Singular Traces Volume 1: Theory [2 ed.] 3110377799, 9783110377798, 9783110378054

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Steven Lord, Fedor Sukochev, Dmitriy Zanin Singular Traces

De Gruyter Studies in Mathematics

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Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Karl-Hermann Neeb, Erlangen, Germany

Volume 46/1

Steven Lord, Fedor Sukochev, Dmitriy Zanin

Singular Traces |

Volume 1: Theory 2nd edition

Mathematics Subject Classification 2020 46L51, 47L20, 58B34, 47B06, 47B10, 46B20, 46E30, 46B45, 47G10, 58J42 Authors Dr. Steven Lord School of Mathematics & Statistics University of New South Wales The Red Centre Sydney, NSW 2052 Australia [email protected]

Dr. Dmitriy Zanin School of Mathematics & Statistics University of New South Wales The Red Centre Sydney, NSW 2052 Australia [email protected]

Prof. Fedor Sukochev School of Mathematics & Statistics University of New South Wales The Red Centre Sydney, NSW 2052 Australia [email protected]

ISBN 978-3-11-037779-8 e-ISBN (PDF) 978-3-11-037805-4 e-ISBN (EPUB) 978-3-11-039231-9 ISSN 0179-0986 Library of Congress Control Number: 2021936627 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. © 2021 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface This new work expands on the survey [172], which in turn built upon the original book about singular traces [170]. Writing [170] instigated the study of logarithmic submajorization closed ideals, and exposed us to the work of Albrecht Pietsch. Pietsch’s technique, described here in Volume I in parallel with the approach inspired by the Figiel– Kalton theorem, provides a complete characterization of traces on a separable Hilbert space. Volume I also provides a complete formulation for positive traces and Dixmier traces on the ideal of weak trace class operators. The spectral description of traces is still largely due to Nigel Kalton’s estimates on quasinilpotent compact operators. Writing an accessible but largely self-contained text on Kalton’s and Pietsch’s approaches, and formulas for traces on weak trace class operators, has split our exposition about singular traces into three volumes. Volume II concentrates on applications of singular traces on a separable Hilbert space. The applications feature the work done with Alain Connes, or originate from his inspired use of singular traces in geometry and physics. Volume III describes the semifinite theory of singular traces and some applications. Our intention is that much of the material from [170] will be absorbed into Volume III. Comments from the preface of the original book [170] carry over to the three volumes. We are still motivated by the fact that the results discovered by the many contributors to the field deserve a wider audience amongst operator algebraists, functional analysts, noncommutative geometers, and mathematical physicists. We hope that we have made the topic accessible, as well as displaying what we think to be the vital and interesting features of singular traces of compact operators, particularly on the ideal of weak trace class operators. The authors thank Albrecht Pietsch and Alain Connes for direction and historical comments. We thank Jacques Dixmier for his permission to quote from the letter he wrote to the conference “Singular Traces and Their Applications” (Luminy, January 2012). We thank Ed McDonald for assistance with the text. End notes to each chapter give historical background and credit of results. We apologize in advance for possible omissions. Steven Lord Fedor Sukochev Dmitriy Zanin 31 December 2020, Sydney, Australia

https://doi.org/10.1515/9783110378054-201

Notations ℕ ℤ ℤ+ ℝ ℝ+ ℝp |⋅| ℂ MN (ℂ) Tr(A) det(A)

set of natural numbers set of integers set of nonnegative integers field of real numbers set of positive real numbers Euclidean space of dimension p Euclidean norm on ℝp field of complex numbers algebra of square N × N complex matrices trace of a matrix A determinant of a matrix A

⨁ ⨂ ⋆ ⊖ ≺≺ ≺≺log s → a+ s → a− aα ↑ a (aα ↓ a)

direct sum of Banach spaces, traces, or operators tensor product of von Neumann algebras, traces, or operators convolution orthocomplement submajorization logarithmic submajorization limit approaching from s > a limit approaching from s < a increasing (decreasing) net with respect to a partial order

