵄩󵄩󵄩vuv∗ 󵄩󵄩󵄩2 = 󵄩󵄩󵄩PPK (x)󵄩󵄩󵄩2 = ∑ 󵄩󵄩󵄩EP (vuv∗ ug −1 )󵄩󵄩󵄩2 . g∈K

Using Theorem 2.4(3), this implies that A ∨ (A′ ∩ pMp) ≺M P. By Lemma 2.5(2), there is a nonzero projection z ∈ 𝒵 (A′ ∩ pMp) such that (A ∨ (A′ ∩ pMp))z ≺sM P.

(5.13)

Using that A ∨ B ⊆ pMp has finite index and A ∨ B ⊆ A ∨ (A′ ∩ pMp), it follows that A ∨ (A′ ∩ pMp) ⊆ pMp also has finite index. By Lemma 2.6(2), we deduce that pMp ≺pMp (A ∨ (A′ ∩ pMp))z. In combination with (5.13), we can apply Lemma 2.5(4) and deduce that pMp ≺M P, which by Lemma 2.7 entails that Γ is finite, a contradiction.

6 A class of existentially closed II1 factors A group G is called existentially closed if every finite set of equations and inequalities defined over G and is soluble in an extension of G is actually soluble in G. Existentially closed groups actually coincide with the class of nontrivial algebraically closed groups. These groups display remarkable structural properties and have been intensively studied over the years; the reader may consult [37] for a comprehensive account on this direction. More recently, in the field of continuous model theory, Farah–Goldbring–Hart–Sherman [28] introduced a natural von Neumann algebraic counterpart of these objects.

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Roughly speaking, a von Neumann algebra M is called existentially closed if whenever a system of equations with coefficients in M has a solution in an extension of M, then it has an approximate solution in M. In more rigorous terms, let M ⊆ N be an inclusion of separable tracial von Neumann algebras. Following [28], one says that M is existentially closed in N if there exists an embedding j : N 󳨅→ M ω , whose restriction on N is the diagonal embedding M ⊂ M ω . A separable tracial von Neumann algebra M is called existentially closed if it is existentially closed in any separable extension, M ⊆ N. As in [2, 31], to avoid any set-theoretic subtleties, in the remaining part of the paper we assume the Continuum Hypothesis. Next, we recall several fundamental facts of existentially closed von Neumann algebras that will be used freely throughout this paper. The first three are well known and were proved in [28, 34], the fourth item follows directly from definitions, while the last has been established in [32]. Theorem 6.1 ([28, 32, 34]). The following hold true: (1) Every existentially closed tracial von Neumann algebra is a II1 factor satisfying McDuff’s property. (2) Every separable tracial von Neumann algebra embeds into a separable existentially closed factor. (3) Every separable tracial von Neumann algebra embeds into any ultrapower M ω of any existentially closed factor M. (4) If M1 ⊆ M2 ⊆ ⋅ ⋅ ⋅ ⊆ Mn ⊆ ⋅ ⋅ ⋅ is a chain of existentially closed factors, then their union wot

is also existentially closed. M = ⋃n Mn (5) If M is an existentially closed factor, then for any property (T) subfactor N ⊂ M, its double commutant satisfies (N ′ ∩ M)′ ∩ M = N.

Throughout the paper, we will denote by ℰ the class of all existentially closed factors. Next, we highlight a family of existentially closed factors which in spirit resembles the inductive limit factors considered in [74] and occurs naturally by combining the aforementioned model theoretic properties with the more recent von Neumann algebraic techniques from [12]. In fact, a very similar construction has been already considered in [12] and various structural results have been obtained there; the interested reader may consult [12, Section 6]. Henceforth we will denote by ℰ𝒯 the class of all existentially closed factors that can be presented as inductive limits Q = ⋃ Nn n∈ℕ

wot

,

where (Nn )n∈ℕ is an increasing sequence of property (T) s-prime II1 factors satisfying that for every n there is r > n such that Nn ⊂ Nr has infinite Jones index.

292 � I. Chifan et al. This family of factors is quite vast. In fact, using the results in Sections 4–5 and the facts stated in Theorem 6.1 along with the interlacing argument from the proof of [12, Proposition 8.1] we obtain the following. Theorem 6.2. The class ℰ𝒯 is embedding universal, i. e., every separable II1 factor embeds into an element of ℰ𝒯 . In particular, ℰ𝒯 is uncountable. Proof. Fix P a separable von Neumann algebra. Then pick Q1 ∈ ℰ such that P ⊂ Q1 . Using Theorem 4.12(2) and Theorem 5.6, one can find a property (T), s-prime II1 factor N1 such that Q1 ⊂ N1 . Since ℰ is embedding universal there exists a separable II1 factor Q2 ∈ ℰ such that N1 ⊂ Q2 . Since Q2 is existentially closed then by part (1) in Theorem 6.1 it is McDuff; in particular, they do not have property (T). Since property (T) passes to finite index suprafactors and N has property (T) it follows that the inclusion N1 ⊂ Q2 has infinite index. Continuing on this fashion, by induction, one can find an increasing sequence (Nn )n∈ℕ of property (T), s-prime II1 factors and an increasing sequence (Qn )n∈ℕ of separable existentially closed II1 factors satisfying P ⊂ Q1 ⊂ N1 ⊂ Q2 ⊂ N2 ⊂ ⋅ ⋅ ⋅ ⊂ Qn ⊂ Nn ⊂ ⋅ ⋅ ⋅ .

(6.1)

Let M be the inductive limit II1 factor arising from the sequence (6.1). By construction, wot

wot

= ⋃n Qn . Since Qn ∈ ℰ for all n ∈ ℕ, then using part (4) in we have M = ⋃n Nn Theorem 6.1 we get that M ∈ ℰ . Since in our construction Nn ⊂ Qn+1 has infinite index then so is Nn ⊂ Nn+1 , for all n ∈ ℕ. Thus ℰ𝒯 is embedding universal and, using [56, Corollary 3], we conclude that ℰ𝒯 is uncountable. We conclude this section by recalling the class of infinitely generic II1 factors introduced in [28, Propositions 5.7, 5.10 and 5.14] (see also [2, Fact 6.3.14]). Proposition 6.3 ([28]). There is a class of separable II1 factors 𝒢 satisfying the following: (1) 𝒢 is embedding universal, (2) any embedding π : Q1 󳨅→ Q2 , for some Q1 , Q2 ∈ 𝒢 is elementary, i. e., it extends to an isomorphism Q1ω ≅ Q2ω , and (3) 𝒢 is the maximum class with properties (1) and (2). The elements of 𝒢 are called infinitely generic II1 factors.

Remark 6.4. The proof of Theorem 6.2 can be used to construct inductive limits of property (T) factors within any family 𝒢 of existentially closed factors that is both embedding universal and closed under inductive limits. For example, by [28], this applies when 𝒢 is the family of all infinitely generic factors.

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7 Tensor indecomposability results for factors in class ℰ𝒯 It is well known that existentially closed groups display strong structural properties, e. g., they are simple [37, Theorem 1.8]. Therefore, they do not admit any nontrivial direct or semidirect product decompositions. Since existentially closed factors are von Neuman algebraic counterparts of these objects it is reasonable to expect that they share similar indecomposability properties. To stimulate the development of new technology to tackle such properties, it would be natural to first understand if there are certain types of tensor decompositions that existentially closed factors cannot have. In this direction, by combining [47, Corollary 2.6] with a result from [2] (see, for instance, [2, Proposition 6.2.11 and Proposition 6.3.2]), one can see that existentially closed factors P do not admit any diffuse tensor decompositions of the form P = R⊗B, where R is the hyperfinite II1 factor and B is an arbitrary full II1 factor; in other words, P is not strongly McDuff. However, besides this tensor indecomposability result, little is known regarding the possible tensor decompositions of existentially closed factors. In this section we make new progress on this problem by showing that the existentially closed factors in the class ℰ𝒯 satisfy an even stronger tensor product indecomposability statement. Specifically, using a mélange of methods which combines spectral gap arguments and various intertwining techniques from [12, 15–17, 32, 42, 67, 74], we obtain the following. Theorem 7.1. For any P ∈ ℰ𝒯 , we have that P ≇ A⊗B, for all II1 factors A and B, with B full. wot

Proof. Suppose that P = ⋃n Nn where (Nn )n is an increasing sequence of property (T), s-prime II1 factors satisfying that for every n there is r > n such that Nn ⊂ Nr has infinite index. Now assume by contradiction that P = A⊗B is a decomposition into II1 factors with B full. By applying [2, Proposition 6.2.11] we get that A is non-amenable. As B is a full factor, by [19] we can find b1 , . . . , bk ∈ U (B) and C > 0 such that for every x ∈ P we have k

󵄩 󵄩 ∑ ‖xbi − bi x‖2 ≥ C 󵄩󵄩󵄩x − EA (x)󵄩󵄩󵄩2 . i=1

Fix 0 < ε < 1. Since bi ∈ P = ⋃n Nn

wot

(7.1)

one can find n ∈ ℕ and p1 , . . . , pk ∈ Nn such that

‖bi − pi ‖2 ≤

εC k

for all i = 1, k.

(7.2)

Fix y ∈ (Nn′ ∩ P)1 . Using inequalities (7.1), (7.2), along with relations ypi = pi y for all i = 1, k, we get

294 � I. Chifan et al. 1 k 󵄩 󵄩󵄩 󵄩󵄩y − EA (y)󵄩󵄩󵄩2 ≤ ∑ ‖ybi − bi y‖2 ≤ ε. C i=1 Using the triangle inequality, this further implies for all unitaries y ∈ Nn′ ∩P we have ‖EA (y)‖2 ≥ 1 − ε > 0. Thus part (3) in Theorem 2.4 shows that Nn′ ∩ P ≺M A. Since Nn has property (T), it has w-spectral gap in the sense of [74] in any extension (in fact, this characterizes property (T), see [77]). As P is existentially closed, by Theorem 6.1(5) we get that (Nn′ ∩ P)′ ∩ P = Nn . Hence, by passing to relative commutants using Lemma 2.5(3) that B ≺P Nn .

(7.3)

In the rest of the proof, we show that (7.3) will lead to a contradiction. For simplicity, denote Q := Nn . As B ≺P Q, using [17, Proposition 2.4] and its proof, one can find projections b ∈ B, q ∈ Q, a nonzero partial isometry v ∈ qPb, a von Neumann subalgebra D ⊆ qQq, and a ∗-isomorphism onto its image ϕ : bBb → D satisfying the following relations: (1) D ∨ (D′ ∩ qQq) ⊆ qQq has finite index; (2) ϕ(x)v = vx for all x ∈ bBb; (3) vv∗ ∈ D′ ∩ qPq and p := v∗ v = a ⊗ b for some projection a ∈ A. Since Q is s-prime, relation (1) implies that D′ ∩ qQq is finite dimensional. Fix 0 ≠ z ∈ D′ ∩ qQq, a minimal central projection. Thus one can find n ∈ ℕ such that (D′ ∩ qOq)z ≅ Mn (ℂ). Since Dz is a factor commuting with (D′ ∩ qQq)z, this further implies the index [(D ∨ D′ ∩ qQq)z : Dz] < ∞. Lemma 2.1 and relation (1) also imply the index [zQz : (D ∨ (D′ ∩ qQq))z] < ∞. Using the transitivity property of finite index inclusions, these relations yield that Dz ⊂ zQz is a finite index inclusion of II1 factors. Moreover, replacing q by qz, D by Dz, v by vz, and ϕ(⋅) by ϕ(⋅)z, relations (2) and (3) still hold and furthermore, instead of (1) we actually have (1’) D ⊆ qQq is a finite index inclusion of II1 factors. Choosing u ∈ U (P) such that v = up, relation (2) entails that Dvv∗ = vBv∗ = u(pBp)u∗ .

(7.4)

Passing to relative commutants above and using (3), we also have vv∗ (D′ ∩ qPq)vv∗ = u(pAp)u∗ .

(7.5)

Altogether, relations (7.4)–(7.5) show that vv∗ (D ∨ (D′ ∩ qPq))vv∗ = upPpu∗ . If t denotes the central support of vv∗ in D ∨ (D′ ∩ qPq), this further implies that (D ∨ (D′ ∩ qPq))t = tPt.

(7.6)

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Using relation (1’), Lemma 2.2, and Proposition 2.1(1), we deduce that (qQq′ ∩ qPq)t ⊆ (D ∩ qPq)t has finite index. Since Dt is a factor and commutes with (D′ ∩ qPq)t, then it is in tensor position with respect to (D′ ∩ qPq)t and (qQq′ ∩ qPq)t; thus, using Lemma 2.1(4) it follows that (D ∨ (qQ′ q ∩ qPq))t ⊆ (D ∨ (D′ ∩ qPq))t = tPt has finite index as well. By Lemma 2.6, we deduce that P ≺P D ∨ (qQq′ ∩ qPq) and, since D ⊆ qQq, we further have P ≺P qQq∨(qQq′ ∩qPq) = q(Q∨(Q′ ∩P))q. Since (Q′ ∩P)′ ∩P = Q, we get that Q∨(Q′ ∩P) has trivial relative commutant inside P. Hence, by [9, Proposition 2.3], we further derive that ′

Nn ∨ (Nn′ ∩ P) ⊆ P is a finite index inclusion of II1 factors.

(7.7)

Now, fix r > n such that Nn ⊂ Nr has infinite index. Let ENr : P → Nr be the canonical conditional expectation and denote by S = ENr (Nn′ ∩ P)′′ the von Neumann algebra generated by the image of Nn′ ∩ P under the expectation ENr inside Nr . Using the Nr -bimodularity of ENr , we can see that ENr (Nn′ ∩ P) ⊂ Nn′ ∩ Nr . In particular, S commutes with Nn . Denote also T = ENr (Nn ∨ (Nn′ ∩ P))′′ . Since for every x ∈ Nn and y ∈ Nn′ ∩ P we have that ENr (xy) = xENr (y), one can see that T = Nn ∨ S. Finally, notice that (7.7) and Lemma 2.3 (for M = Nn ∨ (Nn′ ∩ P) and N = Nr ) imply that Nn ∨ S = T ⊆ Nr has finite index. Since Nr is s-prime it follows that S is finite dimensional. Thus, Nn ⊂ Nr has finite index, which is a contradiction. Remark 7.2. The above proof applies verbatim to any existentially closed factor of the wot

form P = ⋃n Nn , where Nn are s-prime factors that do not necessarily have property (T) but satisfy the bicommutant property (Nn′ ∩ P)′ ∩ P = Nn , for all n ∈ ℕ. It would be interesting to determine if there are such existentially closed factors P when Nn are free products. Tackling this case requires significant development of new technology, e. g., finding suitable replacements for the spectral gap property along with the embedding results from [12] which were used in an essential way in the current approach. Next, we record some immediate consequences of Theorem 7.1. First, we observe that while factors in class ℰ𝒯 are inner asymptotically central (see [2, Definition 6.2.8 and Proposition 6.3.2]), they cannot be written as infinite tensor products of full factors (see also [2, Question 6.4.1]).

Corollary 7.3. For any P ∈ ℰ𝒯 , we have that P ≇ ⊗n∈ℕ Pn for any infinite collection of {Pn : n ∈ ℕ} of full II1 factors. The second application concerns the structure of the central sequence algebra of a certain class of existentially closed factors. A factor P is called super McDuff if its central sequence algebra P′ ∩ Pω is a II1 factor. In [2, Question 6.3.1] it was asked whether all existentially closed factors are super McDuff. In [12, Theorem 6.4] this question was answered positively for all infinitely generic factors. Using Theorem 7.1, we can moreover show that if P is an infinitely generic factor which is also in class ℰ𝒯 , then all of its tensor factors are super McDuff.

296 � I. Chifan et al. Corollary 7.4. Let P ∈ ℰ𝒯 be an infinitely generic factor. Then for any diffuse tensor decomposition P = A⊗B, both factors A and B are super McDuff. Proof. Since P is infinitely generic, it follows from [12, Theorem 6.4] that P is super McDuff. Let P = A⊗B be a diffuse tensor decomposition. Using Theorem 7.1, it follows that A and B have property Gamma. Hence, their central sequence algebras A′ ∩ Aω and B′ ∩ Bω are diffuse [53]. In addition, as P′ ∩ Pω is a factor, [51, Theorem F] further implies that A′ ∩ Aω and B′ ∩ Bω are also factors, which yields the desired conclusion. We believe that in fact all existentially closed factors satisfy the statement of Theorem 7.1 and thus conjecture the following: Conjecture 7.5. For any P ∈ ℰ , we have that P ≇ A⊗B, for all II1 factors A and B with B full. At the time of writing, we do not have an approach for this conjecture in its full generality. However, we would like to mention that the conjecture holds true if one assumes in addition that B is a group von Neumann algebra arising from a non-inner amenable group; for the definition, see Exercise 6.20 in Ioana’s article in this volume. In fact, we have the following more general indecomposability result. Theorem 7.6. If Q ∈ ℰ , then Q ≇ A ⋊σ,α Γ for any trace-preserving cocycle action Γ ↷σ,α (A, τ) of a non-inner amenable group Γ. Proof. Assume by contradiction that Q = A ⋊σ,α Γ for some cocycle action Γ ↷ (A, τ). As Γ is non-inner amenable, then using a generalization of Exercise 6.20 in Ioana’s article in this volume, one can find g1 , . . . , gk ∈ Γ and C > 0 such that for every x ∈ Q we have k

󵄩 󵄩 ∑ ‖xugi − ugi x‖2 ≥ C 󵄩󵄩󵄩x − EA (x)󵄩󵄩󵄩2 . i=1

(7.8)

In particular, this implies that Q′ ∩ Qω ⊆ Aω . On the other hand, since Q is existentially closed then there exists a unitary (vn )n = v ∈ Qω such that vQv∗ ⊆ Q′ ∩ Qω (see [2, Definition 6.2.8 and Proposition 6.3.2]). Altogether, these relations imply that for every g ∈ Γ there exists a sequence (cn (g))n ∈ 𝒰 (A) such that limn→ω ‖vn ug v∗n − cn (g)‖2 = 0. This further shows that 󵄩󵄩

lim 󵄩v n→ω󵄩 n

󵄩 − cn (g)vn ug −1 󵄩󵄩󵄩2 = 0.

(7.9)

To this end let vn = ∑g vng ug with vng ∈ A be its Fourier expansion. Using this relation together with the triangle inequality and the fact that cn (g) and α(h, g) are unitaries, we can see that 󵄩󵄩 󵄩2 󵄩 n n −1 󵄩2 󵄩󵄩vn − cn (g)vn ug −1 󵄩󵄩󵄩2 = ∑󵄩󵄩󵄩vhg −1 − cn (g)vh α(h, g )󵄩󵄩󵄩2 h

Tensor product indecomposability results for existentially closed factors

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󵄩 2 󵄩 󵄩 󵄩 ≥ ∑(󵄩󵄩󵄩vnhg −1 󵄩󵄩󵄩2 − 󵄩󵄩󵄩cn (g)vnh α(h, g −1 )󵄩󵄩󵄩2 ) h

󵄩 󵄩 󵄩 2 󵄩 = ∑(󵄩󵄩󵄩vnhg −1 󵄩󵄩󵄩2 − 󵄩󵄩󵄩vnh 󵄩󵄩󵄩2 ) .

(7.10)

h

Letting ξn (g) := ‖vng ‖2 for all g ∈ Γ one can see that ξn ∈ ℓ2 Γ with ‖ξn ‖2 = 1. Moreover, relations (7.9)–(7.10) show that limn→ω ‖ρg (ξn ) − ξn ‖2 = 0, where ρ is the right regular representation of Γ. This implies that Γ is amenable, a contradiction. We continue with a few basic observations that rule out other particular types of tensor product decompositions for existentially closed factors. It is possible that some of the results could also shed some light towards a possible approach to Conjecture 7.5. Using embedding results into von Neumann algebraic HHN extensions, as in the group situation, we show that any existentially closed factor M does not decompose as M = A⊗B for any factors A and B that contain isomorphic copies of a given nonamenable factor. Theorem 7.7. If P ∈ ℰ , then P ≇ A⊗B for any factors A, B that both contain isomorphic copies of a given nonamenable factor. Proof. Assume by contradiction that P = A⊗B such that there exist isomorphic nonamenable subfactors S ⊆ A, T ⊆ B. Fix a ∗-isomorphism ϕ : S → T. Following an idea from the proof of [33, Theorem 4], we consider the HNN extension Q = HNN(P, ϕ) associated to ϕ [78] and obtain that P ⊂ Q is an inclusion of tracial von Neumann algebras together with a unitary w ∈ Q such that wϕ(x)w∗ = x, for any x ∈ S (see also [29, Section 3]). Since P is existentially closed and w ∈ Q ⊃ P, it follows that we can represent w = (un )n ∈ Pω , and hence, lim ⟨xun , un ⟩ = τ(x)

n→ω

and

󵄩󵄩

lim 󵄩u ϕ(x) n→ω󵄩 n

󵄩 − xun 󵄩󵄩󵄩2 = 0,

for any x ∈ S.

(7.11)

Consider the Hilbert space L2 (P) and endow it with the S-bimodular structure given by x ⋅ ξ ⋅ y = (x ⊗ 1)ξ(1 ⊗ ϕ(y)) for every x, y ∈ S and ξ ∈ L2 (P). Then L2 (P) is isomorphic to a subbimodule of a multiple of the coarse S-bimodule, (L2 (S)⊗L2 (S))⊕∞ . This follows from ̄ and the fact that any left (respectively, right) S-module is contained in L2 (S)⊕∞ P = A⊗B as a left (respectively, right) S-module (see, for instance, [1, Proposition 8.2.3]). Hence, we derive from (7.11) that there exists a sequence of vectors (ξn )n≥1 ⊂ L2 (S) ⊗ L2 (S) such that lim ⟨xξn , ξn ⟩ = τ(x) and

n→∞

lim ‖xξn − ξn x‖2 = 0

n→∞

for all x ∈ S.

Therefore, using Section 2.4 relation (7.11) implies that S is amenable, which is a contradiction. A well-known conjecture in the theory of von Neumann algebras predicts that the following version of von Neumann’s problem in group theory holds true:

298 � I. Chifan et al. Conjecture 7.8. Every nonamenable II1 factor contains a copy of L(𝔽2 ). A positive answer to this conjecture combined with Theorem 7.7 would rule out the existence of tensor product decomposition into nonamenable factors for all existentially closed factors. Unfortunately, this conjecture is wide open at this time and very difficult to establish in full generality. However, since existentially closed factors are highly rich objects, one expects that they all verify Conjecture 7.8. Such a result would lead to new advances regarding Conjecture 7.5. Indeed, we first notice the following elementary result. Theorem 7.9. If Q ∈ ℰ and Q = A⊗B with B a full factor, then A ∈ ℰ . Proof. First, observe that since B is full and Q = A⊗B then the spectral gap condition (7.1) implies that the inclusion B ⊂ Q has w-spectral gap, i. e., ω

B′ ∩ Qω = (B′ ∩ Q) .

(7.12)

Now let A ⊆ C be any extension. Thus Q = A⊗B ⊆ C⊗B. Since Q = A⊗B is existentially closed we have that Q ⊆ C⊗B ⊆ Qω . However, we have that C ⊆ B′ ∩ Qω and using (7.12) we get A ⊆ C ⊆ B′ ∩ Qω = Aω . The recent refutation of the Connes Embedding Conjecture from [49] implies in particular that the hyperfinite II1 factor is not existentially closed. Thus Conjecture 7.8 could potentially hold true for all existentially closed factors. (We point out the corresponding statement for groups holds true as every group with solvable word problem embeds into every existentially closed group.) If this is the case, then, combining Theorems 7.7 and 7.9, we would obtain that for any existentially closed factor P we have that P ≇ A⊗B where B is any full factor containing a copy of L(𝔽2 ). We end this section with one last conjecture on tensor decompositions of existentially closed factors. To this end, we first recall some terminology and provide some context. If P is a McDuff factor then P ≅ P⊗R, where R is the hyperfinite factor. We say that P admits only the canonical McDuff decomposition if the following holds: for any tensor decomposition P = Q⊗R, Q is isomorphic to P. Currently, only a few classes of McDuff factors admitting only the canonical McDuff decomposition are known. These include: (1) All infinite tensor products ⊗n∈ℕ Pn of full factors Pn [74]; see also [51, Corollary G] for an alternative shorter proof. (2) All McDuff’s group factors L(T0 (Γ)) for any nontrivial icc group Γ, where T0 (Γ) are McDuff’s groups introduced in [52]; this is the main result in [18]. We believe that existentially closed factors also have this property. Conjecture 7.10. Any factor P ∈ ℰ admits only the canonical McDuff tensor decomposition.

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8 Open problems and final comments on the structure of existentially closed factors In this section we propose several additional open problems regarding the structure of existentially closed factors. Much of the technology developed within the framework of Popa’s deformation/rigidity theory is tailored for analyzing von Neumann algebras that are built from countable groups and their actions on von Neumann algebras. In order to understand if these methods can be applied to the study of existentially closed factors, one first needs to understand if existentially closed factors arise from countable groups or their actions. The following fundamental question is wide open: Open Problem 8.1. Are there any group II1 factors in the class ℰ ? A possible approach for this problem, using model-theoretic forcing, has been suggested in Goldbring’s article in this volume. If the answer to Problem 8.1 is positive, then we can further ask if it is possible to describe the groups Γ satisfying L(Γ) ∈ ℰ ? If this is too much to ask for, could one at least identify some properties such groups enjoy? It would be natural to investigate if these groups share any properties with existentially closed groups. Obviously, since existentially closed factors are McDuff, such groups Γ might not be simple but is it at least true that they are infinitely generated? Do they always contain free groups? In the opposite direction, constructing existentially closed factors that do not arise from groups also seems challenging. Open Problem 8.2. Are there any examples of existentially closed factors that are not group factors? We believe that most existentially closed factors are not group factors. The idea would be to exploit the embeddings group factors (e. g., the comultiplication Δ : L(Γ) → L(Γ)⊗L(Γ)) possess. However, thus far, we were not able to construct existentially closed factors which are not group factors. We notice that, even by abstract means, so far we do not know the existence of even a single existentially closed group factor. We end with another open problem regarding indecomposability properties of existentially closed factors. Theorem 7.6 implies that existentially closed factors do not appear as group measure space von Neumann algebras L∞ (X) ⋊ Γ associated with probability measure preserving actions Γ ↷ (X, μ) on diffuse probability spaces, where Γ is a non-inner amenable group. We believe that much more should be true and conjecture the following: Conjecture 8.3. There is no factor P ∈ ℰ which has a Cartan subalgebra.

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Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

C. Anantharaman and S. Popa, An introduction to II1 factors, book in preparation, available on Sorin Popa’s website: https://www.math.ucla.edu/~popa/Books/IIunV15.pdf. S. Atkinson, I. Goldbring and S. Kunawalkam-Elyavalli, Factorial relative commutants and the generalized Jung property for II1 factors, Adv. Math. 396 (2022). Paper No. 108107, 53 pp. B. Bekka, Operator-algebraic superridigity for SLn (ℤ), n ≥ 3, Invent. Math. 169 (2007), 401–425. R. Boutonnet, On solid ergodicity for Gaussian actions, J. Funct. Anal. 263 (2012), no. 4, 1040–1063. R. Boutonnet and C. Houdayer, Stationary characters on lattices of semisimple Lie groups, Publ. Math. IHES 133 (2021), 1–46. R. Boutonnet, A. Ioana and J. Peterson, Properly proximal groups and their von Neumann algebras, Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), no. 2, 445–482. K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer, New York–Berlin, 1982. x+306 pp. N. P. Brown and N. Ozawa, C∗ -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, AMS, Providence, RI. I. Chifan and S. Das, A remark on the ultrapower algebra of the hyperfinite factor, Proc. Am. Math. Soc. in press. arXiv:1802.06628. I. Chifan and S. Das, Primeness results for II1 factors associated with fibered product groups, Preprint 2019. I. Chifan, R. de Santiago and W. Sucpikarnon, Tensor product decompositions of II1 factors arising from extensions of amalgamated free product groups, Commun. Math. Phys. 364 (2018), 1163–1194. I. Chifan, D. Drimbe and A. Ioana, Embedding universality for II1 factors with property (T), Adv. Math. 417 (2023), Paper no. 108934, 24pp. I. Chifan and C. Houdayer, Bass–Serre rigidity results in von Neumann algebras, Duke Math. J. 153 (2010), no. 1, 23–54. I. Chifan and A. Ioana, Ergodic subequivalence relations induced by a Bernoulli action, Geom. Funct. Anal. 20 (2010), no. 1, 53–67. I. Chifan, A. Ioana and Y. Kida, W ∗ -superrigidity for arbitrary actions of central quotients of braid groups, Math. Ann. 361 (2015), 563–582. I. Chifan, A. Ioana, D. Osin and B. Sun, Wreath-like product groups and rigidity of their von Neumann algebras, 2021. arXiv:2111.04708. I. Chifan, Y. Kida and S. Pant, Primeness results for von Neumann algebras associated with surface braid groups, Int. Math. Res. Not. 16 (2016), 4807–4848. I. Chifan and W. Suckpikarnon, Tensor product decompositions of McDuff’s factors, Preprint 2020. A. Connes, Classification of injective factors, Ann. Math. 104 (1976), 73–115. A. Connes and V. F. R. Jones, Property (T) for von Neumann algebras, Bull. Lond. Math. Soc. 17 (1985), 57–62. Y. Dabrowski and A. Ioana, Unbounded derivations, free dilations, and indecomposability results for II1 factors, Trans. Am. Math. Soc. 368 (2016), no. 7, 4525–4560. F. Dahmani, V. Guirardel and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Am. Math. Soc. 245 (2017), no. 1156. T. Delzant, Sous-groupes distingués et quotients des groupes hyperboliques, Duke Math. J. 83 (1996), no. 3, 661–682. (French). (Distinguished subgroups and quotients of hyperbolic groups). D. Drimbe, Prime II1 factors arising from actions of product groups, J. Funct. Anal. 278, (2020), no. 5, 108366. D. Drimbe, Orbit equivalence rigidity for product actions, Commun. Math. Phys. 379 (2020), 41–59. D. Drimbe, Measure equivalence rigidity via s-malleable deformations, 2022. arXiv:2209.13320. D. Drimbe, D. Hoff and A. Ioana, Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups, J. Reine Angew. Math. 757 (2019), 197–246.

Tensor product indecomposability results for existentially closed factors

� 301

[28] I. Farah, I. Goldbring, B. Hart and D. Sherman, Existentially closed II1 factors, Fundam. Math. 233 (2016), 173–196. [29] P. Fima and S. Vaes, HNN extensions and unique group measure space decomposition of II1 factors, Trans. Am. Math. Soc. 364 (2012), 2601–2617. [30] L. Ge, Applications of free entropy to finite von Neumann algebras, II, Ann. Math. 147 (1998), 143–157. [31] I. Goldbring, On Popa’s factorial commutant theorem, Proc. Am. Math. Soc. 148 (2020), 5007–5012. [32] I. Goldbring, Spectral gap and definability, Beyond first-order model theory, Vol. 2, in press. arXiv:1805.02752. [33] I. Goldbring, Non-embeddable II1 factors resembling the hyperfinite II1 factor, J. Noncommut. Geom. in press. arXiv:2101.10467. [34] I. Goldbring, B. Hart and T. Sinclair, The theory of tracial von Neumann algebras does not have a model companion, J. Symb. Log. 78 (2013), 1000–1004. [35] M. Gromov, Hyperbolic groups, Essays in group theory, ed. by S. M. Gersten, Mathematical Sciences Research Institute Publications, vol. 8, Springer, New York, NY, 1987, pp. 75–263. [36] U. Haagerup, An example of a non-nuclear C∗ -algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), 279–293. [37] G. Higman and E. Scott, Existentially closed groups, London Mathematical Society Monographs (New series), vol. 3, Oxford University Press, 1988, 170 pp. Proc. Edinb. Math. Soc. 33, no. 1, 168–168. [38] D. Hoff, Von Neumann algebras of equivalence relations with nontrivial one-cohomology, J. Funct. Anal. 270 (2016), no. 4, 1501–1536. [39] C. Houdayer and S. Vaes, Type III factors with unique Cartan decomposition, J. Math. Pures Appl. 100 (2013), 564–590. [40] A. Ioana, Rigidity results for wreath product II1 factors, J. Funct. Anal. 252 (2007), no. 2, 763–791. [41] A. Ioana, Cocycle Superrigidity for Profinite Actions of Property (T) Groups, Duke Math. J. 157 (2011), no. 2, 337–367. [42] A. Ioana, Uniqueness of the group measure space decomposition for Popa’s ℋ𝒯 factors, Geom. Funct. Anal. 22 (2012), no. 3, 699–732. [43] A. Ioana, Classification and rigidity for von Neumann algebras, European congress of mathematics, EMS, 2013, pp. 601–625. [44] A. Ioana, Rigidity for von Neumann algebras, Proceedings of ICM, Vol. 2 (Rio de Janeiro, 2018), pp. 1635–1668. [45] A. Ioana, J. Peterson and S. Popa, Amalgamated free products of weakly rigid factors and calculation of their symmetry groups, Acta Math. 200 (2008), no. 1, 85–153. [46] A. Ioana, S. Popa and S. Vaes, A class of superrigid group von Neumann algebras, Ann. Math. (2) 178 (2013), no. 1, 231–286. [47] A. Ioana and P. Spaas, A class of II1 factors with a unique McDuff decomposition, Math. Ann. 375 (2019), no. 1–2, 177–212. [48] Y. Isono and A. Marrakchi, Tensor product decompositions and rigidity of full factors, Ann. Sci. Éc. Norm. Supér. in press. [49] Z. Ji, A. Natarajan, T. Vidick, J. Wright and H. Yuen. MIP∗ = RE. arXiv:2001.04383. [50] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1–25. [51] A. Marrakchi, Stability of products of equivalence relations, Compos. Math. 154 (2017), no. 9, 2005–2019. [52] D. McDuff, A countable infinity of II1 factors, Ann. Math. 90 (1969), 361–371. [53] D. McDuff, Central sequences and the hyperfinite factor, Proc. Lond. Math. Soc. 21 (1970), 443–461. [54] F. J. Murray and J. von Neumann, Rings of operators IV, Ann. Math. 44 (1943), 716–808. [55] A. Yu. Olshanskii, SQ-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132. (Russian). Translation in Sb. Math. 186 (1995), no. 8, 1199–1211. [56] N. Ozawa, There is no separable universal II1 factor, Proc. Am. Math. Soc. 132 (2004), 487–490. [57] N. Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111–117.

302 � I. Chifan et al.

[58] N. Ozawa, A Kurosh type theorem for type II1 factors, Int. Math. Res. Not. (2006). Art. ID 97560, 21 pp. [59] N. Ozawa, Weak amenability of hyperbolic groups, Groups Geom. Dyn. 2 (2008), 271–280. [60] N. Ozawa and S. Popa, On a class of II1 factors with at most one Cartan subalgebra, Ann. Math. (2) 172 (2010), no. 1, 713–749. [61] J. Peterson, L2 -rigidity in von Neumann algebras, Invent. Math. 175 (2009), no. 2, 417–433. [62] J. Peterson, Character rigidity for lattices in higher-rank groups, Preprint 2014. [63] M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Éc. Norm. Supér. 19 (1986), 57–106. [64] S. Popa, Orthogonal pairs of ∗-subalgebras in finite von Neumann algebras, J. Oper. Theory 9 (1983), no. 2, 253–268. [65] S. Popa, Correspondences, INCREST Preprint, 56/1986. [66] S. Popa, Classification of subfactors and their endomorphisms, CBMS Regional Conference Series in Mathematics, vol. 86, American Mathematical Society, Providence, RI, 1995. Published for the Conference Board of the Mathematical Sciences, Washington, DC. [67] S. Popa, Universal construction of subfactors, J. Reine Angew. Math. 543 (2002), 39–81. [68] S. Popa, Some rigidity results for non-commutative Bernoulli shifts, J. Funct. Anal. 230 (2005), 1–56. [69] S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. Math. 163b (2006), no. 3, 809–899. [70] S. Popa, Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. I, Invent. Math. 165 (2006), no. 2, 369–408. [71] S. Popa, On Ozawa’s property for free group factors, Int. Math. Res. Not. 11 (2007). Art. ID rnm036, 10 pp. [72] S. Popa, Deformation and rigidity for group actions and von Neumann algebras, International congress of mathematicians, Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 445–477. [73] S. Popa, On the superrigidity of malleable actions with spectral gap, J. Am. Math. Soc. 21 (2008), 981–1000. [74] S. Popa, On the classification of inductive limits of II1 factors with spectral gap, Trans. Am. Math. Soc. 364 (2012), no. 6, 2987–3000. [75] S. Popa and S. Vaes, Unique Cartan decomposition for II1 factors arising from arbitrary actions of hyperbolic groups, J. Reine Angew. Math. 694 (2014), 215–239. [76] T. Sinclair, Strong solidity of group factors from lattices in SO(n, 1) and SU(n, 1), J. Funct. Anal. 260 (2011), no. 11, 3209–3221. [77] H. Tan, Spectral gap characterizations of property (T) for II1 factors, 2022. arXiv:2202.06089. [78] Y. Ueda, HNN extensions of von Neumann algebras, J. Funct. Anal. 225 (2005), 383–426. [79] S. Vaes, Explicit computations of all finite index bimodules for a family of II1 factors, Ann. Sci. Éc. Norm. Supér. 41 (2008), 743–788. [80] S. Vaes, Rigidity for von Neumann algebras and their invariants, Proceedings of the ICM, Vol. III (Hyderabad, India, 2010), Hindustan Book Agency, 2010, pp. 1624–1650. [81] S. Vaes, One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor, Math. Ann. 355 (2013), no. 2, 661–696. [82] R. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser, Basel, 1984.

Hiroshi Ando

Introduction to nontracial ultraproducts of von Neumann algebras Abstract: In these notes we give a brief look at elements of Tomita–Takesaki modular theory. Then we extend the ultraproduct construction, an extremely useful idea from model theory, to the setting of type III (nontracial) von Neumann algebras. Keywords: Type III factors, nontracial ultraproduct MSC 2020: Primary 46L10, Secondary 46M07

1 Notations and preliminaries Except in a few places, we will be working exclusively with σ-finite von Neumann algebras: a von Neumann algebra M is called σ-finite if it admits a faithful normal (fn for short) state φ. The σ-finiteness of M is equivalent to the condition that any mutually orthogonal family of projections in M is at most countable. This should not be confused with the condition of M having separable predual M∗ (by an abuse of language, M is often said to be separable to actually mean this), which is equivalent to M having a faithful representation on a separable Hilbert space. In general, M∗ separable implies M σ-finite, but the converse fails. For example, our favorite Rω , where R is the hyperfinite type II1 factor, is σ-finite, but clearly it does not have separable predual. For an fn state φ ∈ M∗ , its GNS Hilbert space is denoted by L2 (M, φ) with the GNS vector ξφ . For x ∈ M, we define the two norms 1

‖x‖φ = φ(x ∗ x) 2 ,

1

‖x‖♯φ = φ(x ∗ x + xx ∗ ) 2 .

The norm ‖ ⋅ ‖φ (resp. ‖ ⋅ ‖φ ) defines the SOT (resp. S∗ OT) on bounded subsets of M. Here, SOT (resp. S∗ OT) stands for the strong (reps. ∗-strong) operator topology (recall that a net (xi )i∈I of bounded operators converges to x in the ∗-strong topology if both limi xi = x and limi xi∗ = x ∗ holds in the strong operator topology). Let (M, φ) = (Mn , φn )∞ n=1 be a ∗ sequence of σ-finite von Neumann algebras with fn states. Then the C -algebra of all ∞ bounded sequences (xn )∞ n=1 with xn ∈ Mn (n ∈ ℕ) is denoted by ℓ (M). ♯

Acknowledgement: The author would like to thank Isaac Goldbring and the anonymous referee for careful proofreading and for numerous suggestions which improved the readability of the manuscript. Hiroshi Ando, Department of Mathematics and Informatics, Chiba University, Chiba, Japan, e-mail: [email protected] https://doi.org/10.1515/9783110768282-008

304 � H. Ando

2 Introduction: How to define M ω if M is nontracial? In the previous chapters, we have seen the tracial ultraproducts and it should be clear that it is an extremely useful machinery! On the other hand, there are von Neumann algebras without traces, especially type III factors (we will explain what they are later; for now let us just think that they are factors which are very far from having tracial states), which are equally interesting. It is natural to expect that one can define the ultraproduct for them that helps to understand their asymptotic behaviors. So, given a sequence (M, φ) = (Mn , φn )∞ n=1 of σ-finite von Neumann algebras with fn states, how do we define the ultraproduct (M, φ)ω along a free ultrafilter ω ∈ βℕ \ ℕ? Let us recall the tracial ultraproduct construction in order to illustrate the difficulty raised by the very absence of tracial states. So assume for now that all φn are tracial and denote them by τn instead. Then inside the C∗ -algebra ℓ∞ (M), the subset ℐω (M, τ) = {x ∈ ℓ (M) | lim ‖xn ‖τn = 0} ∞

n→ω

forms a closed ideal of ℓ∞ (M) because of the inequality ‖axb‖τ ≤ ‖a‖∞ ‖x‖τ ‖b‖∞ valid for all elements a, b, x in a tracial von Neumann algebra M and its tracial state τ. Thus we can safely define the tracial ultraproduct (M, τ) to be the quotient C∗ -algebra ℓ∞ (M)/ℐω (M, τ). It has an fn tracial state τ ω given by the formula τ ω (x) = lim τn (xn ), n→ω

x = (xn )ω ∈ (M, τ)ω .

Moreover, if all Mn are factors (thus they have unique trace), then (M, τ)ω is also a factor. Now let us go back to the general case where we do not assume that φn ’s are tracial. We certainly need ℐω (M, φ) to be self-adjoint. But the ∗-operation is highly SOT♯ discontinuous. This forces us to replace ‖ ⋅ ‖φn with ‖ ⋅ ‖φn to define ℐω (M, φ): ℐω (M, φ) = {x ∈ ℓ (M) | lim ‖xn ‖φn = 0}. ∞

n→ω

♯

♯ When φn is tracial, then ‖ ⋅ ‖φn = √2‖ ⋅ ‖φ . Therefore this is a generalization of ℐω (M, τ). Unfortunately, however, another problem occurs: ℐω (M, φ) is no longer a closed ideal of ℓ∞ (M) in general.

Example 2.1 (ℐω (M, φ) is not an ideal of ℓ∞ (M)). Take Mn = M := 𝔹(ℓ2 ) and φn to be a constant state φ for all n ∈ ℕ. In this case, ℐω (M) = ℐω (M, φ) is the set of all bounded sequences in M converging to 0 in S∗ OT. Let (en )∞ n=1 be a decreasing sequence of projections in M such that both en and 1 − en are of infinite rank and that en tends to 0 in SOT. ∞ For each n ∈ ℕ, take a unitary un such that un en un∗ = 1 − en . Then x = (en )∞ n=1 , u = (un )n=1 ∞ ∗ satisfy x ∈ ℐω (M), u ∈ ℓ (M), while uxu ∉ ℐω (M).

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Then Ocneanu [54] gave the following definition: Definition 2.2 (Ocneanu ultraproduct). Let (M, φ) and ω be as above. Define ℳω (M, φ) to be the normalizer of ℐω (M, φ): the set of all x ∈ ℓ∞ (M) satisfying x ℐω (M, φ) ⊂ ℐω (M, φ) and ℐω (M, φ)x ⊂ ℐω (M, φ). Thus, by definition ℐω (M, φ) is a closed ideal of ℳω (M, φ). The (generalized) Ocneanu ultraproduct (M, φ)ω is the quotient C∗ -algebra (M, φ)ω = ℳω (M, φ)/ℐω (M, φ). One can show (though it is not obvious) that (M, φ)ω is indeed a von Neumann algebra endowed with an fn state φω given by φω (x) = lim φn (xn ), n→ω

x = (xn )ω ∈ (M, φ)ω .

In the case of a constant sequence (Mn , φn ) = (M, φ), (M, φ)ω is independent of the choice of the fn state φ (in this case, ℐω (M, φ) is the set of all x ∈ ℓ∞ (M) converging to 0 in S∗ OT), and is denoted by M ω . In this case, it is easy to see that a ⋅ ℐω (M, φ) ⊂ ℐω (M, φ) and ℐω (M, φ)⋅a ⊂ ℐω (M, φ) for every a ∈ M. Thus for each a ∈ M, the constant sequence (a, a, . . . ) belongs to ℳω (M, φ). Thus, the diagonal embedding M ∋ a 󳨃→ (a, a, . . . , )ω ∈ M ω is well defined and we may view M as a von Neumann subalgebra of M ω . However, we should keep in mind that even if Mn = M is constant, the structure of the ultraproduct (M, φn )ω depends on the choice of the sequence φ. We might think: OK, Ocneanu’s definition looks natural, so why not just use this? We certainly could. However, if we work only with the Ocneanu ultraproduct, it would be rather difficult even to answer basic questions such as the factoriality of M ω if M is a type III factor. Actually, there is an alternative approach to the nontracial ultraproduct found by Groh [26] and later polished by Raynaud [60] (there are yet other approaches by Golodets and Kirchberg, see Remark 2.4 and [1, Introduction] for further historical account). For each n ∈ ℕ, the predual (Mn )∗ is a Banach space of all normal linear functionals on Mn such that ((Mn )∗ )∗ = Mn isometrically. Now consider the Banach space ultraproduct (M ∗ )ω of the sequence M ∗ = ((Mn )∗ )∞ n=1 of Banach spaces. Groh found that this is indeed isometrically isomorphic to the predual of a von Neumann algebra, which we denote ∏ω M and call it the Groh–Raynaud ultraproduct of M. The Groh–Raynaud ultraproduct is more natural than Ocneanu’s from the viewpoint of noncommutative integration theory, because of the following result. We denote by Lp (M) Haagerup’s noncommutative Lp -space (see [28]) associated with M and Lp (M) = (Lp (Mn ))∞ n=1 is the corresponding p sequence of noncommutative L -spaces. Although we do not explain, there is a natural operator space structure on Lp (M). Theorem 2.3 (Raynaud [60]). For 1 ≤ p < ∞, there is a completely isometric isomorphism between the Banach space ultraproduct Lp (M)ω and Lp (∏ω M). Such an identification is impossible for Ocneanu’s construction. However, the Groh– Raynaud construction may look strange: first of all, ∏ω M is in general not σ-finite

306 � H. Ando even if all Mn′ s have separable predual (recall that (M, φ)ω is always σ-finite). Raynaud showed that ∏ω 𝔹(ℓ2 ) contains a type III summand, while on the other hand it can be shown that 𝔹(ℓ2 )ω = 𝔹(ℓ2 ) (more generally, (M⊗𝔹(ℓ2 ))ω = M ω ⊗𝔹(ℓ2 ) holds; see [52, Lemma 2.4]). So what is the right definition of the ultraproduct? It turns out that both approaches are the right ones and that it is crucial to exploit their relationship to study both of them. Remark 2.4. Due to the limited space, the presentation here is far from being exhaustive. In fact, Golodets [24] and Kirchberg [44] independently defined yet another ultraproduct algebra and another central sequence algebra (Golodets called his the asymptotic algebra). In reality, Kirchberg’s construction, even though it is unfortunately not widely known, is the most general one. However, in the study of von Neumann algebras his construction as well as Golodets’ are essentially the same as Ocneanu’s. Moreover, Ocneanu’s construction is the most accessible one. Therefore, we will not define Kirchberg’s version but briefly mention Golodets’ central sequence algebra and its connection to Connes’ central sequence algebra later, because it will be important to study the tensorial absorption of Powers factors. A detailed analysis of the relationship among all the above mentioned ultraproducts/central sequence algebras is given in [1].

3 Modular theory Here, we explain some basic facts from modular theory. The list of results explained here are not sufficient for the study of type III factors, but only suffice to understand the construction of nontracial ultraproducts. We refer the reader to the books by Takesaki [66, 67, 68] or that by Stratila [61] for further details. In particular, we will not discuss the theory of weights (unbounded positive functionals), so we cannot discuss Takesaki’s duality for crossed products or Connes–Takesaki’ theory of noncommutative flow of weights [14].

3.1 Type III factors First, we recall few facts from basic von Neumann algebra theory. See, e. g., [66] for details. A factor M is of type III if it does not contain nonzero finite projections. Any two infinite projections in a σ-finite factor M are equivalent [66, Proposition V.1.39]. In particular, if M is a σ-finite type III factor, then any nonzero projection e ∈ M is Murray– von Neumann equivalent to 1, whence eMe ≅ M holds. The finiteness of a projection is related to the SOT-continuity of the ∗-operation. Indeed, a projection e ∈ M is finite if and only if eMe admits an fn tracial state, if and only if the ∗-operation is SOTcontinuous on the unit ball of eMe (cf. Remark 3.2). Thus, in a type III factor, ∗ is highly SOT-discontinuous, a phenomenon which is reflected in its structure. Modular theory is

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designed for the analysis of the discontinuity in great detail. Fix an fn state φ on a σ-finite von Neumann algebra M. To avoid technicalities, we only present the theory for normal states and avoid the use of unbounded functionals called weights. However, the use of weights is often crucial in many places, especially when one uses Takesaki’s duality. We regard M ⊂ 𝔹(L2 (M, φ)) via the GNS representation, so that φ(a) = ⟨aξφ , ξφ ⟩(a ∈ M) for a cyclic and separating unit vector ξφ ∈ L2 (M, φ). Then Mξφ and M ′ ξφ are both dense subspaces of H = L2 (M, φ). Define densely defined conjugate linear operator Sφ0 and Fφ0 on H by dom(Sφ0 ) = Mξφ ,

dom(Fφ0 )

Sφ0 aξφ = a∗ ξφ ,

= M ξφ , ′

Fφ0 a′ ξφ

= (a ) ξφ , ′ ∗

a ∈ M; a′ ∈ M ′ .

Recall that for a densely defined conjugate-linear map T on a Hilbert space H, its adjoint T ∗ is defined (analogously to the adjoint of a linear operator but taking into account the complex conjugation) by dom(T ∗ ) := {η ∈ H | ∃ζ ∈ H ∀ξ ∈ dom(T) : ⟨Tξ, η⟩ = ⟨ξ, ζ ⟩}, T ∗ η := ζ ,

η ∈ dom(T ∗ ),

where ζ satisfies ⟨Tξ, η⟩ = ⟨ξ, ζ ⟩ (ξ ∈ dom(T)).

Note that ζ above is unique by the density of dom(T). Let a ∈ M and b′ ∈ M ′ . Then ⟨Fφ0 b′ ξφ , aξφ ⟩ = ⟨(b′ ) ξφ , aξφ ⟩ = ⟨a∗ ξφ , b′ ξφ ⟩ ∗

= ⟨b′ ξφ , Sφ0 aξφ ⟩. This shows that Sφ0 ⊂ (Fφ0 )∗ . Similarly, Fφ0 ⊂ (Sφ0 )∗ holds. In particular, both Sφ0 and Fφ0 are closable operators. Moreover, it can be shown that their closures Sφ and Fφ are adjoint to 1

each other: Sφ∗ = Fφ , Fφ∗ = Sφ . Let Sφ = Jφ Δφ2 be the polar decomposition of the conjugatelinear closed operator Sφ . The operator Δφ = Sφ∗ Sφ (resp. Jφ ) is called the modular operator (resp. modular conjugation operator) associated with φ. Suppose for now that φ = τ is tracial. Then Sτ = Sτ0 = Fτ is isometric, and Fτ Sτ = 1. Thus Δτ = 1 and Jτ = Sτ . Actually, Δφ = 1 if and only if φ is tracial, so the modular operator measures how far φ is different from being tracial. We record some basic of their basic properties which, while not obvious, are not too hard to verify. Proposition 3.1. Operator Δφ is nonsingular, positive, and self-adjoint, and Jφ is a conjugate-linear unitary satisfying Jφ2 = 1. Moreover, Jφ ξφ = ξφ = Δφ ξφ and Jφ Δφ Jφ = Δ−1 φ hold. Remark 3.2. We remark that if Δφ is bounded for some normal faithful state φ, then M must be of finite type. Indeed, one can show that x 󳨃→ x ∗ is strongly continuous on the unit ball in this case: let (xi )i∈I be a bounded net in M converging strongly to x. By assumption, ‖Δφ ‖ ≤ C for some C > 0. Then for y′ ∈ M ′ , we see that

308 � H. Ando 󵄩 ′ ∗ 󵄩 󵄩󵄩 ∗ ∗ ∗ ′ 󵄩 󵄩󵄩(xi − x )y ξφ 󵄩󵄩󵄩 = 󵄩󵄩󵄩y (xi − x )ξφ 󵄩󵄩󵄩 1 󵄩 󵄩 = 󵄩󵄩󵄩y′ Jφ Δφ2 (xi − x)ξφ 󵄩󵄩󵄩 1 󵄩 i→∞ 󵄩 󵄩󵄩 ≤ C 2 󵄩󵄩󵄩y′ 󵄩󵄩󵄩󵄩󵄩󵄩(xi − x)ξφ 󵄩󵄩󵄩 → 0. Since M ′ ξφ is dense in H by the separating property of ξφ and since supi∈I ‖xi ‖ < ∞, it holds that xi∗ → x ∗ strongly. But then this implies that M is finite, because if M were not finite, there would exist a nonzero properly infinite central projection z ∈ 𝒵 (M). Then the corner Mz := Mz|zH satisfying Mz ≅ Mz ⊗𝔹(ℓ2 ) (see, e. g., [66, Proposition V.1.40]) 2 would be of infinite type. However, there exists a sequence (an )∞ n=1 ⊂ ℂ ⊗ 𝔹(ℓ ) converg2 ∗ ∗ ing strongly to an operator a ∈ ℂ ⊗ 𝔹(ℓ ) for which an → a (strongly) does not hold (consider a unilateral shift). Hence we get a contradiction. Therefore, the unboundedness of Δφ is a natural phenomenon when one deals with infinite type von Neumann algebras. The next result is the foundation of the modular theory. (Tomita gave the foundation of modular theory in two unpublished papers. The written proofs we know today were later given by Takesaki in [62].) Theorem 3.3 (Tomita). The equalities Jφ MJφ = M ′ and Δitφ MΔ−it φ = M (t ∈ ℝ) hold. In φ particular, there is a u-continuous one-parameter group σ of automorphisms of M given by φ

σt (x) = Δitφ xΔ−it φ ,

x ∈ M,

t ∈ ℝ.

Group σ φ is called the modular automorphisms group (or modular flow) associated with φ. For a von Neumann algebra M, we endow its automorphism group Aut(M) with the so-called u-topology, whose basis is given by ℬ = {U(α; φ1 , . . . , φn ) | α ∈ Aut(M), φ1 , . . . , φn ∈ M∗ }, where U(α; φ1 , . . . , φn ) = {β ∈ Aut(M) | ‖φi ∘ α − φi ∘ β‖ < 1, i = 1, . . . , n}. Thus, the u-topology is the topology of pointwise norm-convergence on M∗ . Group σ φ is characterized by the so-called Kubo–Martin–Schwinger (KMS) condition. Let D = {z ∈ ℂ | 0 ≤ Imz ≤ 1} be a closed strip in the complex plane, and we denote by 𝒜(D) the set of all bounded continuous functions F: D → ℂ which are analytic in the interior of D. Let α: ℝ → Aut(M) be a continuous one-parameter automorphism group of a von Neumann algebra M and let φ be an fn state on M. Then φ satisfies the KMS-condition (KMS stands for the three physicists Kubo, Martin, and Schwinger) with respect to α, or φ is α-KMS, if the following conditions are satisfied: (i) φ ∘ αt = φ, t ∈ ℝ. (ii) For all x, y ∈ M, there exists an F = Fx,y ∈ 𝒜(D) such that F(t) = φ(αt (x)y),

F(t + i) = φ(yαt (x)),

t ∈ ℝ.

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Theorem 3.4. An fn state φ is σ φ -KMS and if α: ℝ → Aut(M) is a continuous oneparameter group for which φ is α-KMS, then α = σ φ holds. Example 3.5. Let M = Mn (ℂ), and let φ be an fn state on M. Then φ(a) = Tr(ha) for some strictly positive h ∈ Mn (ℂ). By changing a basis, we may assume that h is of the form h=(

λ1 O

..

O .

λ1 , . . . , λn > 0,

), λn

n

∑ λk = 1.

k=1

Note that z 󳨃→ h

1+iz

λ11+iz =( O

..

O .

) λ1+iz n

is an Mn (ℂ)-valued analytic function. For λ > 0, the map ℂ ∋ z = α + iβ 󳨃→ λiz = λiα−β is entire analytic. For x, y ∈ M, we define a function Fx,y by Fx,y (z) := Tr(h1+iz xh−iz y),

z ∈ D.

If 0 ≤ β ≤ 1, then 󵄨󵄨 iz 󵄨󵄨 −β −1 󵄨󵄨λ 󵄨󵄨 = λ ≤ λ . Hence z 󳨃→ λiz is uniformly bounded on D. Since Fx,y (z) is a polynomial of such funcφ iz tions λiz 1 , . . . , λn , it follows that F is bounded and continuous on D. We show that σt (x) = hit xh−it . To see this, observe that the right-hand side is u-continuous. Therefore we check that φ is σ-KMS and apply Theorem 3.4. For t ∈ ℝ and x ∈ M, φ ∘ σt (x) = Tr(hhit xh−it ) = Tr(h−it hhit x) = Tr(hx) = φ(x), so φ is σt -invariant. Fix x, y ∈ M. Then φ(σt (x)y) = Tr(h1+it xh−it y). By definition, Fx,y (t) = φ(σt (x)y) (t ∈ ℝ). Moreover, φ(yσt (x)) = Tr(hyhit xh−it ) = Tr(hit xh1−it y) = Fx,y (t + i), φ

t ∈ ℝ.

Therefore φ is σ-KMS, whence σt = σt (t ∈ ℝ) by Theorem 3.4.

310 � H. Ando For an fn state φ on a von Neumann algebra M, the fixed point subalgebra of σ φ is called the centralizer of φ and is denoted by Mφ . It is known that Mφ = {x ∈ M | φ(xy) = φ(yx), y ∈ M} and it is always a finite von Neumann algebra with an fn trace φ|Mφ . Example 3.6. Let us compute Mφ in some simple cases. Set M = M4 (ℂ), and let φ = Tr(h⋅), where λ1 h = diag(λ1 , λ2 , λ3 , λ4 ) = (

λ2

),

λ3

λ1 , . . . , λ4 > 0,

λ4

4

∑ λi = 1. i=1

φ

Then (see Example 3.5) σt (x) = hit xh−it (t ∈ ℝ, x ∈ M4 (ℂ)), so that if x = [xmn ]1≤m,n≤4 , then it holds that φ

σt (x) = [(

it

λm ) xmn ] . λn n,m

Therefore x ∈ Mφ if and only if it

∀t ∈ ℝ

[(

λm ) xmn = xmn ] λn

(1)

holds for all m, n = 1, . . . , 4. Hence there are two cases to consider: Case (i). λm = λn . In this case, (1) is satisfied by arbitrary xmn ∈ ℂ. λ

Case (ii). λm ≠ λn . In this case, there exists t ∈ ℝ such that ( λm )it ≠ 1, so that the only n solution of (1) is xmn = 0. xmn

Therefore if, for example, λ1 ≠ λ2 = λ3 = λ4 (e. g., λ1 = 21 , λ2 = λ3 = λ4 = 61 ), then = 0 if n = 1, m = 2, 3, 4 or n = 2, 3, 4, m = 1 and xmn is arbitrary for all other m, n, x11 { { { { 0 Mφ = {( { 0 { { { 0

0 x22 x32 x42

0 x23 x33 x43

0 } } } } x24 ) ; xmn ∈ ℂ} = ℂ ⊕ M3 (ℂ). } x34 } } x44 }

Note that the exact values of λn are not important for the structure of Mφ ; only the information of whether one λn is equal to another λm matters. Similarly, if we choose λ1 = λ2 ≠ λ3 = λ4 (e. g., h = diag( 31 , 31 , 61 , 61 )), then Mφ = M2 (ℂ) ⊕ M2 (ℂ). And if λm ≠ λn for all m ≠ n (e. g., h = diag( 31 , 61 , 91 , 187 )), then Mφ = ℂ ⊕ ℂ ⊕ ℂ ⊕ ℂ = ℂ4 .