Γ log+ (x)

Gamma function max{log |x|, 0}

H ξ, η ⟨⋅, ⋅⟩ ‖⋅‖ ‖⋅‖∞ ℒ(H) {en }∞ n=0 ξ ⊗η 1 diag(a) [A, B] A∗ |A| A≥0 A+ (A− )

separable complex Hilbert space elements of an abstract separable complex Hilbert space inner product (complex linear in the first variable) norm (usually the vector norm on a Hilbert space) operator (uniform) norm algebra of bounded operators on H orthonormal basis of H one-dimensional operator on H defined by (ξ ⊗ η)x := ⟨x, ξ ⟩η identity map on H diagonal operator on H associated with a ∈ l∞ commutator of operators A and B adjoint of the operator A absolute value of the operator A operator A is positive positive (negative) part of a self-adjoint operator A

https://doi.org/10.1515/9783110378054-202

VIII | Notations

ℜA (ℑA) A⊕n dom(A) ker(A) σ(A) λ(A) λ(n, A) μ(A) μ(n, A) μ(t, A) J J+ Z(J) 𝒥 𝒥+ [𝒥 , ℒ(H)]

c00 c0 c l∞ l1 l2 lp lp,∞ lΦ m1,∞ 𝒞00 (H) 𝒞0 (H)

real (imaginary) part of the operator A direct sum of n copies of the operator A domain of the operator A kernel of the operator A spectrum of the operator A an eigenvalue sequence of the operator A nth element of an eigenvalue sequence of the operator A singular value sequence, or function, of the operator A nth element of the singular value sequence of the operator A value at t ∈ [0, ∞] of the singular value function of the operator A symmetric sequence space positive part of the space J center of J two-sided ideal of compact operators positive part of the ideal 𝒥 commutator subspace of 𝒥

ℒ1 ℒ2 ℒp ℒp,∞ ℒΦ ℒ1,∞ (ℒ1,∞ )0 ℳ1,∞ (ℳ1,∞ )0

space of eventually vanishing sequences space of converging to zero sequences space of convergent sequences space of bounded sequences space of summable sequences Hilbert space of square summable sequences space of p-summable sequences weak-lp sequence space Orlicz sequence space associated with an Orlicz function Φ Sargent sequence space ideal of finite rank operators on H ideal of compact operators on H ideal of trace class operators ideal of Hilbert–Schmidt operators Schatten–von Neumann ideal of compact operators weak-lp ideal of compact operators ideal of compact operators associated with lΦ ideal of weak trace class operators separable part of ℒ1,∞ Dixmier–Macaev ideal of compact operators separable part of ℳ1,∞

‖⋅‖p ‖⋅‖1,∞

(standard) norm on lp , Lp or ℒp quasinorm on l1,∞ or ℒ1,∞

Notations | IX

Tr(A) Trω (A)

(standard) trace of the trace class operator A Dixmier trace of the compact operator A

C(X)

space of continuous functions on a compact Hausdorff topological space X space of continuous functions vanishing at infinity on a locally compact Hausdorff topological space X σ-finite measure space algebra of (equivalence classes of) bounded functions on X space of (equivalence classes of) integrable functions on X Hilbert space of (equivalence classes of) square integrable functions on X space of (equivalence classes of) p-integrable functions on X characteristic function of a measurable set A ⊆ X Lebesgue Lp -space on (0, ∞) multiplication operator on L2 (X) for f ∈ L∞ (X) support of a sequence x, a complex valued function x, or a distribution x

C0 (X) (X, Σ, ρ) L∞ (X) L1 (X) L2 (X) Lp (X) χA Lp Mf supp(x)

ℒ(H)+ Proj(ℒ(H)) ∨ (∧) EA nA

set of positive elements of ℒ(H) lattice of projections in ℳ supremum (infimum) operation (in a lattice) spectral measure of the self adjoint operator A ∈ ℒ(H) spectral distribution function of the operator A ∈ ℒ(H)

K E ‖⋅‖E E0 E∗

O ‖⋅‖O

concavity modulus of a quasi-norm quasi-Banach symmetric sequence or function space symmetric quasinorm on E separable part of E dual space of a quasi-Banach sequence space E quasi-Banach ideal of compact operators symmetric norm on ℰ separable part of ℰ dual space of a quasi-Banach ideal ℰ shift-invariant monotone ideal of l∞ shift-invariant monotone quasinorm on O

φ φ̂ φ+ ℓ ℓ+

trace, positive trace, or fully symmetric trace symmetric functional associated to a trace φ positive part of a hermitian trace or hermitian symmetric functional φ shift-invariant functional on O positive part of a shift-invariant functional on O

ℰ

‖⋅‖ℰ ℰ0 ℰ∗

X | Notations

ℓφ ψ ψ󸀠 mψ

shift-invariant functional associated to a trace φ

ℳψ

positive concave increasing function on (0, ∞) derivative of ψ Lorentz sequence space associated with ψ Lorentz ideal of compact operators associated with mψ

C M T L󸀠 L R σn σs σ̂ 1

Cesàro operator on l∞ logarithmic mean operator on l∞ right-shift operator on l∞ dyadic averaging operator on l∞ dyadic dilation operator on l∞ bijection from L󸀠 l1,∞ to l∞ dilation operator on l∞ dilation operator on L∞ (0, ∞) associated with s > 0 left inverse of the dilation operator σ2 on l∞ lifted to L∞ (0, ∞)

lim ω γ θ

ordinary limit on c extended limit or dilation invariant extended limit on l∞ extended limit on L∞ (0, 1) supported at 0 Banach limit on l∞

ζA,V ζγ ζγ,A ξω ξω,A

ζ -function associated to the operators A and V ζ -function residue associated to γ ζ -function residue associated to γ and the operator A heat trace functional associated to ω heat trace functional associated to ω and the operator A

𝕊p−1 𝕋p Δ Ω S∗ Ω Δ C ∞ (Ω) ‖⋅‖mod

unit sphere in ℝp flat torus of dimension p Laplacian on 𝕋d or ℝd closed Riemannian manifold of dimension p cosphere bundle on Ω Laplace–Beltrami operator on Ω set of smooth functions on Ω norm on the space of modulated operators

2

Contents Preface | V Notations | VII Introduction | XV

Part I: Preliminary material 1 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.4

What is a singular trace? | 3 Compact operators | 3 Calkin correspondence | 11 Examples of traces | 17 The trace | 17 The Dixmier trace | 18 Spectral formulation of traces | 22 Notes | 23

2 2.1 2.2 2.3 2.4 2.5 2.6

Singular values and submajorization | 29 Projections in ℒ(H) | 29 Singular values | 34 Submajorization | 43 Ideals and traces | 55 Examples of ideals | 65 Notes | 72

Part II: Theory of traces on ideals of ℒ(H) 3 3.1 3.2 3.3 3.4 3.5 3.6

Calkin correspondence for norms and traces | 83 Introduction | 83 Calkin correspondence for quasinorms | 87 Discrete Figiel–Kalton theorem | 92 Calkin correspondence for traces | 99 Existence and structure of traces | 101 Notes | 107

4 4.1 4.2

Pietsch correspondence | 113 Introduction | 113 Pietsch correspondence for ideals | 119

XII | Contents 4.3 4.4 4.5 4.6 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Pietsch correspondence for traces | 123 Pietsch correspondence for quasinorms | 126 Lattice of regular traces | 131 Notes | 138 Spectrality of traces | 141 Introduction | 141 Normal operators in the commutator subspace | 145 Subharmonic functions on matrix algebras | 151 Kalton’s estimate for quasinilpotent operators | 159 Logarithmic submajorization estimates for quasinilpotent operators | 165 Quasi-nilpotent operators are commutators | 170 Notes | 175