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Observe also that all diagonal entries are always fixed by σ φ for any φ. Thus if M = Mn (ℂ) (n > 1), then Mφ cannot be isomorphic to ℂ (in fact it is known (see, e. g., [1, Lemma 5.3]) that no semifinite factor can have a state φ with trivial centralizer Mφ = ℂ). Of course, if we choose φ = 41 Tr, the canonical tracial state, then Mφ = M4 (ℂ). So the centralizer can be a factor. Remark 3.7. To check the KMS condition, the following fact is often implicitly used in the literature. Namely, if the condition (ii) above is satisfied for all elements x, y in a ∗-strongly dense subalgebra 𝒜 of M, then by an approximation argument using the Phragmén–Lindelöf theorem (a version of the theorem we need here is: any F ∈ 𝒜(D) satisfies supz∈D |F(z)| = supz∈𝜕D |F(z)|), (ii) holds for all x, y ∈ M. To see how this can be useful, consider, for example, that we have two σ-finite von Neumann algebras M1 , M2 with fn states φ1 ∈ (M1 )∗ , φ2 ∈ (M2 )∗ . Then φ = φ1 ⊗φ2 is an fn state on M = M1 ⊗M2 . Then φ φ φ what is σ φ ? The obvious guess (and it is indeed true) would be σt = σt 1 ⊗ σt 2 , so one need to check that φ is σ φ1 ⊗ σ φ2 -KMS. Then it is easy to check (ii) for x, y which are taken from the algebraic tensor product M1 ⊙ M2 because this ∗-strongly dense subalgebra of M is linearly spanned by elementary tensors and the condition (ii) is directly checkable for such elements. The usefulness of this observation is even more apparent when we consider infinite tensor products. The modular flow σ φ depends on φ, but any two modular flows are intertwined by a unitary cocycle. Theorem 3.8 (Connes). Let M be a von Neumann algebra and φ, ψ be fn states on M. Then there exists a σ-strongly continuous one-parameter family {ut }t∈ℝ of unitaries in M with the following properties: ψ (i) us+t = us σs (ut ), s, t ∈ ℝ. φ ψ (ii) σt (x) = ut σt (x)ut∗ , t ∈ ℝ, x ∈ M. (iii) For each x, y ∈ M, there exists F ∈ 𝒜(D) such that ψ

F(t) = φ(ut σt (y)x),

ψ

F(t + i) = ψ(xut σt (y)),

t ∈ ℝ.

Moreover, the family {ut } is uniquely determined by the condition (iii). Unitary ut is often denoted by (Dφ: Dψ)t and it is called the Connes’ Radon–Nikodym cocycle. Although we do not give a detailed proof of Theorem 3.8, it should be pointed out that the proof is really a soft and elegant 2 × 2 matrix argument: on M2 (M) = M2 (ℂ) ⊗ M, we define a positive linear form φ by φ ((

x11 x21

x12 )) := φ1 (x11 ) + φ2 (x22 ), x22

(

x11 x21

x12 ) ∈ M2 (M). x22

312 � H. Ando Then the Connes cocycle is uniquely determined by 0 ( ut

0 0 φ ) = σt (( 0 1

0 )) , 0

t ∈ ℝ.

Example 3.9. Let M = Mn (ℂ), and let φ1 , φ2 be positive, normal, and faithful functionals on M. Then there are strictly positive elements h1 , h2 ∈ M such that φ1 = Tr(h1 ⋅),

φ2 = Tr(h2 ⋅).

φ

We have σt 1 (x) = h1it xh1−it (see Example 3.5). Let x φ (( 11 x21

x12 )) := φ1 (x11 ) + φ2 (x22 ), x22

(

x11 x21

x12 ) ∈ M2 (Mn (ℂ)). x22

Then φ = Tr(h ⋅ ) (Tr is defined on M2n (ℂ)), where h1 0

h=(

0 ). h2

We now get 0 ut

(

0 0 φ ) = σt (( 0 1 h1it 0

0 0 )( 1 h2it

=(

=(

0 )) 0

0

h2it h1−it

0 h1−it )( 0 0

0 ) h2−it

0 ), 0

i. e., (Dφ2 : Dφ1 )t = h2it h1−it . A notable difference between tracial and nontracial von Neumann algebras is the existence or absence of normal conditional expectations. Recall that whenever N ⊂ M is an inclusion of finite von Neumann algebras and τ is an fn tracial state on M, then there exists a unique τ-preserving fn conditional expectation E: M → N. For a general inclusion of von Neumann algebras and a general fn state φ ∈ M∗ , there need not exists a φ-preserving fn expectation. Takesaki characterized exactly when there exists a φ-preserving fn expectation in terms of σ φ (in the weight setting using Tomita’s modular Hilbert algebra, but here we restrict to the state case). Theorem 3.10 (Takesaki). Let N ⊂ M be an inclusion of σ-finite von Neumann algebras and let φ be an fn state on M. Define φ0 := φ|N . Then the following conditions are equivalent:

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(i) There exists a unique φ-preserving (hence faithful and normal) conditional expectation E: M → N. φ (ii) For every t ∈ ℝ, σt (N) = N holds. φ0 φ| φ φ (iii) σt (x) = σt (x) for every t ∈ ℝ and x ∈ N, i. e., σt |N = σt N holds. Next we introduce Arveson–Connes’ spectral theory for automorphism groups and Connes’ S- and T-invariants. Most results explained here are from Connes’ thesis [6] and we refer the reader to it for further details. Connes defined other useful invariants such as the Sd -invariant and the τ-invariant, but since they are not needed for our presentation we will omit them. Since we apply the theory only to modular flows, we present the case of one-parameter automorphism group only. In the sequel, we identify the dual ̂ of the additive group ℝ with itself. For f ∈ L1 (ℝ), we define the Fourier transgroup ℝ ̂ form f by f ̂(λ) := ∫ eitλ f (t) dt,

̂ = ℝ. λ∈ℝ

ℝ φ

φ

We also define σf (x) := ∫ℝ f (t)σt (x) dt (x ∈ M). The integration is understood to be in the σ-weak sense: it is the unique element a ∈ M characterized by the condition φ

ψ(a) = ∫ f (t)ψ(σt (x)) dt,

ψ ∈ M∗ .

ℝ

(1) For x ∈ M, Spσ φ (x) is defined by ̂ | f ̂(λ) = 0 for all f ∈ L1 (ℝ) with σ φ (x) = 0}. {λ ∈ ℝ f (2) The Arveson spectrum of σ φ , denoted by Sp(σ φ ) is the set φ

̂ | f ̂(λ) = 0 for all f ∈ L1 (ℝ) with σ = 0}. {λ ∈ ℝ f It is shown that Sp(σ φ ) = log(σ(Δφ ) \ {0}) (cf. Remark 3.11). ̂ = ℝ, the spectral subspace of σ φ corresponding to E is given by (3) For a subset E of ℝ M(σ φ , E) := {x ∈ M | Spσ φ (x) ⊂ E}. The fixed point subalgebra of σ φ can be shown to be equal to M(σ φ , {0}), and it is equal to the centralizer Mφ defined above. Recall also that the predual M∗ is an M–M bimodule by the action a ⋅ φ ⋅ b(x) := φ(bxa),

a, b, x ∈ M,

φ ∈ M∗ .

With this in mind, the centralizer is Mφ = {x ∈ M | xφ = φx}. Likewise, for λ > 0, the space M(σ φ , {log λ}) of λ-eigenvectors of φ (i. e., an element a ∈ M such that aξφ is

314 � H. Ando an eigenvector of Δφ corresponding to the eigenvalue λ) is characterized as follows: φ

M(σ φ , {log λ}) = {a ∈ M | σt (a) = λit a (t ∈ ℝ)} = {a ∈ M | λaφ = φa}. The spectral subspaces have the following properties: (i) M(σ φ , E)∗ = M(σ φ , −E); (ii) M(σ φ , E)M(σ φ , F) ⊂ M(σ φ , E + F); (iii) λ ∈ Sp(σ φ ) if and only if M(σ φ , E) ≠ {0} for any closed neighborhood E of λ. (4) The Connes spectrum of σ φ , denoted by Γ(σ φ ), is given by Γ(σ φ ) =

⋂

e∈Proj(Mφ )

Sp(σ φe ).

Here, for e ∈ Proj(Mφ ), σ φe is the restricted action of σ φ to the reduced algebra Me , which coincides with the modular automorphism group of φ|Me . It holds that Γ(σ φ ) =

⋂

0=e∈Proj(𝒵(M ̸ φ ))

Sp(σ φe ),

whence Γ(σ φ ) = Sp(σ φ ) if Mφ is a factor. (5) Let M be a σ-finite factor. The Connes’ S-invariant is defined by S(M) =

⋂

φ∈Sfn (M)

σ(Δφ ),

where Sfn (M) is the set of all fn states on M. It can be shown that S(M) \ {0} is a closed multiplicative subgroup of ℝ∗+ = (0, ∞), and Γ(σ φ ) = log(S(M) \ {0}). Then we can give Connes’ subclassification of type III factors. A σ-finite type III factor M is called of (i) type III0 if S(M) = {0, 1}; (ii) type IIIλ if S(M) = {λn | n ∈ ℤ} ∪ {0} (0 < λ < 1); (iii) type III1 if S(M) = [0, ∞). For general factors, one needs to use normal faithful semifinite weights to define the S-invariant. However, the above classification of type III factors will not be affected by this change. (6) Let M be a σ-finite factor and φ be an fn state on M. The Connes’ T-invariant T(M) is φ φ the set of all t ∈ ℝ for which σt is an inner automorphism. The condition that σt be inner does not depend on the choice of an fn state φ thanks to the existence of the Connes’ Radon–Nikodym cocycle. If M is semifinite, then T(M) = ℝ holds. For the separable case, the converse also holds. Namely, Kallman [42] showed that if M has separable predual with T(M) = ℝ, then there exists a continuous one-parameter φ group of unitaries {u(t)}t∈ℝ in M such that σt = Ad(u(t)) (t ∈ ℝ). That is, σ φ is an inner flow. One can show that in this case, M admits an fn semifinite trace τ (here we need the theory of weights). Thus M is semifinite.

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Remark 3.11. For (1) and (2), we could simply use the spectrum of the modular operator alone if we only care about fn states. The motivation of the present formulation due to Arveson is suitable to generalize this to more general action of locally compact abelian group which does not necessarily fix any fn state. In this case we no longer have a simple picture of the Sp(α) (α is an action) as the “spectrum of the generating self-adjoint operator.” Although we do not have enough space to discuss many examples, it is important to have some basic examples of type III factors. We will see that they naturally appear in the study of problems where the Connes’ bicentralizer problem for type III1 factors (to be explained later) are hidden. First, we recall the infinite tensor product construction (however we assume that the reader is familiar with the C∗ -algebraic infinite tensor product; see Gábor Szabó’s “Introduction to C∗ -algebras” in this volume). Let (Mn )∞ n=1 be a sequence of σ-finite von Neumann algebras and let M0 = ⨂n∈ℕ Mn (we take the minimal tensor product and its inductive limit). Suppose that we are given an fn state φn on Mn for each n ∈ ℕ. Then the product state φ1 ⊗ ⋅ ⋅ ⋅ ⊗ φn ⊗ φn+1 on M n+1 = M1 ⊗⋅ ⋅ ⋅⊗Mn+1 agrees with φ1 ⊗⋅ ⋅ ⋅⊗φn when restricted to M n = M1 ⊗⋅ ⋅ ⋅⊗Mn ⊗ℂ. Therefore there exists a unique linear functional φ on the union ⋃∞ n=1 M n whose restriction to M n is φ1 ⊗ ⋅ ⋅ ⋅ ⊗ φn for every n ∈ ℕ. Now, since 󵄨󵄨 󵄨 󵄨󵄨(φ1 ⊗ ⋅ ⋅ ⋅ ⊗ φn )(x)󵄨󵄨󵄨 ≤ ‖x‖,

x ∈ M n,

n ∈ ℕ,

φ is bounded on ⋃∞ n=1 M n , whence it is uniquely extended to a positive linear functional on M0 = ⨂∞ n=1 Mn , which we denote by ∞

φ = ⨂ φn . n=1

Then the infinite tensor product von Neumann algebra of (Mn , φn )∞ n=1 , written ∞

⨂(Mn , φn ), n=1

is defined as πφ (M0 )′′ , where πφ is the GNS representation of the state φ. The extension of φ as the GNS-vector state on ⨂n∈ℕ (Mn , φn ) is still denoted by φ = ⨂n∈ℕ φn . State φ is shown to be faithful on M. We also write (M, φ) = ⨂∞ n=1 (Mn , φn ) if M is the infinite tensor ∞ ∞ product of (Mn , φn )n=1 and φ = ⨂n=1 φn . Let t ∈ ℝ. There is a unique automorphism σt of M characterized by its action on elementary tensors φ

φ

σt (x1 ⊗ ⋅ ⋅ ⋅ ⊗ xn ) = σt 1 (x1 ) ⊗ ⋅ ⋅ ⋅ ⊗ σt n (xn ), φ

xi ∈ Mi

(i = 1, . . . , n).

Automorphism σt is denoted by ⨂n∈ℕ σt n . It is straightforward to check that φ is σ-KMS (relative to a ∗-strongly dense subalgebra spanned by elementary tensors, cf. φ φ Remark 3.7), so σt = ⨂n∈ℕ σt n holds.

316 � H. Ando Definition 3.12. An ITPFI factor (infinite tensor product of factors of type I) is an infinite tensor product von Neumann algebra of the form ∞

(M, φ) = ⨂(Mi , φi ), i=1

where is a sequence of type I factors; (φn )∞ n=1 is called a sequence of reference states for M. If Mn ≅ M2 (ℂ) for all n ∈ ℕ, then we call M an ITPFI2 factor or an Araki– Woods factor. (Mn )∞ n=1

Example 3.13 (Powers factors Rλ (0 < λ < 1)). Let 0 < λ ≤ 1 and Mn = M2 (ℂ), φn = Tr(h⋅) for all n > 0, where h :=

1 1 ( 1+λ 0

0 ). λ

If λ = 1, then φn = 21 Tr is tracial for all n > 0 and M is the unique hyperfinite II1 factor, denoted by R. For 0 < λ < 1, M can be shown to be a type IIIλ factor which we denote by Rλ and call it the Powers factor. In general, it is not easy to determine the type of ITPFI factors. Let us remark the criterion by Araki and Woods [5]. Let (M, φ) = ⨂∞ n=1 (Mkn (ℂ), Tr(ρn ⋅)), where we allow 2 kn = ∞ (and interpret Mkn (ℂ) = 𝔹(ℓ )). Let λ1,n ≥ λ2,n ≥ ⋅ ⋅ ⋅ ≥ λkn ,n be the collection of eigenvalues of the density matrix ρn (in decreasing order, multiplicity is taken into kn consideration). Thus each λk,n > 0 and ∑k=1 λk,n = 1. Theorem 3.14 (Araki–Woods). Under the above notation, the following statements hold: (i) M is of type I∞ if and only if ∑∞ n=1 (1 − λ1,n ) < ∞. k

−1

1

2 n 2 2 (ii) M is of type II1 if and only if ∑∞ n=1 ∑k=1 |kn − λk,n | < ∞. (iii) If there exists δ > 0 such that λ1,n ≥ δ for every n ∈ ℕ, then M is of type III if and only if there exists C > 0 such that ∞ kn 󵄨󵄨 λ 󵄨󵄨2 󵄨 k,n 󵄨 − 1󵄨󵄨󵄨 , C} = ∞. ∑ ∑ λk,n min{󵄨󵄨󵄨 󵄨 󵄨󵄨 λ 󵄨 1,n n=1 k=1

(2)

If this is the case, then (2) holds for every C > 0. Araki–Woods [5] showed that for each 0 < λ ≤ 1, there exists exactly one ITPFI factor of type IIIλ , namely the Powers factor Rλ , and there exists continuum many nonisomorphic ITPFI factors of type III0 (in today’s terminology; their work preceded the definition of Connes’ S-invariant, or more precisely, Connes generalized their invariants which were used to classify ITPFI factors to introduce the S- and T-invariants). The unique ITPFI factor of type III1 is typically denoted by R∞ . One realization of R∞ as an infinite tensor product is ⨂n∈ℕ (M3 (ℂ), φ), where φ = log λ 1 Tr(ρ⋅) with ρ = 1+λ+μ diag(1, λ, μ), where 0 < λ, μ < 1 and log ∉ ℚ. μ

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3.2 Standard forms Each von Neumann algebra M admits a canonical representation on a Hilbert space L2 (M), and the Hilbert space is endowed with a special positive cone L2 (M)+ which gives a partial ordering which reflects the structure of M. Such a representation is called the standard form, and it is a central concept in the modular theory. It will be crucial for the analysis of the Groh–Raynaud ultraproducts. The formal definition is as follows. Consider a quadruple (M, L2 (M), J, L2 (M)+ ), where M is a von Neumann algebra, L2 (M) is a Hilbert space on which M acts, J is a conjugate-linear isometry on L2 (M) with J 2 = 1, and L2 (M)+ ⊂ L2 (M) is a closed convex cone which is self-dual, i. e., L2 (M)+ = (L2 (M)+ )0 , where the dual cone (L2 (M)+ )0 is defined by 0

(L2 (M)+ ) := {ξ ∈ L2 (M) | ⟨ξ, η⟩ ≥ 0, η ∈ L2 (M)+ }. Then (M, L2 (M), J, L2 (M)+ ) is called the standard form if the following conditions are satisfied: 1. JMJ = M ′ ; 2. Jξ = ξ, ξ ∈ L2 (M)+ ; 3. xJxJξ ∈ L2 (M)+ , x ∈ M, ξ ∈ L2 (M)+ ; 4. JxJ = x ∗ , x ∈ 𝒵 (M). If φ is an fn state on M, it can be shown that the quadruple (M, L2 (M, φ), Jφ , L2 (M, φ)+ ), where 1

L2 (M, φ)+ = Δφ4 M+ ξφ = {xJφ xJφ ξφ | x ∈ M} is a standard form. If φ = τ is tracial, then L2 (M, τ)+ is nothing but the L2 -closure of M+ . For x, y ∈ M+ , observe that 1

1

1

⟨Δφ4 xξφ , Δφ4 yξφ ⟩ = ⟨xξφ , Δφ2 yξφ ⟩ = ⟨Sφ yξφ , Jφ xξφ ⟩ 1

1

= ⟨yξφ , Jφ xJφ ξφ ⟩ = ⟨(Jφ x 2 Jφ )y(Jφ x 2 Jφ )ξφ , ξφ ⟩ ≥ 0. Thus the power

1 4

of Δφ in the definition of L2 (M, φ)+ is appropriate.

Remark 3.15. In [1, Lemma 3.19] it was shown that the condition 4 of the standard form follows automatically from the other conditions 1, 2, and 3. For each M, its standard form is unique in a strong sense, and this leads to the fact that the automorphism group of M is isomorphic to a closed subgroup of the unitary group 𝒰 (L2 (M)): Theorem 3.16 (Haagerup). Any von Neumann algebra M admits a standard from. If (M1 , H1 , J1 , H1+ )

and (M2 , H2 , J2 , H2+ )

318 � H. Ando are standard forms and if π: M1 → M2 is a ∗-isomorphism of von Neumann algebras, then there exists a unitary u: H1 → H2 satisfying the following conditions: (i) π(x) = uxu∗ , x ∈ M1 ; (ii) J2 u = uJ1 ; (iii) H2+ = uH1+ . Furthermore, the group G of all u ∈ 𝒰 (L2 (M)) satisfying the three conditions uMu∗ = M,

uJu∗ = J,

uL2 (M)+ = L2 (M)+

is isomorphic as a topological group to the automorphism group Aut(M) with the u-topology under the map G ∋ u 󳨃→ αu ∈ Aut(M). Here, G is endowed with the SOT and αu (x) = uxu∗ . Note that the latter claim G ≅ Aut(M) follows from the former part. Another important fact (see [27]) is that for any φ ∈ M∗+ , there exists a unique vector 1

ξφ ∈ L2 (M)+ (or often denoted φ 2 ) such that φ(x) = ⟨xξφ , ξφ ⟩ (x ∈ M) holds. Moreover, the following inequality holds. Theorem 3.17 (Araki–Powers–Størmer inequality). Let (M, L2 (M), J, L2 (M)+ ) be the standard form of M. Then for each φ, ψ ∈ M∗+ , the following inequality holds: ‖ξφ − ξψ ‖2 ≤ ‖φ − ψ‖ ≤ ‖ξφ − ξψ ‖‖ξφ + ξψ ‖.

4 Ocneanu vs Groh–Raynaud, and central sequence algebras by Connes and Ocneanu We are now ready to study all the ultraproduct constructions. Let (M, φ) be as in Definition 2.2 and ω ∈ βℕ \ ℕ.

4.1 Ocneanu’s ultraproduct as a corner of the Groh–Raynaud’ ultraproduct First, we define the Groh–Raynaud ultraproduct. For each n ∈ ℕ, let (Mn , Hn , Jn , Hn+ ) be the standard form of Mn , where Hn = L2 (Mn , φn ), and let H ω be the Hilbert space ultra∗ product of H = (Hn )∞ n=1 . Define a ∗-homomorphism πω from the C -algebra ultraproduct (𝔹(Hn ))ω to 𝔹(H ω ) by πω ((an )ω )(ξn )ω = (an ξn )ω ,

∞ (an )∞ n=1 ∈ ℓ (M),

(ξn )ω ∈ H ω .

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One can show that πω is faithful and its image is a proper SOT-dense ∗-subalgebra of 𝔹(H ω ). Definition 4.1 (Groh–Raynaud). The Groh–Raynaud ultraproduct ∏ω M of M is the von Neumann algebra on H ω generated by the image πω (M ω ) of the C∗ -algebra ultraproduct M ω . The definition of ∏ω M above looks different from the one given in the introduction, but thanks to Raynaud’s work [60], they are equivalent, that is, they are isomorphic as von Neumann algebras. Because the standard forms are unique up to unitary equivalence in the sense of Theorem 3.16, the isomorphism class of ∏ω M is independent of the choice of φ. The following theorem explains the relationship between the Ocneanu ultraproduct and the Groh–Raynaud ultraproduct: Theorem 4.2 ([1]). Let M ω = (M, φ)ω , φω = (φn )ω . Define w : L2 (M ω , φω ) → H ω by w(xn )ω ξφω := (xn ξφn )ω ,

(xn )ω ∈ M ω .

Then w is an isometry, and w∗ (∏ω M)w = M ω . Let p = p(φ) ∈ ∏ω M be the support projection of the normal state ⟨ ⋅ ξω , ξω ⟩ on ω ∏ M, where ξω = (ξφn )ω ∈ H ω , that is, p is the orthogonal projection of H ω onto the closed subspace (∏ω M)′ ξω . Let Jω = (Jn )ω be the ultraproduct map of (Jn )∞ n=1 . Then q := ∗ ww = pJω pJω , and we have the following ∗-isomorphisms: ω

ω

M ω ≅ p(∏ M)p ≅ q(∏ M)q. Thus, for any φ, (M, φ)ω is isomorphic to a corner of ∏ω M. Remark 4.3. Conversely, any σ-finite corner of ∏ω M is isomorphic to (M, φ)ω for some ω φ = (φn )∞ n=1 . To see this, if p is a σ-finite projection in ∏ M, then there exists a normal ω state φ on ∏ M whose support projection is p. We view φ as an element in the ultraproduct ((Mn )∗ )ω , so φ is represented by a sequence of fn states φ = (φn )∞ n=1 . Then the corner p(∏ω M)p is isomorphic to the Ocneanu ultraproduct (M, φ)ω . At this point it is nice to recall that not every x ∈ ℓ∞ (M) defines an element in the Ocneanu ultraproduct, because x ∈ ℳω (M, φ) is required. When M is not tracial, this condition is not vacuous even when (Mn , φn ) = (M, φ) is constant. How can we check that a given bounded sequence is in ℳω (M, φ)? We give two answers to this. The first ∞ is given by the support projection p = p(φ). Below, for (xn )∞ n=1 ∈ ℓ (M), we identify the ultraproduct element (xn )ω ∈ M ω with the image πω ((xn )ω ) ∈ 𝔹(H ω ). ∞ ω Theorem 4.4 ([1]). For x = (xn )∞ n=1 ∈ ℓ (M), x belongs to ℳ (M, φ) if and only if (xn )ω p = p(xn )ω holds in 𝔹(H ω ). Moreover, in this case the map

320 � H. Ando ω

M ω ∋ (xn )ω 󳨃→ (xn )ω p = p(xn )ω ∈ p(∏ M)p defines a ∗-isomorphism. Note that if (Mn , φn ) = (M, τ) is a constant sequence of a fixed tracial von Neumann algebra M with an fn tracial state τ, then ℳω (M, τ) = ℓ∞ (M) and we get our favorite ∞ tracial ultrapower M ω . Thus for every (xn )∞ n=1 ∈ ℓ (M), (xn )ω commutes with p = p(τ). ω Thus p is a central projection in ∏ M. In particular, when M = R is the hyperfinite II1 factor, we have a central projection p(τ) ∈ ∏ω R, which can be shown to be nontrivial by the next example. Example 4.5 ([1]). On ∏ω R, it can also happen that p(φ) is not central: fix λ ∈ (0, 1), and 1 λ put ρλ = diag( 1+λ , 1+λ ) ∈ M2 (ℂ)+ . Let Rλ = ⨂ℕ (M2 (ℂ), Tr(ρλ ⋅)) be the Powers factor of type IIIλ . Define φn ∈ Sfn (R) by n

1 Tr, 2 k=n+1 ∞

φn := ⨂ Tr(ρλ ⋅) ⊗ ⨂ k=1

n ≥ 1.

Then there exists a normal injective ∗-homomorphism π: Rλ → (R, φn )ω whose image π(Rλ ) is the range of an fn conditional expectation ε: (R, φn )ω → π(Rλ ) (see [1, Proposition 6.2]). This implies that (R, φn )ω is not semifinite. Thus ∏ω R is not semifinite either. (On the other hand, the corner corresponding to p(τ) is Rω , a type II1 factor.) Thus ∏ω R is neither a factor nor tracial (hopefully this does not look strange any more!). Still, it is hard to check whether x ∈ ℓ∞ (M) commutes with p = p(φ) because p is still a mysterious object. The second answer gives a better description of ℳω (M, φ) because it is expressed in terms of modular theory, thus it is often verifiable in concrete situations. For this purpose, one needs to prove first that the modular flow behaves extremely nicely on the Ocneanu ultraproduct. This type of theorem was obtained by Golodets, Raynaud, and Kirchberg for their ultraproducts, respectively. In [1], a new proof of this fact for the Ocneanu ultraproduct is given using the description of M ω as a corner of ∏ω M: Theorem 4.6 ([1]). Let M ω = (M, φ)ω , φω = (φn )ω . Then φω

φ

ω

σt ((xn )ω ) = (σt n (xn )) ,

t ∈ ℝ,

(xn )ω ∈ M ω .

φ

In particular, t 󳨃→ (σt n )ω is a continuous flow on M ω . It should be emphasized that this is not at all obvious: there are many topological group actions on von Neumann algebras whose ultrapower action on the ultrapower von Neumann algebras are discontinuous. We can use Theorem 4.6 to prove that elements of ℳω can be characterized by the spectral condition for (σ φn )n . ∞ Proposition 4.7 ([1]). Let (M, φ) be as in Theorem 4.6. Then for (xn )∞ n=1 ∈ ℓ (M), the following conditions are equivalent:

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ω (1) (xn )∞ n=1 ∈ ℳ (M, φ); ω (2) For every ε > 0, there exists a > 0 and (yn )∞ n=1 ∈ ℳ (M, φ) such that

(i)

lim ‖xn − yn ‖♯φ < ε,

n→ω

(ii) yn ∈ Mn (σ φn , [−a, a]),

n ∈ ℕ.

In this case, (yn )n can be chosen to satisfy ‖(yn )ω ‖ ≤ ‖(xn )ω ‖.

4.2 State space diameter and factoriality of the Groh–Raynaud ultraproduct Next we consider how the modular theory of the underlying algebra affects the structure of the ultraproduct. We have seen that ∏ω M need not be a factor even when M itself is. However, if M is a factor of type IIIλ , (0 < λ ≤ 1), ∏ω M is still a factor of the same type. This can be seen by studying the diameter of the state space. Let M be a von Neumann algebra. Then an equivalence relation ∼ on the set Sn (M) of all normal states on M is defined by φ ∼ ψ if they are approximately unitarily equivalent, i. e., there is a ∗ sequence of unitaries (un )∞ n=1 in 𝒰 (M) such that limn→∞ ‖φ − un ψun ‖ = 0. Denote by [φ] the equivalence class in Sn (M) represented by φ ∈ Sn (M). Then Sfn (M)/ ∼ is a metric space by d([φ], [ψ]) :=

󵄩 󵄩 inf 󵄩󵄩󵄩φ − uψu∗ 󵄩󵄩󵄩,

u∈𝒰 (M)

[φ], [ψ] ∈ Sn (M)/ ∼ .

Definition 4.8. The state space diameter of M, denoted d(M), is defined by d(M) :=

sup

φ,ψ∈Sn (M)

d([φ], [ψ]).

It is always the case that d(M) ≤ 2 and d(M) = 2 if M is not a factor. By the results of Connes–Størmer [13], Connes–Haagerup–Størmer [12], and Haagerup–Størmer [31], the explicit form of d(M) is given as follows. Theorem 4.9 ([12, 13, 31]). Let M be a factor. Then d(M) is (1) 2(1 − n1 ) if M is of type In (n ∈ ℕ ∪ {∞}); (2) 2 if M is of type II; 1 2

(3) 2 1−λ 1 if M is of type IIIλ (0 ≤ λ ≤ 1). 1+λ 2

Note that no separability assumption on the predual is assumed (and this is crucial because we want to compute d(∏ω M)). The removal of the separability hypothesis is done in [31]. In particular, if M is a σ-finite type III1 factor, then any two fn states φ, ψ on M are approximately unitarily equivalent. This is called the Connes–Størmer transi-

322 � H. Ando tivity for type III1 factors. In this case, the conjugation action of 𝒰 (M) on Sfn (M) has the property that all orbits are dense. Remark 4.10. Nothing is known about the Borel complexity of this action. It is of interest to estimate its complexity. The state space diameter behaves nicely with respect to taking the Groh–Raynaud ultraproduct. Theorem 4.11 ([1]). Let (M, φ) be a sequence of σ-finite von Neumann algebras and fn states as before. Then d(∏ω M) = limn→ω d(Mn ) holds. In particular, if all Mn are type IIIλ factors for a fixed 0 < λ ≤ 1, then both ∏ω M and (M, φ)ω are factors of type IIIλ . Furthermore, if λ = 1, then (M, φ)ω has strictly homogeneous state space in the sense that any two faithful normal states on it are unitarily equivalent. Remarks 4.12. (1) If 0 < λ ≤ 1, the isomorphism class of (M, φ)ω is independent of the choice of φ. Indeed, ∏ω M is also a type IIIλ factor, and in a type III factor, any two sigma-finite projections are equivalent. Therefore, because each Ocneanu ultraproduct appears as a corner of ∏ω M corresponding to a sigma-finite projection, they are all isomorphic. (2) No von Neumann algebra with separable predual which is not isomorphic to ℂ can have strictly homogeneous state space (see [1, Proposition 4.22]). (3) If Mn is a type IIIλn factor where λn ∈ (0, 1], then ∏ω M is a type IIIλω factor provided λω := limn→ω λn ≠ 0. If we take a sequence (λn )∞ n=1 which is dense in (μ, 1] for a fixed 0 < μ ≤ 1, then the map τ: ℕ ∋ n 󳨃→ λn ∈ [μ, 1] can be extended to a continuous map βℕ ∋ ω 󳨃→ λω ∈ [μ, 1]. Since βℕ is compact and τ has dense image, the extended map is onto. Thus, the type of ∏ω M depends on the choice of ω in a crucial way: it can be of type IIIλ for any λ ∈ [μ, 1]. (4) The preceding theorem has model-theoretic content, namely it shows that if S ⊆ (0, 1] is closed, then the set of type IIIλ factors for λ ∈ S forms an axiomatizable class; see [22, Proposition 6.5.7]. As mentioned above, if M is a σ-finite type III1 factor, then any two faithful normal states on M ω are unitarily equivalent. Recently, Goldbring and Houdayer [23] used this property to show that if M is a hyperfinite factor, then any two embeddings of M into Rω∞ in the range of fn conditional expectations are unitarily equivalent. Note that in the type II1 setting, this unique embedding (up to unitary equivalence) of a given factor M into Rω actually characterizes the hyperfinite factors, a result due to Jung [40]. (See also Section 5 of Goldbring and Hart’s article in this volume.) It is natural to ask if the same is true for the embedding into Rω∞ (with fn expectation). Question 4.13. Is the converse to the Goldbring–Houdayer result true? Namely, does the unique embedding property of a factor M into Rω∞ with an fn expectation imply that M is hyperfinite?

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Also the strict homogeneity leads to the following result. We will come back to this result later. Theorem 4.14 ([1]). Let M be a σ-finite type III1 factor with strictly homogeneous state space and let φ ∈ Sfn (M). Then Mφ is a type II1 factor. In particular, if M is a σ-finite type III1 factor and φ ∈ M∗ is an fn state, then Mφωω is a type II1 factor. We have seen that even if all Mn are finite, (M, φ)ω can be type III. A related question is: Can we find type III factors M and a wild φ such that (M, φ)ω is not type III? We know from Theorem 4.9 that if there is a fixed λ ∈ (0, 1] such that Mn has type IIIλn for some λn ∈ [λ, 1], then this is impossible. We can show that this is possible if we start from type III0 factors. Theorem 4.15 ([1]). Let M be a σ-finite type III0 factor. Then there exists a sequence (φn )∞ n=1 of fn states on M such that (M, φn )ω is isomorphic to the finite von Neumann algebra (Mφn , τn )ω where τn := φn |Mφ . In particular, M does not embed into (M, φn )ω . n

A key ingredient in the proof of the previous theorem is Connes’s results [6, Lemmas 2.3.4 and 5.2.3] on type III0 factors, from which one can show that on a σ-finite type III0 factor M, there exist fn states with arbitrarily large spectral gap at 1 in the spectrum of their modular operators. More precisely, for each n ∈ ℕ, there exists an fn state φn on M such that σ(Δφn ) ∩ (e−n , en ) = {1} holds. Then for φ = (φn )∞ n=1 the ultraproduct state ω ω ω ω φ on (M, φn ) can be shown to satisfy (M, φn ) = (Mφn , τn ) . The previous theorem shows that the type III0 case is therefore quite different from the other type IIIλ cases. This is also reflected in the following result. Recall that, by Theorem 4.11, if M is a type IIIλ (0 < λ ≤ 1) factor, then so is M ω . Theorem 4.16 ([1]). Let M be a σ-finite type III0 factor. Then M ω is not a factor. But why does not the following “easy proof” work? Let M be a σ-finite type III factor and e, f ≠ 0, 1 be two nontrivial projections in M ω . One can show that there exist nontrivial projections en , fn ∈ M (n ∈ ℕ) such that e = (en )ω and f = (fn )ω . Then because M is a σ-finite type III factor, for each n ∈ ℕ there exists a unitary un ∈ 𝒰 (M) such that un en un∗ = fn . Then u = (un )ω ∈ 𝒰 (M ω ) satisfies ueu∗ = f , so any two nontrivial projections in M ω are (unitarily) equivalent, which implies that M is a (type III) factor. ω The argument does not work because we cannot guarantee that (un )∞ n=1 is in ℳ (M). Remark 4.17. This remark is for readers who are familiar with weight theory. The outline of the proof of Theorem 4.16 is as follows. By the structure theorem for type III0 factors [6, Theorem 5.3.1], there is a normal faithful lacunary weight (which is some unbounded positive functional) φ on M such that Mφ is of type II∞ with diffuse center. Let τ := φ|Mφ . There is 0 < λ0 < 1 and U ∈ M(σ φ , (−∞, log λ0 ]) such that θ = Ad(U)|Mφ ∈ Aut(Mφ ) is a centrally ergodic automorphism (that is, it acts ergodically when it is restricted to 𝒵 (Mφ )) satisfying τ ∘ θ ≤ λ0 τ. In this setting, we have M ≅ Mφ ⋊θ ℤ and φ = τ̂ (dual weight of τ) under this isomorphism. We call this a discrete decomposition of M.

324 � H. Ando Similar decompositions are possible for type IIIλ (0 < λ < 1) factors, in which case we have τ ∘ θ = λτ and U ∈ M(σ φ , {log λ}). Then the condition τ ∘ θ ≤ λτ can be used to show that M ω is isomorphic to (Mφ )ω ⋊θω ℤ. Then by an argument using Rokhlin’s theorem, it can be shown that θω is not centrally ergodic, whence the crossed product is not a factor (see [1, Theorem 6.18, Lemma 6.19]). Theorem 4.16 also has model-theoretic consequences, namely it shows that the class of type III0 factors is not axiomatizable, and in fact is not even closed under elementary equivalence; see [22, Corollary 6.5.5].

4.3 Central sequence algebras of Connes, Ocneanu, and Golodets Now we can also clarify how various notions of central sequence algebras are related to each other. Let M be a σ-finite factor, which we regard as a von Neumann subalgebra of M ω by the diagonal embedding. The Ocneanu central sequence algebra is the relative commutant M ′ ∩ M ω . Let us introduce another central sequence algebra Mω due to Connes which sits as a subalgebra of M ′ ∩ M ω . The set ℳω of all bounded sequences x ∈ ℓ∞ (M) satisfying lim ‖xn φ − φxn ‖ = 0,

n→ω

φ ∈ M∗

is a unital C∗ -subalgebra of ℓ∞ (M) (in fact, a subalgebra of the normalizer ℳω ) in which ℐω (as mentioned before, both ℳω (M, φ) and ℐω (M, φ) are independent of the choice of a fixed fn state φ) is a closed two-sided ideal. The Connes’ asymptotic centralizer is the quotient algebra Mω = ℳω /ℐω . Elements of ℳω are called ω-centralizing sequences. Let x = (xn )∞ n=1 ∈ ℳω . Because the unit ball of M is σ-weakly compact, we can take the ω-limit xω = limn→ω xn ∈ M in the σ-weak topology. Because x is an ω-centralizing sequence, we have xω φ = φxω for every φ ∈ M∗ . In particular, xω commutes with all yφ for y ∈ M. Since M∗ separates points of M, we can show from this that xω y = yxω for every y ∈ M, so xω is in the center of M. Therefore we have xω ∈ 𝒵 (M) = ℂ and limn→ω φ(xn ) = φ(xω ) = φ(1)xω . If y = (yn )∞ n=1 ∈ ℳω , then 󵄨󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨󵄨φ(xn yn ) − φ(yn xn )󵄨󵄨󵄨 = 󵄨󵄨󵄨[yn , φ](xn )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩[yn , φ]󵄩󵄩󵄩 ‖xn ‖. Thus |(xy)ω − (yx)ω | = limn→ω |φ(xn yn − yn xn )| = 0. Also, xω = 0 if x ∈ ℐω . Therefore, the formula Mω ∋󳨃→ x + ℐω 󳨃→ xω ∈ ℂ defines an (fn) tracial state on Mω , denoted by τω . In particular, Mω is always a finite von Neumann algebra. Connes extended McDuff’s work [53] on central sequence algebras of a type II1 factor to obtain the following result.

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Theorem 4.18 (Connes [7]). Let M be a factor with separable predual. Then the following conditions are equivalent: (i) M is strongly stable, i. e., M ≅ M⊗R holds. (ii) The quotient group Inn(M)/Inn(M) is noncommutative. Here, Inn(M) is the group of inner automorphisms of M and the closure is taken inside Aut(M). (iii) For some (or equivalently, any) free ultrafilter ω ∈ βℕ \ ℕ, Mω is of type II1 . A factor M satisfying the condition M ≅ M⊗R is often called a McDuff factor. Analogously to the McDuff property, for λ ∈ (0, 1), M is often called λ-McDuff if M ≅ M⊗Rλ holds. Also Rλ is McDuff for any 0 < λ < 1, whence a λ-McDuff factor is necessarily McDuff, but the converse need not hold (e. g., R is McDuff but it is a type II1 factor, so it cannot absorb the type III factor Rλ ). The λ-McDuff condition was studied by Araki α [4] as the property L′α ( 1+α = λ) as a variant of Pukanszky’s property Lα . Connes [11] used Araki’s characterization to describe the λ-McDuff condition in the standard Hilbert space, which he then used for his automorphism approach to the classification of injective factors of type III1 (we will come back to his work later). Theorem 4.19 (Connes [11]). Let M be a factor with separable predual, and let 0 < λ < 1. The following conditions are equivalent: (i) M ≅ M⊗Rλ . (ii) For any ε > 0, n ∈ ℕ and φ1 , . . . , φn ∈ M∗+ , there exists a nonzero element x ∈ M such that n

1 󵄩2 󵄩󵄩 2 󵄩󵄩ξφj x − λ 2 xξφj 󵄩󵄩󵄩 ≤ ∑ ‖xξφj ‖ .

j=1

(3)

Let us now discuss the difference between M ′ ∩ M ω and Mω . It is straightforward to show that Mω is a von Neumann subalgebra of M ′ ∩ M ω and that Mω and M ′ ∩ M ω do coincide if M is a type II1 factor. In order to understand the inclusion Mω ⊂ M ′ ∩ M ω better, it is useful to translate Golodets’ work on the asymptotic algebra [24, Section 3.5] to the Ocneanu setting. Before the Ocneanu ultraproduct was defined, Golodets [24] defined the ultrapower MGω of a given factor M with separable predual along ω ∈ βℕ \ ℕ and M sits naturally as a subalgebra of MGω , so the central sequence algebra M ′ ∩MGω (called the asymptotic algebra of M) is defined. The definition of MGω , which we omit here, is somewhat different from Ocneanu’s (Golodets does not use the normalizer algebra ℳω , in particular). But it can be shown that there exists a isomorphism from MGω onto M ω sending M ′ ∩ MGω onto M ′ ∩ Mω (see [1, Section 3.5]). Under this setting, below is our translation of some of Golodets’ results on the asymptotic algebra (the condition (M ′ ∩ Mω )φ̇ = Mω in the asymptotic algebra setting was proved by Golodets and Nessonov [25]). Theorem 4.20 (Golodets–Nessonov theorem for Mω ⊂ M ′ ∩ M ω ). Let M be a factor with separable predual and φ be an fn state on M. Let ω ∈ βℕ \ ℕ. Then the state on M ′ ∩ M ω

326 � H. Ando defined by 󵄨 φ̇ := φω 󵄨󵄨󵄨M ′ ∩M ω is independent of the choice of fn state φ. Moreover, the centralizer of φ̇ is (M ′ ∩M ω )φ̇ = Mω . Thus Mω = M ′ ∩ M ω if and only if φ̇ is tracial. To repeat, the state φ̇ depends only on M, and we call it the Golodets state. Theorem 4.21 (Golodets). Let M, φ be as in Theorem 4.20. For 0 < λ < 1, the following two conditions are equivalent: (i) M is λ-McDuff, i. e., M ≅ M⊗Rλ . (ii) λ is an eigenvalue of Δφ̇ . Remark 4.22. The condition (ii) can be shown to be equivalent to (ii’): λ is in the spectrum of Δφ̇ . Example 4.23. Let (Rλ , φ) = ⨂n∈ℕ (M2 (ℂ), Tr(ρλ ⋅)) be the Powers factor of type IIIλ (0 < λ 1 , 1+λ ). Let λ < 1), where ρλ = diag( 1+λ 0 un := 1⊗n ⊗ ( 0

1 ) ⊗ 1 ⊗ ⋅ ⋅ ⋅ ∈ Rλ , 0

n ≥ 1.

ω ω ′ ω Then u = (un )∞ n=1 ∈ ℳ (Rλ ) and (un ) ∈ Rλ ∩ Rλ . On the other hand, we have

φun = λun φ,

n ∈ ℕ.

Therefore ‖un φ−φun ‖ = (1−λ) ≠ 0 (n ∈ ℕ) and hence (un )ω ∉ (Rλ )ω . Moreover, R′λ ∩Rωλ is a type IIIλ factor. While we omit the proof of its factoriality, we will show that it is type IIIλ . To see this, first note that, by [65, Proposition 1], (Rλ )ω is a type II1 factor. Therefore the centralizer of the Golodets state φ̇ ω = φω |R′ ∩(Rλ )ω is a factor. Then by [6, Corollary 3.2.7], λ we have Γ(σ φω ) = Sp(σ φω ) = log(σ(Δφ̇ ω ) \ {0}) ̇

̇

⊂ log(σ(Δφω ) \ {0}) = log(σ(Δφ ) \ {0}) = (log λ)ℤ.

(For details about Arveson and Connes’ spectral theory needed for the above arguments, the reader can consult either Connes’ thesis [6] or Takesaki’s vol. 2 [67].) On the other hand, we have Δφ̇ ω uk ξφ̇ ω = λk ξφ̇ ω (k ∈ ℤ), so (log λ)ℤ ⊂ Sp(σ φ̇ ω ). Therefore as Γ(σ φ̇ ω ) = log(S(R′λ ∩ Rωλ ) \ {0}), we have S(R′λ ∩ Rωλ ) = {λn ; n ∈ ℤ} ∪ {0}. This proves that R′λ ∩ Rωλ is a type IIIλ factor.

Introduction to nontracial ultraproducts of von Neumann algebras

� 327

Thus, Mω ⊊ M ′ ∩ M ω in general. However, Ueda [69, Section 5.2] asked if Mω = ℂ (following Connes [7], a factor with this property is said to be full) implies M ′ ∩ M ω = ℂ if M is a factor with separable predual. This is indeed the case. Theorem 4.24 ([1]). Let M be a factor with separable predual. If Mω = ℂ, then M ′ ∩ M ω = ℂ holds. This can be proved by the use of the Golodets state [1, Theorem 5.3]. We also point out that Mω can coincide with M ′ ∩ M ω without both being trivial. For instance, one has: Theorem 4.25 ([1]). If M is a σ-finite type III0 factor, then Mω = M ′ ∩ M ω ≠ ℂ holds. There is a characterization of fullness as a kind of spectral gap condition. For the type II1 case, it was established by Connes [9, Theorem 2.1] (the result presented here and the original statement are different, but the present form can be easily deduced from the original one). Theorem 4.26 (Connes [9]). Let M be a type II1 factor with τ the unique tracial state. Then M is full if and only if there exist a1 , . . . , an ∈ M and a constant C > 0 such that the following inequality holds: n

󵄩󵄩 󵄩 󵄩󵄩x − τ(x)1󵄩󵄩󵄩2 ≤ C ∑ ‖xak − ak x‖2 , k=1

x ∈ M.

Marrakchi [49] succeeded in extending Connes’ characterization of full factors to the type III setting. Theorem 4.27 (Marrakchi [49]). Let M be a σ-finite type III factor. Then M is full if and only if there exists an fn state φ on M and ξ1 , . . . , ξn ∈ L2 (M)+ satisfying ⟨ ⋅ ξk , ξk ⟩ ≤ φ (k = 1, . . . , n) and such that the following inequality holds: n

󵄩󵄩 󵄩2 2 󵄩󵄩x − φ(x)󵄩󵄩󵄩φ ≤ ∑ ‖xξk − ξk x‖ , k=1

x ∈ M.

5 Connes’ embedding problem, QWEP and the Effros–Maréchal topology There is an interesting relationship among nontracial ultraproducts, Connes’ embedding problem, Kirchberg’s QWEP problem, and a certain topology on the space of von Neumann algebras. The QWEP conjecture itself has been refuted by the recent breakthrough work by Ji–Natarajan–Vdick–Wright–Yuen [39] (see the article by Isaac Goldbring in this volume). But the author still believes that the content of this section is of use for a better understanding of the problem from the operator-algebraic viewpoint.

328 � H. Ando So the author decided to give a brief explanation of how a topological property of von Neumann algebras, the modular theory, and the QWEP property are related to each other via ultraproducts. As is already explained in this volume, Kirchberg [43] revealed an unexpected connection among tensor products of C∗ -algebras, the weak expectation property (WEP) of Lance [46], and the Connes embedding conjecture for type II1 factors. A C∗ -algebra A is said to have WEP if for any faithful representation A ⊂ 𝔹(H), there is a unital completely positive (ucp) map Φ: 𝔹(H) → A∗∗ such that Φ|A = idA . A C∗ -algebra A is said to have the quotient weak expectation property (QWEP) if there is a surjective ∗-homomorphism from a C∗ -algebra B with WEP onto A. Among other interesting results, Kirchberg proved that the following conditions are equivalent: (1) C ∗ (𝔽∞ ) ⊗min C ∗ (𝔽∞ ) = C ∗ (𝔽∞ ) ⊗max C ∗ (𝔽∞ ). (2) Every C∗ -algebra has QWEP. (3) Every II1 factor N with separable predual embeds into the tracial ultrapower Rω (for some fixed free ultrafilter ω on ℕ) of the hyperfinite type II1 factor R. Condition (2) is called Kirchberg’s QWEP conjecture, and (3) is called the Connes’ embedding conjecture, which was mentioned for the first time in [9]. See also the excellent survey [55] on the QWEP conjecture. Since then many interesting equivalent conditions of the above conjectures were found. On the other hand, the QWEP conjecture also has another connection to the topological properties of the space of von Neumann algebras. The Effros–Maréchal topology on the space vN(H) of von Neumann algebras acting on a fixed Hilbert space H is the weakest topology on vN(H) for which the map 󵄩 󵄩 N 󳨃→ 󵄩󵄩󵄩φ|N 󵄩󵄩󵄩 is continuous for every φ ∈ 𝔹(H)∗ . The topology is Polish if H is separable. Based on the work of Effros [17, 16] on the Borel structure of vN(H), this topology was defined by Maréchal in [48] so that it generates the Effros–Borel structure. Later it was intensively studied by Haagerup and Winsløw in [32, 33], where it was realized that this topology was indeed well matched with Tomita–Takesaki theory and could be used as a tool for the study of global properties of von Neumann algebras. Among other things, it was proven [33, Theorem 5.8] that when H is separable, a II1 factor N ∈ vN(H) is in the closure of the set ℱinj of injective factors on H if and only if N embeds into Rω . It was also shown [33, Theorem 3.5] that the set ℱII1 of II1 factors on H is dense in vN(H). Consequently, Connes’ embedding conjecture (hence all conditions (1)–(3) above) is equivalent to (4) ℱinj is dense in vN(H). To summarize the work of [32, 33], they answered the following questions (except those marked in bold). Here, ℱ stands for the set of factors on H and ℱX is the set of factors on H of type X; ℱ st is the set of factors which act standardly on H.

Introduction to nontracial ultraproducts of von Neumann algebras

Subset of vN(H)

Dense in vN(H)?

Gδ ?

ℱ ⋃n≤n� ℱIn , n� ∈ ℕ ℱIfin ℱI∞ ℱII� ℱII∞ ℱIII� ℱIIIλ , λ ∈ (�, �) ℱIII� ℱinj ℱ st

Yes No No [39] No [39] Yes Yes Yes Yes Yes No [39] Yes

Yes Yes (closed) No (but Fσ ) No (but Fσ ) No No No No Yes Yes Yes

� 329

Since the QWEP property makes sense for arbitrary C∗ -algebras (and hence for arbitrary von Neumann algebras), it is natural to expect that this property can also be characterized by the Effros–Maréchal topology. This is exactly what was described in [3]. What would be the right formulation of the embedding problem for von Neumann algebras? Obviously, every M with separable predual embeds into 𝔹(H), so whenever we talk about embedding of a (not necessarily finite) von Neumann algebra N into another von Neumann algebra M, it is important to assume that there exists an fn conditional expectation E of M onto the image of N under the embedding. Keeping this in mind, the equivalent embedding problem is formulated as follows: Theorem 5.1 ([3]). Let 0 < λ < 1, let M ∈ vN(H), and let L2 (M)+ be the natural cone in the standard form of M. The following conditions are equivalent: (1) M has QWEP. (2) M ∈ ℱinj . (3) There is an embedding i: M → Rω∞ and a normal faithful conditional expectation ε: Rω∞ → i(M). (4) There is an embedding i: M → Rωλ and a normal faithful conditional expectation ε: Rωλ → i(M). (5) There is {kn }∞ n=1 ⊂ ℕ, a normal faithful state φn on Mkn (ℂ) (n ∈ ℕ), an embedding i: M → (Mkn (ℂ), φn )ω , and a normal faithful conditional expectation ε: (Mkn (ℂ), φn )ω → i(M). (6) For every ε > 0, n ∈ ℕ, and ξ1 , . . . , ξn ∈ L2 (M)+ , there exist k ∈ ℕ and a1 , . . . , an ∈ Mk (ℂ)+ such that 󵄨󵄨 󵄨 󵄨󵄨⟨ξi , ξj ⟩ − trk (ai aj )󵄨󵄨󵄨 < ε

(1 ≤ i, j ≤ n).

Now we know that there exists a non-QWEP von Neumann algebra thanks to [39]. It follows that generic algebras are non-QWEP in vN(H), even though we do not have an explicit example of, e. g., a non-QWEP type III factor.

330 � H. Ando Corollary 5.2 (Corollary to [39]). The set vN(H)¬QWEP of all non-QWEP von Neumann algebras on H is an open dense subset of vN(H). Proof. By Theorem 5.1, the set of all QWEP von Neumann algebras is closed in vN(H). Therefore vN(H)¬QWEP is an open set. We now show density. Fix N ∈ vN(H). By [39], there exists M ∈ vN(H) without QWEP. Then by choosing v0 ∈ 𝒰 (H ⊗ H, H), [33, Lemma 2.4] ∗ ∗ asserts that there is (un )∞ n=1 ∈ 𝒰 (H ⊗ H) such that v0 un (N⊗M)un v0 → N in vN(H). Since it is known that the tensor product of von Neumann algebras is QWEP if and only if all tensor factors are QWEP, v0 un∗ (N⊗M)un v∗0 fails QWEP for each n ∈ ℕ, whence N is in the closure of vN(H)¬QWEP .

6 Bicentralizer flow Here we report on a recent application [2] of nontracial ultarproducts to the study of Connes’ bicentralizer problem. This is a longstanding open problem on type III1 factors. While this problem is well known among those working on type III factors, it is not so outside of the community. This is mainly because even the statement is quite technical, but also it is not so easy to see that a solution to this problem would be of importance in applications. So I would like to add some background information on the problem. A von Neumann algebra M on a Hilbert space H is called hyperfinite if there exists an increasing sequence M1 ⊂ M2 ⊂ ⋅ ⋅ ⋅ ⊂ M of finite-dimensional ∗-subalgebras of M whose union is dense in the strong operator topology (SOT). Already Murray and von Neumann showed that there exists only one hyperfinite factor of type II1 , denoted by R, up to ∗-isomorphism. However, their argument does not apply to prove the uniqueness result for, e. g., type II∞ factors, and it was a long standing open problem to prove that there exist only one hyperfinite II∞ factor, typically denoted R0,1 , which is constructed as an infinite tensor product of matrix algebras. Because hyperfiniteness is hard to check, one needs to find a characterization of hyperfiniteness that does not involve finite-dimensional ∗-subalgebras. Many mathematicians have worked on this problem and several conditions (injectivity, semidiscreteness, Schwarz’ property, …) that imply hyperfiniteness have been introduced. To cut a long story short, Connes [9] showed that all these properties, especially injectivity, are equivalent to hyperfiniteness. Algebra M is said to be injective if there exists a norm one projection E: 𝔹(H) → M. Thanks to Tomiyama’s theorem, E is a conditional expectation although it is typically nonnormal (in fact, E can be chosen to be normal if and only if M is atomic). The equivalence of injectivity and hyperfiniteness immediately leads to the theorem that there exists only one hyperfinite factor of type II1 and of type II∞ respectively up to ∗-isomorphism, and any subfactor of the hyperfinite II1 factor R is again hyperfinite. On the other hand, in the early 1970s, Tomita–Takesaki theory had been invented. This led Connes and Takesaki to their structure theorem for type III factors. By Connes’ structure theorem for type III factors, any type IIIλ (0 ≤ λ < 1) factor is

Introduction to nontracial ultraproducts of von Neumann algebras

� 331

of the form M = N ⋊θ ℤ, where N is of type II∞ (and is a factor if λ ≠ 0) and θ a centrally ergodic action on N which scales down a semifinite trace τ of N. For the type III1 case, thanks to Takesaki’s duality theorem, there is a continuous decomposition M = N ⋊θ ℝ where N is a type II∞ factor and θ is a flow on N scaling the semifinite trace τ. Then the classification of hyperfinite type III factors is reduced to the classification of hyperfinite type II∞ factors and of the actions of ℤ (or ℝ for the III1 case) on them. Then by the classification of automorphisms of R and of R0,1 , Connes [6, 8, 9, 10, 14, 64] showed that for each 0 < λ < 1, there exists only one hyperfinite IIIλ factor, namely Rλ , and together with the work of Krieger [45] on ergodic flows, he showed that the isomorphism classes of hyperfinite III0 factors are in 1–1 correspondence with the isomorphism classes of properly ergodic flows. There only remained the type III1 case. In order to settle the III1 case, he found several strategies to prove the uniqueness. Among them, he discovered the following [11]: let M be an injective III1 factor with separable predual and fix an fn 2π . Then N = M ⋊σ φ ℤ is an injective type state φ ∈ M∗ . Fix λ ∈ (0, 1) and let T = − log λ T

φ

IIIλ factor, hence N ≅ Rλ and if we let θ be the dual action of σT , then M ≅ Rλ ⋊θ 𝕋. Then if one shows the uniqueness of the 𝕋 action on Rλ , the uniqueness result for the hyperfinite III1 factor follows. He then showed that this can be achieved if one can show φ that σT ∈ Inn(M) (the approximately inner automorphisms). For an automorphism α of a factor N with separable predual, consider the following conditions: (i) α ∈ Inn(N); (ii) α ⊙ id ∈ Aut(N ⊙ N op ) extends to an automorphism of the C∗ -algebra C∗λ⋅ρ (N) generated by the standard representation of N ⊙ N op on L2 (N) given by (a ⊗ bop ) ⋅ ξ := aJb∗ Jξ,

a, b ∈ N,

ξ ∈ L2 (N).

Here, we fix a standard form (N, L2 (N), J, L2 (N)+ ) for N. Then always (i) ⇒ (ii), and when N = M is an injective type III1 factor, (ii) is satisfied for every α. Then Connes showed [11] that indeed (ii) ⇒ (i) follows if in addition the bicentralizer B(M, φ) is trivial (= ℂ) for some (equivalently, any) fn state φ on M. Here, the bicentralizer of M with respect to φ is defined by n→∞

B(M, φ) = {x ∈ M | xan − an x → 0 (SOT), ∀(an )n ∈ AC(M, φ)} where AC(M, φ) = {(an )n ∈ ℓ∞ (M) | lim ‖an φ − φan ‖ = 0} n→∞

is the asymptotic centralizer of φ. It was observed by Connes (see [29, Corollary 1.5] for a proof) that, thanks to the Connes–Størmer transitivity, the triviality of the bicentralizer for one fn state φ implies the triviality of the bicentralizer for an arbitrary fn state. The question of the triviality of the bicentralizer was solved affirmatively by Haagerup in [29] for hyperfinite M, thus settling the problem of the classification of hyperfinite factors

332 � H. Ando with separable predual (see [9, 11]). Connes also asked whether or not the bicentralizer is trivial for general type III1 factors with separable predual. It is not obvious to see if the bicentralizer problem is still of interest, but there were already some indications in the early works. Indeed, by [29, Theorem 3.1], for any type III1 factor M with separable predual, M has trivial bicentralizer if and only if there exists an fn state φ ∈ M∗ with an irreducible centralizer, meaning that (Mφ )′ ∩ M = ℂ1. This result, together with the work of Haagerup [29, Theorem 3.1] and Popa [56, Theorem 3.2], allows one to show that M has trivial bicentralizer if and only if there exists a maximal abelian subalgebra A ⊂ M that is the range of a normal conditional expectation (see [63, P320, Question] where the problem of finding such maximal abelian subalgebras is mentioned). For these reasons, Connes’ bicentralizer problem appears naturally when one tries to use Popa’s deformation/rigidity theory in the type III context (see, for instance, [35, Theorem C]). The bicentralizer problem is known to have a positive solution for particular classes of nonamenable type III1 factors: factors with a Cartan subalgebra; Shlyakhtenko’s free Araki–Woods factors [34]; (semi-)solid factors [35]; free product factors [37]. However, the bicentralizer problem is still wide open for arbitrary type III1 factors. In his attempt to solve the bicentralizer problem, Connes observed that for any type III1 factor M, the bicentralizer B(M, φ) does not depend on the choice of the state φ up to a canonical isomorphism. Around 2012–2013, Haagerup found out that the idea of Connes’ isomorphism (denoted by βψ,φ below) can be enhanced to construct a canonical flow (u-continuous action) βφ : R∗+ ↷ B(M, φ) with interesting properties. This flow was independently discovered by Marrakchi and this was the starting point of the work [2]. We describe the flow βφ in a relativised (subfactor) setting as follows. Let N ⊂ M be any inclusion of σ-finite von Neumann algebras with fn expectation, meaning that there exists a faithful normal conditional expectation EN : M → N. Following [51, Definition 4.1], we define the relative bicentralizer B(N ⊂ M, φ) of the inclusion N ⊂ M with respect to the faithful state φ ∈ N∗ by n→∞

B(N ⊂ M, φ) = {x ∈ M | xan − an x → 0 (SOT), ∀(an )n ∈ AC(N, φ)}. Observe that we always have N ′ ∩ M ⊂ B(N ⊂ M, φ) ⊂ (Nφ )′ ∩ M. When N = M, we simply have B(N ⊂ M, φ) = B(M, φ). Theorem 6.1 (Relative bicentralizer flow [2]). Let N ⊂ M be any inclusion of σ-finite von Neumann algebras with fn expectation. Assume that N is a type III1 factor. Then the following assertions hold: (i) For every pair of fn states φ, ψ ∈ N∗ , there exists a canonical isomorphism βψ,φ : B(N ⊂ M, φ) → B(N ⊂ M, ψ) characterized by the following property: for any uniformly bounded sequence (an )n∈ℕ in N and any x ∈ B(N ⊂ M, φ), we have

Introduction to nontracial ultraproducts of von Neumann algebras

n→∞

‖an φ − ψan ‖ → 0

󳨐⇒

� 333

n→∞

an x − βψ,φ (x)an → 0 (S∗ OT).

(ii) There exists a canonical flow βφ : ℝ∗+ ↷ B(N ⊂ M, φ) characterized by the following property: for any uniformly bounded sequence (an )n∈ℕ in N, any x ∈ B(N ⊂ M, φ) and any λ > 0, we have n→∞

‖an φ − λφan ‖ → 0

φ

n→∞

an x − βλ (x)an → 0 (S∗ OT).

󳨐⇒

ψ

(iii) We have βφ3 ,φ2 ∘ βφ2 ,φ1 = βφ3 ,φ1 for all fn states φi ∈ N∗ , i ∈ {1, 2, 3}, and βλ ∘ βψ,φ = φ βψ,φ ∘ βλ for every pair of faithful states ψ, φ ∈ N∗ and every λ > 0. (iv) For every pair of faithful states ψ, φ ∈ N∗ and every λ > 0, we have ψ

φ

φ

φ

EN ′ ∩M ∘ βψ,φ = EN ′ ∩M = EN ′ ∩M ∘ βλ φ

ψ

where EN ′ ∩M : M → N ′ ∩M (resp. EN ′ ∩M ) is the unique normal conditional expectation φ

ψ

such that EN ′ ∩M (x) = φ(x)1 (resp. EN ′ ∩M (x) = ψ(x)1) for all x ∈ N.

The meaning of the compatibility relations given in item (iii) is that the W∗ -dynamical system (B(N ⊂ M, φ), βφ ) does not depend on the choice of φ ∈ N∗ up to the canonical isomorphism βψ,φ . Thus, (B(N ⊂ M, φ), βφ ) is an invariant of the inclusion N ⊂ M. We call it the relative bicentralizer flow of the inclusion N ⊂ M. When N = M, we simply call it the bicentralizer flow of M. The construction of βφ relies on the Ocneanu ultraproduct explained above. To construct βφ , fix λ > 0. First of all, for any free ultrafilter ω, we can describe the relative bicentralizer as B(N ⊂ M, φ) = (Nφωω )′ ∩ M (so the right hand side does not depend on ω). Then using the fact that Nφωω is a type II1 factor, one can find partial isometries v1 , . . . , vn ∈ N ω such that vk φω = λφω vk and ∑nk=1 vk v∗k = 1 in N ω [2, Lemma 3.4]. Then for x ∈ B(N ⊂ M, φ), it can be shown that the element φ

n

βλ (x) := ∑ vk xv∗k k=1

actually lies in B(N ⊂ M, φ) = (Nφωω )′ ∩M ⊂ M ω and is independent of the choice of ω and partial isometries v1 , . . . , vn . Then one can show that βφ is indeed a continuous flow on B(N ⊂ M, φ) with the required properties. While it is certainly possible to construct βφ without ever mentioning the ultraproduct (this was in fact the original approach), the argument would have been way more involved (almost commuting, almost partial isometry, almost centralized by φ, almost σ φ -eigenvalues, etc., we rely on too many relations that are true only approximately in the original algebra). We can relate some properties of the flow to structural properties of the inclusion N ⊂ M. Here we mention two such results from [2]: (i) Kadison’s problem on the exis-

334 � H. Ando tence of common maximal abelian subalgebras (masa) and (ii) existence of irreducible hyperfinite subfactors. Let N ⊂ M be an inclusion of factors. The inclusion is called irreducible if N ′ ∩M = ℂ holds. Irreducible inclusions appear naturally. For example, if a properly outer action of a group Γ on a factor N is given, then N ⊂ N ⋊ Γ is irreducible. So it is of interest to find a criterion for the inclusion to be irreducible. One well-known criterion is the existence of a common masa. Here, a masa A ⊂ N is called a common masa if A is still maximal abelian as a subalgebra of M, i. e., A′ ∩ M = A holds. If N ⊂ M admits a common masa, then it is irreducible. Kadison [41] asked whether the converse is also true. Problem 6.2 (Kadison). Let N ⊂ M be an irreducible inclusion of factors. Does it admit a common masa? In the celebrated work of Popa [56], it was shown that when N ⊂ M has separable predual, N is semifinite, and there exists an fn conditional expectation from M onto N, then the answer is affirmative. Since the free group factor L(𝔽n ) does not have property Gamma, the inclusion L(𝔽n ) ⊂ L(𝔽n )ω is irreducible but it does not admit a common masa (because Popa [56, Proposition 4.3] showed that no masa of M ω , where M is a II1 factor, can have separable predual). Therefore the separability of the predual is necessary. Also, Ge–Popa [21] constructed an irreducible inclusion of type II factors with separable predual without common masa, but without fn conditional expectation M → N. Thus, the existence of an fn expectation is also necessary and consequently the only case to be settled is when N ⊂ M are type III, have separable predual, and with fn expectation. While this case is still open, one can answer the remaining case if we moreover assume the masa is with an fn expectation. Definition 6.3. We say that an irreducible inclusion N ⊂ M of factors with an fn expectation M → N satisfies the strong Kadison property if it admits a common masa A and, moreover, if there is an fn expectation N → A. The answer to the case of type IIIλ (0 < λ ≤ 1) can be given using the relative bicentralizer flow as follows: Theorem 6.4 ([2]). Let N ⊂ M be separable irreducible factors with normal expectation. (1) If N is type III1 , then (i) ⇔ (ii) ⇔ (iii) ⇒ (iv). (i) N ⊂ M has strong Kadison’s property; (ii) B(N ⊂ M, φ) = ℂ for some (any) φ ∈ Sfn (N); (iii) There exists φ ∈ Snf (N) such that (Nφ )′ ∩ M = ℂ; (iv) There exists a dominant weight (see [14, Chapter II.1] for the definition and detailed discussion) ψ such that ∀x ∈ M w

co {uxu∗ | u ∈ 𝒰 (Nψ )} ∩ ℂ ≠ 0. If the inclusion N ⊂ M is discrete in the sense of Izumi–Longo–Popa [38], then (iv) ⇒ (i).

Introduction to nontracial ultraproducts of von Neumann algebras

� 335

2π periodic state (i. e., (2) If N is type IIIλ (0 < λ < 1) factor and φ ∈ Snf (N) a T = − log λ φ

a state with σT = id), then the following conditions are equivalent: (i) N ⊂ M has strong Kadison’s property; (ii) (Nφ )′ ∩ M = ℂ.

The equivalence (ii) ⇐⇒ (iii) ⇐⇒ (iv) in the case M = N is simply Haagerup’s bicentralizer theorem [29]. The generalization of this equivalence to the relative setting in the finite index case is established by Popa [59], and we generalized their result to the current form. For the type III0 case, we do not have a clear picture. Problem 6.5. What happens if N is type III0 ? In the above mentioned work of Ge–Popa, it is crucial that any type II1 factor admits an irreducible hyperfinite II1 subfactor. This is established by Popa in the same Kadison’s problem paper [56]. One may ask if every non-type I factor admits an irreducible hyperfinite subfactor (of type II1 ). Popa answered the question in the affirmative for all factors with separable predual except the type III1 case. Theorem 6.6 (Popa [56, 58]). (i) Let N ⊂ M be an irreducible inclusion of factors with separable predual with fn expectation such that N is semifinite. Then there exists a hyperfinite subfactor P ⊂ N with an fn expectation N → P such that P′ ∩ M = ℂ. In particular, any semifinite factor with separable predual admits an irreducible hyperfinite subfactor with an fn expectation. (ii) Let M be a factor with separable predual which is not of type III1 . Then M admits an irreducible hyperfinite subfactor P with an fn expectation M → P. Given a type III1 factor M, can we find an irreducible hyperfinite subfactor P? If we do not require P to be with an fn expectation M → P, then the answer is affirmative: Theorem 6.7 (Popa [57], Longo [47] (infinite factor case)). Let M be a factor with separable predual. Then M admits an irreducible hyperfinite subfactor. We also remark Haagerup’s result [30, Lemma 4.4] that for any type III1 factor M with separable predual, one can always find a hyperfinite III1 subfactor (a copy of R∞ ) with an fn expectation. This subfactor need not be irreducible. However, if we want an fn expectation, then the type III1 case is very different. But why is the III1 case so different? Before stating the result, let us observe that the bicentralizer flow is relevant to the problem if M is a type III1 factor. Namely, if P ⊂ M is an irreducible hyperfinite subfactor with an fn expectation E: M → P, then we may choose any fn almost periodic state φ ∈ P∗ (recall that φ is called almost periodic in the sense of Connes [7] if the modular operator Δφ is diagonalizable; such a state is known to exist in any hyperfinite factor. In particular, if P is ITPFI, the tensor product state used to define P is almost periodic) and extend it via E to an fn state ψ := φ ∘ E ∈ M∗ . Let a ∈ P ⊂ Pω ⊂ M ω be such that aφ = λφa (i. e., a is a λ−1 -eigenvector for φ). Because

336 � H. Ando ψ

σ ψ |P = σ φ , aψ = λψa holds. Then for each x ∈ B(M, ψ), ax = βλ (x)a holds. Now assume ψ

that x ∈ B(M, ψ)β (the fixed point subalgebra of βψ ). Then ax = xa for every a ∈ P which is an φ-eigenvector. By the almost periodicity of φ, φ-eigenvectors generate P, and because P ⊂ M is irreducible, we obtain x ∈ P′ ∩ M = ℂ. Thus, the bicentralizer flow βψ must be ergodic. It turns out that the ergodicity is exactly what we need. Theorem 6.8 ([2]). Let N ⊂ M be an inclusion of factors with separable predual and with an fn expectation and assume that N is type III1 . Then the following conditions are equivalent. (i) There exists a hyperfinite subfactor P ⊂ N with normal expectation such that P′ ∩M = N ′ ∩ M. φ (ii) B(N ⊂ M, φ)β = N ′ ∩ M for some (any) φ ∈ Snf (N). Moreover, in (i) we can always take P = R∞ . φ P can be taken to be Rλ (0 < λ < 1) if and only if B(N ⊂ M, φ)βλ = N ′ ∩ M. P can be R if and only if B(N ⊂ M, φ) = N ′ ∩ M. In a recent breakthrough work, Marrakchi was able to show that the bicentralizer flow is indeed always ergodic. More precisely: Theorem 6.9 (Marrakchi [50]). Let M be a σ-finite type III1 factor and φ an fn state on M. φ Then for each λ > 0, βλ is ergodic on B(M, φ). In particular, if M is separable, then for each λ ∈ (0, 1], there exists an irreducible hyperfinite subfactor P ⊂ M of type IIIλ which is the range of an fn expectation M → P. We recommend the reader to consult [50] for further important results. Thus we can always find an irreducible copy of R∞ with an fn expectation, and we can find an irreducible copy of R with an fn expectation exactly when the bicentralizer B(M, φ) is trivial. This is yet another evidence that the bicentralizer problem is still of great importance. Finally, toward a better understanding of the bicentralizer, assume hypothetically that there does exist a counterexample to the problem, i. e., a type III1 factor M with separable predual and φ ∈ Sfn (M) such that B(M, φ) ≠ ℂ. Set N = B(M, φ) = (Mφωω )′ ∩ M and ψ = φ|N . Also N is a σ φ -invariant subalgebra of M, so Nψ = N ∩ Mφ holds. Let x ∈ Nψ . Then x ∈ (Mφωω ) ∩ M ∩ Mφ ′

⊂ (Mφωω ) ∩ Mφωω = 𝒵 (Mφωω ) = ℂ, ′

where in the last equality we used the fact (Theorem 4.14) that Mφωω is a (type II1 ) factor. Thus, the centralizer Nψ must be trivial. It is known that a σ-finite factor has an fn state with trivial centralizer if and only if either it is trivial or a type III1 factor. Also it is easy to see that B(N, ψ) = N. Thus N is a type III1 factor which is equal to its bicentralizer B(N, ψ).

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Definition 6.10. Let M be a σ-finite factor and φ ∈ Sfn (M). We say that φ is a bicentralizing state if B(M, φ) = M holds. Assume that there exists a type III1 factor M with bicentralizing state φ. Can we say something about M? First, M must have many central sequences. Theorem 6.11 (Houdayer–Isono [36]). A σ-finite type III1 factor with bicentralizing state φ satisfies Mω = Mφωω . Thus Mω is noncommutative, and M ≅ M⊗R holds if M has separable predual (M is a McDuff factor). Indeed, M = B(M, φ) = (Mφωω )′ ∩ M would imply that Mω = Mφωω holds. The inclusion Mω ⊂ Mφωω is true in general, and if x ∈ Mφωω , then by M = (Mφωω )′ ∩ M, we have x ∈ (Mφωω ) ∩ M ′ : because {aφ | a ∈ M} is norm-dense in M∗ , it follows that xχ ω = χ ω x for any χ ∈ M∗ . Therefore, x ∈ Mω holds. Thus Mω is a type II1 factor (thus non-commutative), which implies that M is McDuff. Can M absorb Rλ or R∞ tensorially as well? This is related to the triviality of the bicentralizer flow. Theorem 6.12 ([2]). Let M be any type III1 factor with separable predual and with a bicentralizing state φ ∈ M∗ . Then the following properties hold: (i) Let Δ(M) be the set of all bicentralizing states of M. Then the map Inn(M) ∋ α 󳨃→ α(φ) ∈ Δ(M) is a homeomorphism and its inverse is given by Δ(M) ∋ ψ 󳨃→ βψ,φ ∈ Inn(M). (ii) Define Autφ (M) = {α ∈ Aut(M) | α(φ) = φ} and consider the conjugation action Autφ (M) ↷ Inn(M). Then the natural homomorphism ι : Inn(M) ⋊ Autφ (M) ∋ (g, h) 󳨃→ g ∘ h ∈ Aut(M) φ

is an isomorphism of topological groups. In particular, σt ∉ Inn(M) for all t ≠ 0. (iii) For every 0 < λ < 1, we have M ≅ M⊗Rλ

⇐⇒

φ

βλ = id

⇐⇒

φ

βλ ∈ Inn(M).

In particular, we have M ≅ M⊗R∞ if and only if the bicentralizer flow βφ is trivial. φ (iv) For every λ > 0, the automorphism βλ ⊙ id of M ⊙ M op extends to the C∗ -algebra ∗ Cλ⋅ρ (M) generated by the standard representation of M ⊙ M op on L2 (M). There is also a study of the bicentralizer of tensor product factors and it has an application to unique prime factorization results. We also made a conjecture that the dynamical system βφ : ℝ∗+ ↷ B(N ⊂ M, φ) is isomorphic to the non-commutative flow of

338 � H. Ando weights studied by Connes and Takesaki [14], but since we avoided the theory of weights in these notes, we will not describe it here. See [2, Section 7] for further information.

7 Further readings In these notes we have only touched upon a tiny fragment of modular theory needed for the definition of nontracial ultraproducts. But even just for the study of nontracial products, it will be necessary to use weight theory, and an understanding of the flow of weights. We refer the reader to Takesaki’s books for further information. The study of operator algebra ultraproducts from the model theoretic viewpoint has been an active area of research. For the model theory of tracial ultraproduct, we refer the reader to the series of works by Farah–Hart–Shelman [18, 19, 20] and also Goldbring and Hart’s survey in this volume. However, because the ultraproduct construction itself has not received much attention and is not that well understood, it is not surprising that the appropriate model theory for nontracial ultraproducts was not available until recently. The first work in this direction was carried out by Dabrowski [15], where he axiomatized using model theory for metric structures the Ocneanu theory via σ-finite W∗ -probability space (a pair (M, φ) of a σ-finite von Neumann algebra and its fn state is called a W ∗ -probability space) and the Groh–Raynaud theory via preduals of von Neumann algebras (recall that the Groh–Raynaud ultraproduct is the dual of the Banach space ultraproduct of the preduals). While Dabrowski theory is successful, it is rather technical; even the description of the language used is complicated. Another approach via correspondences is given by Goldbring–Hart–Sinclair [22], where they use a language that does not incorporate modular theoretic objects directly. Further modeltheoretic properties of type III factors were established by Goldbring and Houdayer in [23]. The readers interested in the model theory corresponding to nontracial ultraproducts are recommended to study their works.

Bibliography [1] [2] [3] [4] [5] [6] [7]

H. Ando and U. Haagerup, Ultraproducts of von Neumann algebras, J. Funct. Anal. 266 (2014), 6842–6913. H. Ando, U. Haagerup, C. Houdayer and A. Marrakchi, Structure of bicentralizer algebras and inclusions of type III factors, Math. Ann. 376 (2020), 1145–1194. H. Ando, U. Haagerup and C. Winsløw, Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology, J. Reine Angew. Math. 715 (2016), 231–250. H. Araki, Asymptotic ratio set and property L′λ , Publ. Res. Inst. Math. Sci. 6 (1970/71), 443–460. H. Araki and E. J. Woods, A classification of factors, Publ. Res. Inst. Math. Sci., Ser. A 4 (1968/1969), 51–130. A. Connes, Une classification des facteurs de type III, Ann. Sci. Éc. Norm. Supér. (4) 6 (1973), 133–252. A. Connes, Almost periodic states and factors of type III1 , J. Funct. Anal. 16 (1974), 415–445.

Introduction to nontracial ultraproducts of von Neumann algebras

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

� 339

A. Connes, On hyperfinite factors of type III0 and Krieger’s factors, J. Funct. Anal. 18 (1975), 318–327. A. Connes, Classification of injective factors. Cases II 1 , II ∞ , III λ , λ ≠ 1, Ann. Math. (2) 104 (1976), 73–115. A. Connes, Periodic automorphisms of the hyperfinite factor of type II1 , Acta Sci. Math. (Szeged) 39 (1977), 39–66. A. Connes, Factors of type III1 , property L′λ and closure of inner automorphisms, J. Oper. Theory 14 (1985), 189–211. A. Connes, U. Haagerup and E. Størmer, Diameters of state spaces of type III factors, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 91–116. A. Connes and E. Størmer, Homogeneity of the state space of factors of type III1 , J. Funct. Anal. 28 (1978), 187–196. A. Connes and M. Takesaki, The flow of weights on factors of type III, Tohoku Math. J. (2) 29 (1977), 473–575. Y. Dabrowski, Continuous model theories for von Neumann algebras, J. Funct. Anal. 277 (2019), 108308, 82. E. G. Effros, The Borel space of von Neumann algebras on a separable Hilbert space, Pac. J. Math. 15 (1965), 1153–1164. E. G. Effros, Convergence of closed subsets in a topological space, Proc. Am. Math. Soc. 16 (1965), 929–931. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras I: Stability, Bull. Lond. Math. Soc. 45 (2013), 825–838. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras II: Model theory, Isr. J. Math. 201 (2014), 477–505. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras III: Elementary equivalence and II1 factors, Bull. Lond. Math. Soc. 46 (2014), 609–628. L. Ge and S. Popa, On some decomposition properties for factors of type II1 , Duke Math. J. 94 (1998), 79–101. I. Goldbring, B. Hart and T. Sinclair, Correspondences, ultraproducts and model theory, 2018. arXiv:1809.00049. I. Goldbring and C. Houdayer, Existentially closed w∗ -probability space, Math. Z. (2022), in press. V. J. Golodec, Spectral properties of modular operators, and the asymptotic ratio set, Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), 635–656, 704. V. J. Golodec and N. I. Nessonov, Asymptotic algebra and outer conjugacy classes of automorphisms of factors, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 510–532, 719. U. Groh, Uniform ergodic theorems for identity preserving Schwarz maps on W ∗ -algebras, J. Oper. Theory 11 (1984), 395–404. U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271–283. U. Haagerup, Lp -spaces associated with an arbitrary von Neumann algebra, Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), Colloq. Internat. CNRS, vol. 274, CNRS, Paris, 1979, pp. 175–184. U. Haagerup, Connes’ bicentralizer problem and uniqueness of the injective factor of type III1 , Acta Math. 158 (1987), 95–148. U. Haagerup, On the uniqueness of the injective III1 factor, Doc. Math. 21 (2016), 1193–1226. U. Haagerup and E. Størmer, Equivalence of normal states on von Neumann algebras and the flow of weights, Adv. Math. 83 (1990), 180–262. U. Haagerup and C. Winsløw, The Effros–Maréchal topology in the space of von Neumann algebras, Am. J. Math. 120 (1998), 567–617. U. Haagerup and C. Winsløw, The Effros–Maréchal topology in the space of von Neumann algebras. II, J. Funct. Anal. 171 (2000), 401–431.

340 � H. Ando

[34] C. Houdayer, Free Araki–Woods factors and Connes’ bicentralizer problem, Proc. Am. Math. Soc. 137 (2009), 3749–3755. [35] C. Houdayer and Y. Isono, Free independence in ultraproduct von Neumann algebras and applications, J. Lond. Math. Soc. (2) 92 (2015), 163–177. [36] C. Houdayer and Y. Isono, Unique prime factorization and bicentralizer problem for a class of type III factors, Adv. Math. 305 (2017), 402–455. [37] C. Houdayer and Y. Ueda, Asymptotic structure of free product von Neumann algebras, Math. Proc. Camb. Philos. Soc. 161 (2016), 489–516. [38] M. Izumi, R. Longo and S. Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 155 (1998), 25–63. [39] Z. Ji, A. Natarajan, T. Vidick, J. Wright and H. Yuen, MIP∗ = RE, 2020. arXiv:2001.04383. [40] K. Jung, Amenability, tubularity, and embeddings into Rω , Math. Ann. 338 (2007), 241–248. [41] R. Kadison, Problems on von Neumann algebras, Baton Rouge Conference, 1967. [42] R. R. Kallman, Groups of inner automorphisms of von Neumann algebras, J. Funct. Anal. 7 (1971), 43–60. [43] E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C ∗ -algebras, Invent. Math. 112 (1993), 449–489. [44] E. Kirchberg, Commutants of unitaries in UHF algebras and functorial properties of exactness, J. Reine Angew. Math. 452 (1994), 39–77. [45] W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976), 19–70. [46] C. Lance, On nuclear C ∗ -algebras, J. Funct. Anal. 12 (1973), 157–176. [47] R. Longo, Solution of the factorial Stone–Weierstrass conjecture. An application of the theory of standard split W ∗ -inclusions, Invent. Math. 76 (1984), 145–155. [48] O. Maréchal, Topologie et structure borélienne sur l’ensemble des algèbres de von Neumann, C. R. Acad. Sci. Paris, Sér. A-B 276 (1973), A847–A850. [49] A. Marrakchi, Spectral gap characterization of full type III factors, J. Reine Angew. Math. 753 (2019), 193–210. [50] A. Marrakchi, Full factors, bicentralizer flow and approximately inner automorphisms, Invent. Math. 222 (2020), 375–398. [51] T. Masuda, An analogue of Connes–Haagerup approach for classification of subfactors of type III1 , J. Math. Soc. Jpn. 57 (2005), 959–1001. [52] T. Masuda and R. Tomatsu, Rohlin flows on von Neumann algebras, Mem. Am. Math. Soc. 244 (2016), ix+111. [53] D. McDuff, Central sequences and the hyperfinite factor, Proc. Lond. Math. Soc. (3) 21 (1970), 443–461. [54] A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, vol. 1138, Springer, Berlin, 1985. [55] N. Ozawa, About the QWEP conjecture, Int. J. Math. 15 (2004), 501–530. [56] S. Popa, On a problem of R. V. Kadison on maximal abelian ∗-subalgebras in factors, Invent. Math. 65 (1981/82), 269–281. [57] S. Popa, Semiregular maximal abelian ∗ -subalgebras and the solution to the factor state Stone–Weierstrass problem, Invent. Math. 76 (1984), 157–161. [58] S. Popa, Hyperfinite subalgebras normalized by a given automorphism and related problems, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 421–433. [59] S. Popa, Classification of subfactors and their endomorphisms, CBMS Regional Conference Series in Mathematics, vol. 86, American Mathematical Society, Providence, RI, 1995. Published for the Conference Board of the Mathematical Sciences, Washington, DC. [60] Y. Raynaud, On ultrapowers of noncommutative Lp spaces, J. Oper. Theory 48 (2002), 41–68. [61] Ş. V. Strătilă, Modular theory in operator algebras, 2nd ed. [of 696172], Cambridge IISc Series, Cambridge University Press, Delhi, 2020.

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� 341

[62] M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, vol. 128, Springer, Berlin–New York, 1970. [63] M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal. 9 (1972), 306–321. [64] M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249–310. [65] M. Takesaki, The structure of the automorphism group of an AFD factor, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 19–44. [66] M. Takesaki, Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, vol. 124, Springer, Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5. [67] M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Springer, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6. [68] M. Takesaki, Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, vol. 127, Springer, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 8. [69] Y. Ueda, Factoriality, type classification and fullness for free product von Neumann algebras, Adv. Math. 228 (2011), 2647–2671.

Thomas Sinclair

Model theory of operator systems and C∗-algebras Abstract: We survey the model theory of operator systems and C∗ -algebras. We begin with a brief introduction to operator system theory. We then describe the languages and model-theoretic axiomatizations of the classes of operator systems and C∗ -algebras, including discussions of ultraproducts and definability in both contexts. Finally, we show how various well-studied approximation properties of operator systems and C∗ -algebras may be viewed from the perspective of continuous model theory. Along the way new approaches to several results in the literature are developed. Keywords: Continuous model theory, operator algebras, operator spaces, operator systems, completely positive mappings, nuclearity, lifting property, weak expectation property, semidefinite programming, linear matrix inequalities, free spectrahedra MSC 2020: 03C20, 03C66, 03C98, 15A39, 15A60, 46L05, 46L07, 46L10, 47L25, 90C22

1 Introduction At the heart of the theory of operator algebras is the “noncommutative order” imposed on the algebra ℬ(H) of bounded linear operators on some Hilbert space H by the cone of positive semidefinite operators, that is, operators T which admit a sum-of-squares decomposition T = S1∗ S1 + ⋅ ⋅ ⋅ + Sn∗ Sn for some n and S1 , . . . , Sn ∈ ℬ(H) (equivalently, n = 1). This partial order structure, even in the finite-dimensional case, is incredibly rich and complex and its understanding is deeply connected with many important and outstanding open problems in fields as diverse as quantum information theory and quantum computing, numerical linear algebra and optimization, combinatorics, computer science, and random matrix theory. Beginning in the 1960s and 1970s with work of Arveson, Choi, Effros, Lance, Kirchberg, and Stinespring, it became apparent that the full power of the noncommutative order is captured not just by the “level one” order structure but the higher-level order structure imposed by taking matrix amplifications. In fact, the higher-order structure on a C∗ -algebra is powerful enough to capture the norm-structure as well as many properties which would be considered algebraic in nature. This conception of noncommutative order crystallized in the notion of an Acknowledgement: The author was supported by NSF grant DMS-2055155. The author thanks Isaac Goldbring, Connor Thompson, and the anonymous referee for suggesting many corrections and improvements. Thomas Sinclair, Mathematics Department, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA, e-mail: [email protected]; URL: http://www.math.purdue.edu/~tsincla/ https://doi.org/10.1515/9783110768282-009

344 � T. Sinclair operator system first systematically studied in the groundbreaking work of Choi and Effros [9]. In any semisimple category of algebras, the finite-dimensional, simple objects are of primary importance. In the case of C∗ -algebras, these are the (complex) matrix algebras. Thus, it was a natural theme from the very beginning of the subject to try to understand the structure of a C∗ -algebra by quantifying how well (or how poorly) it was algebraically approximated by direct sums of matrix algebras. With the theory of operator systems, it was realized that finite-dimensional order approximation through matrix algebras was an equally vital aspect of the theory, and such properties as nuclearity and exactness have become standard tools of the trade for working operator algebraists. The surprising relations discovered by celebrated work of Kirchberg between two other well-studied finite-dimensional order approximation properties, the local lifting and weak expectation properties, and a famous conjecture of Connes have helped catalyze an entirely new chapter in operator algebras through their connection with a deep problem of Tsirelson arising in quantum information theory; see Goldbring’s article in this volume. The goal of this set of notes is to introduce the reader to some aspects of the continuous model theory of operator systems and C∗ -algebras mainly through the model-theoretic properties of the noncommutative order and their relation with finitedimensional approximation properties. This article is organized as follows. – Section 2 gives a brief overview of the theory of operator systems. While not totally self-contained, the section aims at presenting enough material for the reader to gain a basic working familiarity with operator systems, including such topics at the Choi–Effros representation theorem, Arveson’s extension theorem, duality and the double dual, and quotients. The latter part of the section provides several examples of canonical ways of constructing operator systems. The reader may wish to consult the excellent books of Brown and Ozawa [6] and Paulsen [37] for a fuller account of the theory. The reader may also wish to read the article of Szabo in this volume for an introduction to C∗ -algebras before proceeding beyond this section. – Section 3 introduces the model theory of operator systems and C∗ -algebras. The first part of the section constructs the language for operator systems, while the next few sections serve to introduce basic model-theoretic concepts such as theories, ultraproducts, and definability, illustrating these ideas in the operator systems context before concluding with a discussion of the model theory of C∗ -algebras. This section is largely based on the accounts found in the paper [20] of Goldbring and the author and the monograph of Farah et al. [15], though many of the proofs of results given here are new and the centering of the account from the perspective of matrix completion problems is also somewhat novel. A reader wishing to explore these topics further would do good to consult [5] and [15]. – Section 4 surveys some applications of the model theory developed in the Section 3 to the theory of finite-dimensional approximation properties for operator systems

Model theory of operator systems and C∗ -algebras

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and C∗ -algebras. The discussion of exactness and nuclearity is largely sourced from [15] and [21], while that of the lifting property is from [20, 21, 43]. The last part of this section on the weak expectation property is sourced from [21] and [36], though the treatment here is substantially new. We refer the reader to Goldbring’s article in this volume for the model-theoretic aspects of the local lifting and weak expectation properties in relation to the famous conjectures of Connes and Kirchberg, and the reader may also wish to consult the recent monograph of Pisier [41] on this subject.

2 General background on operator systems Definition 2.1. An operator system E is a closed subspace of ℬ(H) which is closed under taking adjoints and contains the unit. Remark 2.2. Let E h be the set of hermitian elements of E. From the property that E is closed under adjoints, it is immediate that E h is a real, unital subspace with E = ℂE h . Since E contains the unit, we have that for each x ∈ E h we may decompose x as a difference of positive elements in E, namely as x = (‖x‖1 + x) − ‖x‖1, which shows that E + := E ∩ ℬ(H)+ is a cone in E. In this way E h is an ordered (real) vector space under x ⪯ y ←→ y − x ∈ E + . Moreover, 1 is an order unit, that is, for each x ∈ E h −r1 ⪯ x ⪯ r1 for some r > 0 which is archimedean in the sense that −ϵ1 ⪯ x ⪯ ϵ1 for all ϵ > 0 implies that x = 0. So far we have just established that an operator system is a Banach space with a pleasant real, ordered structure. In the real case, these are referred to as function systems and were studied and classified by Kadison [24] (see [2, Chapter II] as well), while a study of the complex case was undertaken by Paulsen and Tomforde [39]. What distinguishes operator systems from function systems is the following enrichment of structure of the objects or, to put it in a glass-half-empty way, restriction of the category maps between the objects. The enrichment comes from the following, seemingly modest, observation: if E ⊂ ℬ(H) is an operator system, then so is En := Mn (E) ⊂ Mn (ℬ(H)) ≅ ℬ(H ⊕n ), where H ⊕n is the direct sum of n copies of H. Thus, from each operator system there may be derived a sequence of order unit spaces (En , En+ , 1n ). Here, and throughout, 1n is the tensor of the identity matrix In in Mn with the unit 1 in E. Moreover, there is a natural family of connecting maps defined as follows. For a ∈ Mn,k , define ad(a) : En → Ek by ad(a) : x 󳨃→ a∗ xa. Notice that ad(a) preserves the order structure. For v ∈ Mn,k with v∗ v = 1k , we will refer to ad(v) as the compression induced by v. (Notice that vv∗ is a rank k projection in Mn , so n ≥ k.) Given a ∗-linear map φ : E → F between operator systems E and F, we can define a ∗-linear map φn : En → Fn coordinate-wise by φn ([xij ]) := [φ(xij )]. Thinking about Mn (E) ≅ Mn ⊗ E, φn is nothing other than idMn ⊗ φ. Note that the φn ’s commute with the connecting maps (thus, compressions), ad(a) ∘ φn = φk ∘ ad(a).

346 � T. Sinclair Definition 2.3. A map φ : E → F is said to be n-positive (n = 1, 2, 3, . . .) if φn (En+ ) ⊂ Fn+ and completely positive if φ is n-positive for all n. We say that φ is a complete order embedding if it is unital and both φ and φ−1 : φ(E) → E are completely positive. It is easy to see that n-positivity implies k-positivity for all k ≤ n and that positive maps are ∗-linear. For an operator system E ⊂ ℬ(H), we denote by ‖ ⋅ ‖n the restriction of the operator norm on ℬ(H ⊕n ) to En . Definition 2.4. A linear map φ : E → F is said to be n-bounded if φn : (En , ‖ ⋅ ‖n ) → (Fn , ‖ ⋅ ‖n ) is bounded, in which case we denote the norm of φn by ‖φ‖n . The map φ is said to be completely bounded if supn ‖φ‖n < ∞, in which case we write ‖φ‖cb := supn ‖φ‖n . (Note that ‖φ‖k ≤ ‖φ‖n for all k ≤ n.) Perhaps the most important foundational fact in the theory of operator systems is that the higher-order norm structure is totally determined by the structure of the positive cones. Proposition 2.5. Let E ⊂ ℬ(H) be an operator system. For x ∈ En , we have that t1 ‖x‖n = inf {t > 0 : [ ∗ x

x + ] ∈ E2n }. t1

Proof. It clearly suffices to check the case n = 1. We have that [ ba∗ bc ] ∈ M2 (ℬ(H))+ if and only if ⟨aξ, ξ⟩ + 2 Re⟨bξ, η⟩ + ⟨cη, η⟩ ≥ 0 for all ξ, η ∈ H with ‖ξ‖2 + ‖η‖2 = 1. It follows that [ xt1∗ t1x ] is positive if and only if 2t+2 Re⟨xξ, η⟩ ≥ 0 for all unit-norm vectors ξ, η ∈ H. This, in turn, is seen to be equivalent to t ≥ |⟨xξ, η⟩| for all such ξ, η by making substitutions of the form η 󳨃→ eis η for s ∈ ℝ chosen suitably. Since we have that 󵄨 󵄨 ‖x‖ = sup{󵄨󵄨󵄨⟨xξ, η⟩󵄨󵄨󵄨 : ‖ξ‖ = ‖η‖ = 1}, this completes the proof. Exercise 2.6. A unital, 2-positive map φ : E → F is contractive. Proposition 2.7. If φ : E → F is a unital, ∗-linear contractive map between operator systems, then φ is positive. Proof. Let x ∈ E + with 0 ⪯ x ⪯ 1, and define y = 2x − 1 so that −1 ⪯ y ⪯ 1, equivalently ‖y‖ ≤ 1. Since φ is unital and ∗-linear φ(y) = 2φ(x) − 1 is hermitian, and ‖φ(y)‖ ≤ 1 since φ is contractive. Thus, −1 ⪯ 2φ(x) − 1 ⪯ 1, which implies φ(x) ⪰ 0. Proposition 2.8. Any unital, contractive linear map between operator systems is ∗-linear.

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Proof. Let φ : E → F be unital and contractive with E ⊂ ℬ(H) and F ⊂ ℬ(K). Let us fix a unit vector ξ ∈ K and define ϕξ (x) := ⟨ϕ(x)ξ, ξ⟩. For x ∈ E h , a contraction, and t ∈ ℝ, using the identity ‖z∗ z‖ = ‖z‖2 for all z ∈ ℬ(K) we have 󵄩1/2 󵄩 2 2 󵄩1/2 󵄩 󵄨 󵄨󵄨 2 1/2 ∗ 󵄨󵄨ϕξ (x) + it1󵄨󵄨󵄨 ≤ ‖x + it1‖ = 󵄩󵄩󵄩(x + it1) (x + it1)󵄩󵄩󵄩 = 󵄩󵄩󵄩x + t 1󵄩󵄩󵄩 ≤ (1 + t ) , since x 2 + t 2 1 ⪯ (‖x‖2 + t 2 )1. It is now easy to see that 󵄨󵄨 󵄨 2 1/2 󵄨󵄨Im φξ (x)󵄨󵄨󵄨 ≤ (1 − t ) − |t| → 0

as t → ±∞.

Since φξ (x) is then real for all x ∈ E h , we have that φξ is ∗-linear. It follows that φ is ∗-linear since for a (complex) Hilbert space K an operator z ∈ ℬ(K) is hermitian if and only if ⟨zξ, ξ⟩ ∈ ℝ for all ξ ∈ K [13, Proposition 2.12]. We derive the following easily as a consequence of the previous two results. Corollary 2.9. Every complete order embedding of operator systems φ : E → F is a completely isometric embedding, that is, φn : En → Fn and (φ−1 )n : φ(E)n → En are isometries for all n = 1, 2, . . . Every unital completely isometric embedding is a complete order embedding. Exercise 2.10 (Kadison–Cauchy–Schwarz). Let E ⊂ ℬ(H) be an operator system. If φ : E → ℬ(K) is unital and 2-positive, then φ(x)φ(x ∗ ) ⪯ φ(xx ∗ ) for all x ∈ E. See [37, Chapter 3] for this and many other basic properties satisfied by completely positive maps. Exercise 2.11. If φ : E → F is completely positive, then ‖φ‖cb = ‖φ(1)‖. See [37, Proposition 3.6]. Proposition 2.12 (Choi’s theorem [7]). Every completely positive map φ : Mn → Mk is of ∗ the form φ(x) = ∑nk i=1 ai xai for some a1 , . . . , ank ∈ Mn,k . The converse is also seen to be true as x 󳨃→ a∗ xa is completely positive. See [3, Theorem 2.21] for a proof. The matrices ai above are said to form a Kraus decomposition of the map φ. Exercise 2.13. Use Choi’s theorem to deduce that if φ : Mn → Mk is completely positive, then the adjoint map φ† : Mk → Mn defined by the functional equation trk (φ(A)B) = trn (Aφ† (B)) for all A ∈ Mn , B ∈ Mk is again completely positive. If φ is unital, then φ† is trace-preserving, trk ∘ φ = trn , and vice versa. Exercise 2.14. For E = M2 , show that a φ:[ c

b (a + d)/2 ] 󳨃→ [ d c

b ] (a + d)/2

is unital and 1-positive, but not 2-positive. Show that |λ| = 1/2 is the optimal constant so that

348 � T. Sinclair a φ:[ c

b (a + d)/2 ] 󳨃→ [ d λc

λb ] (a + d)/2

is completely positive. Exercise 2.15. Let v1 , v2 be the generators of the Cuntz algebra 𝒪2 (see Szabo’s article in this volume), and consider E := span{1, v1 , v2 , v∗1 , v∗2 }. Show that φ : E → E given by φ(a1 + bv1 + cv2 + dv∗1 + ev∗2 ) = a1 + bv∗1 + cv∗2 + dv1 + ev2 is unital and positive, but not 2-positive, as φ2 is not contractive. Exercise 2.16. We have that φ : M2 → M2 given by transposition, φ(x) := x t , is positive but not 2-positive. Remarkably, the structure of higher-ordered cones and connecting maps is totally sufficient to abstractly characterize operator systems, as discovered in the seminal work of Choi and Effros [9]. We will say that (E, E + , e) is a real ordered ∗-vector space if E is a ∗-vector space with a cone E + ⊂ E h and order unit e ∈ E + . Definition 2.17. An abstract operator system (E, E + , 1) is a real ordered ∗-vector space with archimedean order unit 1 along with a real ordered ∗-structure (En , En+ , 1n ) for each n = 1, 2, . . ., where En ≅ Mn ⊗ E, (a ⊗ x)∗ = a∗ ⊗ x ∗ , and 1n = In ⊗ 1, so that En+ ⊂ Enh and a∗ En+ a ⊂ Ek+ for all a ∈ Mn,k . Remark 2.18. Note that the condition a∗ En+ a ⊂ Ek+ for n = 1 yields that Mk+ ⊗ E + ⊂ Ek+ for all k by the spectral theorem. Remark 2.19. Omitting the requirement of an order unit in the previous definition leads to the definition of a matrix ordered ∗-vector space Theorem 2.20 (Choi and Effros). Every abstract operator system E admits a complete order embedding into ℬ(H) for some Hilbert space H. The representation obtained by the Choi–Effros theorem is canonical, but, like the Gelfand–Naimark theorem, it is impractical to work with. Nonetheless, the abstract characterization of operator provides us with a way to axiomatize the class of operator systems as metric structures. Additionally, the proof of this theorem introduces some important ideas, as we will see in the following sketch. To begin, we introduce the notion of a state φ : E → ℂ to just mean that φ is a unital, positive map. It follows essentially by an application of the Hahn–Banach theorem for archimedean ordered ∗-vector spaces that the states separate points in E and completely determine the positivity structure; to wit, Lemma 2.21. For an abstract operator system E and x ∈ E, we have that x ∈ E + if and only if φ(x) ≥ 0 for every state φ. See [39, Proposition 3.12] for a proof.

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Remark 2.22. Even if (E, E + , e) is a real ordered ∗-vector space with e being an not necessarily archimedean order unit, one may still consider the set of all states on E and define E ′ to be their closed, linear span in E ∗ . For each x ∈ E, we can define the evaluâ tion map x̂ : E ′ → ℂ by x(φ) := φ(x) and set Ê := {x̂ : x ∈ E} to be the image of E under ̂+ , e)̂ is now an archimedean ordered, ∗-vector space ̂ E this map. It can be seen that (E, which is known as the archimedeanization of E. We refer the reader to [39, Section 3.2] for details. The crucial point is that e is an archimedean order unit exactly when the evaluation map x 󳨃→ x̂ is faithful, that is, has trivial kernel. In many of the constructions going forward, we will define some sort of higher-level positivity/order structure on a archimedean ordered ∗-vector space, making it an operator system, by describing the set of higher-order positive maps. At a technical level the positive maps really induce an operator system structure on Ê which we can identify with E canonically as long as we know the evaluation map is faithful. Lemma 2.23. For an abstract operator system E, every positive map φ : E → ℂ is completely positive. In particular, every state is completely positive. Proof. Let x ∈ En+ . We want to show that φn (x) ∈ Mn is positive semidefinite. To this end, choose a ∈ Mn,1 , and observe that a∗ φn (x)a = φ(a∗ xa) ≥ 0 since a∗ xa ∈ E + . We are therefore done since a∗ φn (x)a ≥ 0 for all a ∈ Mn,1 characterizes the matrix φn (x) as being positive semidefinite. Notation 2.24. For (abstract) operator systems E and F, we use CP(E, F) and UCP(E, F), respectively, to denote the set of all completely positive maps from E to F and the set of all unital, completely positive maps from E to F. Lemma 2.25. Let E be an abstract operator system. For every φ ∈ CP(E, Mn ), there is φ′ ∈ UCP(E, Mk ) for some k ≤ n so that ker(φ′ ) = ker(φ). Proof. If φ is unital, then there is nothing to prove. Let z := φ(1) ∈ Mn+ and let p be its support projection, that is, the largest projection p so that pxp is invertible pMn p. Let x ∈ E. Since 12 is an order unit in E2 we may assume without loss of generality that z φ(x) [ x1∗ x1 ] ⪰ 0, so [ φ(x)∗ z ] ⪰ 0 as well. It follows that [

0 p φ(x ∗ ) ⊥

φ(x)p⊥ p⊥ zp⊥ ]=[ ⊥ z p φ(x ∗ )

φ(x)p⊥ ] ⪰ 0, z

hence φ(x)p⊥ = 0 for all x ∈ E and p⊥ φ(x) = 0 as well by ∗-linearity of φ. If p is of rank k, we find a ∈ Mn,k so that pa = a and a∗ za = Ik ∈ Mk . Setting φ′ (x) := a∗ φ(x)a = a∗ pφ(x)pa ∈ UCP(E, Mk ), it is clear that φ′ (x) = 0 if and only if φ(x) = 0. Lemma 2.26. Let E and F be abstract operator systems. There is an affine bijection between CP(En , F) and CP(E, Fn ), CP(En , F) ∋ φ ←→ φ̃ ∈ CP(E, Fn ), given by

350 � T. Sinclair ̃ φ(x) := [φ(eij ⊗ x)]ij ,

̃ ij )ij , φ([xij ]) := ∑ φ(x i,j

where eij are the standard matrix units for Mn . Proof. Let us start with the assumption that φ : En → F is completely positive. It is clear that the map Δn : Mn → Mn ⊗ Mn given by Δn ([xij ]) := ∑ eij ⊗ xij eij i,j

is a (nonunital) ∗-homomorphism, hence is completely positive. Therefore by Proposition 2.12, we have that Δn (x) = ∑i ad(vi )(x), vi ∈ Mn,n2 ; hence, for all x ∈ En+ we have (Δn ⊗ idE )(x) = ∑i (ad(vi ) ⊗ idE )(x) ∈ En+2 . Since φ is completely positive, so is idMn ⊗ φ : Mn ⊗ En → Fn . Let jn ∈ Mn,1 be the matrix of all 1’s. We have that the connecting map ad(jn∗ ) : E → En as defined at the beginning of this section is completely positive. We now compute that ̃ φ(x) = ((idMn ⊗ φn ) ∘ (Δn ⊗ idE ) ∘ ad(jn∗ ))(x), so φ̃ : E → Fn is completely positive. We now turn to the case that φ̃ : E → Fn is completely positive, so idMn ⊗ φ̃ : En → Mn ⊗ Fn is as well. We have that ̃ φ([xij ]) = ad(jn ) ∘ (Δ†n ⊗ idF ) ∘ (idMn ⊗ φ)([x ij ]) is completely positive by Exercise 2.13. We now sketch a proof of Theorem 2.20. Sketch of a proof of Theorem 2.20. Consider all φ ∈ UCP(E, Mn ), over all n = 1, 2, . . ., where we will write n(φ) := n for clarity. By Lemma 2.26, we have that CP(E, Mn ) ≅ CP(En , ℂ), which by Lemmas 2.25, 2.23, and 2.21 shows that ⋃ UCP(E, Mk )

1≤k≤n

completely determines the positive cone in En . For concision, let us denote ∞

I := ⋃ UCP(E, Mn ). n=1

It is now relatively straightforward to check that setting H = ⨁φ∈I ℂn(φ) we have that Φ : E → ℬ(H) given by Φ(x) = ∏ φ(x) ∈ ∏ Mn(φ) ⊂ ℬ(H) φ∈I

is a complete order embedding.

φ∈I

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351

Remark 2.27. It is useful to note that the proof shows that every operator system is a subsystem of a direct product of matrix algebras. One of the main upshots of the Choi–Effros representation theorem from our vantage is that it describes the following “dual” way of viewing an operator system structure via a coherent collection of admissible positive maps into each matrix algebra. Let (E, E + , 1) be a real ordered ∗-vector space with archimedean order unit, and let 𝒮 (E) denote the set of states of E. Proposition 2.28. Suppose that for each n = 1, 2, . . . we define a closed set 𝒮n of unital, linear maps from E to Mn so that 𝒮1 = 𝒮 (E) and for each f ∈ 𝒮n and φ ∈ UCP(Mn , Mk ) we have that φ ∘ f ∈ 𝒮k . Then the collection (𝒮n ) completely determines an operator system structure on E so that 𝒮n = UCP(E, Mn ) for all n. Sketch of proof. For each φ ∈ 𝒮n ⊂ ℒ(E, Mn ), write φ(x) = [φij (x)]. Define En+ ⊂ En to be all x ∈ En so that ∑i,j φij (xij ) ≥ 0 for all φ ∈ 𝒮n . We leave it as an exercise that this defines a compatible family of real ordered structures on each En satisfying Definition 2.17. Perhaps the most important basic result in the theory of operator systems is Arveson’s extension theorem. Theorem 2.29 (Arveson’s extension theorem). For every unital inclusion E ⊂ F of operator systems and every unital, completely positive map φ : E → ℬ(H), there exists a unital, completely positive map φ̃ : F → ℬ(H) extending φ. We will outline a proof here: for a detailed proof, see [6, Theorem 1.6.1] or [37, Theorem 7.5]. Sketch of proof. We first note that any positive map φ : E → ℂ extends to a positive (hence, completely positive) map φ̃ : F → ℂ by the Hahn–Banach extension theorem for ordered vector spaces [13, Corollary 9.12]. We now consider the case when dim(H) = n < ∞. We have that CP(E, ℬ(H)) ≅ CP(E, Mn ) ≅ CP(Mn (E), ℂ). As before, we can now extend any φ ∈ CP(Mn (E), ℂ) to φ̃ ∈ CP(Mn (F), ℂ) ≅ CP(F, Mn ). Now let H be arbitrary and let (Hi )i∈I be the directed set of all finite-dimensional subspaces of H, ordered by inclusion. Let pi : H → Hi be the orthogonal projection onto Hi and consider the unital completely positive maps φi : E → ℬ(Hi ) defined by φi (x) = pi φ(x)pi . We may extend each to φ̃ i : F → ℬ(Hi ) unitally and completely positively. Since all maps are contractive, we can now take a pointwise-ultraweak cluster point of the net φ̃ i to obtain a unital, completely positive map φ̃ : F → ℬ(H). It is straightforward to check that φ̃ extends φ. We now turn our attention to describing the dual of an operator system. We begin with the following proposition.

352 � T. Sinclair Lemma 2.30. Let E be an operator system. Every bounded linear functional φ : E → ℂ is a linear combination of four states. Every completely bounded map φ : E → Mn is a linear combination of four completely positive maps. Proof. Suppose E ⊂ ℬ(H). By the Hahn–Banach theorem, every bounded linear contraction φ : E → ℂ extends to a linear contraction φ̃ : ℬ(H) → ℂ. Now φ̃ is a linear combination of four states by the Jordan decomposition theorem for C∗ -algebras [44, Proposition 2.1]. The rest of the result follows from this by essentially the same proof as Lemma 2.26 by noting the correspondence described therein converts any completely bounded linear map φ : E → Mn to a bounded linear functional φ̃ : Mn (E) → ℂ. From this we see that E ∗ is equipped with a matricial ordered ∗-vector space structure, where the positive cone in Mn (E ∗ ) is identified with the completely positive maps E → Mn . This does not give E ∗ the structure of an operator system, however, as there is no clear choice of an archimedean order unit. In the case that E is a finite-dimensional operator system, there is a way of placing an operator system structure on E ∗ which is noncanonical as it will depend on choosing an order unit. This is essentially a consequence of the following lemma; see [26, Lemma 2.5] for a proof. Lemma 2.31. For any finite-dimensional operator system E, we can find a basis {1 = x1 , . . . , xn } of self-adjoint elements so that the dual basis {x1∗ , . . . , xn∗ }, defined by xi∗ (xj ) = δij , is a hermitian basis for E ∗ with x1∗ an order unit. A related, useful lemma appears as [6, Lemma B.10]. Lemma 2.32. For any finite-dimensional operator system E, we can find a spanning set 1 = x1 , . . . , xn consisting of hermitian elements of unit norm so that the dual basis x1∗ , . . . , xn∗ defined by xi∗ (xj ) = δij satisfies ‖xi∗ ‖ = 1 as well for all i = 1, . . . , n. Moreover, x1∗ may be taken to be an order unit in the cone 𝒮 (E). Even though E ∗ is not an operator system, E ∗∗ is, with the matricial order structure obtained by dualizing twice being given by the weak*-closures of the positive cones En+ under the evaluation map ⋅.̂ The tricky part is checking that 1̂ : E ∗ → ℂ is an archimedean order unit for this matricial ordered structure on E ∗∗ . In fact, the following is true. Proposition 2.33. The natural embedding ⋅ ̂ : E 󳨅→ E ∗∗ given by sending x 󳨃→ x̂ is a complete order embedding of operator systems. If E is finite-dimensional, then it is a complete order isomorphism. Proof. We begin by showing that ℬ(H)∗∗ has the structure of a unital C∗ -algebra whose positive cone agrees with that of the induced matricial order structure and whose unit is 1.̂ It follows that 1̂ is an archimedean order unit in ℬ(H)∗∗ since C∗ -algebras are operator systems. (The author is not aware of any proof of this proposition which avoids this reasoning.) Indeed, by [44, Section III.2] there is a unital ∗-algebra embedding π : ℬ(H) → ℬ(K) so that there is a unique extension π ′ : ℬ(H)∗∗ → ℬ(K) which is a homeomorphism from the weak*-topology on ℬ(H)∗∗ to the ultraweak topology on ℬ(K). This

Model theory of operator systems and C∗ -algebras

� 353

implies that x ∈ Mn (ℬ(H)∗∗ ) is positive in the matricial order structure defined on ℬ(H)∗∗ if and only if π ′ (x) is positive in Mn (ℬ(K)) as both are determined from taking the closures of Mn (ℬ(H))+ in the respective weak topologies. Now, we have that E ⊂ ℬ(H) for some Hilbert space H, and it is straightforward to see that E ∗∗ 󳨅→ ℬ(H)∗∗ is a complete order embedding at the level of the induced matricial ordered structures. Since 1 = π(1) ∈ ℬ(K) is archimedean, we have that 1̂ ∈ E ∗∗ is archimedean, thus E ∗∗ is an operator system. Definition 2.34. For an operator system E, a subspace J ⊂ E is said to be a kernel if J = ⋂i∈I ker(φi ) where {φi : i ∈ I} is a collection of states on E. ̂ for the Given a kernel J ⊂ E, we may put a quotient operator system structure on E/J ordered ∗-vector space quotient E/J by identifying UCP(E/J, Mn ) with the weak∗ -closed, convex subset UCP(E, Mn ; J) of UCP(E, Mn ) consisting of all φ with J ⊂ ker(φ). Note that ̂ and not E/J itself as the class in general the quotient structure must be placed on E/J of 1 in E/J is an order unit which is possibly not archimedean, so we must pass through the evaluation map to obtain the archimedeanization. Quotients of operator systems were first defined by Choi and Effros [9]. We refer the reader to [26, 29] for a in-depth treatment of quotients of operator systems: See also [14, 18].

2.1 Examples of operator systems Example 2.35. Every unital C∗ -algebra A is canonically an operator system where A+n is just the cone of positive elements in the C∗ -algebra Mn (A). In particular, the matrix algebra Mk is naturally an operator system under (Mk )+n = (Mn ⊗ Mk )+ . Note that not every element of (Mn ⊗ Mk )+ can be written as a sum of simple tensors x ⊗ y with x ∈ Mn+ and y ∈ Mk+ as soon as n, k ≥ 2! In quantum information theory, the elements in (Mn ⊗ Mk )+ which are not in the span of simple tensors of positive elements are said to be entangled. Example 2.36. Let X be a finite set. For a vector space V , let V [X] denote the vector space of all functions f : X → V . We have that ℂ[X] is a real, ordered ∗-vector space under pointwise conjugation and the cone ℝ≥0 [X] of all nonnegatively-valued functions f : X → ℝ≥0 , with order unit being the characteristic function 1X . We can define the following “free” canonical operator system structure on ℂ[X] as follows. We have that Mn (ℂ[X]) ≅ Mn [X] canonically, so we can define ℂ[X]+n as the cone of all functions f : X → Mn+ . In the dual picture, consider a map g : X → Mk+ , which extends to a positive linear map g̃ : ℂ[X] → Mk . We see that g̃n : Mn [X] → Mn (Mk ) ≅ Mn ⊗ Mk is given by g̃n (f ) = ∑x∈X f (x) ⊗ g(x); hence, every positive map from ℂ[X] to Mk is completely positive. On the other hand, consider a positive map g : X → Mk+ . For f ∈ Mn [X], we have the pairing ∑i,j g̃ij (fij ) as given in the proof of Proposition 2.28 can be computed as

354 � T. Sinclair ∑ g̃ij (fij ) = ∑ tr(f (x)g(x)). i,j

x∈X

Since A ∈ Mn is positive semidefinite if and only if tr(AB) ≥ 0 for all B positive semidefinite, it follows that ∑i,j g̃ij (fij ) ≥ 0 for all g : X → Mn+ if and only if f (x) ∈ Mn+ for all x ∈ X. These arguments are seen to apply equally well replacing Mn with an arbitrary direct product of matrix algebras as the direct sum is dense in the strong operator topology; thus, the reasoning applies to all operator systems by Remark 2.27. In summary, we see that this operator system structure on ℂ[X] has the following two universal properties: for every operator system E, every positive map φ : ℂ[X] → E is completely positive and every function f : X → E + induces a completely positive map f ̃ : ℂ[X] → E. Let us also observe that a map ψ : E → ℂ[X] is positive if and only if δx ∘ ψ : E → ℂ is positive for all x ∈ X, where δx is the point evaluation map. Thus for any operator system E any positive map ψ : E → ℂ[X] is completely positive. Example 2.37. We now describe the construction of universal operator systems from sets of linear relations. We equip ℂ[X] with the usual inner product structure ⟨f , g⟩ := ̄ Consider a set of elements r1 , . . . , rk ∈ ℝ[X] so that ⟨ri , 1X ⟩ = ∑x ri (x) = 0 ∑x∈X f (x)g(x). for all i = 1, . . . , k. Let R = span{r1 , . . . , rk }. We define an operator system structure on the quotient space ℂ[X]/R by defining the completely positive maps from ℂ[X]/R to Mn to be exactly the set of linear maps induced by functions f : X → Mn+ so that ∑x∈X ri (x)f (x) = 0 1 for all i = 1, . . . , k. Since 1X ∈ R⊥ , we have that the constant function f (x) ≡ |X| In is a unital, completely positive map, so that UCP(ℂ[X]/R, Mn ) is nonempty for each n. This is a special case of the quotienting construction detailed above, as it is easy to check that R ⊂ ℂ[X] is a kernel. This construction seems rather simple, but this belies a very rich structure. For example, let us take |X| = 2n and label the elements of X as s1 , . . . , sn , t1 , . . . , tn . Consider the single element r defined by r(si ) = 1 and r(tj ) = −1 for all i, j = 1, . . . , n. It is a result of Kavruk [27] that in this case ℂ[X]/R is canonically completely order isomorphic to the operator subsystem of C ∗ (ℤ/nℤ ∗ ℤ/nℤ) spanned by the generating set (ℤ/nℤ ∗ 1) ∪ (1 ∗ ℤ/nℤ). Example 2.38. Let P ⊂ ℝk be a closed cone with archimedean order unit e. We can define an operator system structure on V = (ℂk , P, e) by declaring UCP(V : Mn ) to be the set of all linear maps f : ℂk → Mn so that f (ℝk ) ⊂ Mnh , f (e) = In , and f (p) ∈ Mn+ for all p ∈ P. This is the maximal operator system structure MAX(V ) over V as defined in [38]; the following is proved therein. Proposition 2.39. The operator system MAX(V ) is characterized by the universal property that φ : MAX(V ) → ℬ(H) is completely positive if and only if φ(P) ⊂ ℬ(H)+ . We record this property by introducing new terminology. Definition 2.40. We say that an operator system E is k-maximal if every k-positive map φ : E → ℬ(H) is completely positive. For ease of reference, we will say that E is maximal if it is 1-maximal.

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Remark 2.41. If K ⊂ ℝℓ is a compact, convex set with 0 as an interior point, we can form the cone PK ⊂ ℝℓ+1 to consist of all vectors of the form te − tv where e = (1, 0, . . . , 0), v ∈ 0 ⊕ K, and t ≥ 0. It can be checked that e is an archimedean order unit for PK and PK − PK = ℝℓ+1 . Proposition 2.42. For the maximal operator system structure on VK := (ℂℓ+1 , PK , e), we have that UCP(VK , Mn ) is identified with the set of all ℓ-tuples X1 , . . . , Xℓ of hermitian elements in Mn satisfying the linear matrix inequalities In ⪰ v1 X1 + ⋅ ⋅ ⋅ + vℓ Xℓ for all v ∈ K. We leave the proof as an exercise to the reader. Conversely, for a closed cone P with archimedean order unit e belonging to the interior, consider the “slice” K = {p ∈ P : ⟨p, e⟩ = 1}. It can be shown that PK is affinely isomorphic to P, so that all maximal operator systems essentially arise in this way. This is essentially a special case of the Webster–Winkler duality theorem [45]. Example 2.43. Again, let P ⊂ ℝk be a closed cone with archimedean order unit e. We can define an operator system structure on V = (ℂn , P, e) by embedding V via the evaluation map into the algebra C(𝒮 (V )) of continuous functions on the set 𝒮 (V ) of all states, which is a closed and bounded, hence compact, subset of ℂn . This is the minimal operator system structure MIN(V ) defined on V as described in [38]. The following is proved therein. Proposition 2.44. The minimal operator system structure on V is characterized by the universal property that for every operator system E every positive map φ : E → V extends to a completely positive map φ : E → MIN(V ).