Part III: Formulas for traces on ℒ1,∞ 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Dixmier traces and positive traces | 185 Introduction | 185 Extended limits | 190 Positive traces on ℒ1,∞ | 196 Dixmier traces on ℒ1,∞ | 205 Extension of traces to ℳ1,∞ | 208 Dixmier traces on ℳ1,∞ | 213 Notes | 219

7 7.1 7.2 7.3 7.4 7.5

Diagonal formulas for traces | 225 Introduction | 225 Diagonal formulation of a singular trace fails in general | 232 Modulated operators | 234 Weyl asymptotics and expectations | 241 Notes | 245

8 8.1 8.2 8.3 8.4 8.5 8.6

Heat trace and ζ-function formulas | 249 Introduction | 249 Heat trace functionals and Dixmier traces | 256 General heat trace functionals | 266 ζ -function residues and Dixmier traces | 276 Not every Dixmier trace is a ζ -function residue | 283 Notes | 288

Contents | XIII

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Criteria for measurability | 295 Introduction | 295 Spectral description of measurability | 310 Examples of measurability | 315 Measurability criteria and heat traces | 322 Measurability criteria and ζ -functions | 327 Measurability criteria in ℳ1,∞ | 332 A Fubini Theorem | 338 Notes | 349

A A.1 A.2 A.3

Miscellaneous results | 355 Subhankulov Tauberian theorem | 355 Operator inequalities | 361 Matrix inequalities | 364

Bibliography | 367 Index | 381

Introduction The work of famous mathematicians on the foundation of functional analysis, Erik Ivar Fredholm (1886–1927), David Hilbert (1862–1943), Hermann Weyl (1885–1955), and John von Neumann (1903–1957), led to the study of the trace on a separable Hilbert space as an extension of the matrix trace, and the study of the ideal of trace class operators. In the 1950s Paul Halmos showed that the algebra of bounded operators of the separable Hilbert space to itself and the two-sided ideal of compact operators do not possess a nonzero trace. A fundamental question became whether the trace on the set of trace class operators was the only nonzero trace on a two-sided ideal of bounded operators. Jacques Dixmier constructed a trace on an ideal larger than the set of trace class operators in the late 1960s. The Dixmier trace, as it was called by Alain Connes, founder of noncommutative geometry and Dixmier’s doctoral student, had the property that it vanished on trace class operators. The Dixmier trace was not an extension of the matrix trace. Dixmier’s construction began the study of singular traces. From 1988, Alain Connes developed Dixmier’s trace and used it centrally in his noncommutative geometry, demonstrating its role in a unique and remarkable theory of noncommutative integration based on differential geometry. Connes applied the Dixmier trace to extensions of classical Yang–Mills and Polyakov actions, extended the noncommutative residue of Adler, Manin, Wodzicki, and Guillemin, showed an equivalent expression for the Hochschild class of the Chern character in terms of the Dixmier trace, recovered geometric measures on quasi-Fuschian groups using his quantum calculus, and introduced the fundamental relation between the noncommutative integral defined by the Dixmier trace and Voiculescu’s obstruction to the Berg–Weyl–von Neumann discretization theorem for tuples. Many aspects of spectral geometry, which is the analysis and use of the eigenvalues and eigenfunctions of the Laplacian to classify or prove geometric properties of compact manifolds, translate to Connes’ operator-based noncommutative differential geometry. The Minakshisundaram and Pleijel ζ -function for the Laplacian, and Minakshisundaram’s heat kernel and heat trace, are central features of spectral geometry. Connes pioneered the association between Dixmier traces, residues of the ζ -function, and the leading term in the asymptotic expansion of the heat trace. Connes’ work created new interest in traces on the ideal of weak trace class operators, and interest in formulations of those traces. Connes’ work also created renewed interest for the general theory of traces on two-sided ideals of compact operators on a separable Hilbert space, an area which had developed separately from the 1970s. Independent of Dixmier’s construction and Connes development, Albrecht Pietsch identified a bijective correspondence between traces on two-sided ideals and shift-invariant functionals in the 1980s. At the same time Nigel Kalton and Tadeusz Figiel identified the commutator subspace of trace class operators, showing that there exists traces different from “the trace” on the trace class ideal. The commutator https://doi.org/10.1515/9783110378054-203