3 Model theory of operator systems A linear matrix ∗-polynomial in n-variables x = (x1 , . . . , xn ) is an expression of the form p(x) = a1 ⊗ x1 + ⋅ ⋅ ⋅ + an ⊗ xn + b1 ⊗ x1∗ + ⋅ ⋅ ⋅ + bn ⊗ xn∗ + c ⊗ 1 where (a1 , . . . , an , b1 , . . . , bn , c) are elements in Mk for some k. Equivalently, we may think of p(x) as a k × k matrix where each entry is a linear ∗-polynomial in 1, x1 , . . . , xn . Definition 3.1. We say that a linear matrix ∗-polynomial is hermitian if p(x ∗ )∗ = p(x) which is to say that c is hermitian and ai∗ = bi for all i = 1, . . . , n. A linear matrix ∗-polynomial is said to be homogeneous if p(0) = 0, that is, if c = 0. We say that p(x) has degree k if a1 , . . . , an , b1 , . . . , bn , c ∈ Mk , and we write deg(p) = k. We define a linear matrix inequality to be an expression of the form p(x) ⪰ 0 where p(x) is a hermitian linear matrix ∗-polynomial. Note that given a system p1 (x) ⪰ 0, . . . , pm (x) ⪰ 0 of linear matrix inequalities we may without loss of generality combine

356 � T. Sinclair them into a single linear matrix inequality via a block diagonal embedding of the matrix coefficients of each pi into a larger matrix. Many problems in functional analysis can be essentially phrased as the following general problem: Problem 3.2 (Matrix completion problem). Does there exist X = (X1 , . . . , Xn ) ∈ 𝒟n for some domain 𝒟 ⊂ ℬ(H) so that p(X) ⪰ 0? (Here 1 is interpreted as the unit in ℬ(H).) From the perspective of logic, the matrix completion problem can be seen as whether some existential sentence is true over some (hopefully) suitable domain of quantification 𝒟. Notice that by Proposition 2.5, this framework captures problems involving norm estimates as well: if we are interested in the statement 󵄩 󵄩 ∃x ∈ 𝒟n : 󵄩󵄩󵄩p(x)󵄩󵄩󵄩 ≤ 1, then setting p∗ (x) := p(x ∗ )∗ we may write this equivalently as ∃x ∈ 𝒟n : [

1 p∗ (x)

p(x) ] ⪰ 0. 1

If H is a finite-dimensional Hilbert space, the matrix completion problem frequently occurs in the context of semidefinite programming in terms of whether a given semidefinite program is feasible. (We will not go any further into this here, but we refer the reader to [19, 35] for an introduction to modern developments in the theory of semidefinite programming.)

3.1 Building the language One could say that the broad goal of building a model theory for matrix operator systems is to devise a general framework where problems such as the matrix completion problem can be systematically studied. Following [20, Appendix B], we give a description of how to axiomatize operator systems in the context of first-order continuous logic for metric structures. (We refer the reader Hart’s article in this volume or to [15, Section 2.1] for the basics of first-order continuous logic for metric structures.) In terms of building the language, this immediately brings to attention the following considerations. (1) We will need a collection of sorts E1 , E2 , E3 , . . . intended to capture the operator system E = E1 as well as its matrix amplifications so that En should be interpreted as Mn (E) when our work is done. Each of these sorts will need domains of quantification 𝒟r (En ) corresponding to the norm r-balls about the origin. Since we are about to give each En a real vector space structure, it will suffice to consider 𝒟1 (En ) alone. (2) We will need another collection of sorts C1 , C2 , C3 , . . . for which each Cn will need to be interpreted as the cone of positive elements En+ in En with domains of quantification to be interpreted as the restriction of the r-balls in En to Cn .

Model theory of operator systems and C∗ -algebras �

357

(3) Sorts Mn,k for the complex n × k matrices with each domains of quantification the operator norm r-balls. (4) Sorts for ℂ, ℝ, and the nonnegative reals ℝ≥0 with the domains of quantification being the standard r-balls. For each of these sorts we will need the corresponding relational symbols. (1) We will need constant symbols 0, 1 in 𝒟1 (E1 ) for 0 and the order unit. We require function symbols ⋅n : ℂ × En → En ,

+n : En × En → En ,

and ∗n : En → En

to be used for scalar multiplication, addition, and involution, respectively. Additionally, one further set of function symbols fn,k : Mn,k × En → Ek is needed for (a, x) 󳨃→ a∗ xa and another pijn : En → E1 for projections giving the matrix coordinates. There are predicate symbols ‖ ⋅ ‖n : En → ℝ≥0 which we will want to interpret as the norms induced from the matrix ordering as in Proposition 2.5. (2) We define constant symbols 0′ , 1′ in 𝒟1 (C1 ) and function symbols ⋅′n : ℝ≥0 × Cn → Cn ,

+′n : Cn × Cn → Cn ,

and

gn,k : Mn,k × Cn → Ck

to be used for scalar multiplication, addition, and (a, x) 󳨃→ a∗ xa, respectively. (3) Similarly to the first two items, there will need to be a constant symbol 0 and function symbols for scalar multiplication, addition, and taking adjoints ∗ : Mn,k → Mk,n for each Mn,k . (4) Again, we must define symbols for 0, 1 in ℝ≥0 , ℝ, and ℂ with an additional symbol i in ℂ, all in the unit domain of quantification. Function symbols are needed for addition, scalar multiplication, and, in the case of ℂ, conjugation. (5) Besides these we need a couple of other sets of function symbols in : Cn → En , obviously for the inclusion of the positive cone in each En and hn : En → E2n

358 � T. Sinclair which we want to interpret as 1 hn : x 󳨃→ [ n∗ x

x ] 1n

to be used to guarantee our predicate symbols are interpreted as intended. Pertaining to the use of the hn ’s, we require the following result. Lemma 3.3. For an operator system E and x ∈ En , we have that ‖x‖n ∸ 1 = dist ([

1 x∗

x + ] , E2n ). 1

Here x ∸ y := max{x − y, 0}. Proof. We may assume without loss of generality that n = 1 and r := ‖x‖ > 1. By Proposition 2.5, we have that 1 ∗ x r

[1

1 x r ]

1

∈ E2+ ;

hence, 1 h(x) := dist ([ ∗ x

x 0 ] , E2+ ) = dist ([ 1 (1 − 1/r)x ∗

(1 − 1/r)x ] , E2+ ) . 0

Noting that 󵄩󵄩 󵄩󵄩 a 󵄩󵄩[ ∗ 󵄩󵄩 b 󵄩

󵄩 󵄩 b 󵄩󵄩󵄩 󵄩󵄩󵄩 1 ]󵄩󵄩󵄩 = 󵄩󵄩󵄩[ c 󵄩󵄩 󵄩󵄩 0

0 a ][ −1 b∗

b 1 ][ c 0

󵄩 󵄩 0 󵄩󵄩󵄩 󵄩󵄩󵄩 a ]󵄩󵄩󵄩 = 󵄩󵄩󵄩[ ∗ −1 󵄩󵄩 󵄩󵄩 −b

󵄩 −b 󵄩󵄩󵄩 ]󵄩󵄩󵄩 , c 󵄩󵄩

we see by simple arithmetic that 󵄩󵄩 󵄩󵄩 0 󵄩󵄩[ ∗ 󵄩󵄩 y 󵄩

y a ]−[ ∗ 0 b

󵄩 󵄩 b 󵄩󵄩󵄩 󵄩󵄩󵄩 0 ]󵄩󵄩󵄩 = 󵄩󵄩󵄩[ ∗ c 󵄩󵄩 󵄩󵄩 y

y a ]−[ ∗ 0 −b

󵄩 −b 󵄩󵄩󵄩 ]󵄩󵄩󵄩 . c 󵄩󵄩

Hence, by averaging the two expressions and using the triangle inequality, we see that h(x) must be approached by elements in E2+ of the form [ a0 0c ]. However, by the same argument as before 󵄩󵄩 󵄩󵄩 0 󵄩󵄩[ ∗ 󵄩󵄩 y 󵄩

y a ]−[ 0 0

󵄩 󵄩 0 󵄩󵄩󵄩 󵄩󵄩󵄩 0 ]󵄩󵄩󵄩 = 󵄩󵄩󵄩[ ∗ c 󵄩󵄩 󵄩󵄩 −y

−y a ]−[ 0 0

which shows by averaging that a = c = 0 is optimal. Therefore

󵄩 0 󵄩󵄩󵄩 ]󵄩󵄩󵄩 c 󵄩󵄩

Model theory of operator systems and C∗ -algebras

󵄩󵄩 0 󵄩 h(x) = 󵄩󵄩󵄩󵄩[ 󵄩󵄩 (1 − 1/r)x ∗

� 359

󵄩 (1 − 1/r)x 󵄩󵄩󵄩 1 ]󵄩󵄩󵄩 = (1 − )‖x‖ 0 ‖x‖ 󵄩󵄩2

since x [ 0

0 0 ∗] = [ ∗ x x

x 0 ]⋅[ 0 1

1 ]. 0

We require one further technical result; see [20, Lemma B.1]. Exercise 3.4. For an operator system E and x ∈ En+ , we have that dist(−x, En+ ) = ‖x‖n . To complete the axiomatization, we need axioms that tell us that the symbols enumerated above are interpreted as they should be. These would include, for instance, axioms stating that each En has the structure of a ∗-vector space, En is isomorphic to Mn (E1 ), and the ∗-vector space structure on En is the natural one induced from the ∗-vector space structure on E1 via matrix amplification. In addition to these, we need the following, more specialized, axioms: (1) There is an axiom that each ‖ ⋅ ‖n is seminorm on En , and that this predicate is the metric predicate, that is, sup

󵄨󵄨 󵄨 󵄨󵄨dist(x, y) − ‖x − y‖n 󵄨󵄨󵄨 = 0.

x,y∈𝒟1 (En )

This implies that each ‖ ⋅ ‖n is, in fact, a norm. (2) We need an axiom stating that in : Cn → En is an isometric inclusion, an axiom for ensuring that the range of in lies in the hermitian part of En , sup dist(in (x), in (x)∗ ) = 0,

x∈𝒟1 (Cn )

and an axiom (by way of Exercise 3.4) ensuring that −in (Cn ) ∩ in (Cn ) = {0}, sup

󵄨󵄨 󵄨 󵄨󵄨‖x‖n ∸ dist(in (x), −in (y))󵄨󵄨󵄨 = 0.

x,y∈𝒟1 (Cn )

(3) Finally, there is an axiom stating the content of Lemma 3.3, that is, 󵄨 󵄨 sup 󵄨󵄨󵄨(‖x‖n ∸ 1) − dist(hn (x), C2n )󵄨󵄨󵄨 = 0.

x∈𝒟r (En )

Remark 3.5. The last axiom is important as it guarantees that 1 is interpreted as a (complete) order unit. To see this, let x ∈ En with x = x ∗ and set r := ‖x‖n . We have that ‖ r1 x‖n = 1, thus 1 1 hn ( x) = [ 1 n r x r

1 x r ]

1n

∈ C2n .

We then have that v∗ hn (x)v = 1n + r1 x ∈ Cn where v = [ √1 In , √1 In ]t . 2

2

360 � T. Sinclair Remark 3.6. As a consequence of the previous remark, we have that in (Cn ) spans En for all n. Indeed, for all x ∈ En with ‖x‖n ≤ 1, we have that x = ((1n + (x + x ∗ )/2) − 1n ) + √−1((1n − √−1(x − x ∗ )/2) − 1n ), which expresses x as a linear combination of four elements in the positive cone, each with norm at most 2. So far the axiomatization says that any model can be interpreted as a matrix ordered ∗-vector space with order unit (see Remark 2.19), the only difference from being an abstract operator system then is the issue of whether 1 is an archimedean order unit. To bridge this last gap, we will rely crucially on the fact that the metric structure gives rise to a norm ‖ ⋅ ‖n on each En . Lemma 3.7. Let E be an matrix ordered ∗-vector space with order unit 1. Setting t1 ‖x‖′n := inf {t > 0 : [ ∗n x

x + ] ∈ E2n }, t1n

we have that E is an abstract operator system if and only if one of the following equivalent conditions holds: (1) 1 is an archimedean order unit for E; (2) 1n is an archimedean order unit for En for all n; (3) ‖ ⋅ ‖′n is a norm for all n; (4) ‖ ⋅ ‖′1 is norm. Proof. (1) ⇔ (2) Suppose that 1n is not archimedean for some n, that is, there exists a nonzero x ∈ Enh so that r1n ± x ∈ En+ for all r > 0. We must have that y = v∗ xv ≠ 0 for some unit vector v ∈ ℝn . Seeing that r1 ± y = v∗ (r1n ± x)v ∈ E1+ , we verify that 1 is not archimedean. The converse is trivial. (2) ⇒ (3) We leave it as an exercise to check that ‖ ⋅ ‖′n is always a seminorm, that is, ′ ‖x‖n = ‖ − x‖′n , ‖λx‖′n = |λ| ‖x‖′n , and ‖x + y‖′n ≤ ‖x‖′n + ‖y‖′n for all λ ∈ ℂ and x, y ∈ En . For a nonzero x ∈ En , we have that 0 y := [ ∗ x

x h ] ∈ E2n ; 0

+ thus, there is a t > 0 so that for all s < t it is not the case that s12n ± y ∈ E2n . It follows ′ that ‖x‖n = t. (3) ⇒ (2) We can assume without loss of generality that 1 is not archimedean as witnessed by x ∈ E h . Now since x + r1 ∈ E + ,

[

x + r1 x + r1

x + r1 ] = [1, 1]t (x + r1)[1, 1] ∈ E2+ x + r1

Model theory of operator systems and C∗ -algebras �

361

and (2r)1 [ x + r1

x + r1 x + r1 ]−[ (2r)1 x + r1

x + r1 r1 − x ]=[ x + r1 0

0 ] ∈ E2+ . r1 − x

Since |‖x + r1‖′1 − ‖x‖′1 | ≤ ‖r1‖′1 ≤ r, we conclude that ‖x‖′1 = 0, so ‖ ⋅ ‖′1 is not a norm. Finally, (4) ⇒ (3) is trivial and (1) ⇒ (4) is a special case of (2) ⇒ (3). With this lemma now in hand, we see that the axiomatization of abstract operator systems as metric structures is complete. We note in passing that by the Choi–Effros representation theorem for abstract operator systems each En is metrically complete in the ‖ ⋅ ‖′n -norm, so there was nothing lost in insisting upon this in the metric structure language, though nothing is gained by doing so.

3.2 Formulas, theories, and models Now that the language is built, we briefly discuss the basic aspects of the continuous model theory of operator systems. We refer the reader to [15, Chapter 2] and [5] for a more in-depth treatment. Let ℒ be a language for metric structures as described in [15, Section 2.1]. To each domain of quantification in each sort, we assign an infinite number of variables. Definition 3.8. (1) An atomic formula φ(x1 , . . . , xn ) is an expression in the language, built using finitely many variables and function symbols, which terminates with the application of a predicate. (We will assume all predicates take values in ℝ+ .) Since each variable xi has an assigned domain of quantification 𝒟i , we have that f has domain domain(φ) := 𝒟1 × ⋅ ⋅ ⋅ × 𝒟n . Since each variable comes with a (bounded) domain of quantification and each function symbol has a modulus of uniform continuity we can assign a bounded range [0, K] ⊂ ℝ+ to φ. (2) A connective is a uniformly continuous function f : (ℝ+ )k → ℝ+ . (3) If φ1 , . . . , φk are atomic formulas over a common set of variables x = (x1 , . . . , xn ), we can define a quantifier-free formula as an expression of the form g(x) = f (φ1 (x), . . . , φk (x)) where f is a connective. (4) A formula h(x) is an expression of the form h(x) = Qℓ ⋅ ⋅ ⋅ Q1 g(x1 , . . . , xn ) where ℓ ≤ n, g is a quantifier-free formula, and Qi is either the existential quantifier infxi ∈𝒟i or the universal quantifier supxi ∈𝒟i . A formula is a sentence if all variables

362 � T. Sinclair are quantified, that is, if ℓ = n. Similarly to atomic formulas, each formula has an associated domain and bounded range. (5) To each formula h(x) = h(x1 , . . . , xn ) and each metric ℒ-structure M, we can associate an interpretation h(x)M of h which is the function from 𝒟1 (M) × ⋅ ⋅ ⋅ × Dn (M) to ℝ+ determined by interpreting all variables, function symbols, domains of quantification, etc., in the language ℒ in M. If h has range [0, K], then this means that h(x)M ∈ [0, K] for all ℒ-structures M and all x1 , . . . , xn ∈ M. Example 3.9. Let ℒ be the language of operator systems described above. As we will explain below in Remark 3.28, the sorts corresponding to the positive cones do not add any expressive power in terms of defining formulas, so it suffices to build formulas only using the sorts (En ). Since the matrix entries of any variable in En are interpreted as being in the sort E = E1 , it suffices to only consider formulas with variables in this sort alone. Thus, in the language of operator systems, atomic formulas are effectively expressions of the form ‖p(x)‖d where p(x) is a linear matrix ∗-polynomial of degree d, where each variable is restricted to a domain of quantification in E1 . It is a bit annoying to have to deal with the domains of quantification at this level, rather than just assigning each variable to the (unbounded) sort E1 . We will take this view, so each ‖p(x)‖d is technically what we will term an unbounded atomic formula. Definition 3.10. Let ℒ be a language for metric structures. (1) Any collection T of ℒ-sentences is called a theory. (2) An ℒ-structure M models T, written M 󳀀󳨐 T, if hM = 0 for all sentences h ∈ T. (3) A theory T is said to be consistent if there is an ℒ-structure M so that M 󳀀󳨐 T. (4) The theory of an ℒ-structure M, denoted Theory(M), is the collection of all ℒ-sentences h so that hM = 0. Clearly, M 󳀀󳨐 Theory(M). (5) We write Model(T) for the class of all models of a theory. We say that a class 𝒞 of ℒ-structures is elementary if 𝒞 = Model(T) for some theory T. We will discuss more about elementary classes at the end of the next section.

3.3 Ultraproducts Let I be an arbitrary set, and let (Ei )i∈I be a collection of (concrete) operator systems indexed by I. We define the direct product ∏i∈I Ei to be the set of all bounded functions x : I → ⨆i∈I Ei , the disjoint union, with xi ∈ Ei for all i ∈ I. This is a Banach space ∗-vector space under pointwise addition, scalar multiplication, and involution and norm ‖x‖ = supi∈I ‖xi ‖Ei . For representations Ei ⊂ B(Hi ), we see that ∏i∈I Ei is a concrete operator system isometrically represented on the Hilbert space direct sum ⨁i∈I Hi via x(⨁ ξi ) := ⨁ xi ξi . i∈I

i∈I

Model theory of operator systems and C∗ -algebras

� 363

It is left as an exercise to check that under this concrete representation we have that (∏i∈I Ei )+ = ∏i∈I Ei+ with unit 1 = (1i )i∈I where 1i ∈ Ei is the unit. For the rest of the section, I will be a fixed directed set and 𝒰 an ultrafilter on I. Our task will be to give a “concrete” definition of the ultraproduct of the operator systems (Ei )i∈I and then explain how this ultraproduct is the ultraproduct at the level of the language of operator systems that we have developed. Consider an arbitrary collection ϕ = (ϕi )i∈I of matrix states ϕi ∈ 𝒮n (Ei ). Each such ϕ induces a matrix state ϕ𝒰 : ∏ Ei → Mn , i∈I

defined by ϕ𝒰 (x) := lim ϕi (xi ). 𝒰

Exercise 3.11. Show that for ϕ = (ϕi ) ∈ ∏i∈I 𝒮 (Ei ), ⋂ ker(ϕ𝒰 ) = {(xi ) ∈ ∏ Ei : lim ‖xi ‖ = 0} =: 𝒥 . i∈I

ϕ=(ϕi )

𝒰

Hence, 𝒥 is a kernel. Exercise 3.12. Show that the class of 1 is an archimedean order unit for the ordered ∗-vector space quotient; hence, the evaluation map ⋅ ̂ : ∏i∈I Ei /𝒥 → (∏i∈I Ei /𝒥 )∗∗ is faithful and ∏i∈I Ei /𝒥 itself is equipped with a quotient operator system structure. Definition 3.13. We define the ultraproduct ∏𝒰 Ei of the operator systems (Ei )i∈I to be the operator system quotient ∏i∈I Ei /𝒥 . In other words, ∏𝒰 Ei is the operator system structure defined by the matrix states UCP(∏ Ei , Mn ) = {ϕ𝒰 : ϕ ∈ ∏ UCP(Ei , Mn )}. 𝒰

i∈I

To check that this ultraproduct is an ultraproduct of metric structures in the language of operator systems by [5, Chapter 5], it suffices to show that this construction is compatible with the metric ultraproduct for each sort, that is, 𝒟r (∏ Mn (Ei )) = ∏ 𝒟r (Mn (Ei )), 𝒰

𝒰

+

𝒟r ((Mn (∏ Ei )) ) = ∏ 𝒟r (Mn (Ei ) ) 𝒰

+

𝒰

for all r > 0 and n = 1, 2, . . . In fact, checking this for r = 1 suffices due to the existence of a metric compatible scalar multiplication operation, and we may further assume without loss of generality that n = 1. For the first equation, this follows immediately from Exercises 3.11 and 3.12. For the second equation, suppose, by way of contradiction, that there is x = (xi ) ∈ 𝒟1 (∏𝒰 Ei )+ so that lim𝒰 dist(xi , 𝒟1 (Ei+ )) =: α > 0. We can clearly assume that each xi is hermitian. Since ‖xi − 1‖ ≥ α/2 for i ∈ 𝒰 generic, we have that ‖xi ‖ ≥ 1 + α/2 for i ∈ 𝒰 generic. However, this contradicts that 1 = ‖x‖ = lim𝒰 ‖xi ‖, and we have established the second equation holds.

364 � T. Sinclair Notation 3.14. In the case that each Ei is identified with E, we write E 𝒰 := ∏𝒰 Ei and refer to E 𝒰 as the 𝒰 -ultrapower of E. Let 𝒞 be a class of ℒ-structures. We say that 𝒞 is closed under taking ultraroots if E ∈ 𝒞 whenever E 𝒰 for some ultrafilter 𝒰 . The following appears as [15, Theorem 2.4.1]. Proposition 3.15. A class 𝒞 of ℒ-structures is elementary (in the sense of Definition 3.10) if and only if 𝒞 is closed under isomorphisms, ultraproducts, and taking ultraroots. Example 3.16. By essentially the same reasoning as the proof of the Choi–Effros representation theorem (Theorem 2.20), we may see that every operator system E admits a complete order embedding into ∏𝒰 Mni for some nonprincipal ultrafilter on some directed set I. Thus, the class of all operator systems is the smallest elementary class containing all matrix algebras which is closed under taking substructures. Example 3.17. We say an operator system E is minimal if it is a subsystem of an abelian C∗ -algebra; equivalently, if Ê ⊂ C(𝒮 (E)) is a complete order embedding. Since the operator system ultraproduct of unital (abelian) C∗ -algebras is again a unital (abelian) C∗ -algebra under the natural multiplicative structure, we have that the class of minimal operator systems is elementary. Problem 3.18. Find an axiomatization of the class of minimal operator systems. By the same token, if 𝒞 is an elementary class of unital C∗ -algebras, then the class of all subsystems of elements of 𝒞 is an elementary class of operator systems. Question 3.19. Is there an elementary class of operator systems which is closed under taking subsystems which is not the class of all subsystems of some elementary class of unital C∗ -algebras?

3.4 Definability From a practical viewpoint, the most important task at hand once a theory T is constructed is to begin to explore the definable sets in the models of T. In short, this is because the definable sets are exactly those which it is permissible to quantify over, so having a large class of natural sets being definable allows for a great range of intuitive constructions of formulas, “breathing life” into the theory. From the point of view of a working analyst, the concept of definability is perhaps the key feature of continuous model theory as to say that a subset of a structure is definable is to say that it possesses stability/rigidity under small perturbations. In this light many important results in the theory of operator systems and C∗ -algebras can be seen as establishing the definability of sets of elements satisfying certain formulas. Our treatment of definability here is directly taken from [15, Chapter 3], though we will center our discussion here on the practical definition of definability for sets which is afforded by the Beth definability theorem; see [15, Section 4.2] or [5, Theorem 9.32].

Model theory of operator systems and C∗ -algebras

� 365

Let T be a theory and 𝒞 an elementary class of models of T. Let F be an assignment to each A ∈ 𝒞 a closed subset F(A) ⊂ ∏di=1 𝒟ri (A), where 𝒟r1 (A), . . . , 𝒟rd (A) are domains in A for d, r1 , . . . , rd fixed. We write d

codomain(F) := ∏ 𝒟ri i=1

so that the interpretation of codomain(F) over A ∈ 𝒞 is d

codomain(F)A = ∏ 𝒟ri (A). i=1

Definition 3.20. We say that F is a uniform assignment if F is a functor, that is, for every homomorphism φ : A → B with A, B ∈ ℳ we have that φ(F(A)) ⊂ F(B). Remark 3.21. In the language of operator systems defined above, we have that the morphisms are always the unital, completely positive maps which are, rather conveniently, all contractions. When considering the model theory of operator systems (or C∗ -algebras), it makes sense to introduce the following unbounded variant of a uniform assignment. Let 𝒞 be an elementary class in the theory of operator systems. Suppose that F assigns to every A ∈ 𝒞 a closed subset F(A) ⊂ Mr1 (A) × ⋅ ⋅ ⋅ × Mrk (A) × Ms1 (A)+ × ⋅ ⋅ ⋅ × Msl (A)+ × ℂd . We say that F is then an unbounded uniform assignment if its restriction to every product of domains is a uniform assignment. Definition 3.22. Let 𝒞 be an elementary class of operator systems and F be an unbounded uniform assignment on 𝒞 . We say that F is definable if for every ultraproduct ∏𝒰 Ei of elements Ei ∈ 𝒞 we have that F(∏ Ei ) = ∏ F(Ei ). 𝒰

𝒰

To every (unbounded) formula f (x), there is a natural (unbounded) uniform assignment Zf which assigns to every E ∈ 𝒞 the zero set Zf (E) = {x ∈ domain(f )E : f (x) = 0}. Exercise 3.23. Let f be a formula. We have that Zf is definable over 𝒞 if and only if for every ϵ > 0 there is δ > 0 so that for all A ∈ 𝒞 and x ∈ domain(f )A , we have that f (x)A ≤ δ implies that dist(x, Zf (A)) ≤ ϵ.

366 � T. Sinclair Theorem 3.24 (Beth definability theorem). Let 𝒞 be the class of models of a theory T, and let F be a uniform assignment. Writing pF (x) := dist(x, F),

x ∈ codomain(F),

we interpret pF (x)A as dist(x, F(A)) for x ∈ codomain(F)A . We have that F is definable if and only if there is a sequence of formulas f1 , f2 , . . . with domain(fi ) = codomain(F) so that 󵄨 󵄨 sup sup{󵄨󵄨󵄨pF (x)A − fn (x)A 󵄨󵄨󵄨 : x ∈ codomain(F)A } ≤ 1/n

A∈ℳ

for all n = 1, 2, 3, . . . We refer the reader to [15, Section 4.2] or [5, Chapter 9] for a proof. Definition 3.25. If ℳ is the class of models for a theory T and p is a predicate, we say that p is definable if there is a sequence of formulas f1 , f2 , . . . with domain(fi ) = domain(p) so that 󵄨 󵄨 sup sup{󵄨󵄨󵄨p(x)A − fn (x)A 󵄨󵄨󵄨 : x ∈ domain(p)A } ≤ 1/n.

A∈ℳ

The content of the Beth definability theorem is then to say that if F is a definable functor, then pF (x) = dist(x, F) is a definable predicate. In the unbounded case, this means pF restricted to any product of domains of quantification in codomain(F) is a definable predicate. Example 3.26. The hermitian elements in any operator system form a definable set. Example 3.27. For each n, we have that dn+ (x) := dist(x, in (Cn )), x ∈ En , is a definable predicate. Remark 3.28. In fact, something even stronger may be said. Consider the (unbounded) quantifier-free formula pt,n (x) := ‖2x − t1n ‖n ∸ t for t ≥ 0 with domain Enh . We have that pt,n (x) = 0 if and only if ‖x‖n ≤ t and x ∈ Cn . This is because for a hermitian element x ∈ En we have that ‖2x − t1n ‖n ≤ t if and only if 0 ⪯ x ⪯ t1n . Therefore, every formula in the language of operators systems can be replaced with an equivalent formula where all quantifiers have domain in En for some n. That is, it is never necessary to quantify over the positive cones. Proposition 3.29. For each homogeneous, hermitian linear matrix ∗-polynomial p(x) of degree d in x = (x1 , . . . , xn ), the set of tuples X = (X1 , . . . , Xn ) in Ek or Ekh satisfying 1d ⊗ 1k ⪰ p(X) is a definable set.

Model theory of operator systems and C∗ -algebras �

367

Proof. For ease of notation, we will write 1 for 1d ⊗1k . Suppose that dist(1−p(X), Cdk ) < ϵ. Since 1 − p(X) is hermitian, we claim (1 + ϵ)1 ⪰ p(X). Indeed, since there exists Y ∈ Cdk so that ‖1 − p(X) − Y ‖dk ≤ ϵ, we have that ϵ1 ⪰ 1 − p(X) − Y ⪰ −ϵ1,

so 1 − p(X) ⪰ Y − ϵ1 ⪰ −ϵ1.

1 Setting X ′ = 1+ϵ X, by linearity and homogeneity, 1 ⪰ p(X ′ ) and ‖Xi′ − Xi ‖k ≤ ϵ‖Xi ‖k for i = 1, . . . , n. The result now follows by Exercise 3.23.

Exercise 3.30. Let p(x) be a homogeneous, hermitian linear matrix ∗-polynomial. If p(x) ⪰ cI has a solution over ℝ for some c > 0, then the set of tuples X = (X1 , . . . , Xn ) in E or E h satisfying p(X) ⪰ 0 is definable. Question 3.31. If p(x) is a homogeneous linear matrix ∗-polynomial, is the set of all tuples X1 , . . . , Xn in Ek satisfying 1 ⪰ p(X) definable? If not, is there a natural class of operator systems or C∗ -algebras where all such sets are definable? Remark 3.32. We may define a linear operator ∗-polynomial to be an expression of the form q(x) = A1 ⊗x1 +⋅ ⋅ ⋅+An ⊗xn +B1 ⊗x1∗ +⋅ ⋅ ⋅+Bn ⊗xn∗ +C⊗1 where A1 , . . . , An , B1 , . . . , Bn , C ∈ ℬ(H). All terminology from linear matrix ∗-polynomials carries over to this more general context equally well. Just as an finite family of linear matrix ∗-polynomials can be combined into a single linear matrix ∗-polynomial, any infinite family of linear matrix ∗-polynomials can be combined into a single linear operator ∗-polynomial. The main point is that by the same proof as Proposition 3.29, we have the following. Proposition 3.33. For each homogeneous, hermitian linear operator ∗-polynomial q(x) in x = (x1 , . . . , xn ), the set of tuples X1 , . . . , Xn in Ek or Ekh satisfying 1H ⊗ 1k ⪰ q(X) is definable. Via block diagonal embedding, we have the following consequence. Corollary 3.34. Let {pi : i ∈ I} be a collection of homogeneous, hermitian linear matrix ∗-polynomials in x = (x1 , . . . , xn ). The set of tuples X1 , . . . , Xn in Ek or Ekh satisfying 1di ⊗1k ⪰ pi (X) for all i ∈ I is definable. For an operator system E, we may regard the set UCP(Mn , E) as a closed subset of 2 𝒟1 (E)n via the correspondence φ ↔ (φ(eij ))ij . The following result was first observed in [15, Section 5.8]. Proposition 3.35. For each n, we have that UCP(Mn , E) is a definable set for the class of all operator systems. Proof. We have by Lemma 2.26 that CP(Mn , E) ≅ CP(ℂ, Mn (E)). This correspondence identifies UCP(Mn , E) with the set of all X ∈ Mn (E)+ with tr(X) = ∑i Xii = 1. Suppose

368 � T. Sinclair E = ∏𝒰 Ei . We have by the remarks after Definition 3.13 that for any X ∈ Mn (E)+ there is a net Xi ∈ Mn (Ei )+ with X = (Xi ). Given any ϵ > 0, we have that ‖ tr(Xi ) − 1‖ ≤ ϵ for i ∈ 𝒰 generic. Let bi := tr(Xi ) ∈ E + . Since ‖bi − 1‖ ≤ ϵ, we have that b is invertible with ‖b−1/2 − 1‖ < ϵ1/2 . Setting i [Xi′ ]kl := b−1/2 [Xi ]kl b−1/2 i i we have that Xi′ ∈ Mn (Ei )+ with tr(Xi′ ) = 1 and ‖Xi − Xi′ ‖ ≤ 2n2 ϵ1/2 . Definition 3.36. We say that a formula f is positive if it is built from atomic formulas only using connectives θ(x1 , . . . , xn ) satisfying θ(x1 , . . . , xn ) ≤ θ(y1 , . . . , yn ) when xi ≤ yi , i = 1, . . . , n. Remark 3.37. Note that the proof above shows that UCP(Mn , E) is explicitly the zero set of the quantifier-free, positive formula f (X) := ‖ tr(X) − 1‖ where domain(f ) = 𝒟n2 (Cn ). Let Sn be the uniform assignment with codomain 𝒟1 (E)n × ℂn which assigns to each operator system the (closed) subset of all (x1 , . . . , xn , λ1 , . . . , λn ) so that xi 󳨃→ λi extends to a state on E. The following result is essentially contained in [15, Section 5.8], though we give a different proof here. Proposition 3.38. We have that Sn is the zero set of a uniform limit of quantifier-free formulas. (We refer to this condition on Sn as being quantifier-free definable.) Proof. Let 𝔻 ⊂ ℂ be the closed unit disk. For convenience, we write x = (x−n , . . . , xn ) with x0 = 1 and x−i = xi∗ and λ = (λ−n , . . . , λn ) with λ−i = λ̄i and λ0 = 1. Let 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 fn (x, λ) := sup (󵄩󵄩󵄩∑ ai λi 󵄩󵄩󵄩 ∸ 󵄩󵄩󵄩∑ ai xi 󵄩󵄩󵄩) 󵄩 󵄩 󵄩 󵄩󵄩 ai ∈𝔻 󵄩 i 󵄩 󵄩i and note that fn is a uniform limit of quantifier-free formulas. (Recall x ∸ y = max{x − y, 0}.) Indeed, let (Kp ) be a finite 1/p-net in 𝔻, and note that fn is a uniform limit of the formulas 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 fn,p (x, λ) = max(󵄩󵄩󵄩∑ ai λi 󵄩󵄩󵄩 ∸ 󵄩󵄩󵄩∑ ai xi 󵄩󵄩󵄩) 󵄩 󵄩 󵄩󵄩 ai ∈Kp 󵄩 󵄩i 󵄩 󵄩i by an application of the triangle inequality. From now on, let us fix an operator system E. We claim that fn (x, λ)E = 0 if and only if xi 󳨃→ λi extends to a state on E. By Proposition 2.7, Lemma 2.23, and Theorem 2.29, it suffices to check that fn (x, λ)E = 0 implies that φ : xi 󳨃→ λi defines a unital, ∗-linear contraction on F := span{x−n , . . . , xn }. Observe that fn (x, λ)E = 0 is equivalent to 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩∑ ai xi 󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩∑ ai λi 󵄩󵄩󵄩 for all a−n , . . . , an ∈ 𝔻. 󵄩󵄩 󵄩󵄩 i 󵄩󵄩 󵄩󵄩 i

Model theory of operator systems and C∗ -algebras

� 369

Suppose that ∑i ai xi = ∑i bi xi so that ∑i 21 (ai − bi )xi = 0; thus, ∑i 21 (ai − bi )λi = 0. This shows that φ : F → ℂ is a well-defined, unital, ∗-linear map. Finally, {∑i ai xi : ai ∈ 𝔻} contains a sufficiently small ball about the origin in F, so φ is a contraction. Since Mn (ℂ) ⊂ Mn (E) canonically ∗-isometrically for any operator system E, we have the following corollary. Corollary 3.39. Let Sn,k be the uniform assignment with codomain 𝒟1 (E)n × 𝒟1 (Mk (E))n which assigns to each operator system the (closed) subset of all (x1 , . . . , xn , y1 , . . . , yn ) so that y1 , . . . , yn ∈ Mk (ℂ) and xi 󳨃→ yi extends to a matrix state. We have that Sn,k is quantifier-free definable. Remark 3.40. The previous proofs do not show that the Sn,k are positive quantifier-free definable since ∸ is not a positive connective. In fact, it is not possible for n ≥ 5 for Sn,k to be positive quantifier-free definable for all k. This is because such a result would show that any quotient of a nuclear operator system (see Section 4.1 below) would be nuclear, essentially following the reasoning of [15, Section 5.14], but there is a five-dimensional nuclear operator system with a nonnuclear quotient which is a variant of Example 2.37; see [27].

3.5 Model theory of C∗ -algebras We will treat the model theory of C∗ -algebras in a much more abbreviated fashion than operator systems. We refer the reader to Szabo’s article in this volume for background on the basic theory of C∗ -algebras and to [15, 16] for an in-depth treatment of the model theory. In brief, the language of C∗ -algebras is the language of a normed, complex algebra A with a unary involution function ∗ satisfying the axiom 󵄩󵄩 ∗ 󵄩󵄩 2 󵄩󵄩x x 󵄩󵄩 = ‖x‖

for all x ∈ A.

The sorts are, naturally, axiomatized to be the balls around the origin. This is all spelled out in [16, Section 3.1] or [15, Section 2.2]. The atomic formulas in the theory of C∗ -algebras are of the form ‖p(x)‖ where p is a noncommutative ∗-polynomial in the variables x = (x1 , . . . , xn ). Let A be a C∗ -algebra. We say that an element x ∈ A is positive if x = y∗ y for some y ∈ A. (Importantly, y can be chosen with ‖y‖ = √‖x‖.) It is well known that the set A+ of positive elements forms a closed cone. In the case that A is a unital C∗ -algebra then the unit is an archimedean order unit with respect to A+ . As in the case of operator systems, this positivity structure is crucial to the theory of C∗ -algebras. Exercise 3.41. We have that A+ is an (unbounded) definable set in the language of C∗ -algebras; thus, dist(x, A+ ) is a definable predicate.

370 � T. Sinclair As noted above, every unital C∗ -algebra is an operator system under the natural positivity structure imposed on Mn (A) by the fact that Mn (A) is again a C∗ -algebra. We would now like to establish that every property about A that can be expressed in the language of operator systems can be expressed in the language of C∗ -algebras. This reduces to proving the following result, due to Lupini, which can be found in [20, Appendix C]. Proposition 3.42. Let 𝒞 be the elementary class of unital C∗ -algebras. The uniform assignment which sends A ∈ 𝒞 to the norm unit ball of Mn (A) considered as a subset of 2 𝒟1 (A)n is definable. Since by Remark 3.28 every formula in the language of operator systems is equivalent to one quantified only over balls in En for some n, it follows that the operator system structure on A is definably interpretable in the language of C∗ -algebras. We begin by noting the following lemma. Lemma 3.43. The set of all tuples (x1 , . . . , xn ) so that ‖ ∑ni=1 xi∗ xi ‖ ≤ 1 is definable. Proof. Since ∑ni=1 xi∗ xi is positive, we have that ‖ ∑ni=1 xi∗ xi ‖ ≤ 1 if and only if n

∑ xi∗ xi ⪯ 1. i=1

The proof now follows along the similar lines to the proof of Proposition 3.29. Proof of Proposition 3.42. For tuples a, b ∈ An , define ⟨a, b⟩ := ∑ni=1 ai∗ bi ∈ A and ‖a‖ := ‖⟨a, a⟩‖1/2 . This gives An the structure of a left Hilbert C∗ -module in the sense of [34]. By [34, Proposition 1.1], the following variant of the Cauchy–Schwarz inequality is valid: 󵄩󵄩 󵄩 󵄩󵄩⟨a, b⟩󵄩󵄩󵄩 ≤ ‖a‖ ‖b‖. Note that there is a natural action of Mn (A) on An by left multiplication and that ⟨xa, b⟩ = ⟨a, x ∗ b⟩ where x ∗ is the usual adjoint of x in Mn (A). For x ∈ Mn (A), we define the norm 󵄩 󵄩 ‖x‖A := sup{󵄩󵄩󵄩⟨xa, b⟩󵄩󵄩󵄩 : ‖a‖, ‖b‖ ≤ 1} = sup{‖xa‖ : ‖a‖ ≤ 1}, where the last equality is a consequence of the Cauchy–Schwarz inequality above. It is therefore the case that 󵄩󵄩 ∗ 󵄩󵄩 󵄩 ∗ 󵄩 󵄩󵄩x x 󵄩󵄩A = sup{󵄩󵄩󵄩⟨x xa, b⟩󵄩󵄩󵄩 : ‖a‖, ‖b‖ ≤ 1} 󵄩 󵄩 = sup{󵄩󵄩󵄩⟨xa, xb⟩󵄩󵄩󵄩 : ‖a‖, ‖b‖ ≤ 1} ≥ sup{‖xa‖2 : ‖a‖ ≤ 1} = ‖x‖2A .

Model theory of operator systems and C∗ -algebras

� 371

It is straightforward to check that ‖xy‖A ≤ ‖x‖A ‖y‖A for all x, y ∈ Mn (A), hence ‖x ∗ x‖A = ‖x‖2A . That Mn (A) is complete in the ‖ ⋅ ‖A -norm is just an application of the triangle inequality and completeness of A. Thus, we see that ‖ ⋅ ‖A is a C∗ -norm on Mn (A). It must then be the case that ‖ ⋅ ‖A is identical to the operator norm on Mn (A) as C∗ -norms are unique as a consequence of the fact that every ∗-homomorphism of C∗ -algebras is contractive. Since ‖ ⋅ ‖A is definable by Lemma 3.43, the result now follows. Remark 3.44. Having just established that every formula in the language of operator systems is definably interpretable in the language of C∗ -algebras, it is natural to ask whether the converse is true. The following result shown in [22] is a step in this direction. Proposition 3.45. The elementary class 𝒞 of all unital C ∗ -algebras is an elementary class in the language of operator systems. Question 3.46. Is the multiplication operation on C∗ -algebras definable in the language of operator systems? By Proposition 3.15, establishing that 𝒞 forms an elementary class is equivalent to establishing that if E is an operator system so that the operator system structure on some ultrapower E 𝒰 is a reduct of some (unital) C∗ -algebra structure on E 𝒰 , then the same is true for E. We note that while a positive solution to Question 3.46 would show that the languages of operator systems and unital C∗ -algebras are biinterpretable, the quantifier complexity of equivalent formulas may differ depending on the language chosen.

4 Model theory and finite-dimensional approximation properties The importance of finite-rank approximations to the theory of Banach spaces and its connection with the theory of Banach-space tensor products was first made apparent by the seminal work of Grothendieck [23]. In the case of C∗ -algebras, the theory of finitedimensional (matrix) approximations and the largely parallel theory of tensor products of C∗ -algebras were developed in foundational works of Arveson, Choi, Connes, Effros, Kirchberg, Lance, Takesaki, and Tomiyama, among others, and these ideas and concepts continue to form a central part in the theory of operator algebras and operator systems. In this last section, we survey some of the connections that have been made between the model theory of operator systems and their finite-dimensional approximation properties. We refer the reader to the books of Brown and Ozawa [6] and Pisier [40, 41] for an in-depth treatment of the connections between finite-dimensional approximation properties and tensor products of C∗ -algebras and operator systems.

372 � T. Sinclair

4.1 Nuclear and exact operator systems Definition 4.1. An operator system E is said to be nuclear if there are nets φi : E → Mni and ψi : Mni → E of unital, completely positive maps so that ψ ∘ φ converges in the pointwise-norm topology to the identity, that is, limi ‖ψi ∘ φi (x) − x‖ = 0 for all x ∈ E. Remark 4.2. Technically, nuclearity of a C∗ -algebra A refers to the condition that for every C∗ -algebra B there is exactly one cross-norm on the algebraic tensor A ⊙ B which can be completed to a C∗ -algebra; see Szabo’s article in this volume. The equivalence of nuclearity and local completely positive factorization of the identity map through matrix algebras is due to Choi and Effros [11] and Kirchberg [30] and was extended in the appropriate way to the operator system category in [29]. Remark 4.3. (1) It is trivial that Mn is nuclear for each n = 1, 2, . . . and that Mn (E) is nuclear if and only if E is. (2) The classes of AF and UHF C∗ -algebras as described in Szabo’s article in this volume are nuclear. The Cuntz algebras 𝒪n defined therein are also nuclear. (3) ℬ(H) is not nuclear for H infinite dimensional; see [6, Proposition 2.4.9]. (4) It is false in general that C∗ -subalgebras of nuclear C∗ -algebras are nuclear. Remark 4.4. There is a subsystem of M3 which is not nuclear. Indeed, let E be the subsystem of all matrices A ∈ M3 with A13 = A31 = 0. If E was nuclear there would be sequences of unital completely positive maps φn : E → Mn and ψn : Mn → E so that ψn ∘ φn → idE . By Arveson’s extension theorem, we may extend each φn to a unital completely positive map φ̃ n : M3 → Mn , and we see that a cluster point of ψn ∘ φ̃ n gives a completely positive projection p : M3 → E. By [9, Theorem 3.1], this would define a C∗ -algebra structure on E under the multiplication x ⋅ y := p(xy). There are two ∗-homomorphisms M2 → E, A 󳨃→ [

A 0

0 0 ],[ 0 0

0 ] A

with distinct ranges, which are respected by the C∗ -algebra structure on E. However, since every finite-dimensional C∗ -algebra is a direct sum of matrix algebras this is a contradiction since E is seven-dimensional but contains two distinct (nonunital) ∗-subalgebras isomorphic to M2 , so would need to have dimension at least eight. Exercise 4.5. Let X be a compact, Hausdorff space. Show that C(X), the algebra of continuous functions f : X → ℂ under the norm ‖f ‖∞ := supx |f (x)| is nuclear. As a hint, use the fact that for every finite open cover 𝒪 = {O1 , . . . , On } of X there is a partition of unity subordinate to 𝒪. That is, there are continuous functions fi : X → [0, 1] with fi supported in Oi and ∑ni=1 fi (x) = 1 for all x ∈ X. With the definition and basic examples established, we know turn to understanding the model theory of nuclear operator systems.

Model theory of operator systems and C∗ -algebras �

373

Definition 4.6. Let T be a theory of ℒ-structures, and let x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) be finite collections of symbols. Let ℱ be a family of (definable) formulas for T, each depending only on x. We say that ℱ is uniform if there is a continuous function u : ℝ+ → ℝ+ with u(0) = 0 so that 󵄨 󵄨 T 󳀀󳨐 󵄨󵄨󵄨f (x) − f (y)󵄨󵄨󵄨 ≤ u(max d(xi , yi )), i

for all f ∈ ℱ .

For a class 𝒞 of ℒ-structures so that C 󳀀󳨐 T for all C ∈ 𝒞 , we say that 𝒞 is definable by uniform families of formulas if there is a collection of formulas {fn,k : n, k = 1, 2, . . .} defined on a countable number of symbols x = (x1 , x2 , . . .) so that ℱk := {fn,k : n = 1, 2, . . .} is uniform for each k and [E 󳀀󳨐 T and sup inf fn,k (x)E ≡ 0] if and only if E ∈ 𝒞 . k

n

We refer [15, Section 5.7] for the connection between definability by uniform families of formulas and being definable by omitting (partial) types. Proposition 4.7. The property that an operator system E is nuclear is definable by uniform families of existential formulas. Proof. The formulas fn,k are given by 󵄩 󵄩 fn,k (x1 , . . . , xk ) := inf{max󵄩󵄩󵄩xi − ψ ∘ φ(xi )󵄩󵄩󵄩 : φ ∈ UCP(E, Mn ), ψ ∈ UCP(Mn , E)} i

which are existential and definable by Propositions 3.35 and 3.38 and Corollary 3.39. Since ψ ∘ φ is a contraction, we have that each fn,k is 2-Lipschitz, so the families ℱk are uniform. Finally, if infn fn,k (x)E = 0 for all tuples x in E, we can easily construct nets (φi ) and (ψi ) indexed over the directed set of pairs (F, ϵ) where F is a finite subset of E and ϵ > 0 so that ψi ∘ φi converges to the identity on E in the pointwise-norm topology, so E E is nuclear. Conversely, the existence of such a net easily implies that infn fn,k ≡ 0 for all k. Question 4.8. For the elementary class of unital C∗ -algebras, is nuclearity defined by a uniform family of positive existential formulas? Since the values of positive existential formulas decrease under surjective morphisms, this would recover the result due to Choi and Effros, [8, Corollary 3.3] and [10, Corollary 4], that nuclear C∗ -algebras are closed under taking C∗ -algebra quotients avoiding Connes’ [12] celebrated but difficult work on the classification of injective factors. However, as noted in Remark 3.40, this seems like it would require substantially different ideas to avoid the false implication in the operator system category. We refer to [15, Section 5.9] for one such attempt. Definition 4.9. Let E be a finite-dimensional operator system. We say that E is exact operator system, exact if for every ϵ > 0 there is a unital, completely positive embedding

374 � T. Sinclair φ : E → Mn for some n so that ‖φ−1 |φ(E) ‖cb ≤ 1 + ϵ. We say that an operator system is exact if every finite-dimensional subsystem is exact. Remark 4.10. (1) Every nuclear operator system is exact. (2) It is clear from the definition that any subsystem of an exact operator system is exact. As a special case, every subsystem of Mn is exact. (3) Since every subsystem of any unital abelian C ∗ -algebra A is exact, MIN(V ) is exact for any archimedean ordered ∗-vector space V . (4) A difficult theorem of Kirchberg [32] establishes that C∗ -algebraic quotients of exact C∗ -algebras are exact. This is again false in the operator system category by the same counterexample as given in Remark 3.40. Remark 4.11. In Vignati’s article in this volume, a finite-dimensional operator space, that is, a closed subspace of ℬ(H), is said to be c-exact for some c ≥ 1 if for every ϵ > 0 there is a completely contractive embedding φ : E → Mn for some n so that ‖φ−1 |φ(E) ‖cb ≤ c + ϵ. Let E ⊂ ℬ(H) be a finite-dimensional operator space. Define an operator system Ẽ ⊂ M2 (ℬ(H)) by λ1 Ẽ := {[ ∗ y

x ] : x, y ∈ E, λ, μ ∈ ℂ} . μ1

We have that E is 1-exact as an operator space if and only if Ẽ is an exact operator system. We leave this as an exercise based on the following “2 × 2-matrix trick” due to Paulsen: φ : E → ℬ(K) is completely contractive if and only if φ̃ : Ẽ → M2 (ℬ(K)) given by φ̃ ([

λ1 y∗

x λ1 ]) := [ μ1 φ(y)∗

φ(x) ] μ1

is (unital) completely positive. We refer the reader to [6, Theorem B.5] or [37, Lemma 8.1] for a proof of this fact. We point out that exactness, similarly to nuclearity, can be phrased as in terms of local completely positive factorization of some (equivalently, every) inclusion E ⊂ ℬ(H) through matrix algebras, the crucial difference being that the images of the approximating sequence need not land back in the system itself. Proposition 4.12. A finite-dimensional operator system E ⊂ ℬ(H) is exact if and only if for each ϵ > 0 there are unital, completely positive maps φ : E → Mn and ψ : φ(E) → ℬ(H) so that ‖ψ ∘ φ − idE ‖ < ϵ where idE is the restriction of the identity map on ℬ(H) to E. Proof. We have that φ(E) ⊂ Mn and that φ−1 : φ(E) → ℬ(H) is unital and self-adjoint. By [6, Corollary B.9], there is a unital, completely positive map ψ : φ(E) → ℬ(H) with ‖ψ − φ−1 ‖cb ≤ 2(‖φ−1 ‖cb − 1); hence,

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375

󵄩 󵄩 󵄩 󵄩 ‖ψ ∘ φ − idE ‖ ≤ ‖ψ ∘ φ − idE ‖cb = 󵄩󵄩󵄩ψ ∘ φ − φ−1 ∘ φ󵄩󵄩󵄩cb ≤ 󵄩󵄩󵄩ψ − φ−1 󵄩󵄩󵄩cb ‖φ‖cb ≤ 2ϵ. In the other direction, suppose there are unital, completely positive maps φ, ψ as above, and let y1 , . . . , yn , y∗1 , . . . , y∗n be such a basis/dual basis pair for φ(E) as given in Lemma 2.32. We have that n

(ψ − φ−1 )(z) = ∑ y∗i (z)(ψ(yi ) − φ−1 (yi )), i=1

so 󵄩󵄩 󵄩 󵄩 −1 󵄩 −1 󵄩󵄩ψ − φ 󵄩󵄩󵄩 ≤ dim(E) max󵄩󵄩󵄩ψ(yi ) − φ (yi )󵄩󵄩󵄩. i

Setting yi = φ(xi ), we have that 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ‖xi ‖ ≤ 󵄩󵄩󵄩ψ ∘ φ(xi ) − xi 󵄩󵄩󵄩 + 󵄩󵄩󵄩ψ ∘ φ(xi )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩φ(xi )󵄩󵄩󵄩 + ϵ = 1 + ϵ; hence, maxi ‖ψ(yi ) − φ−1 (yi )‖ = maxi ‖ψ ∘ φ(xi ) − xi ‖ ≤ (1 + ϵ)ϵ. Given the similarities between nuclearity and exactness, it is natural to wonder if exact operator systems have a similar low-complexity description. A result of Goldbring and the author [21] shows that this is, however, not the case. Proposition 4.13. The class of exact operator systems is not definable by uniform families of existential formulas. Question 4.14. Is the class of exact C∗ -algebras definable by uniform families of (positive, existential) formulas?

4.2 The lifting property Definition 4.15. An operator system E has the lifting property (LP) if for every unital C∗ -algebra A and every ideal I, for every unital, completely positive map φ : E → A/I there exists a unital, completely positive map φ̃ : E → A lifting φ, that is, q ∘ φ̃ = φ where q : A → A/I is the quotient map. The following result is due to Robertson and Smith [42]. Proposition 4.16. Let E be a finite-dimensional operator system and B/J be a quotient C∗ algebra. For each n = 1, 2, . . ., every unital, completely positive map φ : E → B/J admits a unital n-positive lifting φ̃ : E → B. The following is immediate from Definition 2.40. Corollary 4.17. Every finite-dimensional, k-maximal operator system has the lifting property.

376 � T. Sinclair Definition 4.18. A finite-dimensional operator system E is said to be CP-stable if for every ϵ > 0 there exist δ > 0 and n so that for any unital map φ : E → A into any C∗ -algebra A satisfying ‖φ‖n ≤ 1 + δ, there is a unital, completely positive map φ′ : E → A so that ‖φ′ − φ‖ ≤ ϵ. Parts of the following result are due to Kavruk [26, Theorem 6.6] and Goldbring and the author [20, Proposition 2.42], [21, Section 7], as well as [43]. Theorem 4.19. For a finite-dimensional operator system E, the following are equivalent: (1) E has the lifting property; (2) E ∗ is exact; (3) E is CP-stable; (4) for every sequence (Ai ) of unital C∗ -algebras and every nonprincipal ultrafilter 𝒰 , every unital, completely positive map φ : E → ∏𝒰 Ai admits a unital, completely positive lifting φ̃ : E → ∏ Ai ; (5) for every sequence (Mni ) of matrix algebras and every nonprincipal ultrafilter, 𝒰 every unital, completely positive map φ : E → ∏𝒰 Mni admits a unital, completely positive lifting φ̃ : E → ∏ Mni . Remark 4.20. Let E be a finite-dimensional operator system with a hermitian basis {1 = x0 , x1 , . . . , xn }, and consider the elementary class 𝒞 of all operator systems which are unital C∗ -algebras [22]. Consider the uniform assignment on 𝒞 given by n

LE (A) := {(a1 , . . . , an ) ∈ (Ah ) : φ(xi ) = ai , i = 1, . . . , n, for some φ ∈ UCP(E, A)}. Then the equivalence (1) ⇔ (4) shows that E has the lifting property if and only if LE is definable. Theorem 4.19 allows one to give an interesting interpretation of the lifting property for certain maximal-type operator systems in terms of definability via Proposition 3.29. Example 4.21. Let p(x) with x = (x1 , . . . , xn ) be a homogeneous, hermitian linear matrix ∗-polynomial of degree d. Consider the closed convex set K := {x ∈ ℝn : 1d ⪰ p(x)}. (Any convex set of this form is known as a spectrahedron.) Suppose that K is bounded with 0 as an interior point. Recalling the construction from Remark 2.41 of the archimedean ordered ∗-vector space VK = (ℂn+1 , PK , e) from K, we define an operator system structure on VK by defining Kℓ = UCP(VK , Mℓ ) to be given by all linear maps f : ℝn+1 → Mℓh with f (e1 ) = Iℓ so that the tuple X = (X1 , . . . , Xn ) given by Xi := f (ei+1 ) satisfies 1d ⊗ 1ℓ ⪰ p(X)

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where e1 , e2 , . . . , en+1 form the standard basis for ℝn+1 . (Note the order unit e is e1 .) The collection of sets (Kℓ )ℓ∈ℕ is known as the free spectrahedron defined by p(x) [33]. Consequently, we refer to the operator system structure we have just constructed on VK as the free operator system generated by p, which we will denote by ℱ (p). Proposition 4.22. Using the same assumptions and notation as Example 4.21, the free operator system ℱ (p) has the lifting property. Proof. Let φ : ℱ (p) → ℬ(H) be a unital, completely positive map, and define Xi := φ(ei+1 ) for i = 1, . . . , n as above. Since ℱ (p) is finite-dimensional, we may assume without loss of generality that H is separable. (This is just to avoid the slight inconvenience of working with nets rather than sequences, but the subsequent arguments translate to nets practically verbatim.) Let H1 ⊂ H2 ⊂ ⋅ ⋅ ⋅ ⊂ H be an increasing sequence of finite-dimensional subspaces whose union is dense in H, and let pj : H → Hj be the orthogonal projection onto Hj . We have that φ(j) : ℱ (p) → Mmj defined by φ(j) (x) := pj φ(x)pj is unital, completely positive; thus, for the tuple X (j) = (X1 , . . . , Xn ) given by Xi := φ(j) (ei+1 ) = pj φ(ei+1 )pj we have that X (j) is hermitian and 1d ⊗ 1mj ⪰ p(X (j) ) holds. Since X (j) con(j)

(j)

(j)

verges to X in the strong operator topology, p(X (j) ) also converges strongly to p(X), so 1d ⊗ 1H ⪰ p(X) as well. Now let φ : ℱ (p) → ∏𝒰 Mmj be a unital, completely positive map. Since ℱ (p) is finite-dimensional, we may assume that φ = (φj )𝒰 where φj : ℱ (p) → Mmj is unital, h ∗-linear, and sends ℝn+1 into Mm . Setting Yi := φj (ei+1 ), we have that j (j)

+ dist(1d ⊗ 1mj − p(Y (j) ), Mdm ) 0, there is a n-contractive ∗-linear map φ : A → E so that ‖φ(x) − x‖ ≤ ϵ‖x‖ for all x ∈ E ∩ A. Moreover, it suffices to check that the second statement holds only for all A = A0 + ℂb where A0 ⊂ E is a finite-dimensional subsystem and b ∈ Bh . Proof. By the proof of Proposition 2.33, we have that Mn (E ∗∗ ) is completely order isomorphic to Mn (E)∗∗ ; hence, if φ : B → E ∗∗ is a unital, completely positive map witnessing the weak expectation property, by the principle of local reflexivity [1, Theorem 12.2.4] for any finite-dimensional subsystem A ⊂ B and ϵ > 0 there is a contraction θ : Mn (A) → Mn (E) so that ‖θ(x) − φn (x)‖ < ϵ‖x‖ for all x ∈ Mn (E ∩ A). Let 𝒢 be the group of all signed permutation matrices and set

Model theory of operator systems and C∗ -algebras �

θ′ (x) =

379

1 ∑ g ∗ θ(gxh)h∗ . |𝒢 |2 g,h∈𝒢

Since g ∗ φn (gxh)h∗ = φn (x) for all g, h ∈ 𝒢 , we have that ‖θ′ (x) − φn (x)‖ < ϵ‖x‖ for all x ∈ Mn (E ∩ A) as well. It is not hard to check that θ′ (eij ⊗ x) = eij ⊗ ψ(x), i, j = 1, . . . , n, for a unique linear map ψ : A → E, so that ψ is n-contractive. Since ψ is linear and n-contractive, so is ψ∗ (x) := ψ(x ∗ )∗ ; thus, the average of the two is n-contractive and ∗-linear, with the other properties being preserved. In the other direction, let ℱ bet the directed set of all finite-dimensional subsystems of B ordered by inclusion, and let I be the directed set of triples (A, n, ϵ) where A ∈ ℱ , n is a positive integer, and ϵ > 0, ordered by (A′ , n′ , ϵ′ ) ≥ (A, n, ϵ) if A′ ⊃ A, n′ ≥ n, and ϵ′ ≤ ϵ. For each (A, n, ϵ) ∈ I, find φA,n,ϵ : A → E matching the conditions of the second statement. Equip E ∗∗ with the weak* topology, and take φ : B → E ∗∗ to be a pointwise-weak* cluster point of the net (φA,n,ϵ ), where we identify E with Ê ⊂ E ∗∗ . We have that φ is unital, ∗-linear, and completely contractive with φ(x) = x̂ for all x ∈ E. By Proposition 2.7, φ is also completely positive. For the last part, the previous argument can be easily adapted to verify that for each hermitian b ∈ Bh , there is a unital, completely positive map φ : E + ℂb → E ∗∗ so that φ(x) = x̂ for all x ∈ E. Thus, the result follows by Lemma 4.25. Recall from Definition 3.36 that a formula f is positive if it is built from atomic formulas only using connectives θ(x1 , . . . , xn ) satisfying θ(x1 , . . . , xn ) ≤ θ(y1 , . . . , yn ) when xi ≤ yi , i = 1, . . . , n. Definition 4.27. We say that an operator system E is positively existentially closed if for every unital inclusion E ⊂ B of operator systems and every positive, quantifier-free formula f (x, y) we have that infy f (x, y)E = infy f (x, y)B for all tuples x chosen from E. Remark 4.28. We can as easily talk about a specific inclusion E ⊂ B being positively existential if the condition in the definition is met. In the operator system category, this is equivalent to the existence of an ultrafilter 𝒰 on some set I so that there is a unital, completely positive map φ : B → E 𝒰 so that φ(x) = ι(x) for all x ∈ E, where ι : E → E 𝒰 is the constant embedding, cf. [4, Remark 4.20]. The proof of Lemma 4.26 thus shows the following. Proposition 4.29. The canonical inclusion ⋅ ̂ : E → E ∗∗ is positively existential for any operator system E. Remark 4.30. Since infy f (x, y)E ≥ infy f (x, y)B automatically, checking positive existential closure is equivalent to checking infy f (x, y)E ≤ infy f (x, y)B where the tuple x is chosen from E. It is straightforward to verify that this in turn is equivalent to checking that for any finite collection f1 (x, y), . . . , fk (x, y) of atomic formulas (that is, fi (x, y) = ‖pi (x, y)‖di for pi (x, y) a linear matrix ∗-polynomial of degree di ), any ϵ > 0, and any tuples a in E and b in B, there is a tuple b′ in E so that fi (a, b′ ) ≤ fi (a, b) + ϵ for all i = 1, . . . , k.