XVI | Introduction approach was subsequently developed in the 1990s for arbitrary ideals by Kenneth Dykema, Tadeusz Figiel, Gary Weiss, and Marius Wodzicki. Most users in noncommutative geometry, which is now a substantial field, still refer to the original presentation of Dixmier’s trace by Connes from 1989, or the monograph “Noncommutative Geometry” from 1994. However, from Pietsch’s work on dyadic decomposition and traces and Kalton’s work on symmetric functionals and spectral characterization of commutators, with the contributions of others, there is now a complete description of traces on two-sided ideals of compact operators. This work provides, in turn, a complete characterization of Dixmier traces on the ideal of weak trace class operators, and the ability to complete the partial formulations of Dixmier traces used in the literature. The purpose of this book is to describe the general theory of traces on ideals of bounded operators, which is done in Part II, and derive the implications for formulas for Dixmier traces as a reference for users, which is done in Part III, including formulas involving eigenvalues, formulas involving the diagonal matrix elements of an operator with respect to an orthonormal basis, and ζ -function residue and heat trace formulas. Each chapter of the book has its own introduction, and its own notes section with historical comments and detailed references. Instead of repeating all the results here, we give an overview of some the main results of interest for Dixmier traces and the underlying theory of traces on separable Hilbert spaces. Dixmier’s trace on the ideal of weak trace class operators Denote by ℒ(H) the set of continuous linear operators of a separable complex infinitedimensional Hilbert space H to itself. A two-sided ideal of the algebra ℒ(H) is a subspace 𝒥 such that AB, BA ∈ 𝒥 ,

A ∈ 𝒥 , B ∈ ℒ(H).

The completely continuous operators, or compact operators, of H to itself form an ideal inside ℒ(H) denoted by 𝒞0 (H). Unitary operators U ∈ ℒ(H), U ∗ U = UU ∗ = 1, where U ∗ is the adjoint of U, describe every isometry of the Hilbert space H to itself. An operator A ∈ ℒ(H) is positive, A ≥ 0, if A is equal to its own adjoint A∗ and ⟨Aη, η⟩ ≥ 0 for all η ∈ H. Here ⟨⋅, ⋅⟩ denotes the inner product on H and ‖ ⋅ ‖ will denote the associated norm. Definition. Let 𝒥 be a two-sided ideal of ℒ(H). A trace φ : 𝒥 → ℂ is a linear functional φ on 𝒥 that is unitarily invariant, φ(A) = φ(U ∗ AU),

A ∈ 𝒥 , U ∗ = U −1 ∈ ℒ(H).

A trace φ : 𝒥 → ℂ is positive if φ(A) ≥ 0 when A ≥ 0.

Introduction

| XVII

A compact operator A ∈ 𝒞0 (H) has a sequence of singular values defined in the same way as the singular values of a matrix of complex numbers in linear algebra. Recall that the spectrum of a compact operator is a discrete set composed of eigenvalues, each associated with a finite-dimensional eigenspace of eigenvectors, and with 0 as the only limit point. The dimension of an eigenspace is called the multiplicity of the associated eigenvalue. Definition. An eigenvalue sequence of A ∈ 𝒞0 (H) is a sequence λ(A) = {λ(n, A)}n=0 ∈ c0 ∞

of the eigenvalues of A listed with multiplicity, with zeros appended if A has only a finite number of eigenvalues, and such that the sequence |λ(n, A)|, n ∈ ℤ+ , is nonincreasing. The absolute value |A| of a compact operator A ∈ 𝒞0 (H) can be formed, like the absolute value of a matrix, from the product A∗ A of A and its adjoint A∗ , and using the spectral theorem to take the square root of the positive operator A∗ A. The singular value sequence μ(A) of the compact operator A ∈ 𝒞0 (H) is defined by the eigenvalue sequence of |A|, μ(n, A) = λ(n, |A|),

n ∈ ℤ+ .