380 � T. Sinclair We say that an element x ∈ ℬ(H) is strictly positive if x − ϵ1 is positive for some ϵ > 0. We write x ≻ 0 to denote that x is strictly positive. Lemma 4.31. An operator system is positively existentially closed if and only if for any unital inclusion E ⊂ B and any finite collection p1 (x, y), . . . , pk (x, y) of hermitian linear matrix ∗-polynomials, the following property holds: (R) for any tuples a in E and b in B so that p1 (a, b) ≻ 0, . . . , pk (a, b) ≻ 0, there is a tuple b′ in E so that p1 (a, b′ ) ≻ 0, . . . , pk (a, b′ ) ≻ 0. Proof. Assume that (R) holds. Let p1 (x, y), . . . , pk (x, y) be a collection of linear matrix ∗-polynomials of degree d1 , . . . , dk . Fixing tuples a in E and b in B and ϵ > 0, define ri := ‖pi (a, b)‖di and consider the hermitian, linear matrix ∗-polynomials q1 (x, y), . . . , qk (x, y) where qi (x, y) := [

(ri + ϵ)1di p∗i (x, y)

pi (x, y) ]. (ri + ϵ)1di

By Proposition 2.5, we have that qi (a, b) ≻ 0 for all i = 1, . . . , k. Therefore, by (R) there exists a tuple b′ in E so that qi (a, b′ ) ≻ 0 for all i = 1, . . . , k, and again by Proposition 2.5 we have that ‖pi (a, b′ )‖di < ri + ϵ for i = 1, . . . , k. That E is positively existentially closed now follows from Remark 4.30. For the converse, let p1 (x, y), . . . , pk (x, y) be a collection of hermitian linear matrix ∗-polynomials of degrees d1 , . . . , dk , and fix tuples a in E and b in B so that pi (a, b) ⪰ ϵ1di for all i = 1, . . . , k for some ϵ > 0 sufficiently small. Setting ri = ‖pi (a, b)‖di , we define qi (x, y) := 2pi (x, y) − (ri + ϵ)1di . By some basic arithmetic, we have that −(ri − ϵ)1di ⪯ qi (a, b) ⪯ (ri − ϵ)1di , which is equivalent to ‖qi (a, b)‖di ≤ ri − ϵ. Thus E being positively existentially closed guarantees that there is a tuple b′ in E so that ‖qi (a, b′ )‖di ≤ ri for all i = 1, . . . , k. Since qi (x, y) is hermitian it only takes values in the hermitian elements; thus, −ri 1di ⪯ qi (a, b′ ) = 2pi (a, b′ ) − ri 1di − ϵ1di for all i = 1, . . . , k which shows that pi (a, b′ ) ≻ 0 for i = 1, . . . , k. Since a positivity check on a collection of linear matrix ∗-polynomials is equivalent to a positivity check on a single linear matrix ∗-polynomial obtained from the collection via block diagonal embedding, the following is immediate from the proof of the previous result.

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Corollary 4.32. An operator system is positively existentially closed if and only if for any unital inclusion E ⊂ B and any hermitian linear matrix ∗-polynomial p(x, y) of degree d, one of the following equivalent properties holds: (R1 ) for any tuples a in E and b in B so that p(a, b) ≻ 0 there is a tuple b′ in E so that p(a, b′ ) ≻ 0; (R2 ) for any tuples a in E and b in B so that ‖p(a, b)‖d < 1 there is a tuple b′ in E so that ‖p(a, b′ )‖ < 1. The following result was proved by Goldbring and the author, [20, Section 2.4] and [21, Proposition 5.1], and by Lupini [36, Proposition 3.2] in full generality. We note that in light of Remark 4.28 a characterization of the weak expectation property very much in the same spirit appears as [41, Theorem 9.22]. Proposition 4.33. An operator system has the weak expectation property if and only if it is positively existentially closed. Proof. Assume that E has the weak expectation property and that E ⊂ B is an inclusion of operator systems. Let p(x, y) be a hermitian linear matrix ∗-polynomial. Suppose there are tuples a = (a1 , . . . , an ) in E and b = (b1 , . . . , bk ) in B so that ‖p(a, b)‖d < 1, and let A ⊂ B be the subsystem spanned by a1 , . . . , an , b1 , . . . , bk . As guaranteed by Lemma 4.26, choose φ : A → E to be ∗-linear, d-contractive, and have the property that maxi ‖φ(ai ) − ai ‖ < ϵ for ϵ > 0 suitably small, to be determined later. We have that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩p(a, φ(b))󵄩󵄩󵄩d ≤ 󵄩󵄩󵄩p(φ(a), φ(b))󵄩󵄩󵄩d + Lϵ = 󵄩󵄩󵄩φ(p(a, b))󵄩󵄩󵄩d + Lϵ ≤ 󵄩󵄩󵄩p(a, b)󵄩󵄩󵄩d + Lϵ, where L is the Lipschitz constant for the map x 󳨃→ ‖p(x, φ(b))‖d . We now choose ϵ > 0 so that ‖p(a, b)‖d + Lϵ < 1. In the other direction, it is clear that for any tuples a in E, b in B, and c in E ∗∗ that the statement, “the ∗-linear map φ on the span of a and b induced by sending a to a and b to c is n-contractive” can be verified by the (countably) infinite collection of expressions of the form ‖pi (a, b)‖n < ri ⇒ ‖pi (a, c)‖n < ri for pi linear matrix ∗-polynomials with complex rational coefficients; thus, fixing a and b, by taking weak*-limits of tuples cm in E = Ê ⊂ E ∗∗ satisfying the first m statements we can define an n-contractive ∗-linear map φa,b from the span of a and b to E ∗∗ satisfying φ(ai ) = â i . The required map φ : B → E ∗∗ is then obtained by taking a pointwise-weak* limit of the φa,b ’s over the directed set of all finite tuples. In light of Lemmas 4.26 and 4.31, the proof of the previous proposition can be easily adapted to prove the following. Corollary 4.34. An operator system is positively existentially closed if and only if for any unital inclusion E ⊂ B and any hermitian linear matrix ∗-polynomial p(x, y) = p(x1 , . . . , xn , y) of degree d, one of the following equivalent properties holds: (R3 ) for any tuple a in E and b ∈ Bh so that p(a, b) ≻ 0 there is b′ ∈ E h so that p(a, b′ ) ≻ 0.

382 � T. Sinclair (R4 ) for any tuple a in E and b ∈ Bh so that ‖p(a, b)‖d < 1 there is b′ ∈ E h so that ‖p(a, b′ )‖ < 1. Question 4.35. Is there a proof of the previous corollary which uses positive existential closure directly without passing through the equivalence with the weak expectation property? Remark 4.36. For the (elementary) class of operator systems which are C∗ -algebras, there is an explicit, countable collection of existentially quantified linear matrix ∗-polynomial inequalities which verifies the weak expectation property due to Farenick, Paulsen, and Kavruk [17, Theorem 6.1]. Consider the hermitian linear matrix ∗-polynomial y1 [ ∗ p(x1 , x2 , y1 , y2 ) := [x1 [0

x1 y2 x2∗

0 ] x2 ]. 1 − y1 − y2 ]

A unital C∗ -algebra A has the weak expectation property if and only if for every unital inclusion A ⊂ B of operator systems and every n = 1, 2, . . . whenever a1 , a2 ∈ Mn (A) and b1 , b2 ∈ Mn (B) are so that p(a1 , a2 , b1 , b2 ) ≻ 0, there are b′1 , b′2 ∈ Mn (A) so that p(a1 , a2 , b′1 , b′2 ) ≻ 0. Remark 4.37. Note that since strict inequalities are used in the preceding, the existential conditions verifying the weak expectation property given are not first-order expressible. By replacing the entries of p in the Farenick–Kavruk–Paulsen result with matrices, we actually increase the number of variables as each variable is replaced by a collection of variables standing in for the matrix units. Question 4.38. Is there a countable sequence pi (x, y) of hermitian linear matrix ∗-polynomials in a uniformly bounded number of variables so that a C∗ -algebra has the weak expectation property if and only if for all embeddings A ⊂ ℬ(H) and all tuples a in A and b in ℬ(H) so that pi (a, b) ≻ 0 for all i = 1, 2, . . . there is a tuple b′ in A so that pi (a, b′ ) ≻ 0? Would a single p suffice? Remark 4.39. Perhaps the simplest nontrivial hermitian linear matrix ∗-polynomial is 1−y p(x, y) = [ x

x∗ ]. y

For any C∗ -algebra A ⊂ ℬ(H) it is an exercise in functional calculus to show that if for any a ∈ A there is some b ∈ ℬ(H) so that p(a, b) ≻ 0, then there is a b′ ∈ A so that p(a, b′ ) ≻ 0. (It is a bit easier to see that there is b′ ∈ A∗∗ since A∗∗ is a von Neumann algebra, then use Proposition 4.29.) As is pointed out in the introduction of [17], we have that ∃b : p(a, b) ≻ 0 is equivalent to the numerical radius ω(a) of a being less than 1/2. It

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would be interesting to know if the weak expectation property could be characterized in terms of this simple expression. Question 4.40. With p as in the previous remark, does a C∗ -algebra A have the weak expectation property if for some (for all) n for all a1 , . . . , an ∈ A for which there exists b ∈ ℬ(H) so that p(ai , b) ≻ 0 for all i = 1, . . . , n there is b′ ∈ A so that p(ai , b′ ) ≻ 0 for all i = 1, . . . , n? We close this section with a model-theoretic proof of a characterization of the weak expectation property which is an unpublished result due to Kavruk [25]. Kavruk’s proof uses the theory of operator system tensor products [28, 29] along with Kirchberg’s tensorial characterization of the weak expectation property [31]. Notation 4.41. In the following, we write ⋈ to indicate a fixed choice of ≺ or ≻, so that, for instance, a1 ⋈ b1 , a2 ⋈ b2 means one of four possible expressions with one choice of ≺ or ≻ for each pair ai , bi which remains fixed throughout. Definition 4.42. We say that any operator system E has the complete tight Riesz interpolation property if for any unital inclusion of operator systems E ⊂ B, any n = 1, 2, . . . , and any tuple a = (a1 . . . , ak ) in Mn (E)h and b ∈ Mn (B)h if b ⋈ ai for all i = 1, . . . , k, then there is a b′ ∈ Mn (E)h so that b′ ⋈ ai for all i = 1, . . . , k. Theorem 4.43 (Kavruk [25, Theorem 7.4]). An operator system E has the complete tight Riesz interpolation property if and only if it has the weak expectation property. Proof. Proposition 4.33 and Lemma 4.31 show that the weak expectation property implies the complete tight Riesz interpolation property. For the converse, we will show that the complete tight Riesz interpolation property implies property (R3 ), therefore the weak expectation property by Proposition 4.33 and Corollary 4.34. Let E ⊂ B be a unital inclusion of operator systems and p(x1 , . . . , xn , y) be a hermitian linear matrix ∗-polynomial of degree d. Assume we have chosen a1 , . . . , an ∈ E and b ∈ Bh so that p(a1 , . . . , an , b) ≻ 0. Since b is hermitian, we have that p(a1 , . . . , an , b) = X ⊗ b − A ≻ 0 where X ∈ Mdh and A ∈ Mn (E)h . By perturbing X by ϵId for ϵ sufficiently small, we may assume without loss of generality that X is invertible. Therefore, we have that X = Y ΣY ∗ for Y invertible and Σ a diagonal matrix with ±1 entries on the main diagonal. We may assume the first f entries of the main diagonal of Σ are 1 with the remaining being −1. Let 𝒢 be the group of signed permutation matrices in Md preserving the set {±e1 , . . . , ±ef } where e1 , . . . , ed is the standard basis for ℂd . For each g ∈ 𝒢 , define Ag := gY −1 A(Y −1 )∗ g ∗ ∈ Md (E)h , and for z ∈ Bh consider the system of inequalities {Σ ⊗ z ≻ Ag : g ∈ 𝒢 },

384 � T. Sinclair noting that this system is consistent if and only if X ⊗ z ≻ A since g(Σ ⊗ b)g ∗ = Σ ⊗ b for all g ∈ 𝒢 . By the complete tight Riesz interpolation property, there is b′ ∈ Md (E)h so that b′ ≻ Ag for all g ∈ 𝒢 . We have that gb′ g ∗ ≻ Y −1 A(Y −1 )∗ for all g ∈ 𝒢 ; thus, b′′ :=

1 ∗ ∑ gb′ g ∗ ≻ Y −1 A(Y −1 ) . |𝒢 | g∈𝒢

It is not hard to check that b′′ is block diagonal of the form I ⊗ b1 b′′ = [ f 0

0 ] Id−f ⊗ b2

for some b1 , b2 ∈ E h . It now suffices to find c ∈ E h with c ≻ b1 − ϵ1 and −c ≻ b2 − ϵ1 for ϵ > 0 sufficiently small as this would imply Σ ⊗ c ≻ b′′ − ϵ1d ≻ Y −1 A(Y −1 ) , ∗

therefore X ⊗ c ≻ A.

In order to do so, we choose ϵ > 0 witnessing the strict positivity of b′′ as in the right-hand side of the preceding equation line and set δ = ϵ/2. Realizing E ⊂ ℬ(H) and setting bδi = bi − δ1, we have that bδi = [bδi ]+ − [bδi ]− for some [bδi ]+ , [bδi ]− ∈ ℬ(H)+ , i = 1, 2. Setting c′ := [bδ2 ]+ − [bδ1 ]+ ∈ ℬ(H)h , we can check that b1 − ϵ1 ≺ b1 − δ1 ⪯ c′ ⪯ −b2 + δ1 ≺ −b2 + ϵ1. Hence by the complete tight Riesz interpolation property there is c ∈ E h so that b1 − ϵ1 ≺ c ≺ −(b2 − ϵ1) verifying the claim.

Bibliography [1] [2] [3]

[4] [5]

[6]

F. Albiac and N. J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR2192298. E. M. Alfsen, Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 57, Springer, New York–Heidelberg, 1971. MR0445271. G. Aubrun and S. J. Szarek, Alice and Bob meet Banach. The interface of asymptotic geometric analysis and quantum information theory, Mathematical Surveys and Monographs, vol. 223, American Mathematical Society, Providence, RI, 2017. MR3699754. S. Barlak and G. Szabó, Sequentially split ∗-homomorphisms between C∗ -algebras, Int. J. Math. 27 (2016), no. 13, 1650105, 48. https://doi.org/10.1142/S0129167X16501056. MR3589655. I. Ben Yaacov, A. Berenstein, C. Ward Henson and A. Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008. pp. 315–427. https://doi.org/10.1017/CBO9780511735219.011. MR2436146. N. P. Brown and N. Ozawa, C ∗ -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR2391387.

Model theory of operator systems and C∗ -algebras

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

� 385

M. D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285–290. https://doi.org/10.1016/0024-3795(75)90075-0. M. D. Choi and E. G. Effros, Separable nuclear C ∗ -algebras and injectivity, Duke Math. J. 43 (1976), no. 2, 309–322. MR0405117. M. D. Choi and E. G. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1977), no. 2, 156–209. M. D. Choi and E. G. Effros, Nuclear C ∗ -algebras and injectivity: the general case, Indiana Univ. Math. J. 26 (1977), no. 3, 443–446. https://doi.org/10.1512/iumj.1977.26.26034. MR430794. M. D. Choi and E. G. Effros, Nuclear C ∗ -algebras and the approximation property, Am. J. Math. 100 (1978), no. 1, 61–79. https://doi.org/10.2307/2373876. MR482238. A. Connes, Classification of injective factors. Cases II1 , II∞ , IIIλ , λ ≠ 1, Ann. Math. (2) 104 (1976), no. 1, 73–115. https://doi.org/10.2307/1971057. MR454659. J. B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer, New York, 1990. MR1070713. E. G. Effros, Aspects of noncommutative order, C∗ -algebras and applications to physics (Proc. Second Japan–USA Sem., Los Angeles, Calif., 1977), Lecture Notes in Math., vol. 650, Springer, Berlin, 1978, pp. 1–40. MR0504750. I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati and W. Winter, Model theory of C∗ -algebras, Mem. Am. Math. Soc. 271 (2021), no. 1324, viii+127. https://doi.org/10.1090/memo/1324. MR4279915. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras II: Model theory, Isr. J. Math. 201 (2014), no. 1, 477–505. https://doi.org/10.1007/s11856-014-1046-7. MR3265292. D. Farenick, A. S. Kavruk and V. I. Paulsen, C ∗ -algebras with the weak expectation property and a multivariable analogue of Ando’s theorem on the numerical radius, J. Oper. Theory 70 (2013), no. 2, 573–590. https://doi.org/10.7900/jot.2011oct07.1938. MR3138370. D. Farenick and V. I. Paulsen, Operator system quotients of matrix algebras and their tensor products, Math. Scand. 111 (2012), no. 2, 210–243. https://doi.org/10.7146/math.scand.a-15225. MR3023524. B. Gärtner and J. Matoušek, Approximation algorithms and semidefinite programming, Springer, Heidelberg, 2012. MR3015090. I. Goldbring and T. Sinclair, On Kirchberg’s embedding problem, J. Funct. Anal. 269 (2015), no. 1, 155–198. https://doi.org/10.1016/j.jfa.2015.02.016. MR3345606. I. Goldbring and T. Sinclair, Omitting types in operator systems, Indiana Univ. Math. J. 66 (2017), no. 3, 821–844. https://doi.org/10.1512/iumj.2017.66.6019. MR3663327. I. Goldbring and T. Sinclair, On the axiomatizability of C∗ -algebras as operator systems, Glasg. Math. J. 61 (2019), no. 3, 629–635. https://doi.org/10.1017/s001708951800040x. MR3991361. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Am. Math. Soc. 16 (1955). Chapter 1: 196 pp.; Chapter 2: 140 (French). MR75539. R. V. Kadison, A representation theory for commutative topological algebra, Mem. Am. Math. Soc. 7 (1951), 39. MR0044040. A. S. Kavruk, The weak expectation property and Riesz interpolation, 2012. arXiv:1201.5414. A. S. Kavruk, Nuclearity related properties in operator systems, J. Oper. Theory 71 (2014), no. 1, 95–156. https://doi.org/10.7900/jot.2011nov16.1977. MR3173055. A. S. Kavruk, On a non-commutative analogue of a classical result of Namioka and Phelps, J. Funct. Anal. 269 (2015), no. 10, 3282–3303. https://doi.org/10.1016/j.jfa.2015.09.002. MR3401618. A. S. Kavruk, V. I. Paulsen, I. G. Todorov and M. Tomforde, Tensor products of operator systems, J. Funct. Anal. 261 (2011), no. 2, 267–299. https://doi.org/10.1016/j.jfa.2011.03.014. MR2793115. A. S. Kavruk, V. I. Paulsen, I. G. Todorov and M. Tomforde, Quotients, exactness, and nuclearity in the operator system category, Adv. Math. 235 (2013), 321–360. https://doi.org/10.1016/j.aim.2012.05.025. MR3010061. E. Kirchberg, C ∗ -nuclearity implies CPAP, Math. Nachr. 76 (1977), 203–212. https://doi.org/10.1002/mana.19770760115. MR512362.

386 � T. Sinclair

[31] E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C ∗ -algebras, Invent. Math. 112 (1993), no. 3, 449–489. https://doi.org/10.1007/BF01232444. MR1218321. [32] E. Kirchberg, Commutants of unitaries in UHF algebras and functorial properties of exactness, J. Reine Angew. Math. 452 (1994), 39–77. https://doi.org/10.1515/crll.1994.452.39. MR1282196. [33] T.-L. Kriel, An introduction to matrix convex sets and free spectrahedra, Complex Anal. Oper. Theory 13 (2019), no. 7, 3251–3335. https://doi.org/10.1007/s11785-019-00937-8. MR4020034. [34] E. C. Lance, Hilbert C ∗ -modules. A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. MR1325694. [35] L. Lovász, Semidefinite programs and combinatorial optimization, Recent advances in algorithms and combinatorics, CMS Books Math./Ouvrages Math. SMC, vol. 11, Springer, New York, 2003, pp. 137–194. https://doi.org/10.1007/0-387-22444-0_6. MR1952986. [36] M. Lupini, An intrinsic order-theoretic characterization of the weak expectation property, Integral Equ. Oper. Theory 90 (2018), no. 5, Paper No. 55, 17. 10.1007/s00020-018-2479-x. MR3829543. [37] V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. [38] V. I. Paulsen, I. G. Todorov and M. Tomforde, Operator system structures on ordered spaces, Proc. Lond. Math. Soc. (3) 102 (2011), no. 1, 25–49. https://doi.org/10.1112/plms/pdq011. MR2747723. [39] V. I. Paulsen and M. Tomforde, Vector spaces with an order unit, Indiana Univ. Math. J. 58 (2009), no. 3, 1319–1359. https://doi.org/10.1512/iumj.2009.58.3518. MR2542089. [40] G. Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR2006539. [41] G. Pisier, Tensor products of C ∗ -algebras and operator spaces—the Connes–Kirchberg problem, London Mathematical Society Student Texts, vol. 96, Cambridge University Press, Cambridge, 2020. MR4283471. [42] A. G. Robertson and R. R. Smith, Liftings and extensions of maps on C ∗ -algebras, J. Oper. Theory 21 (1989), no. 1, 117–131. MR1002124. [43] T. Sinclair, CP-stability and the local lifting property, N.Y. J. Math. 23 (2017), 739–747. MR3665586. [44] M. Takesaki, Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, vol. 124, Springer, Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5. MR1873025. [45] C. Webster and S. Winkler, The Krein–Milman theorem in operator convexity, Trans. Am. Math. Soc. 351 (1999), no. 1, 307–322. https://doi.org/10.1090/S0002-9947-99-02364-8. MR1615970.

Martino Lupini

Model theory of G-C*-algebras and order zero dimension Abstract: This article offers an overview of applications of model theory to the study of actions of compact groups on C*-algebras, focussing on results related to the notions of Rokhlin property and Rokhlin dimension. Keywords: Continuous logic, logic for metric structure, G-C*-algebra, spectral subspace, Rokhlin dimension, order zero dimension, nuclear dimension, decomposition rank, crossed product MSC 2010: Primary 03C98, 46L55, Secondary 22D05, 46L05

1 Introduction In this article we explain how actions of a fixed compact metrizable group G on C*-algebras (G-C*-algebras) can be regarded as structures in continuous logic. This allows one to provide semantic characterizations of regularity properties for G-C*-algebras such as the Rokhlin property. More generally, we will consider the notion of Rokhlin dimension, recovering the Rokhlin property as Rokhlin dimension zero. The notion of Rokhlin property for G-C*-algebras has its origin in the study of measure-preserving dynamical systems. In this context, the Rokhlin lemma characterizes free measure-preserving ℤ-actions on the standard probability space in terms of the existence of decompositions of the space known as Rokhlin towers [24, Lemma 4.77]. This lemma plays a key role in the classification of free ergodic measure-preserving ℤ-actions due to Ornstein and Weiss, asserting that they are all orbit equivalent [31]; see also [24, Theorem 4.38]. These results were later generalized to actions of arbitrary countable amenable groups [32]. Acknowledgement: The author was partially supported by the Marsden Fund Fast-Start Grant VUW1816 and the Rutherford Discovery Fellowship VUW2002 from the Royal Society of New Zealand, and the Starting Grant “Definable Algebraic Topology” from the European Research Council. We are thankful to Isaac Goldbring and the anonymous referee for a large number of useful comments, and Eusebio Gardella for many helpful discussions and clarifications. Part of this survey was written while the author was visiting the University of Pisa, the University of Bologna, the Erdős Center in Budapest, and the University of Münster. The hospitality and financial support from these institutions and from Newcastle University is gratefully acknowledged. Martino Lupini, Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy, e-mail: [email protected] https://doi.org/10.1515/9783110768282-010

388 � M. Lupini Rokhlin towers in the context of operator algebras first appeared in the work of Connes on classification of free ℤ-actions on the hyperfinite II1 factor [5], asserting that any two such actions are outer conjugate (conjugate modulo inner automorphisms). As a main ingredient in the proof of this classification, Connes characterized freeness for ℤ-actions on the hyperfinite II1 factor in terms of existence of a suitable noncommutative analogue of Rokhlin towers, defined in terms of projections. Again, this work was later generalized to actions of arbitrary countable amenable groups [15, 22, 23, 30]. In the context of C*-algebras, the Rokhin property for ℤ-actions was introduced by Kishimoto for many important classes of “classifiable” C*-algebras (uniformly hyperfinite C*-algebras, approximately finite-dimensional C*-algebras, A𝕋-algebras, Kirchberg algebras), where again it was instrumental in obtaining classification results for free actions on such algebras [26, 27, 28]. The notion of Rokhlin property for ℤ-actions was later introduced in the case of “well-behaved” C*-algebras without nontrivial projections, such as the Jiang–Su algebra [38]. The Rokhlin property was later considered for actions of finite groups [1, 20, 21], compact groups [10, 11, 18], and compact quantum groups [2], having as its main applications classification of actions, and results of preservation of regularity properties to crossed products. In order to extend such results beyond the case of actions with the Rokhlin property, the notion of Rokhlin dimension was introduced as a relaxation of the Rokhlin property (which corresponds to the case of Rokhlin dimension zero) [9, 10, 13, 19, 39]. From the model-theoretic perspective, it is natural to define a syntactic notion of dimension for G-equivariant *-homomorphisms between G-C*-algebras, called order zero dimension, which recovers the general notion of positive existential morphism as the case of dimension zero. Rokhlin dimension can be naturally expressed in terms of order zero dimension. In particular, the Rokhlin property can be expressed in terms of the notion of positive existential G-equivariant *-homomorphism. This viewpoint affords a unified approach and a simultaneous generalization of several preservation results for crossed products and fixed point algebras of G-C*-algebras with finite Rokhlin dimension. These results are applicable to dimension functions for C*-algebras that can be expressed using formulae that are sufficiently simple, such as nuclear dimension and decomposition rank. It is a nontrivial fact that G-C*-algebras can be seen as structures in continuous logic, and in fact form an axiomatizable class in a suitable language. Indeed, while G is a metrizable topological group, it does not have a canonical translation-invariant metric, nor the action regarded as a group homomorphism from G is required to be uniformly continuous with a prescribed modulus. The key idea is to describe the action in terms of its spectral subspaces associated with irreducible representations of G, which is the perspective adopted when generalizing the notion of G-C*-algebra to the case of quantum groups. A similar viewpoint can be used to consider more generally actions of G on Banach spaces.

Model theory of G-C*-algebras and order zero dimension

� 389

Our goal is to give a quick but complete introduction to preservation results for G-C*-algebras from the viewpoint of the logic for metric structures. We will thus begin with introducing the framework of languages with sorts, which is necessary when considering G-C*-algebras. We will then recall the salient facts about G-C*-algebras and their first-order axiomatization. Our analysis will require us to consider weaker languages than the usual language for C*-algebras, which do not include a symbol for multiplication. This is the syntactic analogue of replacing *-homomorphisms with more general maps such as completely positive maps, possibly unital or order zero, and it affords a finer calibration of the complexity of formulas required to express a variety of notions, and particularly dimension functions such as Rokhlin dimension, nuclear dimension, and order zero dimension. The core of this survey is concerned with the notion of order zero dimension for G-equivariant *-homomorphisms, and its relation with Rokhlin dimension and positive existential embeddings. After recalling the definition and main properties of crossed products of G-C*-algebras, the notion of order zero dimension is applied to obtain a general preservation result for regularity properties for crossed products.

2 Semantic of languages with sorts In this section, we recall some fundamental notions from the logic for metric structures. For a general introduction to this topic, we recommend [3], and [7] for its applications to C*-algebras. For the purpose of applications to C*-dynamics, it is convenient to consider languages with sorts and domains of quantification. Each sort represents a possible source for a function or relation, or a cartesian factor for such a source. Each sort is endowed with a distinguished collection of domains of quantification and a special binary relation symbol for the metric. Domains of quantification represent subsets of the given sort where the quantifiers (inf and sup) can range. The collection of domains of quantification associated with a given sort is endowed with an upward directed order, which is required to be preserved when interpreted in a structure. (Subsets are compared by inclusion.) The language contains uniform continuity moduli for the function and relation symbols relative to given domains of quantifications for the input source. The language also contains boundedness constraints for function and relation symbols restricted to domains of quantification of the input sorts. These are defined as domains of quantification of the output sort for functions, and as compact intervals for relations. ∗ For example, a C*-algebra can be seen as structure in a two-sorted language ℒC . One sort 𝒞 is to be interpreted as the complex numbers. The corresponding domains of quantifications (D𝒞n )n∈ℕ associated with 𝒞 are interpreted as discs centered at the origin with integer radius, and linearly ordered by their indices. The other sort 𝒮 is to be interpreted as the C*-algebra itself. The corresponding domains of quantification (D𝒮 n )n∈ℕ

390 � M. Lupini are interpreted as balls centered at the origin with integer radius, again linearly ordered by their indices. The language has function symbols for the C*-algebra operations, and a constant symbol for each complex number. For example, the scalar multiplication function symbol ⋅ has as input sorts the pair (𝒞 , 𝒮 ) and as output sort 𝒮 . For each choice of domains of quantification for the input sorts (D𝒞n , D𝒮 m ) for n, m ∈ ℕ, there are: – a corresponding domain of quantification D∙D𝒞 ,D𝒮 := D𝒮 nm of the output sort, such n m that in any structure the interpretation of the scalar multiplication symbol restricted to the interpretations of D𝒞n and D𝒮 m takes values in the interpretation of D𝒮 ; nm – a corresponding continuity modulus ϖD∙ 𝒞 ,DA , such that in any structure the intern

m

pretation of the multiplication symbol ⋅ restricted to the interpretations of D𝒞n and ∙ D𝒮 m is uniformly continuous with modulus ϖD𝒞 ,D𝒮 . n

m

Similar considerations apply to the other function symbols and the symbol for the met∗ ric. Unital C*-algebras can be seen as structures in the language ℒ1,C , which contains an additional constant symbol for the identity element. We now recall some terminology from model theory about structures and functions between structures. We fix a language ℒ as above and we consider notions relative to this fixed language ℒ. Recall that a basic formula is an formula of the form R(t1 , . . . , tn ), where R is an n-ary relation symbol and t1 , . . . , tn are terms. Let A, B be ℒ-structures. A function f : A → B is a family of functions f𝒮 : 𝒮 A → 𝒮 B where 𝒮 varies among the sorts of ℒ. Such a function f is: – a morphism if φ(f (a)) ≤ φ(a) for every basic formula φ(x) and every tuple a of elements of A of the right sorts; – an embedding if φ(f (a)) = φ(a) for every basic formula φ(x) and every tuple a of elements of A of the right sorts. Let 𝒞 be a class of structures. Then 𝒞 induces a pseudometric d𝒞 between sentences obtained by setting 󵄨 󵄨 d𝒞 (φ, ψ) = sup{󵄨󵄨󵄨φA − ψA 󵄨󵄨󵄨 : A ∈ 𝒞 }. We say that ℒ is separable for 𝒞 if the pseudometric d𝒞 is separable. A type t(x) in the tuple of free variables x is a collection of conditions of the form φ(x) ≤ r, where t ∈ ℝ and φ(x) is a formula in the tuple of free variables x. Given a type t(x) and ℒ-structure M, one says that: – t(x) is realized in M if there exists a tuple a in M of the same sorts as x such that M 󳀀󳨐 φ(a) ≤ r for every condition φ(x) ≤ r in t(x); – t(x) is finitely realized in M if every finite set of conditions in t(x) is realized in M; – t(x) is approximately finitely realized in M if the set of conditions of the form φ(x) ≤ r + ε for ε > 0 and φ(x) ≤ r in t(x) is finitely realized in M.

Model theory of G-C*-algebras and order zero dimension

� 391

Throughout this article, we fix a nonprincipal ultrafilter 𝒰 over ℕ. The choice of 𝒰 will be irrelevant for our considerations. Given a structure A, we let ∏𝒰 A be the ultrapower of A with respect to 𝒰 . We denote by ΔA : A → ∏𝒰 A the canonical diagonal embedding and we identify A with its image under ΔA .

3 Compact groups and representations In this article, we will denote by G a second-countable compact group. We let C(G) be the separable commutative unital C*-algebra of continuous complex-valued functions on G, with the operations of pointwise sum and product. The multiplication operation ⋅G on G induces a unital *-homomorphism Δ : C(G) → C(G × G) ≅ C(G) ⊗ C(G) defined by setting (Δf )(g, h) = f (g ⋅G h) for f ∈ C(G) and g, h ∈ G. Associativity of the multiplication in G yields the identity (Δ ⊗ id) ∘ Δ = (id ⊗ Δ) ∘ Δ. The pair (C(G), Δ) is a C*-algebraic compact quantum group [40, Chapter 5]. (Conversely, any C*-algebraic compact quantum group whose C*-algebra is separable and commutative is of this form.) The Haar probability measure μ on G induces the Haar state h on C(G) defined by h(f ) = ∫ f (G) dμ(G) for f ∈ C(G). The invariance of μ gives the identity (h ⊗ id) ∘ Δ = (id ⊗ h) ∘ Δ = h. Let Rep(G) be a set of representatives of unitary equivalence classes of unitary representations of G on finite-dimensional Hilbert spaces. (Recall that two representations λ, μ of G on Hilbert spaces ℋλ and ℋμ are unitarily equivalent if there exists a surjective linear isometry U : ℋλ → ℋμ such that Uλg = μg U for every g ∈ G.) For λ ∈ Rep(G), we let ℋλ be the corresponding finite-dimensional Hilbert space and dλ be its dimension, so that λ is a continuous group homomorphism G → U(ℋλ ). λ ∈ C(G) to be the function For ξ, η ∈ ℋλ , we define uξ,η g 󳨃→ ⟨ξ, λg η⟩.

392 � M. Lupini One then lets C(G)λ ⊆ C(G) be the subspace λ {uξ,η : ξ, η ∈ ℋλ }.

We let Irr(G) ⊆ Rep(G) be the set of elements of Rep(G) that are irreducible. One has that ∗ μ

μ

λ λ h((uξ,η ) uζ ,χ ) = h(uξ,η (uζ ,χ ) ) = 0 ∗

whenever λ, μ ∈ Irr(G) are different (and, hence, nonequivalent), ξ, η ∈ ℋλ , and ζ , χ ∈

ℋμ . For λ, μ ∈ Irr(G), we set

1 δλ,μ = { 0

if λ, μ are equal; otherwise.

For λ ∈ Irr(G) we also fix an orthonormal basis (e1λ , . . . , edλλ ) of ℋλ and set λ ujk := ueλλ eλ

j k

for 1 ≤ j, k ≤ dλ . For λ ∈ Rep(G) not irreducible, we write λ as direct sum of irreducible unitary representations λ = λ1 ⊕ ⋅ ⋅ ⋅ ⊕ λn and consider the orthonormal basis of ℋλ = ℋλ1 ⊕ ⋅ ⋅ ⋅ ⊕ ℋλn corresponding to the fixed orthonormal bases of ℋλ1 , . . . , ℋλn . Given two representations λ, μ ∈ Rep(G), one sets λ ≤ μ if there exists a linear isometry v : ℋλ → ℋμ that intertwines λ and μ, in the sense that νλg = μg ν for g ∈ G. For λ, μ ∈ Rep(G), letting λ ⊕ μ be their direct sum, λ ⊗ μ be their tensor product, and λ be the conjugate representation of λ, the following inclusions hold C(G)λ + C(G)μ ⊆ C(G)λ⊕μ , C(G)λ C(G)μ ⊆ C(G)λ⊗μ , (C(G)λ ) = C(G)λ . ∗

Furthermore, if λ ≤ μ, then C(G)λ ⊆ C(G)μ . It follows from this that {C(G)λ : λ ∈ Rep(G)}

Model theory of G-C*-algebras and order zero dimension

� 393

is a collection of finite-dimensional subspaces of C(G) that is upward-directed with respect to inclusion. Its union 𝒪(G) is a dense unital self-adjoint subalgebra of C(G) by the Peter–Weyl theorem; see [16, Chapter Seven], [40, Theorem 5.3.11], or [6, Theorem 1.2]. For λ ∈ Irr(G), there exists a (uniquely determined) positive invertible bounded linear operator F λ ∈ B(ℋλ ) such that, letting Tr(F λ ) be the trace of F λ , the following identities hold: λ h(uξ,η (uζλ ,χ ) ) = ∗

λ h((uξ,η ) uζλ ,χ ) = ∗

⟨ξ, ζ ⟩⟨χ, F λ η⟩ , Tr(F λ )

⟨ξ, (F λ )−1 ζ ⟩⟨χ, η⟩ , Tr(F λ )

and Tr(F λ ) = Tr((F λ ) ) > 0; −1

see [40, Proposition 3.2.9, Proposition 5.3.8] or [6, Theorem 1.8]. We set Fijλ := ⟨eiλ , F λ ejλ ⟩, (F λ )ij := ⟨eiλ , (F λ ) ejλ ⟩ −1

−1

for 1 ≤ i, j ≤ dλ . In particular, one has that μ

h(uijλ (ukℓ ) ) = δλμ δik ∗

Fℓjλ

Tr(F λ )

for λ, μ ∈ Irr(G), 1 ≤ i, j ≤ dλ , and 1 ≤ k, ℓ ≤ dμ . One can extend the definition of F λ to an arbitrary λ ∈ Rep(G) by taking direct sums. For 1 ≤ s, m ≤ dλ , define ωλsm to be the continuous linear functional on C(G) defined by dλ

λ ωλsm (x) = Tr(Fλ ) ∑ (F λ )mk h(x(usk ) ) −1

∗

k=1

Then one has that, for λ, μ ∈ Irr(G), 1 ≤ s, m ≤ dλ , and 1 ≤ l, n ≤ dλ , μ

ωλsm (uln ) = δλμ δsl δmn and μ

(ωλsm ⊗ ωln ) ∘ Δ = δml δλμ ωμsn ; see [35, Section 1]. Define λ Pks = (ωλks ⊗ id) ∘ Δ

394 � M. Lupini for 1 ≤ k, s ≤ dλ . It follows from the above that μ

λ λ Psm Pln = δλμ δml Psn λ for λ, μ ∈ Irr(G), 1 ≤ s, m ≤ dλ , and 1 ≤ l, n ≤ dμ . Furthermore, Pkk is a projection operator onto λ C(G)λk := span{uik : 1 ≤ i ≤ dλ }

and dλ

λ Pλ := ∑ Pkk k=1

is a projection operator onto C(G)λ ; see [35, Section 1].

4 Multiplier algebras Recall that the multiplier algebra of a C*-algebra B is a unital C*-algebra M(B) that contains B as an essential ideal (in the sense that, for x ∈ M(B), if xb = 0 for every b ∈ B, then x = 0) and satisfies the following universal property: whenever D is a unital C*-algebra containing B as an ideal, there exists a unique *-homomorphism D → M(B) with kernel B⊥ := {d ∈ D : ∀b ∈ B, db = 0} that is the identity on B [4, II.7.3.11]. The strict topology on M(B) is the locally convex vector space topology generated by the seminorms x 󳨃→ max{‖xb‖, ‖bx‖} for b ∈ B. The unit ball of B is dense in the unit ball of M(B) with respect to the strict topology [29, Proposition 1.3]. When B is separable, the unit ball of M(B) is a Polish space with respect to the strict topology. When B is unital, one has that B = M(B). In the commutative case, when B is the C*-algebra C0 (X) of continuous complex-valued functions vanishing at infinity on some locally compact Hausdorff space X, one has that M(B) is the algebra C(βX) of continuous complex-valued functions on the Čech–Stone compactification βX of X. One can concretely describe M(B) as the C*-algebra of (necessarily bounded) linear maps t : B → B that are adjointable, in the sense that there exists a linear map t ∗ : B → B such that t(a)∗ ⋅ b = a∗ ⋅ (t ∗ )(b) for all a, b ∈ B. For this reason, the adjointable maps t : B → B are also called multipliers of B. The inclusion B → M(B) is given by identifying a ∈ B with the linear map B → B, b 󳨃→ ab. The strict topology is generated by the seminorms t 󳨃→ max{‖t(b)‖, ‖t ∗ (b)‖} for b ∈ B. Lemma 4.1. Suppose that B is a C*-algebra. There exists a canonical injective unital *-homomorphism J : ∏𝒰 M(B) → M(∏𝒰 B) making the following diagram commute:

Model theory of G-C*-algebras and order zero dimension

∏𝒰 M(B) ΔM(B) |B

B

J

ΔB

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M( ∏𝒰 B) ∏𝒰 B

where the vertical arrow in the right is the inclusion map. Proof. Given an element [tn ]𝒰 of ∏𝒰 M(B), define the linear map J([tn ]) : ∏𝒰 B → ∏𝒰 B, [bn ]𝒰 󳨃→ [tn (bn )]𝒰 . It is easy to verify that J([tn ]) is an adjointable linear map with adjoint J([tn∗ ]𝒰 ). Thus, J([tn ]) defines an element of M(∏𝒰 B). It is easily seen that J : ∏𝒰 M(B) → M(∏𝒰 B) is a well-defined unital *-homomorphism that makes the diagram above commute. To see that J is injective, suppose that [tn ]𝒰 ∈ ∏𝒰 M(B) is nonzero. Thus, we have that ‖[tn ]𝒰 ‖ = 𝒰 − limn ‖tn ‖ > c for some c > 0. Thus, U := {n ∈ ℕ : ‖tn ‖ > c} ∈ 𝒰 . For n ∈ U, there exists bn ∈ B such that ‖bn ‖ ≤ 1 and ‖tn (bn )‖ > c. Hence, [bn ]𝒰 ∈ ∏𝒰 B and ‖J([tn ]𝒰 )([bn ]𝒰 )‖ > c. This shows that J([tn ]𝒰 ) is nonzero. If A, B are C*-algebras, then a *-homomorphism ϕ : A → M(B) is nondegenerate if {ϕ(a)b : a ∈ A, b ∈ B} has dense linear span in B. (Notice that ϕ(a)b ∈ B whenever a ∈ A and b ∈ B since B is an ideal in M(B).) When B = K(ℋ) for some separable Hilbert space ℋ, one has that M(B) = B(ℋ). In this case, a *-homomorphism π : A → M(B) = B(ℋ) is nondegenerate if and only if ϕ is a nondegenerate representation, namely {π(a)ξ : a ∈ A, ξ ∈ ℋ} is dense in ℋ. More generally, a completely positive map ϕ : A → M(B) is nondegenerate if (ϕ(ei )) converges strictly to 1 for some approximate unit (ei ) of A. If A ⊆ B(ℋ) is a C*-algebra that acts nondegenerately on ℋ (in the sense that the inclusion A → B(ℋ) is nondegenerate), then M(A) is the idealizer of A in B(ℋ), namely the set of T ∈ B(ℋ) such that Ta and aT belong to A for every a ∈ A [29, Proposition 2.3]. From this, it easily follows that if A, B are C*-algebras, then there exists a unital injective *-homomorphism M(A) ⊗min M(B) → M(A ⊗min B). The multiplier algebra M(A) of a C*-algebra A satisfies the following universal property: if φ : A → M(B) is a nondegenerate completely positive map, then there exists a unique unital completely positive map φ : M(A) → M(B) extending φ that is strictly continuous on the unit ball. If furthermore φ is a *-homomorphism, then φ is a *-homomorphism as well; see [29, Propositions 2.5 and 5.8].

5 Actions of compact groups on C*-algebras Suppose that A is a C*-algebra. An action of G on A is a continuous group homomorphism G → Aut(A), g 󳨃→ αg , where Aut(A) denotes the automorphism group of A endowed with the topology of pointwise convergence. Such an action induces an injective nondegenerate *-homomorphism α : A → C(G, A) ≅ C(G) ⊗ A

396 � M. Lupini defined by setting α(x)(g) = αg −1 (x) for x ∈ A and g ∈ G. Such a nondegenerate *-homomorphism α satisfies: (1) (Δ ⊗ idA ) ∘ α = (idC(G) ⊗ α) ∘ α; (2) [α(A)(C(G) ⊗ 1M(A) )] = C(G) ⊗ A. Indeed, we have that ((Δ ⊗ idA ) ∘ α)(x)(g, h) = (Δ ⊗ idA )(α(x))(g, h) = α(x)(gh) = α(gh)−1 (x) and ((id ⊗ α) ∘ α)(x)(g, h) = (id ⊗ α)(α(x))(g, h) Here, we identify C(G) ⊗ A ⊆ M(C(G)) ⊗ M(A) as a C*-subalgebra of M(C(G) ⊗ A). Furthermore, [α(A)(C(G) ⊗ 1M(A) )] denotes the closure of the linear span in C(G) ⊗ A of the set α(A)(C(G) ⊗ 1M(A) ) := {α(a)(ξ ⊗ 1M(A) ) : a ∈ A, ξ ∈ C(G)} ⊆ C(G) ⊗ A. Conversely, if α : A → C(G, A) = C(G) ⊗ A is an injective *-homomorphism satisfying (1) and (2) above, then it is nondegenerate, and setting αg = evg −1 ∘ α, for g ∈ G, where evg −1 : C(G, A) → A is the linear map given by evaluation at g −1 , gives an action of G on A. These constructions are mutually inverse; see [40, Theorem 9.2.4]. In the following, we will identify the action G → Aut(A) with the corresponding nondegenerate *-homomorphism A → C(G)⊗A. A G-C*-algebra is a pair (A, α) where A is a C*-algebra and α is an action of G on A. We now recall some facts and identities concerning spectral subspaces and spectral projections associated with G-C*-algebras, to be used in their axiomatization in first-order logic for metric structures. Suppose that α is an action of G on A and λ ∈ Rep(G). An intertwiner between λ and α is a linear map v : ℋλ → A such that

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αg (vξ) = vλg (ξ) for g ∈ G and ξ ∈ ℋλ . The space of intertwiners between λ and α is denoted by Int(λ, α). The spectral subspaces of the G-C*-algebra (A, α) associated with λ is Aλ := {vξ : ξ ∈ ℋλ , v ∈ Int(λ, α)}. This is a closed subspace equal to {a ∈ A : α(a) ∈ C(G)λ ⊗ A} = {a ∈ A : α(a) ∈ C(G)λ ⊗ Aλ }. Furthermore, we have that Aλ + Aμ ⊆ Aλ⊕μ , Aλ Aμ ⊆ Aλ⊗μ ,

(Aλ ) = Aλ ∗

for λ, μ ∈ Rep(G). Hence, the collection {Aλ : λ ∈ Rep(A)} of closed subspaces of A is upward directed. Its union 𝒪(A) is a dense self-adjoint subalgebra of A called the algebraic core of the G-C*-algebra A. We now recall notation from [35, Section 1] and identities established in [35, Theoλ rem 1.5]. Define Eks : A → Aλ by setting λ Eks = (ωλks ⊗ id) ∘ α

for 1 ≤ k, s ≤ dλ . Then we have that μ

λ λ Esm Eln = δλμ δml Esn

for λ, μ ∈ Irr(G). Furthermore, dλ

λ E λ := ∑ Ekk k=1

λ is a projection onto Aλ . Moreover, Ekk is the projection onto a closed subspace Aλk of Aλ of (possibly infinite) dimension cλ . We regard cλ as a cardinal number, and hence we identify with the set of ordinals less than cλ . Let (a1iλ )i 0 (that does not depend on A) such that if a ∈ Aλ and g, h ∈ G satisfy d(g, h) < δ, then ‖αg (a) − αh (a)‖ ≤ ε‖a‖. λ Proof. Suppose that a ∈ Aλk for some k ∈ {1, 2, . . . , dλ }. We have that (aki )i 0, there exist tuples c0 , . . . , cd in C such that η(f (a), w, c0 + ⋅ ⋅ ⋅ + cd ) ≤ η(θ(a), w, b) + ε, and for j = 0, . . . , d, φ(w, cj ) ≤ φ(w, b) + ε.

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As a unital cpc order zero map is a *-homomorphism, 0-containment coincides with ∗ positive weak ℒCG -containment. The notion of G-equivariant order zero dimension is obtained by considering d-containment in the identity map. Definition 10.1. Let θ : A → B be a G-equivariant *-homomorphism between G-C*-algebras. The G-equivariant order zero dimension dimGoz (θ) is the least d ∈ ℕ such that θ is G-equivariantly d-contained in idA , if such a d exists, and ∞ otherwise. The case of C*-algebras is obtained as the particular case when G is the trivial group. This gives the notion of order zero dimension dimoz (θ) for a *-homomorphism θ between C*-algebras.

11 Commutant d-containment Suppose that A, B are separable unital G-C*-algebras. Then we say that A is commutant positively weakly contained in B if for every separable unital G-C*-subalgebra C of A′ ∩ ∏G𝒰 A there exists a unital G-equivariant *-homomorphism C → B′ ∩ ∏G𝒰 B. Lemma 11.1. If there exists a unital G-equivariant *-homomorphism A → A′ ∩ ∏G𝒰 A (for example, this holds when A is abelian), then A is commutant positively weakly contained in B if and only if there exists a unital G-equivariant *-homomorphism A → B′ ∩ ∏G𝒰 B. Proof. Suppose that there exists a unital G-equivariant *-homomorphism ϕ : A → A′ ∩ ∏G𝒰 A. If A is commutant positively weakly contained in B, then by definition we have that for every separable unital G-C*-subalgebra C of A′ ∩ ∏G𝒰 A there exists a unital G-equivariant *-homomorphism C → B′ ∩ ∏G𝒰 B. In particular, for C = ϕ(A), there exists a unital G-equivariant *-homomorphism ψ : C → B′ ∩∏G𝒰 B. Hence, ψ∘φ : A → B′ ∩∏G𝒰 B is a unital G-equivariant *-homomorphism. Conversely, suppose that there exists a unital G-equivariant *-homomorphism A → B′ ∩ ∏G𝒰 B. Then we can conclude that for every separable unital G-C*-algebra C of ∏G𝒰 A there exists a unital G-equivariant *-homomorphism C → B′ ∩ ∏G𝒰 B by Łos’ theorem and countable saturation of ultraproducts. For a tuple of variables x = (x1 , . . . , xn ), let tAc (x) be the collection of conditions ‖xj a − axj ‖ ≤ 0 for a ∈ A and 1 ≤ j ≤ n. Then we have that A is commutant positively

c weakly contained in B if for every positive quantifier-free ℒ1,C G -type t(x), if t(x) ∪ tA (x) is approximately finitely realized in A, then t(x) ∪ tBc (x) is approximately finitely realized in B. More generally, we consider the notion of commutant d-containment, which recovers commutant positive weak containment for d = 0. We say that A is commutant d-contained in B, and write A ≾d B, if for every separable unital G-C*-algebra L ⊆ A′ ∩ ∏G𝒰 A there exist G-equivariant cpc order zero maps η0 , . . . , ηd : L → B′ ∩ ∏G𝒰 B ∗

406 � M. Lupini such that η0 + ⋅ ⋅ ⋅ + ηd is unital. Again, for d = 0, this recovers the notion of commutant positive weak containment. Semantically, one can say that A is commutant d-contained osy in B if and only if for every positive quantifier-free ℒG -type t osy (x) and every positive oz oz osy ℒoz (x) ∪ tAc (x) is G -type t (x) in the free variables x = (x0 , . . . , xn ) such that t (x) ∪ t approximately realized in A, the type tBc (x (0) , . . . , x (d) ) ∪ t oz (x (0) ) ∪ ⋅ ⋅ ⋅ ∪ t oz (x (d) ) ∪ t osy (x (0) + ⋅ ⋅ ⋅ + x (d) ) is approximately realized in B, where x (i) = (x0(i) , . . . , xn(i) ) is a distinct tuple of variables for every 0 ≤ i ≤ d and x (0) + ⋅ ⋅ ⋅ + x (d) = (x0(0) + ⋅ ⋅ ⋅ + x0(d) , . . . , xn(0) + ⋅ ⋅ ⋅ + xn(d) ). It is easily seen that for n, m ≥ 1, if A ≾n−1 B and B ≾m−1 C then A ≾nm−1 C. Lemma 11.2. Suppose that A, B, C are separable unital G-C*-algebras. Let θ : A → C be a unital G-equivariant *-homomorphism. (1) Suppose that the map 1C ⊗ θ : A → B ⊗max C has order zero dimension at most d, as witnessed by G-equivariant cpc order zero A-bimodule maps ψ0 , . . . , ψd : B ⊗max C → ∏G𝒰 A such that the restriction of ψ0 + ⋅ ⋅ ⋅ + ψd to A is the diagonal embedding A → ∏G𝒰 A. Then there exist G-equivariant cpc order zero maps η0 , . . . , ηd : B → A′ ∩ ∏G𝒰 A such that η0 + ⋅ ⋅ ⋅ + ηd is unital. (2) In (1), the converse holds if C = A and θ is the identity map of A. Proof. (1) Since ψ0 , . . . , ψd are A-bimodule maps, we have that the functions η0 := ψ0 |B , . . . , ηd := ψd |B have image contained in A′ ∩ ∏G𝒰 A, and hence satisfy the desired conclusion. (2) The function (A′ ∩ ∏G𝒰 A) × A → ∏G𝒰 A, (x, y) 󳨃→ xy induces by the universal property of the maximal tensor product a G-equivariant unital *-homomorphism Ψ : (A′ ∩ ∏G𝒰 A) ⊗max A → ∏G𝒰 A. One can then define ψi := Ψ ∘ (ηi ⊗ idA ) for 0 ≤ i ≤ d. Proposition 11.3. Let A, B be separable unital G-C*-algebras. Assume that there exists a unital G-equivariant *-homomorphism A → A′ ∩ ∏G𝒰 A. Then the following assertions are equivalent: (1) A ≾d B; (2) there exist G-equivariant order zero maps η0 , . . . , ηd : A → B′ ∩ ∏G𝒰 B with unital sum; (3) dimGoz (1B ⊗ idA : A → B ⊗max A) ≤ d. Proof. The equivalence of (2) and (3) is the content of Lemma 11.2(2). It remains to prove that (1) and (2) are equivalent. Let θ : A → ∏G𝒰 A be a unital G-equivariant *-homomorphism. If A ≾d B then there exists G-equivariant cpc order zero maps η0 , . . . , ηd : θ(A) → B′ ∩ ∏G𝒰 B with unital sum. Thus, η0 ∘ θ, . . . , ηd ∘ θ : A → B′ ∩ ∏G𝒰 B are

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G-equivariant cpc order zero maps with unital sum. Thus, (1) implies (2). Conversely, osy suppose that (2) holds. Consider a positive quantifier-free ℒG -type t osy (x) and a positive oz oz ℒG -type t (x) in the free variables x = (x1 , . . . , xn ) such that t oz (x) ∪ t osy (x) ∪ tAc (x) is approximately realized in A. Then by (2) it easily follows that the type tBc (x (0) , . . . , x (d) ) ∪ t oz (x (0) ) ∪ ⋅ ⋅ ⋅ ∪ t oz (x (d) ) ∪ t osy (x (0) + ⋅ ⋅ ⋅ + x (d) ) is approximately finitely realized in B. This shows that A ≾d B. Proposition 11.4. Suppose that A, B are separable unital G-C*-algebras, and θ : A → B is a G-equivariant unital *-homomorphism. If dimGoz (θ) ≤ d, then B ≾d A. Proof. Let C be a separable unital G-C*-subalgebra of B′ ∩ ∏G𝒰 B. Thus, the inclusion map C → B′ ∩ ∏G𝒰 B is a unital G-equivariant *-homomorphism. By (2) in the previous proposition applied in the case when d = 0, we have that 1C ⊗ idB : B → C ⊗max B has order zero dimension equal to zero. Thus, 1C ⊗ θ = (1C ⊗ idB ) ∘ θ : A → C ⊗max B has order zero dimension at most d. Hence, by the previous proposition again, there exist G-equivariant cpc order zero maps η0 , . . . , ηd : C → A′ ∩ ∏G𝒰 A with unital sum.

12 Rokhlin dimension and crossed products Suppose that A is a unital G-C*-algebra. We regard C(G) as a G-C*-algebra with respect to the canonical translation action. One says that A has the Rokhlin property if C(G) is commutant positively weakly contained in A. By Lemma 11.1, this is equivalent to the assertion that there exists a unital G-equivariant *-homomorphism C(G) → A′ ∩ ∏G𝒰 A. Notice that Lemma 11.1 applies in this context because C(G) is abelian. The notion of Rokhlin dimension generalizes the Rokhlin property, which is recovered for d = 0. The Rokhlin dimension dimRok (A) of A is the least d such that C(G) ≾d A. As mentioned in the introduction, the notion of Rokhlin property has its origin in the Rokhlin lemma from ergodic theory. In the context of C*-algebras, it was initially considered for finite groups, in which case it was instrumental in the classification theorems of Izumi from [20, 21]. This was later generalized to arbitrary compact second countable groups in [18]. Since then, the Rokhlin property for compact group actions has been the subject of a number of articles [8, 12, 33], showing the fruitfulness of this notion. The Rokhlin property can be seen as a notion of freeness for a compact group action that, at least for suitably well-behaved C*-algebras, entails strong classification and structural results for actions, as well as preservation results from the given C*-algebra to the crossed product and the fixed-point algebra. In order to extend these results beyond the Rokhlin case, the notion of Rokhlin dimension was introduced in [19] for finite groups and [10] for compact groups. This was in analogy with similar dimension function for C*-algebras such as nuclear dimension and

408 � M. Lupini decomposition rank, which can be seen as relaxations of the strong regularity property of being approximately finite-dimensional, which is recovered in the case of dimension zero. A number of classification results and preservation results have been established in the case of actions with finite Rokhlin dimension [8, 9, 17, 19, 39]. As a particular instance of Proposition 11.3 in the case when A = C(G), we obtain the following. Proposition 12.1. Suppose that B is a unital G-C*-algebra. Let θ : B → C(G) ⊗ B be the second factor embedding. Then dimRok (B) = dimGoz (θ). Suppose that G is a compact group with normalized Haar measure μ. On C(G) consider the L1 -norm defined by setting 󵄨 󵄨 ‖f ‖1 = ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dμ(x). We denote by L1 (G) the completion of C(G) with respect to such a norm. We regard L1 (G) as a *-algebra with respect to the convolution product defined by setting (f0 ∗ f1 )(x) = ∫ f0 (y)f1 (y−1 x) dμ(y) for f0 , f1 ∈ L1 (G), and involution given by f ∗ (x) = f (x −1 ) for f ∈ L1 (G). If ρ is a continuous representation of G on a Hilbert space ℋ, namely a strongly continuous group homomorphism G → U(ℋ), then ρ induces a nondegenerate L1 -norm decreasing *-homomorphism πρ : C(G) → B(ℋ) such that ⟨πρ (f )ξ, η⟩ = ∫⟨f (G)ξ, η⟩ dμ(G) for f ∈ C(G) and ξ, η ∈ ℋ. (L1 -norm decreasing means that πρ is norm-decreasing when C(G) is endowed with the L1 -norm and B(ℋ) is endowed with the operator norm.) Define C ∗ (G) to be the C*-algebra obtained as the completion of C(G) with respect to the norm 󵄩 󵄩 ‖f ‖ = sup󵄩󵄩󵄩πρ (f )󵄩󵄩󵄩 ρ

where ρ ranges among the continuous representations of G on Hilbert spaces. This is called the group C*-algebra of G. There is an injective continuous homomorphism G → M(C ∗ (G)) obtained by mapping g ∈ G to the multiplier C ∗ (G) obtained by extending the linear map C(G) → C(G), f 󳨃→ (x 󳨃→ f (g −1 x)); see Section 4. We will identify G with its image inside M(C ∗ (G)).