If a = [a(n, m)] ∈ Ml (ℂ) is a square l × l complex-valued matrix, then it may be associated to the finite-rank operator Fa ∈ ℒ(H) by setting a(n, m) = 0, n, m > l, and ∞

Fa η = ∑ a(n, m)⟨en , η⟩em , n,m=0

η ∈ H,

where {en }∞ n=0 is an orthonormal basis of H. The first l terms of the sequence μ(Fa ) are the singular values of the matrix a with 0’s appended afterwards. If x ∈ l∞ is a complex-valued bounded sequence, then it may be associated to the diagonal operator diag(x) ∈ ℒ(H) given by ∞

diag(x)η = ∑ x(n)⟨en , η⟩en , n=0

η ∈ H,

where {en }∞ n=0 is an orthonormal basis of H. The sequence μ(diag(x)) is the classical decreasing rearrangement of the sequence x. The set of trace class operators ℒ1 , ℒ1 := {A ∈ 𝒞0 (H) : μ(A) ∈ l1 },

is a two-sided ideal of the algebra ℒ(H). Here l1 is the space of absolutely summable sequences. The set of weak trace class operators ℒ1,∞ , ℒ1,∞ := {A ∈ 𝒞0 (H) : μ(A) ∈ l1,∞ },

XVIII | Introduction is also a two-sided ideal of the algebra ℒ(H). Here l1,∞ is the classical space of weak summable sequences 󵄨 󵄨 l1,∞ := {x ∈ l∞ : sup (n + 1)󵄨󵄨󵄨μ(n, diag(x))󵄨󵄨󵄨 < ∞}. n∈ℤ+

Definition. The trace on ℒ1 can be defined by the formula ∞

Tr(A) = ∑ λ(k, A), k=0

A ∈ ℒ1 ,

(1)

where λ(k, A), k ∈ ℤ+ , is any eigenvalue sequence of the trace class operator A. A Dixmier trace on ℒ1,∞ can be defined by the formula ∞

n 1 Trω (A) = ω({ ∑ λ(k, A)} ), log(n + 2) k=0 n=0

A ∈ ℒ1,∞ ,

(2)

where λ(k, A), k ∈ ℤ+ , is any eigenvalue sequence of the weak trace class operator A. Here the linear functional ω : l∞ → ℂ is a positive extension to l∞ of the limit on convergent sequences, ω(x) = lim x(n), n→∞

x ∈ c,

and is called an extended limit on l∞ . To define a Dixmier trace, the extended limit ω on l∞ is required since the bounded sequence ∞

{

n 1 ∑ λ(k, A)} log(n + 2) k=0

n=0

∈ l∞

need not converge. Eigenvalues and their multiplicity are unitarily invariant, since the operator A−λ1 is invertible in ℒ(H) if and only if U ∗ AU −λ1 is invertible for any unitary U ∈ ℒ(H). So it is easy to see that Tr and Trω are unitarily-invariant functionals. Eigenvalue sequences are not linear, an eigenvalue sequence of the sum of two operators is not given by the sum of any two eigenvalue sequences of those operators, so it is not easy to see that Tr and Trω are linear functionals. That the formula for Tr in (1) is a linear functional on ℒ1 and defines the extension of the matrix trace introduced by von Neumann on ℒ1 is due to a celebrated result of Victor Lidskii in 1959. Section 1.3.2 gives a standard proof, using singular values, that the Dixmier trace Trω in (2) is additive on the positive cone of ℒ1,∞ . Section 6.4, relying on the extension of Victor Lidskii’s result to arbitrary two-sided ideals of ℒ(H) in Chapter 5, shows linearity of Trω on ℒ1,∞ when defined by eigenvalues.