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Suppose that (A, α) is a G-C*-algebra. Then one can consider the corresponding (full) crossed product A ⋊ G; see [41, Section 2.3]. We recall here its construction and main properties. Let C(G, A) be the *-algebra of continuous functions G → A. The algebra operation on C(G, A) is given by the convolution product, defined by setting (f0 ∗ f1 )(x) = ∫ f0 (y)αy (f1 (y−1 x)) dμ(y) for f0 , f1 ∈ C(G, A) and x ∈ G. The integral is taken in the sense of vector-valued integration; see [41, Appendix B.1]. The involution on C(G, A) is defined by f ∗ (x) = αx (f (x −1 ) ) ∗

for f ∈ C(G, A) and x ∈ G. The L1 -norm on C(G, A) is defined by setting 󵄩 󵄩 ‖f ‖1 = ∫󵄩󵄩󵄩f (x)󵄩󵄩󵄩 dμ(x) for ξ ∈ C(G, A). A (nondegenerate) covariant *-homomorphism from a G-C*-algebra A to a C*-algebra B is a pair (jA , jG ) where jA : A → M(B) is a (nondegenerate) *-homomorphism and jG : G → UM(B) is a strictly continuous group homomorphism such that jG (s)jA (A) = jA (αs (a))jG (s) for s ∈ G and a ∈ A, where M(B) is the multiplier algebra of B and UM(B) is its unitary group [41, Definition 2.37]. The strictly continuous homomorphism jG : G → UM(B) induces a *-homomorphism jG : C ∗ (G) → M(B) defined by setting jG (ξ) = ∫ ξ(s)jG (s) dμ(s) for f ∈ C(G) ⊆ C ∗ (G); see [36, Corollary 8]. A (nondegenerate) covariant representation of A on a Hilbert space ℋ is a (nondegenerate) covariant *-homomorphism from A to the algebra K(ℋ) of compact operators on ℋ. Thus, a covariant representation of A on ℋ is a pair (π, U) where π is a representation of A on ℋ and U is a strictly continuous unitary representation of G. Such a covariant representation induces an L1 -norm decreasing *-homomorphism π ⋊ U : C(G, A) → B(ℋ), called the integrated form of (π, U), defined by setting (π ⋊ U)(f ) = ∫ π(f (x))U(x) dμ(x) for f ∈ C(G, A). The universal norm on C(G, A) is the norm defined by the family ℱ of integrated forms of (nondegenerate) covariant representations of A on Hilbert spaces, by setting 󵄩 󵄩 ‖f ‖ = sup󵄩󵄩󵄩π(f )󵄩󵄩󵄩; π∈ℱ

410 � M. Lupini see [41, Lemma 2.27]. The crossed product A ⋊ G is the C*-algebra obtained as the corresponding completion. In the particular case when A = ℂ, the crossed product A ⋊ G coincides with the group C*-algebra C ∗ (G). We have a nondegenerate injective *-homomorphism iA : A → M(A ⋊ G) obtained by letting, for a ∈ A, iA (a) be the multiplier of A ⋊ G induced by the linear map C(G, A) → C(G, A), f 󳨃→ af , where (af )(s) = af (s) for s ∈ G. We also have a nondegenerate *-homomorphism iG : C ∗ (G) → M(A ⋊ G) [41, Lemma 2.35], such that iA (a)iG (ξ) = ξ ⊗ a ∈ C(G, A) ⊆ A ⋊ G for a ∈ A and ξ ∈ C(G). The set of elements of A ⋊ G of the form iA (a)iG (ξ) for a ∈ A and ξ ∈ C(G) has dense linear span; see [41, Corollary 2.36]. Considering the unique unital extension iG : M(C ∗ (G)) → M(A ⋊ G) that is strictly continuous on the unit ball given by the universal property of the multiplier algebra, we have that, for g ∈ G, iG (G) is the multiplier on A ⋊ G obtained by extending the linear map C(G, A) → C(G, A), f 󳨃→ αg (f (g −1 ⋅)). One has that (iA , iG ) is a nondegenerate unital covariant *-homomorphism from A to A ⋊ G that satisfies the following universal property [36, Proposition 2]: if (jA , jG ) is a nondegenerate covariant homomorphism from A to B, then there exists a unique nondegenerate *-homomorphism φ : A ⋊ G → B such that φ ∘ iA = jA and φ ∘ iG = jG , where φ : M(A ⋊ G) → M(B) is the unique unital extension of φ that is strictly continuous on the unit ball obtained from the universal property of M(B). The unital *-homomorphism iG : M(C ∗ (G)) → M(A⋊G) induces a unital *-homomorphism iG𝒰 : M(C ∗ (G)) → ∏𝒰 M(A⋊G). Composing this with the canonical injective unital *-homomorphism J : ∏𝒰 M(A ⋊ G) → M(∏𝒰 (A ⋊ G)) as in Lemma 4.1 we obtain a unital *-homomorphism J ∘ iG : M(C ∗ (G)) → M(∏𝒰 (A ⋊ G)). Suppose that B0 , B1 are G-C*-algebras, and θ : B0 → B1 is a G-equivariant *-homomorphism. Then by the universal property of the crossed product, θ uniquely induces a nondegenerate *-homomorphism θ̂ : B0 ⋊ G → B1 ⋊ G whose unique unital extension θ : M(B0 ⋊ G) → M(B1 ⋊ G) to the multiplier algebras that is strictly continuous on the unit ball satisfies θ ∘ iG = iG and θ ∘ iB0 = iB1 ∘ θ. Explicitly, we have that ̂ (b)i (ξ)) = i (θ(b))i (ξ) θ(i B0 G B1 G for b ∈ B0 and ξ ∈ C(G). More generally, if θ : B0 → B1 is a G-equivariant cpc order zero map, then by [42, Corollary 4.1] there exists a unique *-homomorphism ρθ : C0 ((0, 1]) ⊗ B0 → B1 such that θ(A) = ρθ (id(0,1] ⊗ a) for a ∈ B0 . Furthermore, ρθ is equivariant when C0 ((0, 1]) is regarded as a G-C*-algebra endowed with the trivial G-action. By the universal prop̂θ : erty of the crossed product again, there exists a nondegenerate *-homomorphism ρ (C0 ((0, 1]) ⊗ B0 ) ⋊ G → B1 ⋊ G obtained from ρθ as above. As C0 ((0, 1]) is endowed with the trivial G-action, we have a canonical *-isomorphism (C0 ((0, 1]) ⊗ B0 ) ⋊ G ≅ C0 ((0, 1]) ⊗ (B0 ⋊ G).

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One can define the nondegenerate cpc order zero map θ̂ : B0 ⋊ G → B1 ⋊ G by setting ̂ ̂ θ (idC((0,1]) ⊗ x) for x ∈ B0 ⋊ G; see [9, Proposition 2.3]. Explicitly, for b ∈ B0 and θ(x) =ρ ξ ∈ C(G) we have that ̂ (b)i (ξ)) = ρ ̂ θ (idC((0,1]) ⊗ iB0 (b)iG (ξ)) θ(i B0 G

̂ θ ((idC((0,1]) ⊗ iB0 (b))iG (ξ)) =ρ = ρθ ((idC((0,1]) ⊗ iB0 (b)))iG (ξ)

= iB1 (θ(b))iG (ξ).

If θ : M(B0 ⋊G) → M(B1 ⋊G) is the unique unital extension of θ̂ to the multiplier algebras that is strictly continuous on the unit ball given, then again we have that θ ∘ iG = iG and θ ∘ iB0 = iB1 ∘ θ. Lemma 12.2. Suppose that A is a separable G-C*-algebra. Then the fixed point algebra AG := {x ∈ A : ∀g ∈ G, αg (x) = x} is a corner of A ⋊ G. Proof. We recall the argument of Rosenberg from [37]. Let A† be equal to A if A is unital, and be equal to the unitization of A if A is not unital. Notice that A† has a canonical G-C*-algebra structure such that the inclusion A → A† is G-equivariant. Define ξ ∈ L1 (G, A† ) to be the function constantly equal to 1. Then for x ∈ C(G, A) we have that (x ∗ ξ)(t) = ∫ x(s)αs (ξ(s−1 t)) dμ(s) = ∫ x(s) dμ(s) = x0 and (ξ ∗ x ∗ ξ)(t) = ∫ ξ(s)αs ((x ∗ ξ)(s−1 t)) dμ(s) = ∫ αs (x0 ) dμ(s) = x1 . Thus, ξ ∗ x ∗ ξ ∈ C(G, A) is a constant function with constant value x1 ∈ AG . Thus, we have that ξ defines a multiplier projection iG (ξ) of A⋊G, namely an adjointable operator iG (ξ) : A ⋊ G → A ⋊ G, x 󳨃→ ξ ∗ x ∗ ξ, such that iG (ξ)(A ⋊ G)iG (ξ) ⊆ iA (AG )iG (ξ) = iG (ξ)iA (AG )iG (ξ). For a ∈ AG , we have that ξ ∗ (a ⊗ ξ) = a ⊗ ξ and hence iG (ξ)iA (a)iG (ξ) = iA (a)iG (ξ).

412 � M. Lupini It easily follows that AG → iG (ξ)(A ⋊ G)iG (ξ), a 󳨃→ iA (a)iG (ξ) is an isomorphism of C*-algebras. Lemma 12.3. Suppose that A is a separable G-C*-algebra. Then AG contains an approximate identity for A. Proof. Suppose that (un ) is an approximate identity for A. For n ∈ ℕ, define wn = ∫ αg (un ) dμ(G) ∈ AG . We claim that (wn ) is an approximate identity for A. Indeed, if x ∈ A and ε > 0 then by compactness of G there exists n0 ∈ ℕ such that for n ≥ n0 we have that ‖un αg (x) − αg (x)‖ < ε for every g ∈ G, and hence ‖αg (un )x − x‖ < ε for every g ∈ G. Taking integrals, we have that 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ‖wn x − x‖ = 󵄩󵄩󵄩∫(αg (un )x − x) dμ(G)󵄩󵄩󵄩 ≤ ε 󵄩󵄩 󵄩󵄩 for n ≥ n0 . This concludes the proof. Lemma 12.4. Suppose that A is a separable G-C*-algebra. Let ∏𝒰 (A⋊G) be the C*-algebra ultrapower of A ⋊ G. Then there is a canonical *-homomorphism Φ : (∏G𝒰 A) ⋊ G → ∏𝒰 (A ⋊ G) that makes the following diagram commute ( ∏G𝒰 A) ⋊ G Δ̂ A

A⋊G

ΔA⋊G

∏𝒰 (A ⋊ G)

where Δ̂ A : A ⋊ G → (∏G𝒰 A) ⋊ G is the nondegenerate *-homomorphism induced by ΔA : A → ∏G𝒰 A. Proof. Let J : ∏𝒰 M(A ⋊ G) → M(∏𝒰 A ⋊ G) be the injective unital *-homomorphism as in Lemma 4.1. The injective unital *-homomorphism iA : A → M(A ⋊ G) induces a unital *-homomorphism iA𝒰 : ∏G𝒰 A → ∏𝒰 M(A ⋊ G). Composing these, we obtain an injective unital *-homomorphism J ∘ iA𝒰 : ∏G𝒰 A → M(∏𝒰 A ⋊ G). Likewise, the unital *-homomorphism iG : M(C ∗ (G)) → M(A ⋊ G) induces a unital *-homomorphism M(C ∗ (G)) → ∏𝒰 M(A ⋊ G). Let ℬ ⊆ ∏𝒰 (A ⋊ G) be the C*-algebra comprising those x ∈ ∏𝒰 (A ⋊ G) such that the functions G → ∏𝒰 (A⋊G), g 󳨃→ (J ∘iG𝒰 )(G)x and g 󳨃→ x(J ∘iG𝒰 )(G) are continuous. It is clear that ℬ is indeed a C*-algebra that contains the image of the diagonal embedding ΔA⋊G . Thus, we can regard ΔA⋊G as a *-homomorphism A ⋊ G → ℬ. Furthermore, for g ∈ G, jG (g) : ℬ → ℬ, x 󳨃→ (J ∘ iG𝒰 )(g)x defines an adjointable linear map on ℬ with adjoint x 󳨃→ (J ∘ iG𝒰 )(g −1 )x. Thus, jG (g) ∈ UM(ℬ), and jG : G → UM(ℬ) is a strictly continuous group homomorphism.

Model theory of G-C*-algebras and order zero dimension

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Claim. For a ∈ ∏G𝒰 A, jA (a) : ℬ → ℬ, x 󳨃→ (J ∘ iA𝒰 )(a)x is a well-defined adjointable linear map on ℬ. Proof of claim. Without loss of generality, suppose that ‖a‖ ≤ 1. Suppose that x ∈ ℬ and ‖x‖ ≤ 1. We need to prove that (J ∘ iA𝒰 )(a)x ∈ ℬ. Since x ∈ ℬ, we have that the function G → ∏𝒰 (A ⋊ G), g 󳨃→ (J ∘ iA𝒰 )(a)x(J ∘ iG𝒰 )(g) is continuous. It remains to prove that the function G → ∏𝒰 (A ⋊ G), g 󳨃→ (J ∘ iG𝒰 )(G)(J ∘ iA𝒰 )(A)x, is continuous. Fix a compatible metric d on G. For g ∈ G, we have that (J ∘ iG𝒰 )(g)(J ∘ iA𝒰 )(a)x = (J ∘ iA𝒰 )(αg (a))(J ∘ iG𝒰 )(g)x. Fix ε > 0. Since x ∈ ℬ and a ∈ ∏G𝒰 A, there exists δ > 0 such that if g, h ∈ G satisfy d(g, h) < δ, then 󵄩 󵄩󵄩 𝒰 𝒰 󵄩󵄩(J ∘ iA )(αg (a)) − (J ∘ iA )(αh (a))󵄩󵄩󵄩 < ε/2 and 󵄩󵄩 󵄩 𝒰 𝒰 󵄩󵄩(J ∘ iG )(g)x − (J ∘ iG )(h)x 󵄩󵄩󵄩 < ε/2, and hence 󵄩󵄩 󵄩 𝒰 𝒰 𝒰 𝒰 󵄩󵄩(J ∘ iA )(αg (a))(J ∘ iG )(g)x − (J ∘ iA )(αh (a))(J ∘ iG )(h)x 󵄩󵄩󵄩 < ε. This concludes the proof that (J ∘ iA𝒰 )(a)x ∈ ℬ. It is easy to verify now that x 󳨃→ (J ∘ iA𝒰 )(a)x is an adjointable linear map on ℬ with adjoint the linear map x 󳨃→ (J ∘ iA𝒰 )(a∗ )x. By the claim above, for a ∈ ∏G𝒰 A, jA (a) ∈ M(ℬ). It is easy to verify that jA : ∏G𝒰 A → M(ℬ) is a *-homomorphism. We claim that jA is nondegenerate. Indeed, let (ui ) be an approximate identity for A contained in AG as in Lemma 12.3. If x = [xn ]𝒰 ∈ ℬ and ε > 0, then for every n ∈ ℕ there exists in ∈ ℕ such that ‖iA (uin )xn − xn ‖ < ε. Thus, setting a := [uin ]𝒰 ∈ ∏G𝒰 A, we have that

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩jA (a)x − x 󵄩󵄩󵄩 = 𝒰 − lim󵄩󵄩󵄩iA (uin )xn − xn 󵄩󵄩󵄩 ≤ ε. n As this holds for every x ∈ ℬ and ε > 0, this proves that jA is nondegenerate. We have that (jA , jG ) is a nondegenerate covariant representation of ∏G𝒰 A. Indeed, if a = [an ]𝒰 ∈ ∏G𝒰 A and g ∈ G, the we have that for x = [xn ]𝒰 ∈ ℬ, jA (αg (a))jG (g)x = [iA (αg (a))iG (g)xn ]𝒰 = [iG (g)iA (a)xn ]𝒰 = jG (g)jA (A)x. Thus, jA (αg (a))jG (g) = jG (g)jA (a).

414 � M. Lupini Therefore, the pair (jA , jG ) induces a nondegenerate ∗-homomorphism Φ : (∏G𝒰 A)⋊G → ℬ such that Φ∘i∏G A = jA and Φ∘iG = jG , where Φ is the unique unital extension of Φ to the 𝒰 multiplier algebras that is strictly continuous on the unit ball. We claim that Φ∘Δ̂ A = ΔA⋊G as *-homomorphisms A ⋊ G → ℬ. In order to verify this, we need to show that Φ ∘ Δ̂ A and ΔA⋊G have the same unital extensions Φ ∘ Δ̂ A and ΔA⋊G to functions M(A ⋊ G) → M(ℬ) that are strictly continuous

on the unit ball. To this purpose, it suffices to show that

Φ ∘ Δ̂ A ∘ iGA⋊G = ΔA⋊G ∘ iGA⋊G and Φ ∘ Δ̂ A ∘ iAA⋊G = ΔA⋊G ∘ iAA⋊G We have that, by definition, G

(∏ A)⋊G Φ ∘ Δ̂ A ∘ iGA⋊G = Φ ∘ iG 𝒰 = jG = ΔA⋊G ∘ iGA⋊G

where the last equality holds by definition of jG . Likewise, we have that Φ ∘ Δ̂ A ∘ iAA⋊G = Φ ∘ i

(∏G𝒰 A)⋊G

∏G𝒰 A

∘ ΔA = jA ∘ ΔA = ΔA⋊G ∘ iAA⋊G

where the last equality holds by definition of jA . Lemma 12.5. Let θ : A → B be a G-equivariant ∗-homomorphism between G-C*-algebras. Then, adopting the notations above, dimoz (θ̂ : A ⋊ G → B ⋊ G) ≤ dimGoz (θ) and dimoz (θ|AG : AG → BG ) ≤ dimGoz (θ). Proof. Suppose that dimoz (θ) = d. Thus, there exist G-equivariant cpc order zero A-bimodule maps ψ0 , . . . , ψd : B → ∏G𝒰 A such that ψ := ψ0 + ⋅ ⋅ ⋅ + ψd is contractive satisfies ψ ∘ θ = ΔA . We begin with proving the first assertion. Set C := (∏G𝒰 A) ⋊ G. We have induced ̂ ,...,ψ ̂ : B ⋊ G → C induced by ψ , . . . , ψ . For nondegenerate cpc order zero maps ψ 0 d 0 d ̂ to the multiplier 0 ≤ i ≤ d, if ψi : M(B ⋊ G) → M(C) is the unique extension of ψ i algebras that is strictly continuous on the unit ball, then we have that ψi ∘ iB = i∏G A ∘ ψi 𝒰

and ψi ∘ iG = iG . We denote the G-action on B by (g, b) 󳨃→ g ⋅ b, and similarly for A and for ∏G𝒰 A. We ̂ ,...,ψ ̂ are (A ⋊ G)-bimodule maps. Indeed, for 0 ≤ i ≤ d, a ∈ A, b ∈ B, observe that ψ 0 d

Model theory of G-C*-algebras and order zero dimension

� 415

and f ∈ C(G), and g ∈ G, we have that ̂ (i (a) ⋅ i (b)i (f )) = ψ ̂ (i (θ(a)b)i (f )) ψ i A B G i B G

= iC (ψi (θ(a)b))iG (f )

= iC (ΔA (a)ψi (b))iG (f )

= iC (ΔA (a))iC (ψi (b))iG (f ) ̂ (i (b)i (f )) = iA (a) ⋅ ψ i B G and, using the fact that ψ0 , . . . , ψd are G-equivariant, ̂ (i (g) ⋅ i (b)i (f )) = ψ ̂ (i (α (b))i (g)i (f )) ψ i G B G i B g G G ̂ (i (g ⋅ b)i (g)i (f )) =ψ i B G G

= iC (ψi (g ⋅ b))iG (g)iG (f ) = iC (g ⋅ ψi (b))iG (g)iG (f )

= iG (g) ⋅ iC (ψi (b))iG (f ).

As this holds for every a ∈ A and g ∈ G, by the universal property of the crossed product we obtain that ̂ (x ⋅ i (b)i (f )) = x ⋅ i (ψ (b))i (f ) ψ i B G C i G for every x ∈ A ⋊ G. As this holds for every b ∈ B and f ∈ C(G), we can conclude that ̂ (x ⋅ y) = x ⋅ y ψ i for x ∈ A ⋊ G and y ∈ B ⋊ G. By Lemma 12.4, we have a nondegenerate *-homomorphism Φ : (∏G𝒰 A) ⋊ G → ̂ A = ΔA⋊G , where Δ ̂ A : A ⋊ G → C is the nondegenerate ∏𝒰 (A ⋊ G) such that Φ ∘ Δ ̂ , . . . , Φ∘ ψ ̂ : B⋊G → ∏ (A⋊G) *-homomorphism induced by ΔA . Thus, we have that Φ∘ ψ 0 d 𝒰 are cpc order zero (A ⋊ G)-bimodule maps such that ̂ + ⋅⋅⋅ + Φ ∘ ψ ̂ = Φ ∘ (ψ ̂ + ⋅⋅⋅ + ψ ̂ ) Φ∘ψ 0 d 0 d is contractive. Furthermore, we have that ̂ + ⋅⋅⋅ + ψ ̂ ) ∘ θ̂ = Δ Φ ∘ (ψ 0 d A⋊G : A ⋊ G → ∏𝒰 (A ⋊ G). Indeed, let Φ be the extension of Φ to the multiplier algebras given by the universal ̂ Then we have that ̂ ,...,ψ ̂ and θ. property, and similarly for ψ 0 d Φ ∘ (ψ0 + ⋅ ⋅ ⋅ + ψd ) ∘ θ ∘ iA = Φ ∘ (ψ0 + ⋅ ⋅ ⋅ + ψd ) ∘ iB ∘ θ = Φ ∘ i∏G

𝒰

A

∘ (ψ0 + ⋅ ⋅ ⋅ + ψd ) ∘ θ

416 � M. Lupini = Φ ∘ i∏G

𝒰

A

∘ ΔA

̂ A ∘ iA =Φ∘Δ

= ΔA⋊G ∘ iA and

Φ ∘ (ψ0 + ⋅ ⋅ ⋅ + ψd ) ∘ θ ∘ iG = Φ ∘ (ψ0 + ⋅ ⋅ ⋅ + ψd ) ∘ iG = Φ ∘ iG ̂ A ∘ iG =Φ∘Δ = ΔA⋊G ∘ iG .

This concludes the proof of the first assertion. For the second assertion, notice that ψ0 |BG , . . . , ψd |BG : BG → ∏𝒰 AG are cpc order zero A-bimodule maps such that ψ|BG = ψ0 |BG + ⋅ ⋅ ⋅ + ψd |BG is contractive and satisfies ψ|BG ∘ θ = ΔAG . Let A be a G-C*-algebra. Denote by λ : G → U(L2 (G)) the left regular representation of G. Then λ induces an action of G on 𝒦(L2 (G)) by inner automorphisms that turns 𝒦(L2 (G)) into a G-C*-algebra. With respect to such a G-C*-algebra structure, we have that the crossed product A ⋊ G is isomorphic to the fixed point algebra (A ⊗ 𝒦(L2 (G)))G . We let σ : A ⋊ G → A ⊗ 𝒦(L2 (G)) be the inclusion map obtained from this identification. Proposition 12.6. Let A be a G-C*-algebra with dimRok (A) = d. Then dimoz (ι : AG → A) ≤ d and dimoz (σ : A ⋊ G → A ⊗ 𝒦(L2 (G))) ≤ d. Proof. We regard as before C(G) as a G-C*-algebra with respect to the canonical translation action. By the Stone–von Neumann theorem, we have that C(G) ⋊ G ≅ 𝒦(L2 (G)) [41, Theorem 4.24]. Let θ : A → C(G) ⊗ A be the second factor embedding, which is G-equivariant. Thus, θ induces a *-homomorphism θ̂ : A ⋊ G → (C(G) ⊗ A) ⋊ G. We have that C(G) ⊗ A is G-equivariantly isomorphic to C(G) ⊗ A0 , where A0 is the C*-algebra isomorphic to A as a C*-algebra but endowed with the trivial G-action [8, Proposition 2.3]. Thus, we have a canonical isomorphism of C*-algebras (C(G) ⊗ A) ⋊ G ≅ (C(G) ⊗ A0 ) ⋊ G ≅ A ⊗ 𝒦(L2 (G)) and G

G

(C(G) ⊗ A) ≅ (C(G) ⊗ A0 ) ≅ A.

Model theory of G-C*-algebras and order zero dimension

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With respect to such isomorphisms, θ̂ : A ⋊ G → A ⊗ 𝒦(L2 (G)) corresponds to σ, and θ|AG corresponds to ι : AG → A. Thus, by Proposition 12.1 and Lemma 12.5 we have that ̂ = dim (σ). dimRok (A) = dimGoz (θ) ≥ dimoz (θ) oz Similarly, dimRok (A) = dimGoz (θ) ≥ dimoz (θ|AG ) = dimoz (ι : AG → A). This concludes the proof.

13 Dimensional functions A dimension function for separable (nuclear) G-C*-algebras is a function from the class of (nuclear) G-C*-algebras to {0, 1, 2, . . . , ∞}. Let us say that a dimension function dim for unital G-C*-algebras is positively ∀∃-axiomatizable if there exist a family H of pos∗ itive quantifier-free ℒCG -formulas of the form η(x, z), a family Φ of positive quantifierfree ℒoz G -formulas of the form φ(z, y), where z has a finite-dimensional C*-algebra as sort, and a positive quantifier-free ℒosos G -formula ψ(x, y), such that for a separable unital G-C*-algebra A one has that dim(A) ≤ d if and only if A 󳀀󳨐 sup x

inf

inf

z(0) ,...,z(d) y(0) ,...,y(d)

ξ(x, z(0) , . . . , z(d) , y(0) , . . . , y(d) ) ≤ 0

where the quantifiers range in the unit ball and ξ(x, z0 , . . . , zd , y0 , . . . , yd ) is the formula max{η0 (x, z(0) ), . . . , ηd (x, z(d) ), φ0 (z(0) , y(0) ), . . . , φd (z(d) , y(d) ), ψ(x, y(0) + ⋅ ⋅ ⋅ + y(d) )} for some formulas η ∈ H and φ ∈ Φ. A dimension function for separable nuclear G-C*-algebras is nuclearly positively ∀∃axiomatizable if it satisfies the above where H is a family of positive quantifier-free ∗ ℒCG −nuc -formulas, Φ is a family of positive quantifier-free ℒoz−nuc -formulas, and ψ is a G positive quantifier-free ℒosos−nuc -formula. G Example 13.1. Nuclear dimension is a nuclearly positively ∀∃-axiomatizable dimension function for nuclear C*-algebras. Recall that by definition a (necessarily nuclear) C*-algebra A has nuclear dimension at most d ∈ ℕ if for every finite subset A0 of A and ε > 0 there exist finite-dimensional C*-algebras F (0) , . . . , F (d) , cpc maps f (i) : A → F (i) , and cpc order zero maps g (i) : F (i) → A for 0 ≤ i ≤ d such that ‖g0 f0 (x) + ⋅ ⋅ ⋅ + gd fd (x) − x‖ < ε for every x ∈ A0 ; see [43, Definition 2.1]. Indeed, one can let ∗ – H be the collection of positive quantifier-free ℒC −nuc -formulas η(x, z) of the form

418 � M. Lupini inf

s∈CPC(A,F)

–

k

where x = (xk ) has sort A and z = (zk ) has sort a finite-dimensional C*-algebra F; Φ the collection of positive quantifier-free ℒoz G -formulas φ(z, y) of the form inf

t∈OZ(F,A)

–

󵄩 󵄩 max󵄩󵄩󵄩s(xk ) − zk 󵄩󵄩󵄩

󵄩 󵄩 max󵄩󵄩󵄩t(zk ) − yk 󵄩󵄩󵄩 k

where z = (zk ) has sort a finite-dimensional C*-algebra F and y = (yk ) as sort A; ψ(x, y) to be the ℒosos G -formula max ‖xk − yk ‖ k

where x = (xk ) and y = (yk ) have sort A. Example 13.2. Decomposition rank is a nuclearly positively ∀∃-axiomatizable dimension function for nuclear C*-algebras. This is defined as nuclear dimension, with the additional requirement that the map g := g (0) + ⋅ ⋅ ⋅ + g (d) : F (0) ⊕ ⋅ ⋅ ⋅ ⊕ F (d) → A defined by g(z(0) , . . . , z(d) ) = g (0) (z(0) ) + ⋅ ⋅ ⋅ + g (d) (z(d) ) is contractive. In this case, we let: ∗ – H be the collection ℒC −nuc -formulas as in the previous example; – Φ be the collection of positive quantifier-free ℒoz G -formulas φ(x, y, u) of the form inf

t∈OZ(F,A)

–

󵄩 󵄩󵄩 󵄩 max max{󵄩󵄩󵄩t(zk ) − yk 󵄩󵄩󵄩, 󵄩󵄩󵄩t(1) − u󵄩󵄩󵄩} k

ψ(x, y, u) be ℒosos G -formula max{max ‖xk − yk ‖, 1 − ‖u‖}. k

Proposition 13.3. Let dim be a positively ∀∃-axiomatizable dimension function for separable unital G-C*-algebras. Suppose that A, B are G-C*-algebras, and θ : A → B is a nondegenerate G-equivariant *-homomorphism. Then dim(A) + 1 ≤ (dimGoz (θ) + 1)(dim(B) + 1). The same conclusions hold if dim is nuclearly positively ∀∃-axiomatizable and B is nuclear. Proof. Suppose that dim(B) = n and dimGoz (θ) = d. To simplify the notation, we consider the case d = 1. The proof in the case of an arbitrary d is analogous. We will also focus on the first assertion, the proof of the second assertion being similar. We need to prove that dim(A) + 1 ≤ 2(n + 1). We identify A as a subalgebra of ∏𝒰 A. Since dimGoz (θ) = 1, there exists cpc order zero maps ρ0 , ρ1 : B → ∏G𝒰 A such that setting ρ := ρ0 + ρ1 one has that ρ ∘ θ = ΔA : A → ∏G𝒰 A.

Model theory of G-C*-algebras and order zero dimension

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We adopt the notation as in the definition of nuclearly positively ∀∃-axiomatizable dimension function. Let η0 (x, z(0) ), . . . , ηn (x, z(n) ) ∈ H and φ0 (z(0) , y(0) ), . . . , φn (z(n) , y(n) ) ∈ Φ be formulas that witness that dim(B) ≤ n. For 0 ≤ j ≤ d, let zj have a finite-dimensional C*-algebra F (j) as sort. Suppose that a = (ak )ℓk=0 is an ℓ-tuple in the unit ball of A and ε > 0. By construction, we have that B 󳀀󳨐 sup inf x

inf ξ(x, z0 , . . . , zd , y0 , . . . , yd )

z0 ,...,zd y0 ,...,yd

where ξ(x, z0 , . . . , zd , y0 , . . . , yd ) is the formula max{η0 (x, z(0) ), . . . , ηd (x, z(d) ), φ0 (z(0) , y(0) ), . . . , φd (z(d) , y(d) ), ψ(x, y(0) + ⋅ ⋅ ⋅ + y(d) )} Thus, there exist tuples w(j) = (wk ) in F (j) and b(j) = (bk ) in B such that, for 0 ≤ j ≤ n: (j)

(j)

ηj (θ(a), w(j) ) ≤ ε, φj (w(j) , b(j) ) ≤ ε, and ψ(θ(A), b(0) + ⋅ ⋅ ⋅ + b(0) ) ≤ ε. Consider now the elements ck

= ρ0 (bk )

ck

= ρ1 (bk )

(j,0)

(j)

and (j,1)

(j)

of ∏G𝒰 A. Then we have that ηj (a, w(j) ) = ηj (ρθ(a), w(j) ) ≤ ηj (θ(a), w(j) ) ≤ ε and φj (w(j) , c(j,0) ) = φj (w(j) , ρ0 (b(j) )) ≤ φj (w(j) , b(j) ) ≤ ε, φj (w(j) , c(j,1) ) = φj (w(j) , ρ1 (b(j) )) ≤ φj (w(j) , b(j) ) ≤ ε, as well as ψ(a, c(j,0) + ⋅ ⋅ ⋅ + c(n,0) + c(j,1) + ⋅ ⋅ ⋅ + c(j,1) ) = ψ(ρθ(a), ρ(b(0) + ⋅ ⋅ ⋅ + b(n) )) ≤ ψ(θ(a), b(0) + ⋅ ⋅ ⋅ + b(n) ) ≤ ε. By Łos’ theorem applied to ∏G𝒰 A [3, Theorem 5.4], this concludes the proof.

420 � M. Lupini Theorem 13.4. Let A be a (nuclear) G-C*-algebra, and let dim be a (nuclearly) positively ∀∃-axiomatizable dimension function for (nuclear) C*-algebras. Then dim(AG ) + 1 ≤ (dimRok (A) + 1)(dim(A) + 1) and dim(A ⋊ G) + 1 ≤ (dimRok (A) + 1)(dim(A ⊗ 𝒦(L2 (G))) + 1) Proof. Define d := dimRok (A) + 1. We have that, by Proposition 12.6, dimoz (ι : AG → A) ≤ d. Thus, by Proposition 13.3, we have that dim(AG ) + 1 ≤ (d + 1)(dim(A) + 1). Similarly, by 12.6 we have that dimoz (σ : A ⋊ G → A ⊗ 𝒦(L2 (G))) ≤ d. Thus, by Proposition 13.3, we have that dim(A ⋊ G) + 1 ≤ (d + 1)(dim(A ⊗ 𝒦(L2 (G))) + 1). The following corollary recovers a result that was obtained in [9] in the separable case. Corollary 13.5. Let A be a G-C*-algebra. Then dimnuc (AG ) + 1 ≤ dimnuc (A ⋊ G) + 1 ≤ (dimRok (A) + 1)(dimnuc (A) + 1) and dr(AG ) + 1 ≤ dr(A ⋊ G) + 1 ≤ (dimRok (A) + 1)(dr(A) + 1). Proof. We have that AG is corner of A⋊G by Lemma 12.2. We have that nuclear dimension and decomposition rank are nondecreasing when passing to corner or, more generally, hereditary C*-subalgebras; see [25, Proposition 3.8] and [43, Proposition 2.5]. (It is not clear if the same holds for an arbitrary positive ∀∃-axiomatizable dimension function.) Therefore, we have that dimnuc (AG ) ≤ dimnuc (A ⋊ G) and

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dr(AG ) ≤ dr(A ⋊ G). The other inequalities follow from Theorem 13.4 after observing that dimnuc and dr are nuclearly positively ∀∃-axiomatizable dimension function for nuclear C*-algebras satisfying dr(A) = dr(A ⊗ 𝒦(L2 (G))) and dimnuc (A) = dimnuc (A ⊗ 𝒦(L2 (G))), where one inequality holds since 𝒦(L2 (G)) is an AF C*-algebra [43, Proposition 2.3] [25, Section 3], while the other inequality holds since A is a hereditary subalgebra of A ⊗ 𝒦(L2 (G)) [43, Proposition 2.5] [25, Proposition 3.8]. This concludes the proof.

Bibliography [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

S. Barlak and G. Szabó, Rokhlin actions of finite groups on UHF-absorbing C*-algebras, Trans. Am. Math. Soc. 369 (2017), no. 2, 833–859. MR3572256. S. Barlak, G. Szabó and C. Voigt, The spatial Rokhlin property for actions of compact quantum groups, J. Funct. Anal. 272 (2017), no. 6, 2308–2360. I. Ben Yaacov, A. Berenstein, C. Ward Henson and A. Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2, London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, 2008, pp. 315–427. B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer, Berlin, 2006. A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. Éc. Norm. Supér. (4) 8 (1975), no. 3, 383–419. MR0394228. K. De Commer, Actions of compact quantum groups, Topological quantum groups, Banach Center Publ., vol. 111, Polish Acad. Sci. Inst. Math., Warsaw, 2017, pp. 33–100. MR3675047. I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati and W. Winter, Model theory of C*-algebras, Mem. Am. Math. Soc. 271 (2021), no. 1324, viii+127. E. Gardella, Crossed products by compact group actions with the Rokhlin property, J. Noncommut. Geom. 11 (2017), no. 4, 1593–1626. MR3743232. E. Gardella, Regularity properties and Rokhlin dimension for compact group actions, Houst. J. Math. 43 (2017), no. 3, 861–889. MR3739037. E. Gardella, Rokhlin dimension for compact group actions, Indiana Univ. Math. J. 66 (2017), no. 2, 659–703. MR3641489. E. Gardella, Compact group actions with the Rokhlin property, Trans. Am. Math. Soc. 371 (2019), no. 4, 2837–2874. MR3896099. E. Gardella, I. Hirshberg and L. Santiago, Rokhlin dimension: Duality, tracial properties, and crossed products, Ergod. Theory Dyn. Syst. 41 (2021), no. 2, 408–460. MR4177290. E. Gardella, M. Kalantar and M. Lupini, Model theory and Rokhlin dimension for compact quantum group actions, J. Noncommut. Geom. 13 (2019), no. 2, 711–767. MR3988085.

422 � M. Lupini

[14] I. Goldbring and T. Sinclair, On the axiomatizability of C*-algebras as operator systems, Glasg. Math. J. 61 (2019), no. 3, 629–635. MR3991361. [15] R. H. Herman and V. F. R. Jones, Period two automorphisms of UHF C*-algebras, J. Funct. Anal. 45 (1982), no. 2, 169–176. MR647069. [16] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II, Die Grundlehren der mathematischen Wissenschaften, vol. 152, Springer, New York–Berlin, 1970. MR0262773. [17] I. Hirshberg and N. C. Phillips, Rokhlin dimension: Obstructions and permanence properties, Doc. Math. 20 (2015), 199–236. MR3398712. [18] I. Hirshberg and W. Winter, Rokhlin actions and self-absorbing C*-algebras, Pac. J. Math. 233 (2007), no. 1, 125–143. MR2366371. [19] I. Hirshberg, W. Winter and J. Zacharias, Rokhlin dimension and C*-dynamics, Commun. Math. Phys. 335 (2015), no. 2, 637–670. [20] M. Izumi, Finite group actions on C*-algebras with the Rohlin property, I, Duke Math. J. 122 (2004), no. 2, 233–280. [21] M. Izumi, Finite group actions on C*-algebras with the Rohlin property—II, Adv. Math. 184 (2004), no. 1, 119–160. [22] V. F. R. Jones, Actions of finite groups on the hyperfinite type II1 factor, Mem. Am. Math. Soc. 28 (1980), no. 237, v+70. MR587749. [23] V. F. R. Jones, A converse to Ocneanu’s theorem, J. Oper. Theory 10 (1983), no. 1, 61–63. MR715556. [24] D. Kerr and H. Li, Ergodic theory, Springer Monographs in Mathematics, Springer, Cham, 2016. [25] E. Kirchberg and W. Winter, Covering dimension and quasidiagonality, Int. J. Math. 15 (2004), no. 01, 63–85. [26] A. Kishimoto, The Rohlin property for automorphisms of UHF algebras, J. Reine Angew. Math. 1995 (1995), no. 465, 183–196. [27] A. Kishimoto, A Rohlin property for one-parameter automorphism groups, Commun. Math. Phys. 179 (1996), no. 3, 599–622. MR1400754. [28] A. Kishimoto, Automorphisms of AT algebras with the Rohlin property, J. Oper. Theory 40 (1998), no. 2, 277–294. MR1660386. [29] E. C. Lance, Hilbert C*-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. [30] A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, vol. 1138, Springer, Berlin, 1985. MR807949. [31] D. S. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull., New Ser., Am. Math. Soc. 2 (1980), no. 1, 161–164. MR551753. [32] D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1–141. MR910005. [33] H. Osaka and N. C. Phillips, Crossed products by finite group actions with the Rokhlin property, Math. Z. 270 (2012), no. 1–2, 19–42. MR2875821. [34] V. I. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. [35] P. Podleś, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Commun. Math. Phys. 170 (1995), no. 1, 1–20. MR1331688. [36] I. Raeburn, On crossed products and Takai duality, Proc. Edinb. Math. Soc. (2) 31 (1988), no. 2, 321–330. MR989764. [37] J. Rosenberg, Appendix to: “Crossed products of UHF algebras by product type actions”, [Duke Math. J. 46 (1979), no. 1, 1–23] by O. Bratteli, Duke Math. J. 46 (1979), no. 1, 25–26. MR523599. [38] Y. Sato, The Rohlin property for automorphisms of the Jiang–Su algebra, J. Funct. Anal. 259 (2010), no. 2, 453–476. [39] G. Szabó, Rokhlin dimension: Absorption of model actions, Anal. PDE 12 (2018), no. 5, 1357–1396.

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� 423

[40] T. Timmermann, An invitation to quantum groups and duality, EMS Textbooks in Mathematics, European Mathematical Society, 2008. [41] D. P. Williams, Crossed products of C*-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. [42] W. Winter and J. Zacharias, Completely positive maps of order zero, Münster J. Math. 2 (2009), 311–324. [43] W. Winter and J. Zacharias, The nuclear dimension of C*-algebras, Adv. Math. 224 (2010), no. 2, 461–498.

Isaac Goldbring

Model theory and ultrapower embedding problems in operator algebras Abstract: We survey the different ultrapower embedding problems in operator algebras and explain how model-theoretic techniques can be used to shed light on these problems. Keywords: Ultrapower embedding problems, model-theoretic forcing, Connes embedding problem, Kirchberg embedding problem, quasidiagonality problem, quantum complexity MSC 2020: 03C66, 46L10, 46L05, 46B08, 03C20, 03C25, 68Q12, 68Q15

1 Introduction Suppose that T is an ∀∃-axiomatizable theory in some countable, classical language L. It is often too much to ask that there exists a countable model of T into which all other countable models embed. (This does happen, for example, when the theory T admits an ℵ0 -categorical model companion.) Nevertheless, under the mild assumption that the theory T has the joint embedding property (JEP), meaning that any two models of T mutually embed into a third model of T, we can infer the existence of countable models of T whose ultrapower with respect to any nonprincipal ultrafilter 𝒰 on ℕ embeds all countable models of T; we refer to such models of T as locally universal models of T. Indeed, if M is any existentially closed model of T and N is any countable model of T, then by jointly embedding M and N into a (without loss of generality) countable model P of T, we see that P embeds in M 𝒰 (this follows from the fact that M is e. c. and M 𝒰 is a countably saturated model of its theory) whence N also embeds into M 𝒰 . (By using so-called good ultrafilters, one can obtain ultrapowers of M which embed larger models of T.) This discussion holds verbatim for continuous theories: if T is an ∀∃-axiomatizable theory with JEP in a separable language, then separable locally universal models of T exist. It is clear that any model of T that contains a locally universal model of T is itself locally universal, whence countable (separable) locally universal models of T are ubiquitous. Seemingly unaware of this abstract model-theoretic discussion, operator algebraists have posed a number of problems which ask whether or not concrete operator algebras Acknowledgement: Goldbring was partially supported by NSF grant DMS-2054477. Isaac Goldbring, Department of Mathematics, University of California, 340 Rowland Hall (Bldg.# 400), Irvine, CA 92697-3875, USA, e-mail: [email protected]; URL: http://www.math.uci.edu/~isaac https://doi.org/10.1515/9783110768282-011

426 � I. Goldbring are locally universal for the corresponding classes to which they belong (which are not always elementary classes). The most famous of these problems is the Connes embedding problem, which appeared in Alain Connes’ fundamental work [7] from 1976, in which he showed that any separable injective II1 factor is necessarily hyperfinite (and for which he received the Fields medal in 1982). Part of the proof of his main theorem involved showing that a particular separable II1 factor embedded into an ultrapower of the hyperfinite II1 factor ℛ. He casually remarked that such an embedding “ought to” exist for any separable II1 factor, that is, ℛ should be a locally universal object for the elementary class of II1 factors. (Incidentally, since ℛ embeds into any II1 factor, the CEP is equivalent to the assertion that all II1 factors have the same universal theory.) Connes’ reason why this “ought to” be the case is not entirely compelling: he points out that such an embedding exists for L(𝔽2 ), the group von Neumann algebra associated to the free group on two generators, and it is for this reason that such an embedding should exist for all separable II1 factors. While some operator algebraists refer to Connes’ offhand remark as a “Conjecture,” most prefer to call it a “Problem.” Over the years, the CEP has gained significant interest due its connections with a wide variety of areas of mathematics, including C∗ -algebra theory (in connection with Kirchberg’s QWEP problem), quantum information theory (in connection with Tsirelson’s problem), free probability, group theory, and noncommutative real algebraic geometry, to name a few. Remarkably, in early 2020, a negative resolution was obtained to the CEP via its equivalence with Tsirelson’s problem, which was itself refuted using a remarkable theorem in quantum complexity theory known as MIP∗ = RE. Independent of its connection with the CEP, this latter result is widely considered to be a landmark scientific achievement; the reader interested in understanding the entire story behind these connections can consult the author’s survey [16]. That being said, someone wishing to understand the proof of the negative solution to the CEP using MIP∗ = RE must tread the deep waters connecting these two seemingly distant results. In joint work with Bradd Hart [19], we showed how, using basic ideas from continuous model theory (most notably the completeness theorem for first-order continuous logic and the theory of definable sets in continuous logic), one can obtain a more direct proof of the failure of CEP from MIP∗ = RE. Moreover, the model-theoretic approach offers more insight into this refutation and allows one to prove extra results, such as “many counterexamples” to the CEP, that is, many different universal theories of II1 factors, as well as a Gödelian-style refutation stating that any effectively axiomatizable class of II1 factors will contain a counterexample to the CEP. We present this model-theoretic approach to the negative solution to the CEP in Section 3. In Sections 4 and 5, we consider two C∗ -algebraic analogs of the CEP, the so-called Kirchberg embedding problem and the MF problem, which, in some sense, can be thought of as the “infinite” and “finite” C∗ -versions of the CEP. The former problem asks whether or not the Cuntz algebra 𝒪2 is locally universal for the class of all C∗ -algebras. This problem has eluded model-theoretic techniques thus far and we discuss what we know about

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this still-open problem; the material presented in this section represents joint work of the author and Thomas Sinclair [21]. The latter problem is the so-called MF problem, which asks whether or not the universal UHF algebra 𝒬 is a locally universal object for the elementary class of stably finite C∗ -algebras. While a negative solution to the MF problem can readily be deduced from the failure of the CEP, the model-theoretic approach allows us to establish a similar Gödelian-style refutation of the MF problem. A variant of the MF problem, known as the quasidiagonality (QD) problem, asks if 𝒬 is a locally universal object for the (nonelementary!) class of stably finite nuclear C∗ -algebras. While the QD problem is still open, a major breakthrough was obtained by Tikuisis, Winter, and White [36], who resolved this problem in the positive (for simple such algebras) assuming a technical assumption known as the universal coefficient theorem (UCT). The model-theoretic content of the UCT is not widely understood at the time of the writing of this paper. Nevertheless, we discuss some model-theoretic ideas around the QD problem representing more joint work of the author and Sinclair [22]. In the final subsection of Section 4, we return to the ideas of the author and Hart from [19] and show how the Gödelian-style refutation of the MF problem extends to a larger class of algebras which, in particular, allow us to refute a stably projectionless version of the MF problem which asks if every stably projectionless algebra embeds into an ultrapower of a very important algebra in the classification program for nuclear C∗ -algebras known as the Jiang Su algebra 𝒵 . Currently, this latter result has no purely operator-algebraic proof. In the final section, we consider the simpler case of (unital) abelian C∗ -algebras. There, an ℵ0 -categorical model completion exists, namely the theory of C(2ℕ ). This result is essentially (after some category-theoretic considerations) a restatement of the existence of an ℵ0 -categorical model completion for the classical theory of Boolean algebras, namely the theory of atomless Boolean algebras. However, an interesting phenomenon arises when restricting to the class of projectionless abelian C∗ -algebras, whose models are of the form C(X) for X a continuum (that is, a connected compact Hausdorff space). In this case, a theorem of K. P. Hart [25] states that all such algebras (except for the trivial case of C(a point) ≅ ℂ) have the same universal theory, whence all nontrivial objects of this class are locally universal! We provide a fairly detailed proof of this result below. However, this theorem does not represent the end of the story for this class of algebras, for the question of a model companion for this class is still open. Concerning existentially closed projectionless abelian C∗ -algebras, we discuss the positive solution, due to Christopher Eagle, Alessandro Vignati, and the author [9] of a question of Bankston, who asked if a fairly important (and generic) continuum, the so-called pseudoarc ℙ, is co-existentially closed (which just means that C(ℙ) is an existentially closed projectionless abelian C∗ -algebra). Many of the results to be discussed below involve a particular approach to the Henkin construction known as building models by games (as first popularized by Hodges in his book [26]). We discuss the essential properties of this construction in the next section.

428 � I. Goldbring Throughout this article, I assume the reader is familiar with basic continuous logic as exposited, for example, in Hart’s introductory article in this volume.

2 Building models by games In this section, we recall the basic facts from the theory of model-theoretic forcing needed throughout this paper. The relevant version of model-theoretic forcing for us is the game-theoretic approach, originally presented in Hodges’ classic book [26] and adapted to the continuous setting by the author in [14]. That being said, for some of what is to follows, we need to consider a slightly more general setting and so we take the opportunity to extend the context here. Throughout this section, we fix a countable (continuous) language L. By a ∀ ⋁ ∃-sentence we mean an Lω1 ,ω -sentence of the form sup ⋅ ⋅ ⋅ sup ⋁ φm (x1 , . . . , xkn ), x1

xkn m∈ℕ

where each φm is an existential L-formula and the symbol ⋁ denotes a countable infimum. (We note that there are several different approaches to Lω1 ,ω in the literature; in the above sentences, there is no requirement on a common modulus of uniform continuity for the formulae appearing in the countable infimum). By a ∀ ⋁ ∃-theory we mean a collection of ∀ ⋁ ∃-sentences. We say that a class 𝒦 of L-structures is ∀ ⋁ ∃-axiomatizable if there is a ∀ ⋁ ∃-theory T such that, for all L-structures A, we have A ∈ 𝒦 if and only if σ A = 0 for all σ ∈ T. The need to consider such infinitary theories arises as many important classes of C∗ -algebras are not first-order axiomatizable but are ∀ ⋁ ∃-axiomatizable, such as simple C∗ -algebras and nuclear C∗ -algebras (see, for example, [11], where the class is called definable by a uniform family of formulae). Motivated by this, if T is a ∀ ⋁ ∃-axiomatizable theory and P is a property that may or may not hold of models of T, we say that P is ∀ ⋁ ∃-axiomatizable (relative to T) if the collection of models of T having property P is itself ∀ ⋁ ∃-axiomatizable. In the remainder of this subsection, we fix a ∀ ⋁ ∃-axiomatizable L-theory T. We note that this assumption implies that the class of models of T is an inductive class (that is, is closed under direct limits), whence every model of T is contained in an e. c. model of T of the same density character. We now fix a countably infinite set C of constant symbols enumerated (cn )n σ1 2 , we have that M1 does not embed into an ultrapower

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of M2 . Since M2 does not embed into an ultrapower of ℛ, there is a universal sentence M M σ2 such that σ2ℛ = 0 but σ2 2 > 0. Let r2 ∈ (0, σ2 2 ) be rational and let T2 := T1 ∪ {σ2 ∸ r2 }. Once again, T2 is effectively enumerable, so there is a separable model M3 of T2 that does not embed into an ultrapower of ℛ. It is clear that neither M1 nor M2 embed into an ultrapower of M3 . One constructs the remainder of the sequence analogously. We believe the following question should have a positive answer: Question 3.13. Do there exist continuum many universal theories of II1 factors? Another application of Corollary 3.11 above is the following, which also appears not to follow from the “standard” refutation of the CEP: Corollary 3.14. There are infinitely many distinct universal theories of II1 factors without property Gamma. To see this, let σ be one of the sentences in the axiomatization of property Gamma for which σ L(𝔽2 ) > 0 and let T := TII1 ∪ {σ ∸ r} for some rational number r ∈ (0, σ L(𝔽2 ) ). By Corollary 3.11 above, there is a model of T that does not embed into an ultrapower of ℛ. Then one continues as in the proof of Corollary 3.12.

3.5 The existence of the enforceable II1 factor In this subsection, we mention a model-theoretic variant of the CEP that is still open and, in this author’s opinion, is one of the more interesting open problems in the model theory of operator algebras. As discussed in the article by the author and Hart in this volume, ℛ is an e. c. model of its universal theory and the CEP is thus equivalent to the assertion that ℛ is an e. c. II1 factor. The ideas in this subsection elaborate further on this observation. We first note the following: Lemma 3.15. Being hyperfinite is a ∀ ⋁ ∃-axiomatizable property of II1 factors. The proof of this lemma has not appeared explicitly in the literature but is similar to the proof of the main results in [6]. Armed with this and Proposition 2.1 above, we arrive at the following: Corollary 3.16. ℛ is the enforceable model of its universal theory. In particular, if CEP were to hold, then ℛ would be the enforceable II1 factor. (Of course, the converse is also true, but moot at this point.) Nevertheless, the following question is open and tantalizing: Question 3.17. Does the enforceable II1 factor exist? How likely is it that the enforceable II1 factor exists? That is of course difficult to say. If it did exist, then it would “rival” ℛ for being “the most important II1 factor” for it

436 � I. Goldbring would be generic from the model-theoretic point of view. In [17], some properties of the enforceable II1 factor were established (of course, presuming its existence). It is worth pointing out one notable case when the enforceable object does not exist, namely for the (classical) theory of groups. (This seems implicit in Hodges’ book [26] but is written down explicitly in the article [20] by Kunnawalkam Elayavalli, Lodha, and the author.) However, the ingredients involved in the proof are a blend of recursiontheoretic and combinatorial group-theoretic tools that seem to be currently unavailable to us in the II1 factor setting. Another interesting variant of the CEP is the following: Question 3.18. Is the property of being isomorphic to a group von Neumann algebra an enforceable property?

4 The Kirchberg embedding problem In this and the next section, we consider C∗ -algebra versions of the CEP. For simplicity, henceforth all C∗ -algebras will be assumed to be unital. (Much of what is said below can be adapted to the not necessarily unital situation, but this assumption simplifies the exposition.) If one phrases the CEP as the statement that every tracial von Neumann algebra embeds into the tracial ultrapower of an injective II1 factor, then a natural C∗ -algebra analog of CEP would be to ask whether or not every C∗ -algebra embeds into an ultrapower of a nuclear C∗ -algebra (for a C∗ -algebra A is nuclear if and only if its enveloping von Neumann algebra A∗∗ is injective). By Kirchberg’s celebrated theorem [25], every separable nuclear C∗ -algebra embeds into the Cuntz algebra 𝒪2 (see Szabó’s article in this volume for the definition of 𝒪2 ), whence it is equivalent to ask whether or not 𝒪2 is a locally universal C∗ -algebra. We refer to this problem as the Kirchberg embedding problem (KEP). (We attribute this problem to Kirchberg as we first learned of this problem from Ilijas Farah, who in turn first learned of this problem during a discussion with Kirchberg in 2007. The first mention of this problem in the literature appears to be in the author’s article [21] with Sinclair.) At the moment of the writing of this article, the KEP remains an open problem. In this section, we mention the connection between it and the model theory of C∗ -algebras in a way that parallels the situation with the CEP. Recall that the CEP is equivalent to the statement that ℛ is an e. c. tracial von Neumann algebra. The analogous statement for the KEP holds: Proposition 4.1. The KEP has a positive solution if and only if 𝒪2 is an e. c. C∗ -algebra. The proof is analogous to the proof of the same statement for tracial von Neumann algebras, using the fact that any two embeddings of 𝒪2 into its ultrapower are unitarily

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conjugate (a consequence of the fact that 𝒪2 is a strongly self-absorbing C∗ -algebra) and that the theory of C∗ -algebras has the joint embedding property. The following variation on the preceding proposition is also of interest: Proposition 4.2. Algebra 𝒪2 is the only possible separable C∗ -algebra that is nuclear and e. c. Consequently, a positive solution to the KEP is equivalent to the statement that there is a C∗ -algebra that is both nuclear and e. c. The proof of this proposition is quite interesting. Indeed, suppose that A is a separable C∗ -algebra that is both nuclear and e. c. A consequence of being e. c. is that A is simple (see [21]). By another fundamental result of Kirchberg [25], the fact that A is simple, separable, and nuclear implies that A ⊗ 𝒪2 ≅ 𝒪2 . However, a consequence of A being e. c. is that A is “𝒪2 -stable,” that is, A ⊗ 𝒪2 ≅ A; this follows from the fact that being 𝒪2 -stable is an ∀∃-axiomatizable property of C∗ -algebras (see [11]) together with the fact that every C∗ -algebra embeds into an 𝒪2 -stable algebra (namely by tensoring the algebra with 𝒪2 itself and using the fact, due to Cuntz, that 𝒪2 ⊗ 𝒪2 ≅ 𝒪2 ). It follows that A ≅ 𝒪2 , as desired. An alternative, slightly more elementary proof, can be found in [22, Remark 21]. Another similarity with the CEP concerns enforceability. First, we will need the following result: Proposition 4.3. Being nuclear is a ∀ ⋁ ∃-axiomatizable property of C∗ -algebras. Two proofs for the preceding proposition are offered in [11], one “soft” and modeltheoretic, the other “concrete,” writing down specific axioms for nuclearity. Let us sketch the former argument. Given a C∗ -algebra A and k, n ∈ ℕ, define the predicate nucAk,n : ⃗ where ϕ ranges over all ucp maps Ak1 → ℝ by nucAk,n (a)⃗ = infϕ,ψ ‖(ψ ∘ ϕ)(a)⃗ − a‖, A → Mn (ℂ) and ψ ranges over all ucp maps Mn (ℂ) → A. An argument using the Beth definability theorem shows that the predicates nuck,n are actually existentially definable relative to the theory of C∗ -algebras, that is, there are existential formulae (really, uniform limits of existential formulae) Φk,n (x)⃗ such that, for every C∗ -algebra A, every ⃗ It remains to note that a C∗ -alk, n ∈ ℕ, and every a⃗ ∈ Ak1 , we have nucAk,n (a)⃗ = ΦAk,n (a). ⃗ A = 0. gebra A is nuclear if and only if, for every k ∈ ℕ, we have (supx⃗ ⋁n∈ℕ Φk,n (x)) As with the CEP, we arrive at the following formulation of the KEP, whose proof uses everything we have discussed thus far: Theorem 4.4 (Goldbring [14]). The following are equivalent: (1) The KEP has a positive solution. (2) The property of being nuclear is enforceable. (3) 𝒪2 is the enforceable C∗ -algebra. The previous theorem yields an interesting local, finitary reformulation of the KEP first identified by Sinclair and the author in [21]. First, we say a condition (relative to ⃗ where x⃗ is a k-tuple, has good nuclear witnesses if, the theory of C∗ -algebras) p(x),

438 � I. Goldbring for every ϵ > 0, there is a C∗ -algebra A, a⃗ ∈ Ak , and an n ∈ ℕ such that a⃗ satisfies the condition p(x)⃗ and for which nucAk,n (a)⃗ < ϵ. The previous theorem then yields the following corollary: Corollary 4.5. The KEP has a positive solution if and only if every condition has good nuclear witnesses. The import of the previous corollary is a (seemingly) significant weakening of the demand that every condition be satisfied in a nuclear C∗ -algebra (equivalently, satisfied in 𝒪2 ), for one only asks that the witness admit a good ucp factorization through a matrix algebra, and, moreover, the witness and the dimension of the matrix algebra can vary as the level of approximation varies.