Introduction

| XIX

There are two aspects of the formulation of the Dixmier trace Trω in (2) that were not known to Dixmier in 1966, and are still not commonly known amongst users of noncommutative geometry. The first is that the Dixmier trace has a Lidksii-type formula. The second is that the extended limit ω on l∞ does not require any additional assumptions, such as the condition that ω be invariant under the shift or dilation operators on l∞ . The mapping ω 󳨃→ Trω is not injective. Distinct extended limits on l∞ can result in the same trace on ℒ1,∞ . This problem is solved by Albrecht Pietsch’s formulation of the Dixmier trace in terms of dyadic averages of the eigenvalues sequence. The next theorem collates the results about Dixmier traces on ℒ1,∞ from Theorems 6.1.1 and 6.1.2, as well as Corollary 7.1.4. A trace φ on the two-sided ideal ℒ1,∞ is normalized if φ(diag{

1 } ) = 1. n + 1 n=0 ∞

If T : l∞ → l∞ denotes the right-shift operator, T(x(0), x(1), x(2), . . .) = (0, x(0), x(1), . . .),

x = {x(k)}k=0 ∈ l∞ , ∞

then an extended limit θ on l∞ is called a Banach limit if θ = θ ∘ T. The Cesaro operator C : l∞ → l∞ is defined by (Cx)(n) =

n 1 ∑ x(k), n + 1 k=0

x ∈ l∞ .

Theorem (Dixmier traces on ℒ1,∞ ). (a) Spectral formula and characterization. (i) Let ω be an extended limit on l∞ . The formula Trω (A) = ω(

n 1 ∑ λ(k, A)), log(n + 2) k=0

A ∈ ℒ1,∞ ,

where λ(A) is any eigenvalue sequence of A, is linear and defines a positive normalized trace on ℒ1,∞ . (ii) Let θ be a Banach limit on l∞ . The formula n+1

φθ (A) = θ(

1 2 −2 ∑ λ(k, A)), log 2 k=2n −1

A ∈ ℒ1,∞ ,

where λ(A) is any eigenvalue sequence of A, is linear and defines a positive normalized trace on ℒ1,∞ . The map θ 󳨃→ φθ is a bijective correspondence between Banach limits on l∞ and normalized positive traces on ℒ1,∞ .

XX | Introduction (iii) The positive normalized trace φθ is a Dixmier trace on ℒ1,∞ if and only if the Banach limit θ factorizes as θ =γ∘C for some extended limit γ on l∞ . The map θ 󳨃→ φθ is a bijective correspondence between factorizable Banach limits on l∞ and Dixmier traces on ℒ1,∞ . (b) Diagonal formula. Let {en }∞ n=0 be an orthonormal basis of H, and suppose A ∈ ℒ1,∞ satisfies the decay condition ∞

∑ ‖Aen ‖2 = O(n−1 ),

k=n+1

n ≥ 0.

(i) Then n+1

φθ (A) = θ(

1 2 −2 ∑ ⟨Aek , ek ⟩) log(2) k=2n −1

for every Banach limit θ on l∞ . (ii) In particular, the formula in (i) holds for a factorizable Banach limit and Trω (A) = ω(

n 1 ∑ ⟨Aek , ek ⟩) log(2 + n) k=0

for every extended limit ω on l∞ . The above theorem, particularly the bijective association between Dixmier traces and factorizable Banach limits, has the consequences discussed in Chapters 3, 6, 7, and 9. (a) There are an infinite number of linearly independent Dixmier traces on ℒ1,∞ , which should be contrasted with Tr which is the unique positive trace on ℒ1 . (b) There are positive normalized traces on ℒ1,∞ that are not Dixmier traces, since not every Banach limit is factorizable. (c) A trace on ℒ1,∞ is a Dixmier trace if and only if 0 ≤ φ(A) ≤ φ(B) when 0 ≤ A, B ∈ ℒ1,∞ n

n

k=0

k=0

∑ μ(k, A) ≤ ∑ μ(k, B),

and

∀n ≥ 0.