5 The MF problem and the quasidiagonality problem 5.1 A negative solution to the MF problem Recall that the CEP is also equivalent to the statement that every separable tracial von Neumann algebra embeds into ∏𝒰 Mn (ℂ), a tracial ultraproduct of matrix algebras with respect to a nonprincipal ultrafilter on ℕ. It is thus natural to formulate a C∗ -algebra version of CEP by asking that every separable C∗ -algebra embed into a C∗ -algebra ultraproduct ∏𝒰 Mn (ℂ) of matrix algebras with respect to a nonprincipal ultrafilter on ℕ. Using the same notation for both tracial von Neumann algebra ultraproducts and C∗ -algebra ultraproducts is potentially dangerous (and some authors even use different notations for the two ultraproducts); to prevent confusion, in the remainder of this section, unless explicitly stated otherwise, all ultraproducts will be C∗ -algebra ultraproducts. There is an immediate obstruction to the statement “every separable C∗ -algebra embeds into ∏𝒰 Mn (ℂ)” from being true, namely the C∗ -algebra ∏𝒰 Mn (ℂ) is stably finite, as is any subalgebra. Thus, we may modify the problem as follows: Definition 5.1. The MF problem is the problem of whether or not every separable stably finite C∗ -algebra embeds into ∏𝒰 Mn (ℂ). The terminology MF comes from the fact that a separable C∗ -algebra is called matricially finite (or MF) if it embeds into ∏𝒰 Mn (ℂ). Consequently, the MF problem asks if the notions of stably finite and MF coincide for separable C∗ -algebras. As in the case of the CEP, the MF problem can be reformulated in terms of ultrapowers of a single object. Indeed, the MF problem is equivalent to the problem of whether every separable stably finite C∗ -algebra embeds into a nonprincipal ultrapower 𝒬𝒰 of the universal UHF algebra 𝒬 (see [11, Lemma 4.4.1]). An immediate consequence of the negative solution of the CEP is that the MF problem also has a negative solution:

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Corollary 5.2. The MF problem has a negative solution. To prove the previous corollary, suppose that M is a II1 factor that does not embed into ℛ𝒰 (here we mean the tracial von Neumann algebra ultrapower). We claim then that M does not embed (as a C*-algebra) into a nonprincipal C*-ultrapower of 𝒬; by considering a separable elementary subalgebra of M (in the language of C∗ -algebras), we obtain the desired counterexample to the MF problem. Suppose, towards a contradiction, that i : M 󳨅→ 𝒬𝒰 is an embedding. Recall that 𝒬 has a unique trace τ and the von Neumann algebra generated by 𝒬 with respect to the GNS represenation corresponding to τ is ℛ. Let π : 𝒬𝒰 → ℛ𝒰 denote the composition of the quotient map 𝒬𝒰 → 𝒬𝒰 /I, where I = {(xn )∙ ∈ 𝒬𝒰 : lim ‖xn ‖2 = 0} 𝒰

is the trace ideal, with the natural inclusion 𝒬𝒰 /I 󳨅→ ℛ𝒰 obtained by viewing operator norm bounded balls in 𝒬 as ‖ ⋅ ‖2 -dense subsets of the corresponding balls in ℛ. Since M has a unique trace, which is faithful, we get that the composition π ∘ i : M → ℛ𝒰 is a trace-preserving *-homomorphism, a contradiction. The following question seems wide open: Question 5.3. Does ℛ embed, as a C∗ -algebra, into 𝒬𝒰 ? The negative solution to the MF problem also has a Gödelian-style refutation, that is, the 𝒬EP has a negative solution as well; we postpone the discussion of this fact until Section 5.3 below. Unlike most of the embedding problems discussed in this paper, it is not even clear that there ought to be a locally universal object for the class of stably finite C∗ -algebras as the following question appears to be open: Question 5.4. Does the class of stably finite C∗ -algebras have the JEP? A natural guess would be that the minimal tensor product of two stably finite C -algebras would once again be stably finite. However, the validity of this statement is far from clear. In fact, the question of whether or not the minimal tensor product of two simple stably finite C∗ -algebras is once again stably finite is equivalent to a well-known open problem, namely whether or not every stably finite C∗ -algebra admits a trace (see [22]). It is worth mentioning that the class of C∗ -algebras admitting a trace does have JEP (the tensor product trace on the minimal tensor product witnesses this) and consequently any e. c. object for this class (which exists since the class is inductive) is locally universal. ∗

5.2 The quasidiagonality problem Connes’ original motivation for considering the question of which tracial von Neumann algebras embed into ultrapowers of ℛ came from his striking result proving that injec-

440 � I. Goldbring tive II1 factors were hyperfinite, thus completing the classification of injective II1 factors [7]. A crucial ingredient in his proof was that injective factors did indeed admit embeddings into ultrapowers of ℛ. In C∗ -algebra theory, the analogous problem would be trying to classify simple, nuclear C∗ -algebras, where simple is the analog of being a factor and (as already mentioned) nuclear is the analog of being injective. In trying to mimic Connes’ approach in classifying simple, nuclear C∗ -algebras, it thus becomes natural to try to prove that they admit embeddings into 𝒬𝒰 . As stated in the previous subsection, an immediate obstruction to proving such a result is that the C∗ -algebra in question must be stably finite. Thus, one is naturally led to: Definition 5.5 (Quasidiagonality problem; simple version). Does every simple, stably finite, nuclear C∗ -algebra embed into 𝒬𝒰 ? A word about the nomenclature in the previous definition is in order. A C∗ -algebra A is said to be quasidiagonal if there is an embedding A 󳨅→ ∏𝒰 Mn (ℂ) that admits a ucp lift A → ∏n∈ℕ Mn (ℂ). Thus, quasidiagonal C∗ -algebras form a special subclass of the class of MF-algebras. However, by the Choi–Effros lifting theorem, if a nuclear C∗ algebra is MF, then the aforementioned ucp lift automatically exists, whence there is no difference in the two notions. Halmos defined what it meant for a set of bounded operators to be quasidiagonal and then a C∗ -algebra is called quasidiagonal if it admits a concrete representation for which the operators in the image of the representation form a quasidiagonal set. Voiculescu then proved that this definition of quasidiagonal C∗ -algebra agrees with that given at the beginning of this paragraph (see [37]). As with the MF problem, it is not evident that the class of simple, stably finite nuclear C∗ -algebras should have a locally universal object for this class is not known to have JEP. Amazingly enough, this modified version of the MF problem has almost been shown to be true, modulo one technical assumption: Theorem 5.6 (Tikuisis, Winter, White [36]). Every simple, stably finite, nuclear C∗ -algebra satisfying the UCT is quasidiagonal. Here, the UCT is short for the universal coefficient theorem. Assuming that a C∗ -algebra satisfies the UCT is a technical K-theoretic assumption on the algebra. (See [35] for more information on the UCT.) One of the major open questions in C∗ -algebra theory is: Question 5.7 (UCT problem). Do all separable nuclear C∗ -algebras satisfy the UCT? Of all of the adjectives appearing in the statement of Theorem 5.6, all but the UCT have been shown to have model-theoretic meaning: being stably finite and MF are universally axiomatizable properties whilst being simple and being nuclear are ∀ ⋁ ∃-axiomatizable properties. (It turns out that quasidiagonality in general is also ∀ ⋁ ∃-axiomatizable; see [11, Section 5.13].)

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It is interesting to ask: Question 5.8. Is satisfying the UCT a ∀ ⋁ ∃-axiomatizable property of separable nuclear C∗ -algebras? The previous question notwithstanding, Barlak and Szabó [4] proved the following interesting fact: Theorem 5.9. If A is an e. c. subalgebra of B and B is a nuclear C∗ -algebra satisfying the UCT, then so does A. This theorem allows one to deduce the truth of the simple version of the quasidiagonality problem from a weakening of the UCT problem: Theorem 5.10 (Goldbring and Sinclair [22]). Suppose that every simple, stably finite, nuclear C∗ -algebra embeds into a simple, stably finite, nuclear C∗ -algebra satisfying the UCT. Then the simple version of the quasidiagonality problem is true. Indeed, suppose that A is an e. c. simple, stably finite, nuclear C∗ -algebra (which exists since this class is ∀ ⋁ ∃-axiomatizable). By the assumption of the theorem and Theorem 5.9 above, it follows that A itself satisfies the UCT, whence A is quasidiagonal by Theorem 5.6 above. Now if B is any separable, simple, stably finite, nuclear C∗ -algebra, then A ⊗ B is also stably finite; this uses the quasidiagonality of A (see [22, Lemma 4]). Since A is e. c., we have that A ⊗ B, and thus B, embeds into an ultrapower of A, whence A is locally universal for the class of simple, stably finite, nuclear C∗ -algebras. Since A is quasidiagonal, the result follows. One can remove the simplicity assumption in the quasidiagonality problem, arriving at: Definition 5.11 (Quasidiagonality problem; general version). Does every stably finite, nuclear C∗ -algebra embed into 𝒬𝒰 ? The quasidiagonality problem has several model-theoretic equivalents: Theorem 5.12. The following are equivalent: (1) The quasidiagonality problem has a positive solution. (2) Being UHF is an enforceable property of stably finite, nuclear C∗ -algebras. (3) Algebra 𝒬 is the enforceable stably finite, nuclear C∗ -algebra. (4) Algebra 𝒬 is an e. c. stably finite, nuclear C∗ -algebra. The implication (1) implies (2) in the previous proposition follows from Proposition 2.1 and the fact that being UHF is ∀ ⋁ ∃-axiomatizable (see [6]). The implication (3) implies (4) follows from the fact that being e. c. is enforceable while the implication (4) implies (1) proceeds along the lines of Theorem 5.10 above. Finally, to see (2) implies (3), one first notes that the property of being 𝒬-stable, that is, that A ⊗ 𝒬 ≅ A, is also enforceable. Indeed, this property is ∀∃-axiomatizable (for the same reason as in the case of 𝒪2 )

442 � I. Goldbring and is thus true of any e. c. object in this class as any object in this class is a subalgebra of a 𝒬-stable object in the class (by tensoring with 𝒬). It remains to note that 𝒬 is the only 𝒬-stable UHF algebra. The following question is the natural stably finite analog of Theorem 4.2 above; see [22] for partial progress towards its resolution: Question 5.13. Suppose that A is an e. c. stably finite C∗ -algebra that is also nuclear. Must we have A ≅ 𝒬? At the moment, it is unclear if the general version of the quasidiagonality problem could be deduced from the simple version, even assuming a positive solution to the UCT problem. However, using model-theoretic forcing again, one can prove the following result: Theorem 5.14 (Goldbring and Sinclair [22]). Suppose the following hold: (1) Every stably finite nuclear C∗ -algebra embeds into a stably finite, nuclear C∗ -algebra satisfying the UCT. (2) There is a simple, stably finite, nuclear C∗ -algebra that is locally universal for the class of stably finite nuclear C∗ -algebras. Then the general version of the quasidiagonality problem holds. Note that the second item in the hypotheses of the previous theorem is indeed a weakening of the statement of the quasidiagonality problem as 𝒬 itself is simple. The proof of the preceding theorem proceeds similarly as in the proof of Theorem 5.10 above. Indeed, the second condition, Proposition 2.1, and the fact that being simple is ∀ ⋁ ∃-axiomatizable allows one to construct an e. c. stably finite nuclear C∗ -algebra A that is simple. Moreover, the first condition and Theorem 5.9 above allows one to conclude that A satisfies the UCT. Thus, Theorem 5.6 above allows one to conclude that A is quasidiagonal. It follows that the quasidiagonality problem has a positive answer just as in the conclusion of the proof of Theorem 5.10. By the negative solution to the MF problem, there is a universal sentence σ in the language of C∗ -algebras for which σ 𝒬 = 0 and yet σ A = r > 0 for some stably finite C∗ -algebra A. Let T be the theory of stably finite C∗ -algebras together with the existential condition r ∸ σ = 0. We adapt the terminology from Section 4 above and say that a condition p(x)⃗ (with x⃗ = (x1 , . . . , xk )) relative to the theory T has good nuclear witnesses if, for every ϵ > 0, there is A 󳀀󳨐 T and a⃗ ∈ A satisfying p such that nucAk,n (a)⃗ < ϵ. Theorem 5.15. Using the terminology in the previous paragraph, suppose that every condition has good nuclear witnesses. Then the quasidiagonality problem has a negative solution. Indeed, the assumption that every condition has good nuclear witnesses allows one to construct a model of T that is nuclear; being a model of T, the algebra is also stably finite but not embeddable in an ultrapower of 𝒬 (that is, not quasidiagonal). Considering

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the contrapositive of this theorem, if the quasidiagonality problem has a positive solution, then whenever one has a counterexample to the MF problem as in the paragraph above, then there must be some condition p(x)⃗ relative to the associated theory T which does not have good nuclear witnesses, meaning that there is some ϵ > 0 such that, in every model A of T, every k-tuple a⃗ from A satisfying p must satisfy nucAk,n (a)⃗ ≥ ϵ for all n ∈ ℕ.

5.3 Monotracial C∗ -algebras and the Jiang–Su embedding problem One can adapt the techniques used to show that the ℛEP fails to show that the AEP fails for a large class of C∗ -algebras. We will be concerned with monotracial C∗ -algebras, that is, C∗ -algebras which admit a unique tracial state. For example, the universal UHF algebra 𝒬 is monotracial. Suppose that A is a monotracial C∗ -algebra whose unique trace is τA . Let N denote the von Neumann algebra generated by A via the GNS representation of A associated to τA . Then N is a tracial von Neumann algebra when equipped with the extension τN of the original trace τA to N. Since A has a unique trace, it follows that τN is the unique trace on N, whence N is in fact a tracial factor. If we further assume that A is infinite-dimensional, then we can conclude that N is a II1 factor. Now suppose that σ is a universal sentence in the language of tracial von Neumann algebras. We can also view σ as a sentence in the language of tracial C∗ -algebras (one that simply does not refer to the operator norm in any way) and can thus compare the values σ (A,τA ) and σ (N,τN ) . Since A is ‖ ⋅ ‖2 -dense in any operator norm bounded subset of N, it is clear that σ (N,τN ) ≤ σ (A,τA ) . On the other hand, if A is further assumed to be simple, then A embeds into N and thus, σ (A,τA ) = σ (N,τN ) . Summarizing thus far: if A is a unital, simple, infinite-dimensional, monotracial C∗ -algebra whose weak closure in the GNS representation we denote by N, then for any sentence σ in the language of tracial von Neumann algebras, we have that σ (A,τA ) = σ (N,τN ) . If we further assume that N embeds in an ultrapower of ℛ, then this common value equals σ (ℛ,τR ) . It is thus tempting to try to conclude that the AEP must fail for any C∗ -algebra A satisfying the conditions appearing in the previous paragraph. Indeed, if, towards a contradiction, there was an effectively enumerable subset T ⊆ Th(A), all of whose models embed into an ultrapower of A, then by adding to these axioms the (effectively enumerable) axioms for tracial C∗ -algebras, one might hope that by running proofs from this new theory T ′ , one might be able to obtain effective upper bounds for Th∀ (ℛ) and thus contradict MIP∗ = RE as before. The issue with this is that if (M, τ) 󳀀󳨐 T ′ , one is only guaranteed that M embeds into A𝒰 as a C∗ -algebra, that is, the embedding need not preserve the trace τ on M. Consequently, we would not know that sup{σ (M,τ) : (M, τ) 󳀀󳨐 T ′ } coincides with σ (A,τA ) = σ (ℛ,τR ) and thus our usual completeness theorem argument need not go through. However, if M were itself monotracial, then the above embedding would be guaranteed to be trace-preserving and the above argument would work. Unfortunately,

444 � I. Goldbring being monotracial is not an axiomatizable property of C∗ -algebras. That being said, there is a an axiomatizable property of C∗ -algebras known as the uniform Dixmier property which implies being monotracial. (To be fair, the uniform Dixmier property itself is not axiomatizable. Instead, there are quantitative versions known as the (m, γ)-uniform Dixmier property for some parameters m ∈ ℕ and γ ∈ (0, 1), which are each axiomatizable; having the uniform Dixmier property means having the (m, γ)-Dixmier property for some choice of parameters m and γ. See [1] for details.) In summary, we have: Theorem 5.16 (Goldbring and Hart [19]). Let A be a unital, infinite-dimensional, simple, C∗ -algebra A with the uniform Dixmier property whose associated GNS von Neumann algebra N embeds in an ultrapower of ℛ. Then the AEP has a negative solution. There are many examples of C∗ -algebras satisfying the hypotheses appearing in the previous theorem. In particular, it follows from [24] and [1, Corollary 3.11] that 𝒬 satisfies all of the above hypotheses, leading to the aforementioned strengthened refutation of the MF problem: Corollary 5.17. The 𝒬EP has a negative solution. We can use Theorem 5.16 to prove a new, purely operator algebra-theoretic result, which refutes another natural ultarpower embedding problem. One of the most important algebras in modern C∗ -algebra classification theory is the Jiang–Su algebra 𝒵 . (See Vignati’s article in this volume for more information on 𝒵 .) It follows from the works in [24] and [1, Remark 3.18 and Corollary 3.22] that 𝒵 also satisfies hypotheses of Theorem 5.16, whence we have: Corollary 5.18. The 𝒵 EP has a negative solution. One of the defining features of 𝒵 is that it is stably projectionless, meaning that for any n ∈ ℕ and any projection p ∈ Mn (𝒵 ), there is a projection q ∈ Mn (ℂ) unitarily conjugate to p. Being stably projectionless is axiomatizable by the following (effective) list of axioms, one for each n ∈ ℕ: sup

inf

inf

p∈Mn (A) q∈Mn (ℂ) u∈U(Mn (A))

d(upu∗ , q) = 0.

A couple of words are in order about this axiomatization. First, it is know that the matrix amplifications Mn (A) belong to the imaginary sorts of the theory of C∗ -algebras, whence the first and third quantifiers are not problematic. Similarly, being (locally) compact, adding the matrix algebras Mn (ℂ) to the theory of C∗ -algebras is also harmless. Finally, the above axioms only seem to say that every projection in Mn (A) is approximately unitarily equivalent to a projection in Mn (ℂ); however, it is well known that two projections that are sufficiently close are actually unitarily conjugate, whence the axioms do indeed express that an algebra is stably projectionless.

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Combining Corollary 5.18 with the discussion in the previous paragraph yields the following stably projectionless analog of the negative solution to the MF problem, which currently has no purely operator-algebraic proof: Corollary 5.19 (Goldbring and Hart [19]). There is a stably projectionless C∗ -algebra that does not embed into an ultrapower of 𝒵 . We close this section by noting that for the C∗ -algebras A for which the AEP fails, the analogues of Corollaries 3.11 and 3.12 hold as well.

6 Abelian C∗ -algebras In this section, we move on from the more difficult embedding problems in the previous sections and instead consider the case of abelian C∗ -algebras. Not surprisingly, the ensuing discussion becomes topological in nature. As in the previous two sections, all C∗ -algebras in this section are assumed to be unital.

6.1 Preliminaries on ultracoproducts of compact Hausdorff spaces Some of the model theory of abelian C∗ -algebras to be discussed below follows immediately from classical model-theoretic facts about Boolean algebras together with a categorical understanding of the relevant ultraproduct constructions. The ideas presented here are an elaboration of those in [10, Section 5]. To begin, let ZDComp and Bool denote the categories of zero-dimensional compact Hausdorff spaces and Boolean algebras respectively. Consider the Stone functor ZDComp → Bool given by sending the zero-dimensional compact Hausdorff space X to the Boolean algebra Cl(X) of clopen subsets of X. This functor is contravariant and is a duality of categories whose inverse is given by the functor taking a Boolean algebra 𝔹 to its spectrum, that is, the set of ultrafilters on 𝔹, or, equivalently, the set of Boolean algebra homomorphisms 𝔹 → {0, 1}. Letting Comp and AbC* denote the categories of compact Hausdorff spaces and unital abelian C∗ -algebras, then we may also consider the Gelfand functor Comp → AbC* given by sending the compact Hausdorff space X to the unital abelian C∗ -algebra C(X) of complex-valued continuous functions on X. Like the Stone functor, the Gelfand functor is contravariant and is a duality of categories, this time the inverse given by the functor taking a unital abelian C∗ -algebra A to its spectrum Σ(A) consisting of all unital ∗-homomorphisms A → ℂ. In a sense, the Gelfand functor is an “extension” of the Stone functor. More precisely, recall first that a unital C∗ -algebra is called real rank zero if the set of invertible selfadjoint elements is dense in the set of self-adjoint elements. A unital abelian C∗ -algebra

446 � I. Goldbring C(X) is real rank zero if and only if X is zero-dimensional. Let RRZ denotes the category of real-rank zero unital abelian C∗ -algebras. Then the covariant functor RRZ → Bool given by composing the inverse of the Gelfand functor (restricted to RRZ) with the Stone functor is an equivalence of categories. In this case, for X ∈ ZDComp, C(X) is the closed linear span of its space of projections P(X), which in turn is naturally isomorphic to Cl(X). We refer to this equivalence of categories as the “forgetful functor” as it forgets the C∗ -algebra structure and only remembers the Boolean algebra structure on the set of projections. Next recall that one can present the C∗ -algebra ultraproduct construction in purely categorical language. Indeed, suppose that (Ai )i∈I is a family of C∗ -algebras and 𝒰 is an ultrafilter on I. For each J ∈ 𝒰 , let AJ := ∏j∈J Aj denote the direct product and note that the family (AJ , πJK ) forms a directed family, where πJK : AK → AJ is the canonical projection map when J, K ∈ 𝒰 are such that J ⊆ K. There is then a natural isomorphism ∏𝒰 Ai ≅ lim AJ . This fact is actually completely general and holds for ultraproducts of 󳨀󳨀→ L-structures for any (classical or continuous) language L. In particular, the same observation holds verbatim for Boolean algebras and their ultraproducts. (See [15, Chapter 6, Section 10] for more details.) Now suppose that (Xi )i∈I is a family of compact Hausdorff spaces and 𝒰 is an ultrafilter on I. Since ∏𝒰 C(Xi ) is once again a unital abelian C∗ -algebra, it makes sense to consider the compact Hausdorff space Σ(∏𝒰 C(Xi )). One can give a purely topological description of Σ(∏𝒰 C(Xi )). Towards this end, for J ∈ 𝒰 , set XJ := ∐i∈J Xi = β(⨁i∈J Xi ), the Stone–Cech compactification of the direct sum of the Xi ’s, which is the coproduct construction for the category Comp. Since the Gelfand functor is a duality of categories, it follows that ∏i∈J C(Xi ) ≅ C(XJ ). Applying the inverse of the Gelfand functor to the isomorphism ∏𝒰 C(Xi ) ≅ lim C(XJ ) yields the isomorphism Σ(∏𝒰 C(Xi )) ≅ lim XJ . The 󳨀󳨀→ ←󳨀󳨀 compact Hausdorff space lim XJ is called the ultracoproduct of the family (Xi )i∈I with ←󳨀󳨀 respect to 𝒰 , denoted ∐𝒰 Xi . If each Xi = X, we speak of the ultracopower of X with respect to 𝒰 , denoted X𝒰 . Suppose now that each Xi in the previous paragraph is also assumed to be zerodimensional. One can then apply the forgetful functor to the isomorphisms C(∐ Xi ) ≅ ∏ C(Xi ) ≅ lim C(XJ ) 󳨀󳨀→ 𝒰 𝒰 to get the isomorphisms Cl(∐ Xi ) ≅ ∏ Cl(Xi ) ≅ lim Cl(XJ ). 󳨀󳨀→ 𝒰 𝒰 Arguing in a similar fashion, one sees that the forgetful functor sends the diagonal embedding C(X) 󳨅→ C(X)𝒰 to the corresponding diagonal embedding Cl(X) 󳨅→ Cl(X)𝒰 .

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6.2 The model companion of TAbC* The discussion in the previous subsection, together with classical facts about the model theory of Boolean algebras, will allow us to immediately deduce the existence of an ℵ0 -categorical model companion for the theory TAbC* of unital abelian C∗ -algebras. We first observe: Proposition 6.1. Suppose that X and Y are zero-dimensional compact Hausdorff spaces. Then C(X) ≡ C(Y ) if and only if Cl(X) ≡ Cl(Y ). The shortest proof of the previous proposition appeals to the Keisler–Shelah theorem. Indeed, if 𝒰 and 𝒱 are ultrafilters, then by the results discussed at the end of the previous subsection, C(X)𝒰 ≅ C(Y )𝒱 if and only if X𝒰 ≅ Y𝒱 (homeomorphic) if and only if Cl(X)𝒰 ≅ Cl(Y )𝒱 . Proposition 6.2. Suppose that X is a compact Hausdorff space. Then C(X) is an e. c. model of TAbC* if and only if (i) X is zero-dimensional, and (ii) Cl(X) is an e. c. Boolean algebra. For the forward direction, to prove item (i), we note that being real rank 0 is ∀∃-axiomatizable (see [11, Section 3.6.2]) and every model of TAbC* embeds in a real rank-zero model of TAbC* (since every separable compact metric space is a continuous image of Cantor space). Item (ii) follows from our analysis in the previous subsection: if Cl(X) ⊆ Cl(Y ), then C(X) ⊆ C(Y ) and thus there is an embedding C(Y ) 󳨅→ C(X)𝒰 that restricts to the diagonal embedding on C(X). Applying the forgetful functor shows that Cl(X) is e. c. in Cl(Y ). The backwards direction is proven in a similar manner. We remind the reader that the theory of Boolean algebras has an ℵ0 -categorical model completion, namely the theory of atomless Boolean algebras. In particular, Cl(2ℕ ) is the unique countable model of this model completion. With everything in place, we can now conclude: Theorem 6.3. Theory Th(C(2ℕ )) is ℵ0 -categorical and is the model completion of TAbC* . The ℵ0 -categoricity of Th(C(2ℕ )) follows from Proposition 6.1 above, the fact that being real rank-zero is elementary, and the ℵ0 -categoricity of Th(Cl(2ℕ )). Propositions 6.1 and 6.2, together with the fact that Th(Cl(2ℕ )) is the model companion of the theory of Boolean algebras, shows that Th(C(2ℕ )) axiomatizes the e. c. models of TAbC* . To see that the model companion is in fact a model completion, we simply use the fact that TAbC* has the amalgamation property, which follows from the fiber product construction for compact spaces: if C(X) is embedded in C(Y ) and C(Z), then there are surjections πY : Y → X and πZ : Z → X that induce these embeddings. The corresponding fiber product is the space W = Y ×X Z := {(y, z) ∈ Y × Z : πY (y) = πZ (z)}. The natural projection mappings θY : W → Y and θZ : W → Z satisfy πY ∘ θY = πZ ∘ θZ . It follows that the embeddings of C(Y ) and C(Z) into C(W ) corresponding to θY and θZ yield the desired amalgamation.

448 � I. Goldbring

6.3 The projectionless case While the model theory of the entire class of unital abelian C∗ -algebras is fairly mundane (in the sense that Theorem 6.3 above is basically a classical result in disguise), the situation when one considers the subclass of projectionless unital abelian C∗ -algebras is far more interesting. Note that C(X) is projectionless if and only if X is a connected compact Hausdorff space, otherwise known as a continuum. The collection of projectionless unital abelian C∗ -algebras does indeed form an elementary class, which follows either from the observation that the ultracoproduct of a family of continua is once again a continuum (see [23]) or from writing down concrete axioms in the language of C∗ -algebras (see [9]). We let Tcont denote the theory of projectionless unital abelian C∗ -algebras and note that this theory is universally axiomatizable. Unlike the case of arbitrary abelian C∗ -algebras, surprisingly all (nondegenerate) projectionless abelian C∗ -algebras have the same universal theory: Theorem 6.4 (K. P. Hart [25]). If X and Y are continua with X nondegenerate (that is, X is not a point), then C(X) embeds in an ultrapower of C(Y ). Consequently, all nondegenerate models of Tcont have the same universal theory. We outline the proof here, including extra details not present in the published version communicated to us directly by Hart; we thank him for his permission to include this discussion here. By downward Löweinheim–Skolem theorem, we may assume that C(X) and C(Y ) are separable, that is, that X and Y are metric continua. We fix countable bases ℬ and 𝒞 for their lattices of closed sets and enumerate 𝒞 = (Cn )n∈ω . We note that ℬ𝒰 (the lattice ultrapower) is a base of closed sets for the ultracopower X𝒰 of X. Thus, by [8], in order to construct a surjection X𝒰 → Y (and thus an embedding C(Y ) 󳨅→ C(X)𝒰 ), it suffices to find a map ϕ : 𝒞 → ℬ𝒰 satisfying: (1) For all F ∈ 𝒞 , ϕ(F) = 0 if and only if F = 0; (2) For all F, G ∈ 𝒞 , if F ∪ G = Y , then ϕ(F) ∪ ϕ(G) = (X, X, X, . . .)𝒰 ; and (3) For all F1 , . . . , Fn ∈ 𝒞 , if ⋂ni=1 Fi = 0, then ⋂ni=1 ϕ(Fi ) = 0. Towards this end, we fix a surjection f : X → [0, 1] (this is where the nondegeneracy of X is used) and identify Y with a closed subspace of the Hilbert cube Q := [0, 1]ℕ . We set κ : 𝒫 (Y ) → 𝒫 (Q) to be the function κ(F) := {q ∈ Q : d(q, F) ≤ d(q, Y \ F)}. In [32], it was shown that this map κ has the following properties for all closed F, G ⊆ Y : (a) κ(F) ∩ Y = F; (b) κ(F ∪ G) = κ(F) ∪ κ(G); and (c) κ(Y ) = Q and κ(0) = 0. Moreover, for any F ⊆ Y , κ(F) is a closed subset of Q.

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Set E := {e ⊆ ω : ⋂i∈e Ci = 0} and En := E∩ 𝒫 (n). By (a), we have that Y ∩⋂i∈e κ(Ci ) = 0 for all e ∈ E. For each n ∈ ω, take ϵn > 0 such that ϵn < min{d(Y , ⋂i∈e κ(Ci )) : e ∈ En }. Without loss of generality, we may assume that limn→∞ ϵn = 0. Since Y is compact, there is a finite open cover Un of U by basic open sets contained in the ϵn -fattening Yϵn := {z ∈ Q : d(z, Y ) ≤ ϵn } of Y . Again, since Y is compact, we may fix a finite ϵn -dense set Tn ⊆ Y with the property that Tn ∩ Ci ≠ 0 for all i ≤ n for which Ci ≠ 0. Since Y is connected, any two points in ⋃ Un are connected by a piecewise linear path contained in ⋃ Un . Consequently, we may define a continuous map gn : [0, 1] → ⋃ Un ⊆ Q containing Tn in its range. In particular, this map satisfies: (i) d(gn (t), Y ) < ϵn for all t ∈ [0, 1]; (ii) For each y ∈ Y , there is t ∈ [0, 1] such that d(gn (t), y) < ϵn ; (iii) gn−1 (Ci ) ≠ 0 for all i ≤ n for which Ci ≠ 0. For i < n < ω, set Dni := f −1 (gn−1 (κ(Ci ))), a closed subset of X. By items (b) and (c), whenever Ci ∪ Cj = Y , we have that κ(Ci ) ∪ κ(Cj ) = Q, whence Dni ∪ Dnj = X. Moreover, by (i),

whenever e ∈ En , we have gn−1 (⋂i∈e κ(Ci )) = 0, whence ⋂i∈e Dni = 0. Since ℬ is a lattice base for X, there are elements Bin ∈ ℬ containing Dni such that ⋂i∈e Bin = 0 for all e ∈ En . We are finally ready to define the map ϕ : 𝒞 → ℬ𝒰 by setting ϕ(Ci ) := (Bin )𝒰 . We verify that ϕ has the desired properties. First, if Ci = 0, then κ(Ci ) = 0 by (c), whence Dni = 0 and thus Bin = 0 (set e = {i}) for all n > i, whence ϕ(Ci ) = 0. On the other hand, if Ci ≠ 0, then by (iii), for all n > i, we have gn−1 (Ci ) ≠ 0, whence Dni ≠ 0 (since f is surjective) and thus Bin ≠ 0, as desired. Since Dni ⊆ Bin for all i < n < ω, it follows from our observations above that whenever Ci ∪ Cj = Y , we have Bin ∪ Bjn = X for all i < n < ω, whence ϕ(Ci ) ∪ ϕ(Cj ) = (Bin )𝒰 ∪ (Bjn )𝒰 = (Bin ∪ Bjn )𝒰 = (X, X, X, . . .)𝒰 , establishing (b). Finally, suppose that e ∈ En . Then ⋂i∈e Bin = 0, whence ⋂ ϕ(Ci ) = ⋂(Bin )𝒰 = (⋂ Bin ) = 0, i∈e

i∈e

i∈e

𝒰

establishing (c). This finishes the proof of Theorem 6.4. In a series of papers (see, for example, [2]), Bankston studied the model theory of continua in a fairly semantic way by dualizing most notions from classical model theory, occasionally resorting to syntactic techniques by working with lattices bases for the closed sets (which has the disadvantage of not being canonical). In particular, Bankston introduced the notion of a coexistentially closed (co-e. c.) continuum, which, when formulated in our context, is simply a continuum X for which C(X) is an e. c. model of Tcont . Bankston established many properties of co-e. c. continua, including the fact that they are always hereditarily indecomposable (see [2, Theorem 4.1]). A continuum is indecomposable if it cannot be written as the proper union of two subcontinua and a continuum is hereditarily indecomposable if all subcontinua are indecomposable. In [3], Bankston showed that the class of hereditarily indecomposable continua

450 � I. Goldbring is coelementary, which, after applying the Gelfand functor, implies that the class of models of Tcont of the form C(X) for X a hereditarily indecomposable continuum is elementary. Moreover, since the inverse limit of hereditarily indecomposable continua is again hereditarily indecomposable, it follows that the aforementioned class of models of Tcont is inductive and thus ∀∃-axiomatizable. One of Bankston’s main questions was whether or not the pseudoarc ℙ is a co-e. c. closed continuum (see [2, Remark 4.2(i)]). Recall that ℙ is the unique metric continuum that is hereditarily indecomposable and chainable, which is a condition that ensures that the continuum is “arclike” in an appropriate sense. The pseudoarc is “generic” in the descriptive set-theoretic sense of the word [33] and thus it is also natural to ask if it is generic from the model-theoretic perspective as well. In joint work with Eagle and Vignati [9], we were able to answer Bankston’s question affirmatively: Theorem 6.5. C(ℙ) is an e. c. model of Tcont . Our main contribution was to show that chainability is a ∀ ⋁ ∃ property of models of Tcont (or rather their image under the inverse Gelfand functor). By Theorem 6.4, C(ℙ) is a locally universal model of Tcont , whence one can enforce being chainable by Proposition 2.1. Since being e. c. is also enforceable, it follows that being e. c. and chainable is enforceable. Since e. c. implies hereditariliy indecomposable, by the fact that ℙ is the unique metrizable hereditarily indecomposable, chainable continuum, we see that C(ℙ) is actually the enforceable model of Tcont . One of the main open questions in the model theory of abelian C∗ -algebras is the following: Question 6.6. Does Tcont have a model companion? As pointed out in [9], if this model companion exists, then it will not be a model completion for Tcont does not have the amalgamation property. Of course, if the model companion exists, then it must in fact be Th(C(ℙ)).

Bibliography [1] [2] [3] [4] [5]

R. Archbold, L. Robert and A. Tikuisis, The Dixmier property and tracial states for C∗ -algebras, J. Funct. Anal. 273 (2017), 2655–2718. P. Bankston, Continua and the co-elementary hierarchy of maps, Topol. Proc. 25 (2000), 45–62. P. Bankston, The Chang–Łoś–Suszko theorem in a topological setting, Arch. Math. Log. 45 (2006), 97–112. S. Barlak and G. Szabó, Sequentially split *-homomorphisms between C∗ -algebras, Int. J. Math. 27 (2016), 48 pages. I. Ben Yaacov and A. P. Pederson, A proof of completeness for continuous first order logic, J. Symb. Log. 75 (2010), 168–190.

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[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

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K. Carlson, E. Cheung, I. Farah, A. Gerhardt-Bourke, B. Hart, L. Mezuman, N. Sequeira and A. Sherman, Omitting types in AF algebras, Arch. Math. Log. 53 (2014), 157–169. A. Connes, Classification of injective factors. Cases II1 , II∞ , IIIλ , λ ≠ 1, Ann. Math. 104 (1976), 73–115. A. Dow and K. P. Hart, A universal continuum of weight ℵ, Trans. Am. Math. Soc. 353 (2001), 1819–1838. C. Eagle, I. Goldbring and A. Vignati, The pseudoarc is a co-existentially closed continuum, Topol. Appl. 207 (2016), 1–9. C. Eagle and A. Vignati, Saturation and elementary equivalence of C∗ -algebras, J. Funct. Anal. 269 (2015), 2631–2664. I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati and W. Winter, Model Theory of C∗ -algebras, Mem. Am. Math. Soc. (2021). T. Fritz, Tsirelson’s problem and Kirchberg’s conjecture, Rev. Math. Phys. 24 (2012), Article 1250012. T. Fritz, T. Netzer and A. Thom, Can you compute the operator norm? Proc. Am. Math. Soc. 142, no. 12014, 4265–4276. I. Goldbring, Enforceable operator algebras, J. Inst. Math. Jussieu 20 (2021), 31–63. I. Goldbring, Ultrafilters throughout mathematics, Graduate Studies in Mathematics, vol. 220, American Mathematical Society, 2022. I. Goldbring, The Connes embedding problem: A guided tour, Bull. Am. Meteorol. Soc. in press. I. Goldbring, Non-embeddable II1 factors resembling the hyperfinite II1 factor, J. Noncommut. Geom. in press. I. Goldbring and B. Hart, A computability-theoretic reformulation of the Connes embedding problem, Bull. Symb. Log. 22 (2016), 238–248. I. Goldbring and B. Hart, The universal theory of the hyperfinite II1 factor is not computable. arXiv:2006.05629. I. Goldbring, S. Kunnawalkam Elayavalli and Y. Lodha, Generic algebraic properties in spaces of enumerated groups. arXiv:2006.14048. I. Goldbring and T. Sinclair, On Kirchberg’s embedding problem, J. Funct. Anal. 269 (2015), 155–198. I. Goldbring and T. Sinclair, Robinson forcing and the quasidiagonality problem, Int. J. Math. 28 (2017), Article 1750008. R. Gurevic, On ultracoproducts of compact Hausdorff spaces, J. Symb. Log. 53 (1988), 294–300. U. Haagerup and L. Zsidó, Sur la propriété de Dixmier pour les C∗ -algébres, C. R. Acad. Sci. Paris, Sér. I Math. 298 (1984), 173–176. K. P. Hart, There is no categorical metric continuum, Continuum theory: In honor of Professor David P. Bellamy on the occasion of his 60th birthday, Aportaciones Mat. Investig., vol. 19, Soc. Mat. Mexicana, Mexico, 2007, pp. 39–43. W. Hodges, Building models by games, London Math Soc. Student Texts, vol. 2, Cambridge University Press, 1985. Z. Ji, A. Natarajan, T. Vidick, J. Wright and H. Yuen. MIP∗ = RE. arXiv:2001.04383. M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V. Scholz and R. Werner, Connes’ embedding problem and Tsirelson’s problem, J. Math. Phys. 52 (2011), Article 012102. H. J. Keisler, Fundamentals of model theory, Handbook of Math. Logic, 1978, pp. 47–103. S. Kim, V. Paulsen and C. Schafhauser, A synchronous game for binary constraint systems, J. Math. Phys. 59 (2018), Article 032201. E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C∗ -algebras, Invent. Math. 112 (1993), 449–489. K. Kuratowski, Topology: Volume I, Elsevier, 2014. W. Lewis, The pseudo-arc, Bol. Soc. Mat. Mexicana 5 (1999), 25–77. N. Ozawa, About the Connes embedding conjecture, Jpn. J. Math. 8 (2013), 147–183. M. Rordam, Classification of simple nuclear C∗ -algebras, Encyclopaedia Math. Sci. 126 (2002), 1–145. A. Tikuisis, S. White and W. Winter, Quasidiagonality of nuclear C∗ -algebras, Ann. Math. 185 (2017), 229–284. D. Voiculescu, A note on quasi-diagonal C∗ -algebras and homotopy, Duke Math. J. 62 (1991), 267–271.

Alessandro Vignati

Fraïssé theory in operator algebras Abstract: We overview the development of Fraïssé theory in the setting of continuous model theory, and some of the its recent applications to C∗ -algebra theory and functional analysis. Keywords: Operator algebras, continuous logic, continuous model theory, homogeneous structures, Fraïssé theory, Poulsen simplex, C∗ -algebras, automorphism group, extreme amenability, Ramsey property, Jiang–Su algebra, Choquet simplices MSC 2020: 03C66, 46L05, 46L40, 46M40, 46L52, 46A55

1 Introduction This short note aims to review the recent progress in the study of the applications of Fraïssé theory to continuous model theory, and in particular to operator algebras. Fraïssé theory, after [23], is an area of mathematics at the crossroads of combinatorics and model theory. In the discrete setting, Fraïssé theory studies countable homogeneous structures and their relations with properties of their finitely-generated substructures. Given a countable structure ℳ, its age is the class of all finitely-generated substructures of ℳ. Ages of countable homogeneous structures satisfy certain combinatorial properties (notably amalgamation properties) making them Fraïssé classes. Conversely, given any Fraïssé class, one constructs a countable homogeneous structure, its Fraïssé limit, which is the unique (up to isomorphism) structure with the given class as its age. Many interesting objects across mathematics (in group theory, graph theory, and topology) were identified as Fraïssé limits (see, for example, [34, Chapter 7] and [36]). Fraïssé theory has found applications in several areas of mathematics. An example deserving to be mentioned is topological dynamics: In the seminal [42], Kechris, Pestov, and Todorčevic ̀ established what is now known as the KPT-correspondence, linking Ramsey theoretical property of a Fraïssé class to dynamical properties of the automorphism group of its Fraïssé limit. The KPT correspondence is used to compute, and study properties of, universal minimal flows of automorphism groups of Fraïssé limits via Ramsey-theoretic conditions on the Fraïssé class of interest. (The universal minimal flow of a topological group G is a compact dynamical system of G canonically associated Acknowledgement: The author is partially supported by the ANR project AGRUME (ANR-17-CE40-0026) and by an Emergence en Recherche grant from the Université Paris Cité—IdeX. Alessandro Vignati, Institut de Mathématiques de Jussieu – Paris Rive Gauche (IMJ-PRG), Université Paris Cité, Bâtiment Sophie Germain, 8 Place Aurélie Nemours, 75013 Paris, France, e-mail: [email protected]; URL: http://www.automorph.net/avignati https://doi.org/10.1515/9783110768282-012

454 � A. Vignati with the group which is of great interest in topological dynamics, see Section 5.) The KPT correspondence and its extensions were used to study automorphism groups of Fraïssé limits of certain classes of graphs, directed graphs, posets, lattices, and so on. Moving to the continuous setting, Fraïssé theoretic ideas were discussed in analysing the class of finite-dimensional Banach spaces and its amalgamation properties by Kubis̀ and Solecki in [49]. Before a proper formalization of Fraïssé theory for continuous structures, objects of Fraïssé-theoretic nature were studied via topological and categorical methods (see, e. g., [46] and [4]). A systematic approach to Fraïssé theory in the setting of continuous logic was discussed in [68] and then introduced in [8]. Notably, many structures in functional analysis have been recognized to be of Fraïssé-theoretic nature, for example, in the work of Ben Yaacov and collaborators [7, 10], Kubis̀ and his collaborators [25, 48, 46, 47], Conley and Tørnquist [16], and Lupini [53, 52]. Operator algebra examples were treated first in [18], then in Masumoto’s work [56, 55], and further in [27, 38]. Here, we follow the approach of [8], where Ben Yaacov gives the formal definitions of Fraïssé classes and Fraïssé limits for metric structures. Fix a language for metric structures ℒ. Informally, the Fraïssé theorem for continuous logic asserts that if 𝒦 is a class of finitely-generated ℒ-structures which is hereditary, directed, satisfies reasonable separability conditions, and has the near amalgamation property, then there is a unique separable ℒ-structure M which has 𝒦 as its age and is ultrahomogeneous, meaning that, given a finite tuple ā in M the closure of the automorphism orbit of ā contains all tuples with the same quantifier-free type of a.̄ A version of the Fraïssé theorem for nonhereditary classes, restricting the amount of homogeneity one obtains, also holds. Pivotal objects in operator algebras and functional analysis have been recognized as Fraïssé limits of appropriate classes of structures. We give two examples of crucial importance in their respective areas: the Jiang–Su algebra 𝒵 and the Poulsen simplex P. The Jiang–Su algebra 𝒵 was constructed by Jiang and Su in [39] as a limit of subhomogeneous C∗ -algebras known as dimension drop algebras (see Section 3.3.1). Algebra 𝒵 has been at the center of the classification program for C∗ -algebras. In a precise sense, 𝒵 is the smallest possible C∗ -algebra that tensorially absorbs itself in a very strong way, and so, from this perspective, is the most natural C∗ -analogue of the hyperfinite II1 factor ℛ, at the center of Connes’ Fields Medal work on von Neumann algebras [17]. A C∗ -algebra A which satisfies A ⊗ 𝒵 ≅ A is called 𝒵 -absorbing. By a celebrated result of many hands, 𝒵 -absorption is a regularity condition, capable of detecting those (suitable) C∗ -algebras which can be classified. More precisely, separable 𝒵 -absorbing C∗ -algebras which are in addition unital, simple, and nuclear (and satisfy the technical condition known as UCT) are precisely those which can be classified by their Elliott invariant (see [31, 30] or [13, Corollary D]). Further, the Toms–Winter conjecture asserts that 𝒵 -absorption is linked to other well-behavedness and regularity conditions related to low dimensionality (in terms of the nuclear dimension of Winter and Zacharias [72]) and perforation properties of the Cuntz semigroup (a refined version of K-theory defined by Cuntz). Algebra 𝒵 has been the focus of prominent research in the last 20

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years for its unique and peculiar properties (see, e. g., [71]), and has been approached by Fraïssé-theoretic methods in [18], and then in [56, 55]. Algebra 𝒵 ’s nonunital cousins, 𝒲 and 𝒵0 , play the role of 𝒵 in the classification program of nonunital C∗ -algebras; they were approached by Fraïssé-theoretic methods in [38]. The theory of Choquet simplices lies at the intersection of Banach space theory, convex geometry, and the study of Polish spaces. In fact, Kadison’s representation theorem [1, Theorem II.1.8] gives a contravariant equivalence between the categories of compact convex sets (with continuous affine maps) and that of function systems (with positive linear maps). Choquet simplices correspond to function systems that are, moreover, Lindenstrauss spaces (preduals of L1 spaces), and are therefore connected to measure theory, and consequently the whole of functional analysis. The Poulsen simplex P was introduced in [64] as a metrizable Choquet simplex whose extreme points are dense. Despite the fact that Poulsen’s original construction has many degree of freedom, in [51] (see also [60]), P was shown to be the unique metrizable Choquet simplex whose extreme points are dense (up to affine homeomorphism). The study of P has been inspired by that of Gurarij’s universal and homogeneous Banach space G (see Section 4). In fact, as specifically stated in [51], the idea to prove uniqueness of the Poulsen simplex came directly from Lusky’s proof of the uniqueness of Gurarij’s generic Banach space [54]. P has many homogeneity and universality properties (which can be reconstructed by Fraïssétheoretic methods, as noticed in [53]). Our overview of the applications of Fraïssé theory to the study of operator algebras and functional analysis is structured as follows. In Section 2, we introduce Fraïssé classes and their limits. In Section 3 we describe the findings of [18, 56, 38], where the C∗ -algebras 𝒵 , 𝒲 , 𝒵0 , and the unital separable UHF algebras were recognized as Fraïssé limits. We also see how to construct surjectively universal AF algebras as Fraïssé limits [27]. In Section 4 we overview the applications of Fraïssé theory to functional analysis: we introduce the Gurarij spaces G and NG, as well as the Poulsen simplex P and its noncommutative version, the noncommutative Choquet simplex NP, and we describe how there can be viewed as Fraïssé limits of suitable classes. Lastly, in Section 5, we go through some of the connections between Ramsey theory and topological dynamics of automorphisms groups of Fraïssé structures, describing some of the findings of [6] and [5]. In case the reader lacks the preliminary notions, it is redirected to [12] or Szabò’s article for an introduction to C∗ -algebras, to [9] or Hart’s article for an introduction to continuous model theory, and to [21] or Sinclair’s article for the model theory of C∗ -algebras.

2 Fraïssé classes and their limits In this section, we introduce Fraïssé classes and their limits. We follow an approach similar to that of [8]. We recall the relevant information from Section 3 of Hart’s article in this book.

456 � A. Vignati Definition 2.1. A language for metric structures ℒ is a set of relation symbols (Ri )i∈I and function symbols (fj )j∈J . To each symbol of the language are attached a natural number (its arity) and a modulus of uniform continuity; 0-ary functions are constants. An ℒ-structure is a complete metric space (X, d) together with interpretations for relation and function symbols. We always assume that our languages contain a special binary symbol, which is interpreted by the distance function in all structures, analogously as one interprets equality in the discrete setting. If A is an ℒ-structure, n ∈ ℕ, and ā ∈ An , we denote by ⟨a⟩̄ the smallest ℒ-substructure containing ā (i. e., it is stable under interpretations of all functions in ℒ). Definition 2.2. An ℒ-embedding is a map which commutes with interpretations of the symbols in ℒ. We fix, for the remaining part of this section, a separable language for metric structures ℒ. All structures (substructures, embeddings, …) are implicitly ℒ-structures (ℒ-substructures, ℒ-embeddings, …). Definition 2.3. Let 𝒦 be a class of finitely-generated ℒ-structures. Then 𝒦 is said to have: – the hereditary property (HP) if whenever A ∈ 𝒦 and B is a substructure of A, then B ∈ 𝒦; – the joint embedding property (JEP) if for all A, B ∈ 𝒦 there is C ∈ 𝒦 and two embeddings ϕ: A → C and ψ: B → C; – the near amalgamation property (NAP) if whenever we are given A, B, and C in 𝒦, a finite set F ⊆ A and ε > 0, then for every pair of embeddings ϕ1 : A → B and ϕ2 : A → C there is D ∈ 𝒦 and two embeddings ψ1 : B → D and ψ2 : C → D such that d(ψ1 ∘ ϕ1 (a), ψ2 ∘ ϕ2 (a)) < ε

for all a ∈ F,

where d is the distance predicate as computed in D. B ψ1 ϕ1 A

D

↻F,ε ϕ2

C

ψ2

The final condition one must impose is the analogue of countability in the discrete setting. If ā = (a1 , . . . , an ) and b̄ = (b1 , . . . , bn ) are tuples in a common structure A with distance d, we abuse of notation and let d(a,̄ b)̄ := max d(ai , bi ). i

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Definition 2.4. Let 𝒦 be a class of finitely-generated ℒ-structures satisfying JEP and NAP. We denote by 𝒦n the set of all pairs (A, a)̄ where ā ∈ An generates A, and A ∈ 𝒦. Define a pseudometric dn on 𝒦n by setting ̄ = inf d(ϕ(a), ̄ ̄ (B, b)) ̄ ψ(b)), dn ((A, a), ϕ,ψ

where ϕ and ψ range over all possible embeddings of ⟨a⟩̄ and ⟨b⟩̄ in a common C ∈ 𝒦. Class 𝒦 is said to have the weak Polish property (WPP) if each (𝒦n , dn ) is a separable space. Definition 2.5. A class of finitely-generated structures satisfying JEP, NAP, and WPP is said to be a Fraïssé class. If the class satisfies in addition HP, it is said to be a hereditary Fraïssé class. Definition 2.6. Let 𝒦 be a class of finitely-generated ℒ-structures. A separable ℒ-structure is said to be a 𝒦-structure if it is the inductive limit of structures in 𝒦. A 𝒦-structure M is a Fraïssé limit of 𝒦 if 1. M is 𝒦-universal: each 𝒦-structure admits an embedding into M; 2. M is approximately 𝒦-homogeneous: for every A = ⟨a⟩̄ ∈ 𝒦, ε > 0, and embeddings ϕ1 , ϕ2 : A → M there is σ, an automorphism of M, such that ̄ ϕ2 (a)) ̄ < ε, d(σ(ϕ1 (a)), where d is the distance in M. The following result is the main result relating Fraïssé classes and homogeneous structures. See [8] for a proof. Theorem 2.7 (Fraïssé theorem). Every Fraïssé class has a Fraïssé limit which is unique up to isomorphism. Definition 2.8. Let 𝒦 be a Fraïssé class with Fraïssé limit M. Let (An , φn ) be a sequence where An ∈ 𝒦 and each φn : An → An+1 is an embedding. The sequence (An , φn ) is said to be generic if lim(An , φn ) ≅ M. We conclude this section with a few remarks. Remark 2.9. – Our approach is slightly different from that of [8], which is more technical and involved, as translating notions and theorems from discrete model theory to the continuous setting has to be done always extremely carefully. The setting displayed suffices for all the applications we are going to review. (For example, notice that the classes introduced here are incomplete in the sense of [8]. The completions of our classes will include their Fraïssé limits.) – A second and simpler proof of the Fraïssé theorem is due to Todor Tsankov. The author (and others) would like very much to see such a proof published.

458 � A. Vignati –

–

The Fraïssé classes appearing in Section 3, apart from those defining the algebra of continuous functions on the Cantor space and the hyperfinite II1 factor ℛ, are not hereditary. This is due to the fact that the class of finitely-generated subalgebras of a given C∗ -algebra A is often complicated, and very large. In fact, for the age of a C∗ -algebra A, amalgamation indeed fails most of the time. All examples considered in Section 4 are hereditary classes by their very nature. While in the discrete setting many (though not all) well-known Fraïssé limits have theories with quantifier elimination, this is not true in the operator-algebraic setting. It was proved in [19] that the only separable infinite-dimensional unital C∗ -algebra having quantifier elimination in the usual language for C∗ -algebras is C(2ℕ ), the algebra of continuous functions on the Cantor set. On the other hand, objects from functional analysis arising as Fraïssé limits (such as the Urysohn space, or the Hilbert space ℓ2 (ℕ)), do have quantifier elimination in their natural languages.

3 Fraïssé theory in operator algebras Here, we describe the main applications of Fraïssé theory to operator algebras. Most of the classes we will be concerned with are not hereditary classes, as mentioned in Remark 2.9. This is due to the fact that infinite-dimensional C∗ -algebras are often singly generated (see, e. g., [70]). (In the unital abelian setting, finitely-generated C∗ -algebras correspond to those whose spectrum is of finite covering dimension.) With plenty of finitely-generated structures to take care of, obtaining approximate amalgamation (even allowing natural expansions of the language) is therefore quite difficult. For this reason, we only consider classes that are made of suitably “small” algebras. We will (almost) always work in the language of C∗ -algebras as introduced in [22] (see also [21] or Sinclair’s article in this book), or some of its natural expansions.

3.1 Abelian examples We start by recording easy examples of topological nature. 3.1.1 The Cantor set The following is equivalent to the fact that the class of finite sets together with surjective maps forms a projective Fraïssé class1 whose limit is the Cantor set 2ℕ . 1 Projective Fraïssé theory can be seen as the dual version of the classical ones, where maps are often surjections. Projective Fraïssé theory have been used in topology (e. g., [4, 36]) to recognize that certain spaces are generic.

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Theorem 3.1. Let 𝒦 be the class whose objects are finite-dimensional abelian C∗ -algebras and whose maps are unital injective ∗ -homomorphisms. Then 𝒦 is a Fraïssé class whose Fraïssé limit is C(2ℕ ). 3.1.2 The pseudoarc We dualize the content of [46, Section 4.3], where it was proved that Bing’s pseudoarc ℙ is the projective Fraïssé limits of intervals (even though in the language of metric enriched categories). In particular, we show that the algebra C(ℙ) is a Fraïssé limit in the setting of Section 2. First, a lemma. (See [35] for the original version, and [69] for a proof of the result as stated below.) Lemma 3.2 (Mountain climbing lemma). Let f and g be continuous piecewise linear surjections [0, 1] → [0, 1] such that f (0) = 0 = g(0) and f (1) = 1 = g(1). Then there are surjections f ′ , g ′ : [0, 1] → [0, 1] such that f ∘ f ′ = g ∘ g ′ . Dualizing the content of the mountain climbing lemma, we get the following result. (Since the proof in the current language has not appeared anywhere, we sketch it.) Theorem 3.3. Let 𝒦ℙ be the class whose only object is C([0, 1]) and whose maps are unital injective ∗ -homomorphisms. Then 𝒦ℙ is a Fraïssé class. Sketch of the proof. The only nontrivial property to prove is the near amalgamation property. Fix a finite F ⊆ C([0, 1]), ε > 0, and two injective ∗ -homomorphisms ϕ1 , ϕ2 : C([0, 1]) → C([0, 1]). Since injections between abelian C∗ -algebras correspond to surjections between the corresponding spaces, consider the continuous surjections ξ1 , ξ2 : [0, 1] → [0, 1] such that ϕi (f )(t) = f (ξi (t)) for i = 1, 2 and t ∈ [0, 1]. First, we find two continuous surjections ξ1′ and ξ2′ with the property that ξi ∘ ξi′ (0) = 0 and ξi ∘ ξi′ (1) = 1 for i = 1, 2. Since those ∗ -homomorphisms whose duals are piecewise linear maps are dense in point norm topology, and F is finite, we can assume that each ξi ∘ ξi′ , for i = 1, 2, is piecewise linear. The duals of the amalgamating surjections provided by the mountain climbing lemma give the NAP. To study the Fraïssé limit of the class 𝒦ℙ we use Gelfand duality, and study the topological properties of its spectrum. A connected compact Hausdorff space is a continuum. A continuum X is chainable if every open cover of X can be refined by a finite open cover U1 , . . . , Un with the property that Ui ∩ Uj ≠ 0 if and only if |i − j| ≤ 1. X is indecomposable if it cannot be written as the union of two of its proper subcontinua, and hereditary indecomposable if each of its subcontinua is indecomposable. The only metrizable chainable hereditary indecomposable continuum is (up to homeomorphism) Bing’s pseudoarc ℙ [11]. Theorem 3.4. The Fraïssé limit of 𝒦ℙ is Bing’s pseudoarc ℙ.

460 � A. Vignati Sketch of the proof. Let X be the compact metrizable space such that the Fraïssé limit of 𝒦ℙ is isomorphic to C(X). That X is chainable is obvious, being an inverse limit of chainable continua. Kubis̀’s argument (see [46, Lemma 4.21 and 4.22]), when dualized, gives that X is hereditary indecomposable (in fact, he shows that X is indecomposable and every proper subcontinuum of X is homeomorphic to X itself). The result follows from Bing’s theorem [11].

3.1.3 Other examples Other topological spaces can be viewed as projective Fraïssé limits of suitable classes of spaces, once certain restrictions are given on the maps in consideration. Dually, their algebras of functions could be approached by a Fraïssé-theoretic point of view as limit of suitable classes of abelian C∗ -algebras. Examples are the Menger curve 𝕄 and the Lelek fan 𝕃, treated as projective Fraïssé limits of finite graphs (with connected maps) and fans, both considered as onedimensional continua, see, e. g., [61] and [4]. Neither C(𝕄) nor C(𝕃) has been formally recognized as a Fraïssé limit in the context of continuous logic, and more specifically operator algebras. Problem 3.5. Is there a class of C∗ -algebras 𝒦𝕄 where embeddings are injective ∗ -homomorphisms, having C(𝕄) as its Fraïssé limit? What about C(𝕃)?

3.2 Nonabelian C∗ -algebras In this section we summarize the work of [18, 56, 38], where certain pivotal objects in the classification of nuclear C∗ -algebras were recognized as Fraïssé limits of suitable classes of finitely generated and simpler building blocks. If n ∈ ℕ, Mn denotes the C∗ -algebra of n × n complex valued matrices, Mn (ℂ). 3.2.1 UHF algebras A separable C∗ -algebra A is called uniformly hyperfinite (UHF) if it is a direct limit of matrix algebras (see Section 7 of Szabò’s article in this volume). Thanks to Glimm’s theorem, unital UHF algebras are classified by their supernatural numbers, that is, by formal products p̄ = ∏p prime pℓp , where ℓp ∈ ℕ ∪ {∞}. If A is a unital UHF algebra with A = limi Mni , for each prime p, let ℓp = sup{r | pr divides ni for some i}. This associates a supernatural number p̄ A to A, and one checks that

Fraïssé theory in operator algebras � 461

A ≅ Mp̄A := ⨂ ⨂ Mp . p prime j≤ℓp

Definition 3.6. Let p̄ be a supernatural number. Then 𝒦p̄ is the class of matrix algebras whose size divides p,̄ where maps are unital injective ∗ -homomorphisms. The fact that any two unital ∗ -homomorphisms between matrix algebras are unitarily equivalent immediately gives (exact!) amalgamation for 𝒦p̄ . As the limit of a sequence of objects in 𝒦p̄ is automatically UHF, we have the following. Theorem 3.7. For each supernatural p,̄ the class 𝒦p̄ is a Fraïssé class whose Fraïssé limit is Mp̄ . 3.2.2 The hyperfinite II1 factor ℛ Analyzing UHF algebras, we treated matrix algebras as C∗ -algebras; here we consider them as tracial von Neumann algebras in the appropriate language (see [22]). In this setting, the distance symbol is interpreted as the distance associated with the trace norm ‖x‖τ = τ(xx ∗ )1/2 . We recall that a tracial von Neumann algebra (M, τ) is a factor if it has trivial center. Among separable (for ‖⋅‖τ ) factors, there is a unique amenable object, known as the II1 factor ℛ. In this setting, amenability coincides with hyperfiniteness, where a II1 factor is hyperfinite if it can be approximated, in trace-norm, by matrix algebras (see, e. g., [17]). The same amalgamation one proves for matrix algebras gives the following, which was stated as [18, Theorem 3.5]. Theorem 3.8. In the language of tracial von Neumann algebras, the hyperfinite II1 factor ℛ is the Fraïssé limit of the Fraïssé class of matrix algebras.

3.3 Finite-dimensional C∗ -algebras We have seen (Section 3.2.1) that full matrix algebras have amalgamation, in the category of unital C∗ -algebras. One might wonder whether the same holds for finite-dimensional C∗ -algebras. If F is a finite-dimensional C∗ -algebra, then there are positive naturals n1 , . . . , nk such that F ≅ ⨁i≤k Mni . Limits of finite-dimensional C∗ -algebras are called approximately finite-dimensional (AF) algebras; these algebras were of great interest in the 1970s and 1980s, representing some of the first nontrivial examples of infinitedimensional simple C∗ -algebras. They were classified by Elliott in [20] (see again Szabò’s article in this book). A first guess in trying to construct a universal and somewhat homogeneous AF algebra would be to consider the class of all finite-dimensional C∗ -algebras. Unfortunately, this class does not have amalgamation: Let A = ℂ ⊕ ℂ and B = M3 , and consider the maps φ1 : A → B and φ2 : A → B given by

462 � A. Vignati a φ1 ((a, b)) = (0 0

0 a 0

0 0) b

a and φ2 ((a, b)) = (0 0

0 b 0

0 0) . b

A trace argument shows that one cannot amalgamate these two maps: if ψ1 and ψ2 are two unital ∗ -homomorphisms whose range is a matrix algebra Mk , then, with τ the canonical trace on Mk , one verifies that 2 1 = τ(ψ1 ∘ φ1 ((1, 0))) ≠ τ(ψ2 ∘ φ2 ((1, 0))) = . 3 3 The obstructions to amalgamation above are of tracial nature. By adjoining the trace to the language, and only considering trace preserving ∗ -homomorphisms (see Section 3.3.1), in [18, 3.8 and 3.9] it was shown that certain subclasses of finite-dimensional C∗ -algebras are indeed Fraïssé classes. All of the Fraïssé limits arising this way are simple AF algebras. On the other side of the spectrum are AF algebras which are surjectively universal, and therefore must contain an abundance of ideals. These were treated, by using a different approach not relying on tracial information, by Ghasemi and Kubis̀ in [27]. A definition is as follows: Definition 3.9. Let A and B be C∗ -algebras. A map φ: A → B is left-invertible if there is a ∗ -homomorphism π: B → A such that π ∘ φ is the identity on A. The following is the main content of [27]: Theorem 3.10. Let 𝒞 be a class of finite-dimensional which is closed under taking direct sums and ideals, whose maps are left-invertible embeddings (not necessarily unital). Then 𝒞 is a Fraïssé class. By considering 𝒞 be the class of all finite-dimensional C∗ -algebras, the Fraïssé limit of such class A𝒞 is a separable AF algebra which surjects onto any separable AF algebra. Further, its trace space is affinely homeomorphic to the Poulsen simplex P (see Section 4.3). 3.3.1 Some classes of small algebras Let us introduce certain classes of C∗ -algebras we will use as building blocks. Some notations: 0k denotes the 0 matrix in Mk , likewise 1k is the identity. If a ∈ Mk and b ∈ Mk ′ are matrices, let diag(a, b) := ( a0 0b ) ∈ Mk+k ′ . This naturally generalizes to diag(a1 , . . . , an ). Definition 3.11. If n ∈ ℕ, let Cn := C([0, 1], Mn ). We will refer to these as interval algebras.

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If p and q are natural numbers, by identifying Mpq with Mp ⊗ Mq , we define 𝒵p,q := {f ∈ Cpq | f (0) ∈ 1p ⊗ Mq and f (1) ∈ Mp ⊗ 1q }.

We refer to these as dimension drop algebras. Notice that 𝒵p,q is projectionless if and only if p and q are coprime (see [39, Lemma 2.2]). Coprime dimension drop algebras have K-theory (ℤ, 0), where 1 is the class of the identity.2 Another interesting class of subalgebras of interval algebras was identified in [65]. Definition 3.12. If n, k ∈ ℕ, let An,k := {f ∈ C([0, 1], Mnk ) | ∃a ∈ Mk (f (0) = diag(a, . . . , a) and ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

f (1) = diag(a, . . . , a, 0k ))}. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n−1

The algebras An,k are called Razak blocks. Let Bn,k := {f ∈ C([0, 1], M2nk ) | ∃a, b ∈ Mk (f (0) = diag(a, . . . , a, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ b, . . . , b) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n

n

and f (1) = diag(a, . . . , a, 0k , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ b, . . . , b, 0k ))}. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n−1

n−1

The algebras Bn,k are called generalized Razak blocks. The following summarizes a few of the basic properties of these objects (see [38, Proposition 2.8]). Recall that a C∗ -algebra A is projectionless if it has no proper projections. A is stably projectionless if A ⊗ 𝒦 is projectionless. Proposition 3.13. Let n, k ∈ ℕ. Then 1. Razak blocks and generalized Razak blocks are stably projectionless, but every proper quotient of each of them has a nonzero projection; 2. nonzero ∗ -homomorphisms whose domain and codomain are a Razak block or a generalized Razak block are injective; 3. K0 (An,k ) = 0, K0 (Bn,k ) ≅ ℤ and K1 (An,k ) = K1 (Bn,k ) = 0. By Proposition 3.13, any limit of Razak blocks has trivial K-theory. On the other hand, limits of generalized Razak blocks can have different K0 -groups. Since K0 (Bn,k ) ≅ ℤ, a ∗ -homomorphism between generalized Razak blocks φ induces a group endomorphism φ∗ : ℤ → ℤ. 2 We do not define K-theory in this article. For the basics of K-theory, please see [67].

464 � A. Vignati Definition 3.14. Let φ be a ∗ -homomorphism between generalized Razak blocks, and let k ∈ ℤ. We say that φ has K-theory k if φ∗ (1) = k. Before giving the definition of the classes of interest, let us look at traces on interval algebras, dimension drop algebras, and (generalized) Razak blocks. Definition 3.15. If A is a C∗ -algebra, a state τ on A is a trace if τ(ab) = τ(ba) for all a, b ∈ A. We denote by T(A) the trace space of A. If A and B are C∗ -algebras, σ ∈ T(A) and τ ∈ T(B), we say that a ∗ -homomorphism φ: A → B sends σ to τ, and write φ: (A, σ) → (B, τ), if σ(a) = τ(φ(a)) for all a ∈ A. As Mn has a unique trace τn , traces on Cn correspond to Radon probability measures on [0, 1]: if μ is a measure on [0, 1], let τμ (f ) = ∫ τn (f (t)) dμ(t). Then τμ is a trace on Cn . Conversely, given a trace τ on Cn , we can construct a measure μτ by μτ (O) =

sup

f :supp(f )⊆O,‖f ‖≤1

τ(ff ∗ ),

for any open set O ⊆ [0, 1]. It is routine to check that τ = μτμ and μ = μτμ . With this association, we can borrow measure-theoretic terminology and refer to faithful diffuse traces for those whose associated measure is faithful and diffuse. (Recall that a measure is diffuse if every measurable set of nonzero measure can be split in two measurable subsets each of nonzero measure. A measure is faithful if no nonempty open set has zero measure.) We denote by Tfd (Cn ) the set of faithful diffuse traces on Cn . The same reasoning applies to dimension drop algebras and (generalized) Razak’s blocks; we will write Tfd (𝒵p,q ), Tfd (An,k ) and Tfd (Bn,k ) for the corresponding sets of faithful diffuse traces. Definition 3.16. Let A and B be either interval algebras, dimension drop algebras, Razak blocks, or generalized Razak blocks. Let φ: A → B be a ∗ -homomorphism. We say that φ pulls back faithful diffuse traces if there are σ ∈ Tfd (A) and τ ∈ Tfd (B) such that φ: (A, σ) → (B, τ). Notice that ∗ -homomorphisms pulling back faithful diffuse traces are automatically injective. Furthermore, the choice of τ in the definition above is not relevant: in the above setting, if φ pulls back a faithful diffuse trace to a faithful diffuse trace then the pullback of any faithful diffuse trace is such. Before proceeding, we record the following, which shows that all diffuse faithful traces are the same up to isomorphism. Lemma 3.17. Let n ∈ ℕ and σ, τ ∈ Tfd (Cn ). Then there is an isomorphism (Cn , σ) → (Cn , τ). The same applies to dimension drop algebras, and (generalized) Razak blocks.

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Proof. For t ∈ [0, 1], define φ(f )(t) = f (s), where s ∈ [0, 1] is the unique point such that μσ ([0, s]) = μτ ([0, t]). This is the required isomorphism. Definition 3.18. The language of tracial C∗ -algebras ℒC∗ ,τ is the language of C∗ -algebras expanded by a unary predicate τ whose modulus of uniform continuity is the identity. We are ready to define our classes. Definition 3.19. All classes are considered in the language of tracial C∗ -algebras, that is, structures are pairs (A, τ). Let p̄ be a supernatural number. Then – 𝒦[0,1],p̄ is the class whose objects have the form (Cn , τ), where n divides p̄ and τ ∈ Tfd (Cn ). Embeddings are ∗ -homomorphisms which pull back faithful diffuse traces; – 𝒦𝒵 is the class whose objects are pairs (𝒵p,q , τ) where p and q are coprime and τ ∈ Tfd (𝒵p,q ). Embeddings are ∗ -homomorphisms which pull back faithful diffuse traces; – 𝒦𝒲 is the class whose objects are pairs (An , τ) where An is a Razak block and τ ∈ Tfd (An ). Embeddings are ∗ -homomorphisms which pull back faithful diffuse traces; – 𝒦𝒵0 is the class whose objects are pairs (Bn , τ) where Bn is a generalized Razak block and τ ∈ Tfd (Bn ). Embeddings are ∗ -homomorphisms which pull back faithful diffuse traces; – 𝒦r,1 is the subclass of 𝒦𝒵0 where only embeddings whose K-theory belongs to {−1, 1} are considered; – 𝒦r,p̄ is the subclass of 𝒦𝒵0 where only embeddings whose K-theory divides p̄ are considered. Theorem 3.20. In the language of tracial C∗ -algebras, each of the classes in Definition 3.19 is a Fraïssé class. Moreover, – The Fraïssé limit of 𝒦[0,1],p̄ is Mp̄ , the UHF algebra associated to the supernatural number p;̄ – The Fraïssé limit of 𝒦𝒵 is the Jiang–Su algebra 𝒵 ; – The Fraïssé limit of 𝒦𝒲 is the Jacelon algebra 𝒲 , which is also the Fraïssé limit of the class 𝒦r,1 ; – The Fraïssé limit of 𝒦𝒵0 is the algebra 𝒵0 ; – The Fraïssé limit of 𝒦r,p̄ is the algebra 𝒵0 ⊗Mp̄ , where Mp̄ is the UHF algebra associated to the supernatural number p.̄ Theorem 3.20 is a result of the work of many hands. The class 𝒦𝒵 was treated in [18]. The proof in [18] relies on the classification of morphisms from 1-dimensional NCCW complexes into stable rank-one C∗ -algebras (and in particular, into 𝒵 ) of Robert’s [66], which served to proved the NAP. A by hand proof of the NAP for dimension drop algebras, not relying on any classification result, was written in [56] (see also [55]). The classes 𝒦[0,1],p̄ were treated in [56, Section 3]. The other classes, e. g., those giving rise to nonunital C∗ -algebras, were treated in [38].

466 � A. Vignati The proofs (except that from [18]) follow a similar pattern, which we will now sketch in the case of the classes 𝒦[0,1],p̄ . While the proofs for interval algebras, dimension drop algebras, and Razak blocks carry a similar amount of technical difficulties, this is not the case for generalized Razak blocks, where the proof is quite intricate. They main difficulty is always in proving the near amalgamation property. 3.3.2 The proof This section is dedicated to the proof of Theorem 3.20 for the classes 𝒦[0,1],p̄ . We will not prove the individual lemmas leading to the proof of the theorem; full proofs can be found in [56]. For simplicity, we assume p̄ is the largest supernatural number, i. e., p̄ = ∏p prime p∞ . We write 𝒦 for 𝒦[0,1],p̄ . Recall that Cn = C([0, 1], Mn ), so objects in 𝒦 are pairs (Cn , σ), with σ ∈ Tfd (Cn ), and morphisms are ∗ -homomorphisms which pull back faithful diffuse traces. As already mentioned, all such maps are injective. It is an easy exercise to show that this class has the WPP. Another easy exercise, using Lemma 3.17, gives that 𝒦 has the JEP. We are left to show that 𝒦 has the NAP and to analyze its Fraïssé limit. Definition 3.21. Let n and m be natural numbers, where n divides m. Let j = mn . A ∗ -homomorphism φ: Cn → Cm is diagonal if there are continuous maps ξ1 , . . . , ξj : [0, 1] → [0, 1] with ξ1 ≤ ⋅ ⋅ ⋅ ≤ ξj and a unitary u ∈ Cm such that for every f ∈ Cn and t ∈ [0, 1] we have φ(f )(t) = u(t)diag(f (ξ1 (t)), . . . , f (ξj (t)))u∗ (t). The maps ξ1 , . . . , ξj are said to be associated to φ First, we show that we can reduce to the case of diagonal maps. Lemma 3.22. Let n, m ∈ ℕ and suppose that n divides m. Let F ⊆ Cn be finite and ε > 0. Suppose that φ: Cn → Cm is a unital ∗ -homomorphism. Then there is a diagonal ψ: Cn → Cm such that 󵄩󵄩 󵄩 󵄩󵄩φ(a) − ψ(a)󵄩󵄩󵄩 < ε,

for all a ∈ F.

Moreover, if σ ∈ T(Cn ) and τ ∈ T(Cm ) are such that φ pulls τ back to σ, then so does ψ. A useful technical tool is the definition of diameter. Definition 3.23. Let φ: Cn → Cm be a diagonal map with associated maps {ξi }, for i ≤ The diameter of φ is the quantity 󵄨 󵄨 max sup 󵄨󵄨󵄨ξi (s) − ξi (t)󵄨󵄨󵄨. i s,t∈[0,1]

m . n

Fraïssé theory in operator algebras � 467

The proof of amalgamation passes through the following two propositions. The first uses the fact that continuous maps [0, 1] → [0, 1] are uniformly continuous, while the second uses that elements of Cn are uniformly continuous (when viewed as map from [0, 1] to a bounded ball of Mn .) Proposition 3.24. Let n, m ∈ ℕ and suppose that n divides m. Let φ: Cn → Cm be a diagonal map. Fix ε > 0. Then there is δ > 0 such that for all k and all diagonal maps ψ: Cm → Ck , if the diameter of ψ is < δ, then the diameter of ψ ∘ φ is < ε. Proposition 3.25. Let n ∈ ℕ, F ⊆ Cn be finite, and ε > 0. Then there is δ > 0 such that for all m ∈ ℕ with n dividing m, σ ∈ Tfd (Cn ) and τ ∈ Tfd (Cm ), if φ1 , φ2 : (Cn , σ) → (Cm , τ) are diagonal maps of diameter < δ, then there is a unitary in Cm such that 󵄩󵄩 󵄩 ∗ 󵄩󵄩uφ1 (a)u − φ2 (a)󵄩󵄩󵄩 < ε,

for all a ∈ F.

Proposition 3.25 is the key that opens the door to amalgamation. This is also where, in the case of other classes of algebras, technical issues arise. In fact, the main problem there is choosing (for example, in case φ1 and φ2 are maps between dimension drop algebras A and B) the unitary u in such a way that uφ1 [A]u∗ remains inside B. The amount of work required for this step in case of dimension drop algebras and Razak blocks is acceptable, but technicalities become a real issue in case of generalized Razak blocks. Theorem 3.26. The class 𝒦 has the near amalgamation property. Proof. Let λ be the Lebesgue measure on [0, 1]. We denote by τλ its associated trace on C([0, 1]). This gives a trace (again denoted by τλ ) on each Cn . By Lemma 3.17 and the JEP, it is enough to show that when given n, m ∈ ℕ with n dividing m, a finite F ⊆ Cn , ε > 0 and two ∗ -homomorphisms φ1 , φ2 : (Cn , τλ ) → (Cm , τλ ), then one can find a large enough k and two ∗ -homomorphisms ψ1 , ψ2 : (Cm , τλ ) → (Ck , τλ ) such that 󵄩󵄩 󵄩 󵄩󵄩ψ1 ∘ φ1 (a) − ψ2 ∘ φ2 (a)󵄩󵄩󵄩 < ε,

for all a ∈ F.

First, one finds δ so that it satisfies Proposition 3.25 for F and ε. Secondly, one finds δ′ so small that whenever we have a map of diameter < δ′ , then the diameter of both ψ ∘ φ1 and ψ ∘ φ2 is < δ. Finding maps of arbitrarily small diameter which pull back faithful diffuse traces is easy for interval algebras, but requires some work for dimension drop and (generalized) Razak blocks; in this case, the thesis follows by applying again Lemma 3.17. Let M be the Fraïssé limit of 𝒦. To show that M is isomorphic to the universal UHF algebra Mp̄ , first one shows that M is simple, monotracial, and AF (that is, a limit of finitedimensional C∗ -algebras). All such verifications are not difficult, assuming the reader is comfortable with C∗ -algebra theory (one can follow the ideas of [56, Section 4]). Then, one computes K0 (M), the K0 -group of M, and shows that such group must be divisible,

468 � A. Vignati and therefore equal to ℚ. The result then follows from Elliott’s classification theorem [20], asserting that AF algebras are classified by their K0 group. For the other classes (and limits) treated in Theorem 3.20, one can either appeal to classification results for amenable C∗ -algebras, or show that the sequences explicitly defining the objects 𝒵 , 𝒲 , and 𝒵0 (as given in [39, 37, 38]) are generic (i. e., their limit is the Fraïssé limit).

3.3.3 Strong self-absorption of 𝒵 by Fraïssé -theoretic methods One of the most important features of 𝒵 is that 𝒵 is isomorphic to 𝒵 ⊗ 𝒵 in a very strong sense (this is called strong self-absorption, see [39]). Can one prove strong self-absorption of 𝒵 with Fraïssé-theoretic methods? If done in a clumsy way (e. g., by adding to the class 𝒦𝒵 all possible tensor products of dimension drop algebras and considering all maps which pull back faithful diffuse trace), amalgamation fails, as one is capable to build K1 -related obstructions in [0, 1]2 which do not appear in [0, 1]. Ghasemi, in [26], was more careful and, restricting the class of maps considered, showed that one can construct a Fraïssé class which is rich enough to have two generic sequences, one having 𝒵 as its limit and the other having 𝒵 ⊗ 𝒵 . While in general, even without the use of Fraïssé theory, it is not difficult to show that 𝒵 tensorially absorbs itself, the same cannot be said about Jacelon’s 𝒲 . It is known that 𝒲 and 𝒲 ⊗ 𝒲 are isomorphic (by deep, long, and convoluted classification methods, e. g., [29]), but a direct proof has yet to be found. There have been unsuccessful, yet serious, attempts to use Fraïssé theory to show that 𝒲 ≅ 𝒲 ⊗ 𝒲 without the aid of classification tools. This problem seems difficult and deserves to be stated. Similar problems can be stated for the algebra 𝒵0 . Problem 3.27. Use Fraïssé theory to show that 𝒲 ≅ 𝒲 ⊗ 𝒲 , that is, find a Fraïssé class which contains enough Razak blocks and their tensor products, which has two generic sequences naturally giving 𝒲 and 𝒲 ⊗ 𝒲 .

4 Fraïssé limits in functional analysis In this section we record Fraïssé theoretic results in functional analysis. We will show below that the Urysohn space, the Hilbert space ℓ2 (ℕ), the Gurarij space, the Poulsen simplex, and the noncommutative versions of the latter two are Fraïssé limits of suitable classes of objects. The Urysohn space, ℓ2 (ℕ), and the Gurarij space were treated in [8] (see also [49]), while the Poulsen simplex, its noncommutative version, and the noncommutative Gurarij space were treated in [53] and [52]. The Poulsen simplex was also studied in [16].

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4.1 The Urysohn space A remarkable date for topologists is August 3rd, 1924, when, in a letter to Hausdorff, Urysohn announced the construction of a separable complete metric space containing isometrically any other separable metric space and satisfying a strong homogeneity property.3 Definition 4.1. A separable metric space X is ultrahomogeneous if every isometry between finite subsets of X can be extended to a self-isometry of X. Definition 4.2. A Urysohn space is an ultrahomogeneous separable complete metric space in which every separable metric space isometrically embeds. Urysohn shows that a Urysohn space exists, and further, that it is unique up to isometry. It does make sense therefore to talk about the Urysohn space U. The Urysohn space was somewhat rediscovered by Katetov in [41], who constructed a space which is universal for metric spaces of arbitrary density character. With ultrahomogeneity for the class of finite metric spaces in our pocket, it remains to formalize the proper class of objects we are interested in, and their maps, to recognize U as a Fraïssé limit. This was done in [8]. Theorem 4.3. Let 𝒦U be the class of finite metric spaces with maps being isometric embeddings. Then 𝒦U is a Fraïssé class whose Fraïssé limit is U. Similarly to the Urysohn space, the Urysohn sphere, which plays the same role as U does for metric spaces whose diameter is bounded by 1, can be obtained as a Fraïssé limit.

4.2 The Gurarij spaces The language of metric spaces is the simplest possible language of metric structures, as only the metric is in the language. In the discrete setting, this correspond to the fact that a countable infinite set is the Fraïssé limit of finite sets. Further results are obtained by adding a bit more structure to the language. Definition 4.4. A Gurarij space is a separable Banach space G having the property that, for any ε > 0, any finite-dimensional Banach spaces E ⊆ F, any isometric linear embedding E → G can be extended to a linear embedding φ: F → G satisfying ‖φ‖⋅‖φ−1 ‖ ≤ 1+ε. The existence of a Gurarij space was proved in [33], but its uniqueness had to wait for [54], where it was shown by the use of deep techniques of Lazar and Lindenstrauss [50]. Let G be the unique (up to linear isometries) Gurarij space. After an

3 The letter does not contain any detail of the construction, which Hausdorff redid himself.

470 � A. Vignati unpublished model theoretic proof of Henson, Kubis̀, and Solecki [49] proved uniqueness of G by essentially showing that it is a Fraïssé limit. Their argument was formalized in [8] when the general theory was developed. Theorem 4.5. Let 𝒦G be the class of finite-dimensional Banach spaces with maps being isometric linear embeddings. Then 𝒦G is a Fraïssé class whose Fraïssé limit is G. We further add structure, and consider Hilbert spaces. The class of finite-dimensional Hilbert spaces together with Hilbert space isometries has the amalgamation property, hence it is a Fraïssé class. The Fraïssé limit is the unique separable infinitedimensional Hilbert space, ℓ2 (ℕ). We now consider a noncommutative analog of Gurarij’s G. (For more on operator spaces, see Sinclair’s article in this volume.) Definition 4.6. An operator space is a Banach subspace of ℬ(H) for some Hilbert space H. Pertinent for linear maps between operator spaces is their completely bounded norm. If E and F are operator spaces and φ: E → F is a linear map, we denote by φn : Mn (E) → Mn (F) its amplification, where Mn (E) is the operator space of n×n E-valued matrices. The completely bounded norm of φ is given by ‖φ‖cb := sup‖φn ‖. n

Notions of amenability often pass through internal approximations via finite (or finite-dimensional) objects in the category. The following is the appropriate notion for operator spaces; it is strongly linked to the notion of exactness of C∗ -algebras [63]. Definition 4.7. Let c ∈ ℝ. An operator space X is c-exact if for all ε > 0 and any finitedimensional E ⊆ X, there is n and an operator subspace F ⊆ Mn such that there is an isomorphism φ: E → F with ‖φ‖cb ⋅ ‖φ−1 ‖cb < c + ε. An operator space is exact if it is c-exact for some c ∈ ℝ. Definition 4.8. A separable operator space NG is a noncommutative Gurarij space if it satisfies the following: suppose a finite-dimensional operator space F is 1-exact, E is a subspace of F, and E ′ is a subspace of NG which is isomorphic to E via a completely bounded φ whose inverse is completely bounded. Then, for every ε > 0, there is a subspace F ′ ⊆ NG which contains E ′ , and an isomorphism φ:̃ F → F ′ which extends φ and satisfies ‖φ‖̃ cb ⋅ ‖φ̃ −1 ‖cb ≤ (1 + ε)‖φ‖cb ⋅ ‖φ−1 ‖cb . A noncommutative Gurarij space exists by [59], where a weaker version of universality, for those operator spaces which can be locally approximated by finitedimensional C∗ -algebras, was proved. Again in [59], a weak form of uniqueness was established: it was shown that any two noncommutative Gurarij spaces are c-isomorphic, for every c > 1, that is, one can find an isomorphism φ satisfying ‖φ‖cb ⋅‖φ−1 ‖cb < c. Later, uniqueness (up to linear complete isometries) was shown in [52] with Fraïssé-theoretic methods, while also obtaining the appropriate homogeneity in this setting.

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Theorem 4.9. Let 𝒦OS be the class of finite-dimensional 1-exact operator spaces with maps being completely isometric embeddings. Then 𝒦OS is a Fraïssé class whose Fraïssé limit is the noncommutative Gurarij space NG, which is therefore unique (up to linear complete isometries) and universal for all separable 1-exact operator spaces.

4.3 The Poulsen simplex and the Poulsen operator system Before getting into the study of the Poulsen simplex, we review a few definitions from Choquet’s theory. We direct the reader to [62] for an introduction to this topic. Definition 4.10. Let X be a compact subset of a locally convex space E. Let μ be a Borel probability measure on X and let x ∈ X. We say that x is the barycenter of μ if for every continuous linear functional f on E we have that f (x) = ∫X f dμ. The following question was instrumental for the origins and developments of Choquet’s theory: If X is a compact convex subset of a locally convex space E and x ∈ X, does there exist a probability measure μ on X which is supported on the extreme points of X and which has x as its barycenter? If it exists, is it unique? In case X is metrizable, the first question has a positive answer [14]. The following is one of the equivalent definitions of Choquet simplex one finds in literature (see [15]). Definition 4.11. Let X be a metrizable compact convex subset of a locally convex space E. Then X is said to be a Choquet simplex if for every x ∈ X there is a unique probability measure μ on X which is supported on the extreme points of X and has x as its barycenter.4 Compact convex sets are dual to function systems (see Section 2 of Sinclair’s article): Definition 4.12. A function system is a pointed ordered vector space (V , u) where u is a positive Archimedean order unit. If (V , u) is a function system, V can be endowed with a norm defined by ‖x‖ = inf{−ru ≤ x ≤ ru}. r>0

Isomorphisms in this class are surjective linear isometries preserving the order unit. From the definition of the norm, one can define the space of states on a function system (V , u), by considering all of those positive linear functionals on V which have norm 1. This is a compact convex set. Conversely, if K is a compact convex set, let A(K) be the space of real affine continuous functions on K. A(K) is a function system, with the order

4 In case of nonmetrizable compact convex spaces, the extreme boundary does not have to be Borel, so we ask for uniqueness among measures vanishing on any Baire subset of X \ δX.

472 � A. Vignati given by function domination and the order unit being the identity function. Kadison’s representation theorem, stated below, gives the required duality (see [1, Theorem II.1.8]): Theorem 4.13. The assignment which associated a compact convex set K to the function system A(K) is a controvariant equivalence of categories. Under this equivalence, metrizable Choquet simplices correspond to those separable function systems which are in addition Lindenstrauss spaces (preduals of L1 spaces). Definition 4.14. A Poulsen simplex is a metrizable Choquet simplex whose extreme points are dense. A Poulsen simplex P was constructed in [64]. In [51], it was shown that P is the unique Poulsen simplex (up to affine homeomorphism). Methods used in approaching the study of P are closely related to those used for the Gurarij space G (see, e. g., [60]). In [16], and later [53], the Poulsen simplex P was recognized as the space of states of a Fraïssé limit of the class of function systems. In particular, in [16], it was showed that A(P) is the unique separable function system that is approximately homogeneous and universal for separable function systems. In a sense, A(P) has the same properties as Gurarij’s G does, but in the category of function systems. Theorem 4.15 ([16], see also [53, Sections 6 and 7]). The class of finite-dimensional function systems where maps are order unit preserving, linear, isometric embeddings is a Fraïssé class whose Fraïssé limit is A(P). The universality properties (for function spaces) one obtains for A(P) translate in the language of Choquet simplices in that a metrizable compact convex set is a Choquet simplex if and only if it is affinely homeomorphic to a closed face of P. We study now the noncommutative version of Poulsen’s P. Definition 4.16. An operator system is an operator subspace of ℬ(H) which is closed by adjoints and contains the unit of ℬ(H). As operator spaces can be viewed as noncommutative Banach spaces, the work of Arveson ([2] and [3]) shows that operator systems provide the natural noncommutative analogues of function systems. In fact, function systems are those operator systems contained in a unital abelian C∗ -algebra. For this reason, the space introduced below is considered to be the noncommutative version of Paulsen’s P. In the same way to a function system one can associate its state space, operator systems give rise to noncommutative Choquet simplices. Noncommutative Choquet simplices are special cases of noncommutative compact convex spaces. The latter were introduced in [43] and extensively studied in [44]. We will not give the precise definition of a noncommutative Choquet simplex here, as it is fairly technical, and we direct the reader to [44, Section 4] for the details. The important correspondence to retain is the existence of an equivalence of categories between the category of noncommutative com-

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pact convex spaces and that of operator systems, associating to each noncommutative compact convex space X an operator system A(X). Again noncommutative Choquet simplices correspond to those operator systems satisfying Lindestrauss-like conditions. In this setting, Kennedy and Shamovich proved in [44] the existence of a metrizable noncommutative Poulsen simplex NP having dense extreme points (again, we decided not to be precise here to avoid too many technicalities). Even though it is not yet known whether this denseness-like property alone gives uniqueness of NP among metrizable noncommutative Choquet simplices, the corresponding (via the equivalence of categories of [43]) operator system associated to NP has been recognized as a Fraïssé limit. Theorem 4.17 ([53, Section 7]). The class of finite-dimensional exact operator systems where maps are unital completely isometric linear embeddings5 is a Fraïssé class, whose Fraïssé limit is A(NP), the unique operator system having the noncommutative Poulsen simplex NP as its dual. The space A(NP) is the unique separable nuclear operator systems that is universal in the sense of Kirchberg and Wassermann [45], that is, it contains a completely isometric copy of any separable exact operator system. Also, in the same fashion as for the Poulsen simplex, universality is expressed in that compact metrizable noncommutative Choquet simplices are exactly those homeomorphic to faces of NP.

5 Connections with Ramsey theory In this section we review the continuous version of the KPT correspondence [42], and we see how this was applied to study the automorphisms of some of the Fraïssé limits described in Section 4, summarizing some of the results of [6] and [5]. The main idea is the following: if 𝒦 is a Fraïssé class of finitely-generated structures with Fraïssé limit M, then Ramsey-theoretic properties of 𝒦 correspond to dynamical information about the group Aut(M). As any Polish group can be seen as the automorphism group of a Fraïssé limit (in an appropriately defined language, see [24, Theorem 2.4.5] and [57, Theorem 6], but also [40, Chapter 5]), Fraïssé theory has the potential of accessing certain dynamical features of Polish groups. Definition 5.1. A Polish group G is extremely amenable if every continuous action of G on a compact metric space X has a fixed point. As in Section 2, a language for metric structure ℒ and a class of finitely-generated ℒ-structures 𝒦 are fixed. Recall that if A belongs to 𝒦, and ā generates A, we use the notation (A, a)̄ to record the tuple of generators. 5 Notice that these automatically preserve the adjoint.

474 � A. Vignati Fix (A, a)̄ and B in 𝒦. Let A B be the set of morphisms of A into B. If φ, ψ ∈ A B, set ̄ ψ(a)) ̄ dā (φ, ψ) := dB (φ(a), where dB is the metric on B. A coloring of A B is a 1-Lipschitz map A B → [0, 1], where A B is considered with the metric dā . Definition 5.2. A Fraïssé class 𝒦 has the approximate Ramsey property if the following happens: for all A and B in 𝒦, all finite F ⊆ A B, and all ε > 0, there is C ∈ 𝒦 with the property that for every coloring γ of A C there is φ ∈ B C such that 󵄨󵄨 󵄨 󵄨󵄨γ(φ ∘ ψ1 ) − γ(φ ∘ ψ2 )󵄨󵄨󵄨 < ε,

for all ψ1 , ψ2 ∈ F.

(First, notice that the definition above depends on the choice of generators of A. Second, in order the above definition is not vacuous, we assume that the empty function is 1-Lipschitz, so that B C is nonempty.) The following was proved in [58] for hereditary classes. A proof for non hereditary classes can be extracted from the proof contained in [58], even though it has never been written down formally. Theorem 5.3. Let 𝒦 be a Fraïssé class with Fraïssé limit M. The following are equivalent: – 𝒦 has the approximate Ramsey property; – the group Aut(M) is extremely amenable. To a Polish group G, one associates its universal minimal flow. Definition 5.4. Let G be a Polish group. A G-flow is a compact Hausdorff space X equipped with a continuous action of G on X. A G-flow is minimal if it has no proper subflows. The universal minimal flow M(G) of G is a minimal G-flow which maps continuously and equivariantly onto every minimal G-flow. The universal minimal flow is of great interest in topological dynamics; in fact, extreme amenability of G is equivalent to the universal minimal flow M(G) being a singleton. In the case of automorphism groups of Fraïssé limits, another “smallness” property, namely metrizability of M(G), can be associated to Ramsey-like conditions on the class of interest. This was studied extensively by Zucker (see, e. g., [73]). We quickly overview to which of the structures mentioned in Section 3 and Section 4 Theorem 5.3 applies. Let us first focus on the Fraïssé classes mentioned in Section 3. In the abelian setting, while the dual Ramsey theorem of Graham and Rothschild6 can be used to show that the automorphism group of C(2ℕ ) (which is, the homeomorphism group of the Cantor set) 6 The dual Ramsey theorem is a powerful pigeonhole principle which is equivalent to a factorization result for colorings of Boolean matrices, see [32].

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is metrizable (and indeed homeomorphic to the Cantor set itself, see [28]), the same is not known for C(ℙ), ℙ being Bing’s pseudoarc (even though Aut(C(ℙ)) is not extremely amenable, as the action of Aut(C(ℙ)) on ℙ itself has no fixed points, by homogeneity of ℙ.) In the nonabelian setting, not much in known. Using Ramsey-type results for the class of matrix algebras (either with the operator norm or the trace norm), it is shown in [18] that the automorphism groups of UHF algebras and of the hyperfinite II1 factor ℛ are extremely amenable. (Indeed, even more, their automorphism groups are Lévy, see [18, Section 5 and Section 6].) The situation of the automorphism group of 𝒵 , 𝒲 , and 𝒵0 is yet unclear; the reason for this is that rephrasing Ramsey like conditions on diagonal maps between dimension drop algebras or (generalized) Razak blocks turns out to be extremely technical. Problem 5.5. Are the automorphism groups of 𝒵 , 𝒲 , or 𝒵0 extremely amenable? Do they have metrizable universal minimal flow? Much more is known for the structures described in Section 4, mainly thanks to the work of [6] and [5]. In [6], the authors study the Ramsey property for Banach spaces and Choquet simplices. Establishing the approximate Ramsey property for the class of finite-dimensional Banach spaces and function systems give the following: Theorem 5.6. The following hold: – The automorphism group of the Gurarij space G is extremely amenable. – If F is a face of the Poulsen simplex P, the group of automorphisms of P stabilizing F is extremely amenable. The main tool for establishing the above mentioned results is again the dual Ramsey theorem. The same arguments, adapted to the proper setting, were replicated in [5] to show corresponding results for the noncommutative Gurarij space NG and the noncommutative Poulsen simplex NP.

Bibliography [1] [2] [3] [4] [5] [6]

E. M. Alfsen, Compact convex sets and boundary integrals, Ergeb. Math. Grenzgeb., vol. 57, Springer, Berlin, 1971. W. B. Arveson, Subalgebras of C ∗ -algebras, Acta Math. 123 (1969), 141–224. W. B. Arveson, Subalgebras of C ∗ -algebras. II, Acta Math. 128 (1972), no. 3–4, 271–308. D. Bartošová and A. Kwiatkowska, Lelek fan from a projective Fraïssé limit, Fundam. Math. 231 (2015), no. 1, 57–79. D. Bartošová, J. López-Abad, M. Lupini and B. Mbombo, The Ramsey property for operator spaces and noncommutative Choquet simplices, J. Funct. Anal. 281 (2021), no. 9, Paper No. 109192, 39. D. Bartošová, J. Lopez-Abad, M. Lupini and B. Mbombo, The Ramsey property for Banach spaces and Choquet simplices, J. Eur. Math. Soc. 24 (2022), no. 4, 1353–1388.

476 � A. Vignati

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

I. Ben Yaacov, The linear isometry group of the Gurarij space is universal, Proc. Am. Math. Soc. 142 (2014), no. 7, 2459–2467. I. Ben Yaacov, Fraïssé limits of metric structures, J. Symb. Log. 80 (2015), no. 1, 100–115. I. Ben Yaacov, A. Berenstein, C. Ward Henson and A. Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315–427. I. Ben Yaacov and C. Ward Henson, Generic orbits and type isolation in the Gurarij space, Fundam. Math. 237 (2017), no. 1, 47–82. R. H. Bing, Concerning hereditarily indecomposable continua, Pac. J. Math. 1 (1951), 43–51. B. Blackadar, Operator algebras, Theory of C ∗ -algebras and von Neumann algebras, operator algebras and non-commutative geometry, III, Encyclopaedia of Mathematical Sciences, vol. 122, Springer, Berlin, 2006. J. Castillejos, S. Evington, A. Tikuisis, S. White and W. Winter, Nuclear dimension of simple C∗ -algebras, Invent. Math. 224 (2021), no. 1, 245–290. G. Choquet, Existence et unicité des représentations intégrales au moyen des points extremaux dans les cônes convexes, Semin. Bourbaki 9 (1959), no. (1956/57), No. 139, 15 p., 1959. G. Choquet and P. A. Meyer, Existence et unicité des représentations intégrales dans les convexes compacts quelconques, Ann. Inst. Fourier 13 (1963), no. 1, 139–154. C. Conley and A. Törnquist, A Fraïssé approach to the Poulsen simplex, Sets and computations, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 33, World Sci. Publ., Hackensack, NJ, 2018, pp. 11–24. A. Connes, Classification of injective factors. Cases II1 , II∞ , IIIλ , λ ≠ 1, Ann. Math. (2) 104 (1976), no. 1, 73–115. C. J. Eagle, I. Farah, B. Hart, B. Kadets, V. Kalashnyk and M. Lupini, Fraïssé limits of C ∗ -algebras, J. Symb. Log. 81 (2016), no. 2, 755–773. C. J. Eagle, I. Farah, E. Kirchberg and A. Vignati, Quantifier elimination in C∗ -algebras, Int. Math. Res. Not. 2017 (2017), no. 24, 7580–7606. G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati and W. Winter, Model theory of C∗ -algebras, Mem. Am. Math. Soc. 271 (2021), no. 1324, viii+127. I. Farah, B. Hart and D. Sherman, Model theory of operator algebras II: Model theory, Isr. J. Math. 201 (2014), 477–505. R. Fraïssé, Sur l’extension aux relations de quelques propriétés des ordres, Ann. Sci. Éc. Norm. Supér. (3) 71 (1954), 363–388. S. Gao, Invariant descriptive set theory, Pure Appl. Math. (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009. J. Garbulińska-Wȩgrzyn and W. Kubiś, A universal operator on the Gurariĭ space, J. Oper. Theory 73 (2015), no. 1, 143–158. S. Ghasemi, Strongly self-absorbing C∗ -algebras and Fraïssé limits, Bull. Lond. Math. Soc. 53 (2021), no. 3, 937–955. S. Ghasemi and W. Kubiś, Universal AF-algebras, J. Funct. Anal. 279 (2020), no. 5, 108590, 32. E. Glasner and B. Weiss, The universal minimal system for the group of homeomorphisms of the Cantor set, Fundam. Math. 176 (2003), no. 3, 277–289. G. Gong and H. Lin, On classification of non-unital amenable simple C∗ -algebras, II, J. Geom. Phys. 158 (2020), 103865, 102. G. Gong, H. Lin and Z. Niu, A classification of finite simple amenable 𝒵-stable C∗ -algebras, I: C∗ -algebras with generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can. 42 (2020), no. 3, 63–450. G. Gong, H. Lin and Z. Niu, A classification of finite simple amenable 𝒵-stable C∗ -algebras, II: C∗ -algebras with rational generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can. 42 (2020), no. 4, 451–539.

Fraïssé theory in operator algebras

� 477

[32] R. L. Graham and B. L. Rothschild, Ramsey’s theorem for n-parameter sets, Trans. Am. Math. Soc. 159 (1971), 257–292. [33] V. I. Gurariĭ, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sib. Mat. Zh. 7 (1966), 1002–1013. [34] W. Hodges, Model theory, Cambridge University Press, 1993. [35] T. Homma, A theorem on continuous functions, Kōdai Math. Semin. Rep. 1952 (1952), 13–16. [36] T. Irwin and S. Solecki, Projective Fraïssé limits and the pseudo-arc, Trans. Am. Math. Soc. 358 (2006), no. 7, 3077–3096. [37] B. Jacelon, A simple, monotracial, stably projectionless C∗ -algebra, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 365–383. [38] B. Jacelon and A. Vignati, Stably projectionless Fraïssé limits, Stud. Math. 267 (2022), no. 2, 161–199. [39] X. Jiang and H. Su, On a simple unital projectionless C∗ -algebra, Am. J. Math. 121 (1999), no. 2, 359–413. [40] A. Kaïchouh, Metric structures and their automorphism groups: Reconstruction, homogeneity, amenability and automatic continuity, PhD thesis, Université Clude Bernard, Lyon 1, 2015. [41] M. Katětov, On universal metric spaces, General topology and its relations to modern analysis and algebra, VI (Prague, 1986), Res. Exp. Math., vol. 16, Heldermann, Berlin, 1988, pp. 323–330. [42] A. S. Kechris, V. G. Pestov and S. Todorcevic, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005), no. 1, 106–189. [43] M. Kennedy and K. R. Davidson, Noncommutative Choquet theory, 2019. arXiv:1905.08436. [44] M. Kennedy and E. Shamovich, Noncommutative Choquet simplices, Math. Ann. 382 (2022), no. 3–4, 1591–1629. [45] E. Kirchberg and S. Wassermann, C ∗ -algebras generated by operator systems, J. Funct. Anal. 155 (1998), no. 2, 324–351. [46] W. Kubiś, Metric-enriched categories and approximate Fraïssé limits, 2012. arXiv:1210.6506. [47] W. Kubiś, Injective objects and retracts of Fraïssé limits, Forum Math. 27 (2015), no. 2, 807–842. [48] W. Kubiś and A. Kwiatkowska, The Lelek fan and the Poulsen simplex as Fraïssé limits, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 111 (2017), no. 4, 967–981. [49] W. Kubiś and S. Solecki, A proof of uniqueness of the Gurariĭ space, Isr. J. Math. 195 (2013), no. 1, 449–456. [50] A. J. Lazar and J. Lindenstrauss, On Banach spaces whose duals are L1 spaces, Isr. J. Math. 4 (1966), 205–207. [51] J. Lindenstrauss, G. Olsen and Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, vi, 91–114. [52] M. Lupini, Uniqueness, universality, and homogeneity of the noncommutative Gurarij space, Adv. Math. 298 (2016), 286–324. [53] M. Lupini, Fraïssé limits in functional analysis, Adv. Math. 338 (2018), 93–174. [54] W. Lusky, The Gurarij spaces are unique, Arch. Math. (Basel) 27 (1976), no. 6, 627–635. [55] S. Masumoto, A Fraïssé theoretic approach to the Jiang–Su algebra, December 2016. arXiv:1612.00646 [math.OA]. [56] S. Masumoto, The Jiang–Su algebra as a Fraïssé limit, J. Symb. Log. 82 (2017), no. 4, 1541–1559. [57] J. Melleray, A note on Hjorth’s oscillation theorem, J. Symb. Log. 75 (2010), no. 4, 1359–1365. [58] J. Melleray and T. Tsankov, Extremely amenable groups via continuous logic, 2014. arXiv:1404.4590. [59] T. Oikhberg, The non-commutative Gurarii space, Arch. Math. (Basel) 86 (2006), no. 4, 356–364. [60] G. Olsen, On simplices and the Poulsen simplex, Functional analysis: Surveys and recent results, II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979), Notas Mat., vol. 68, North-Holland, Amsterdam–New York, 1980, pp. 31–52. [61] A. Panagiotopoulos and S. Solecki, A combinatorial model for the Menger curve, J. Topol. Anal. 14 (2022), no. 1, 203–229.

478 � A. Vignati

[62] R. R. Phelps, Lectures on Choquet’s theorem, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer, Berlin, 2001. [63] G. Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. [64] E. T. Poulsen, A simplex with dense extreme points, Ann. Inst. Fourier (Grenoble) 11 (1961), 83–87, XIV. [65] S. Razak, On the classification of simple stably projectionless C∗ -algebras, Can. J. Math. 54 (2002), no. 1, 138–224. [66] L. Robert, Classification of inductive limits of 1-dimensional NCCW complexes, Adv. Math. 231 (2012), no. 5, 2802–2836. [67] M. Rørdam, F. Larsen and N. Laustsen, An introduction to K -theory for C ∗ -algebras, Lond. Math. Soc. Stud. Texts, vol. 49, Cambridge University Press, Cambridge, 2000. [68] K. Schoretsanitis, Fraisse theory for metric structures, Thesis (PhD), University of Illinois at Urbana-Champaign, 2007. [69] R. Sikorski and K. Zarankiewicz, On uniformization of functions. I, Fundam. Math. 41 (1955), 339–344. [70] H. Thiel and W. Winter, The generator problem for 𝒵-stable C∗ -algebras, Trans. Am. Math. Soc. 366 (2014), no. 5, 2327–2343. [71] W. Winter, Structure of nuclear C∗ -algebras: from quasidiagonality to classification and back again, Proceedings of the international congress of mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 1801–1823. [72] W. Winter and J. Zacharias, The nuclear dimension of C∗ -algebras, Adv. Math. 224 (2010), no. 2, 461–498. [73] A. Zucker, Big Ramsey degrees and topological dynamics, Groups Geom. Dyn. 13 (2019), no. 1, 235–276.

Index

https://doi.org/10.1515/9783110768282-013

Pr 169 Pr∗ 174 Φℳ 176 φω 305 π(E1 , . . . , E2n+1 ) 181 165 ∏ω M 305, 319 R 303 ω R∞ 322 Rλ 316 ρn (E, F) 179 RV, RVn 179 Sfn (M) 314 Sφ 307 S(M) 314 Sp(σ φ ) 313 σ-finite 303 σ φ 308 ̂ρℳ n (e, f ) 180 T meq , ℳmeq 167 T (M) 314 𝒱Cb , τCb 203 (X, ℬ, μ) 159 𝒵0 465

󳀀󳨐

∗-algebra 1, 45 ∗-closed 45 ∗-homomorphism 45 ∗-isomorphism 45 ∗-representations of C∗ -algebras 32 1-bounded entropy 144 𝒜(D) 308 as , x s 174 ℵ0 -categorical 257 ℬ ↾ b 193 βφ 333 B(M, φ) 331 B(N ⊂ M, φ) 332 ̂ 160 ̂ μ̂, d) (ℬ, ∞ ⨂n=1 (Mn , φn ) 315 C # , ⟨S⟩ 162 dCb 204 dφ (p, q) 203 dP (E, F) 179 p dx φ(y) 203 d(M) 321 d(p, q) 189 d([φ], [ψ]) 321 (Dφ: Dψ)t 311 Δφ 307 ℓ∞ (M) 303, 304 fE 179 ∀ ⋁ ∃-sentence 428 Γ(σ φ ) 314 ℐω (M, τ) 304 ℐω (M, φ) 304 | 199 ⌣ Jφ 307 𝒦ℳ , Ψℳ 194 κ-saturated 113 Lp (M) 305 Lpr 168 Mω 324 ℳω 324 Mφ 310 ℳω (M, φ) 305 M(σ φ , E) 313 (M, L2 (M), J, L2 (M)+ ) 317 (M, φ)ω 305 p ↿ A 206 ℙ(A|𝒞), 𝔼(f |𝒞) 163 ℙ(a|D) 163

additive functional 187 AF algebra 37 algebraic core 397 almost central 79 almost orthogonal 206 amenable group 74 amenable von Neumann algebra 76 APA 184 approximate Ramsey property 474 approximate unit 24 approximately finitely satisfiable 107 Arveson spectrum 313 Arveson’s extension theorem 351 asymptotic algebra 325 asymptotic centralizer 324 atom 161 – relative 196 atomic diagram 100 atomic element 161 atomic formula 95 atomic formula, unbounded 362 atomic model 129

480 � Index

atomless – element 161 – probability space 161 atomless over 196 axiomatizable 97 Banach A-bimodule 403 Banach algebra 1 Banach left A-module 403 basic construction 64 basic formula 94, 390 Bernoulli action 70 Beth definability theorem 366 bicentralizer 331 bicentralizing state 337 C∗ -algebra 9, 45, 47 canonical base topology 203 canonical bases 201 canonical parameters 124, 167 Cantor set 458 Cartan subalgebra 66 categorical 127 centralizer 310 chainable continuum 450 character on a C∗ -algebra 5 Choquet simplex 471 co-existentially closed continuum 449 coarse bimodule 63 cocycle action of a group on a tracial von Neumann algebra 282 cocycle actions of groups 281 cocycle crossed product von Neumann algebra 282 cocycle semidirect product of groups 281 coherent sequence 181, 182 coloring 474 common MASA 334 commutant 45 commutant d-containment 405 commutant positively weakly contained 405 compact quantum group 391 compiled structure 429 complete isometry 347 complete order embedding 346 complete theory 98 complete type 111 completely additive 59 completely bounded map 346 completely positive contractive 402

completely positive map 346 conditional expectation 63, 163 conditional probability 163 connective 361 Connes embedding problem 143, 147, 257, 426, 429 Connes’ embedding problem 327 Connes’ Radon–Nikodym cocycle 311 Connes spectrum 314 Connes–Størmer transitivity 322, 331 consistency 362 continuous functional calculus 14, 23, 48 convolution product 408 covariant *-homomorphism 409 covariant representation 409 cpc order zero maps 402 crossed product 409 Cuntz algebra 30 d-containment 404 d-metric 257 decomposition rank 418 definable function 227 – quantifier-free 227 definable functor 365 definable predicate 225 – quantifier-free 226 definable set 117 deformation of a von Neumann algebra 275 density character 99 diagonal embedding 86 diffuse 52 dimension drop algebras 463 dimension function 417 discrete decomposition 323 dissection 92 dissection of a tracial von Neumann algebra 135 domains of quantification 389 Effros–Maréchal topology 328 Ehrenfeucht–Fraïsse games 145, 154 elementary 97 elementary class 362 elementary diagram 101 elementary embedding 98 elementary equivalence 221, 245 elementary equivalent 98 elementary substructure 98 embedding 89, 390 – automorphically conjugate 250

Index � 481

– into matrix ultraproduct 220, 236 – unitarily conjugate 247 embedding universality 280 enforceable model 435 enforceable property 429 enforceable structure 429 entangled strategy 430 entangled value of a nonlocal game 431 entropy 210 equivalent 54 ergodic 58 essential ideal 394 (essentially) free 58 event 161 exact operator space 470 exactness 373 existentially closed 233 existentially closed model 147, 425 existentially closed von Neumann algebra 291 expansion 89 extremely amenable 473 factorial commutant embedding condition 149, 153 factorizable map 258 faithful 59 faithful diffuse trace 464 filter 85 finite 56 finitely generic model 148 finitely satisfiable 107 fn (faithful normal) state 303 formula 361 formula, atomic 361 formula, positive 368 formula, quantifier-free 361 Fraïssé class 457 Fräissé limit 457 Fréchet filter 85 free entropy 218 – for existential types 239 – for full types 233 – in the presence 219, 241 – variational principle 235, 239, 245 free independence 217, 245 – independent join 250 – with amalgamation 255 free product – of tracial von Neumann algebras 245 free product von Neumann algebra 72, 73

full crossed product 30 full factor 327 full group C∗ -algebra 29 function system 471 fundamental group 78, 143 – first order 143 fundamental group of a II1 factor 283 G-equivariant order zero dimension 405 Gelfand spectrum 5, 22 Gelfand transform 9 Gelfand–Naimark 48 Gelfand–Naimark theorem 13, 23 Gelfand–Naimark–Segal 48 generalized Razak blocks 463 generic 457 GNS construction 33 GNS theorem 33 Golodets state 326 good nuclear witnesses 437, 442 Groh–Raynaud ultraproduct 305, 319 group C*-algebra 410 group measure space von Neumann algebra 58 group von Neumann algebra 67 Gurarij space 469 heir 154 hereditarily indecomposable continuum 449 Hilbert M-bimodule 63 homogeneous – algebra 194 – element 194 homomorphism 89 hyperfinite 67, 330 hyperfinite II1 factor 67, 461 independence 254 – conditional 165 – extension 254 – full existence 254 – model theoretic 199 – probabilistic 165 induced metric (on spaces of types) 189 inductive limit of C∗ -algebras 31 infinite conjugacy class 68 infinite tensor product von Neumann algebra 315 infinitely generic model 151 inner amenable 80 interpretation 125, 167

482 � Index

intertwining-by-bimodules 65 interval algebras 462 involution 1 irreducible (subfactor) 334 isometry 44 isomorphism 90 ITPFI factor 316 Jiang–Su algebra 444 joint embedding property 425, 456 Kazhdan’s property (T) 77 Keisler–Shelah theorem 145 Kirchberg embedding problem 426, 436 Kirchberg’s QWEP problem 426, 432 KMS condition 308 ℒ-formula 98 L1 -distance 179 λ-McDuff 325 left Hilbert M-module 61 left-regular representation 36 lifting 257 lifting property, operator system 375 linear matrix ∗-polynomial 355 linear matrix ∗-polynomial, hermitian 355 linear matrix ∗-polynomial, homogeneous 355 linear matrix inequality 355 locally universal model 150, 425 logic topology 111, 225 Łoś’ theorem 106 M∗ predual of M 303 Maharam invariants 194 Maharam’s lemma 197 Maharam’s theorem 198 malleable deformation 276 matricially finite C∗ -algebra 438 matrix completion problem 356 maximal tensor product 40 McDuff 80 McDuff factor 325 meq expansion 167 metric imaginary 167 metric structure 87 metric topology 111 MF problem 426, 438 microstate space 218, 233, 250 minimal 52

minimal tensor product 40 model 362 model companion 147 model completion 147 modular automorphism group (modular flow) 308 modular conjugation operator 307 modular operator 307 monotracial C∗ -algebra 443 morphism 390 near amalgamation property 456 Neumann series 3 noncommutative Gurarij space 470 noncommutative Poulsen simplex 473 nonlocal game 430 nonmultidimensional 207 nonsingular 57 norm topology 44 normal 44, 59 normal elements 10 nuclear C∗ -algebra 40 nuclear dimension 417 nuclearity 372 Ocneanu ultraproduct 305 omitting type 125 operator norm 44 operator space 470 operator system 345, 472 – quotient 353 operator system, abstract 348 operator system, maximal 354 operator system, minimal 355 operator system, nuclear 372 operator system, ultraproduct 363 operator system language 402 optimal coupling 257 orbit equivalence 71 order zero language 402 ordered self-adjoint operator space language 402 orthogonal 206 outer automorphism group 78 parallelism of types 202 partial isometry 53 partial type 111, 113 partition 161 – associated to a tuple 174

Index

Popa’s factorial commutant embedding problem 150 – property (T) version 150 Popa’s intertwining-by-bimodule technique 273 positive 59 positive elements 16 positive existential closure 379 positive functional 18 positive quantifier-free ℒ-formula 404 positively weakly ℒ-contained 404 Poulsen simplex 472 POVM 430 Powers factor 316 prenex formula 95 prime model 129 principal type 118 principal ultrafilter 85 probabilistic index 272 probability algebra 161 – abstract 169 probability measure preserving 69 probability space 161 projection 44 projectionless 463 property Gamma 79, 221 property (T) II1 factor 150 pseudoarc 450, 459 quantifier, existential 361 quantifier, universal 361 quantifier elimination 120, 146, 223, 228, 257 quantifier-free formula 95 quasidiagonality problem 427, 440 QWEP 327 Radon–Nikodym theorem 162 random matrix 246 Razak blocks 463 real algebra system 99 real rank zero C∗ -algebra 445 reduced group C∗ -algebra 36 reduct 89 regular wreath-like product of groups 281 relative amenability 274 relative bicentralizer 332 relative bicentralizer flow 333 relative property (T) for inclusions of von Neumann algebras 277 Riesz interpolation property 383

� 483

Rodon–Nikodym 62 Rokhlin dimension 407 Rokhlin property 407 Ryll–Nardzewski theorem 257 S-invariant 314 s-prime von Neumann algebra 287 S∗ OT 303 satisfaction 97 saturated 113 self-adjoint 44 self-adjoint elements 10 semidefinite program 356 sentence 361 separably categorical 127 small set 188 small theory 130 solid von Neumann algebra 278 sorts 389 SOT (Strong Operator Topology) 303 spectrahedron 376 spectral gap subalgebra 151 spectral measure 48 spectral radius 8 spectral radius formula 8 spectral subspace 397 spectral subspace (Arveson) 313 spectrum of an element 3 stable structure 138, 140 stably finite C∗ -algebra 35 stably projectionless C∗ -algebra 444 standard form 317 standard representation 60 state 47 state space 20 state space diameter 321 strict positivity 380 strict topology 394 strong operator topology 44 strongly finitely based (SFB) 204 strongly homogeneous 188 strongly self-absorbing 468 strongly self-absorbing C∗ -algebra 437 substructure 89 synchronous strategy 432 synchronous value of a nonlocal game 433 T -definable 117 T -equivalent 98

484 � Index

T -formula 98 T -functor 115 T-invariant 314 tensor product von Neumann algebra 56 the spectral theorem 46, 50 the strong Kadison property 334 theory 97, 362 – stable 254 Toeplitz algebra 30 trace 464 tracial 59 tracial state 35 tracial wreath-like product von Neumann algebra 282 trivial bimodule 63 Tsirelson’s problem 426, 432 type 223, 224, 390 – existential 231 – principal 259 – quantifier-free 223, 226 – space of types 224 type I 56 type Ifin 56 type I∞ 56 type II 56 type II1 56 type II∞ 56 type III 56 type III0 314 type III1 314 type IIIλ 314 type III factor 306 type space 110 u-topology 308 UHF algebra 37, 460 ultracopower 446 ultracoproduct 446 ultrafilter 85

ultralimit 86 ultrapower 86, 106 ultrapower of operator systems 364 ultrapower von Neumann algebra 74 ultraproduct 86, 106 – of matrix algebras 220, 247 ultraproduct of operator systems 363 ultraproduct of von Neumann algebras 74 ultraroot 109, 364 unidimensional 206 uniform continuity moduli 389 uniform continuity modulus 88 uniform Dixmier property 444 uniform family of formulas 373 uniformly bounded 86 unitary 44 unitary elements 10 universal C∗ -algebra 28 universal coefficient theorem 440 universal crossed product 30 universal domain 188 universal group C∗ -algebra 29 universal minimal flow 474 universal norm 409 Urysohn space 469 von Neumann algebra 45 von Neumann’s double commutant theorem 45 𝒲, the Jacelon algebra 465 w-spectral gap subalgebra 150 Wasserstein distance 256 weak elimination of metric imaginaries 202 weak expectation property (WEP) 328, 378 weak operator topology 44 wreath-like product notation 281 𝒵, the Jiang–Su algebra 465 zero-set 115

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