The latter condition on A and B is Hardy–Littlewood submajorization. (d) An operator A ∈ ℒ1,∞ has a unique Dixmier trace, meaning Trω (A) = c takes the same value c for every Dixmier trace Trω , if and only if n 1 ∑ λ(k, A) = c. n→∞ log(n + 2) k=0

lim

Introduction

| XXI

Dixmier traces and asymptotics of measures, heat traces, and ζ-functions Let 0 ≤ V ∈ ℒ1 be a trace class operator. The positive linear functional on the algebra ℒ(H) defined by A 󳨃→ Tr(AV),

A ∈ ℒ(H),

provides the form of all normal linear functionals on ℒ(H). Irving Segal, Jacques Dixmier, and other authors in the 1950s, extended measure theory on the commutative algebra L∞ (Ω) for a finite measure space Ω to a theory of normal linear functionals on the noncommutative algebra ℒ(L2 (Ω)). Let Ω be a p-dimensional compact Riemannian manifold Ω with Laplacian Δ. By p Weyl’s law, the operator (1 − Δ)− 2 has asymptotic spectral behavior p

λ(n, (1 − Δ)− 2 ) ∼

1 , n+1

n → ∞.

Alain Connes, in his noncommutative differential geometry, observed that the operator p

0 ≤ (1 − Δ)− 2 ∈ ℒ1,∞ , and that p

A 󳨃→ Trω (A(1 − Δ)− 2 ),

A ∈ ℒ(L2 (Ω)),

defined an noncommutative extension of the integral on the manifold Ω, from the commutative algebra L∞ (Ω) embedded in ℒ(L2 (Ω)) by the operation of left multiplication of an essentially bounded function on a square integrable function, to the noncommutative algebra ℒ(L2 (Ω)). The next theorem collates the formulas from Theorems 7.1.5, 8.1.2, and 8.1.5, characterizing Connes’ noncommutative integral and relating Dixmier traces to the leading asymptotics of spectral measures, heat traces, and the residue at the first pole of spectral ζ -functions. The logarithmic mean operator M : l∞ → l∞ is defined by (Mx)(n) =

n x(k) 1 , ∑ log(n + 2) k=0 k + 1

x ∈ l∞ , n ≥ 0.

Theorem. Let 0 ≤ V ∈ ℒ1,∞ . (a) Limit of normal states. Let ω be an extended limit on l∞ . (i) Then A 󳨃→ Trω (AV),

A ∈ ℒ(H),

is a positive linear functional on the algebra ℒ(H) that vanishes on 𝒞0 (H).

XXII | Introduction (ii) If V satisfies λ(n, V) ∼

1 , n+1

n → ∞,

and {en }n≥0 is an orthonormal basis of H such that Ven = λ(n, V)en , n ≥ 0, then Trω (AV) = (ω ∘ M)(⟨Aen , en ⟩),

A ∈ ℒ(H),

where ω ∘ M is the extended limit on l∞ obtained by composing ω with M. (b) Heat trace asymptotic formula. Let ω be an extended limit on l∞ . Then Trω (AV) = (ω ∘ M)(

−1 1 Tr(Ae−(nV) )), n+1

where ω∘M is the extended limit on l∞ obtained by composing ω with M and e−(nV) , n ∈ ℕ, is obtained from the bounded functional calculus applied to the function 1 e− ns , s ≥ 0. (c) ζ -function residue formula. (i) For each state γ on L∞ (0, 1) such that −1

γ(f ) = 0

when lim+ f (s) = 0, f ∈ Cb (0, 1), s→0

there is a factorizable Banach limit θγ on l∞ , θγ (x) := log(2) ⋅ γ(s ∑ 2−ks x(k)), k≥0

x ∈ l∞ ,

such that φθγ (AV) = γ(sTr(AV 1+s )) for the Dixmier trace φθγ on ℒ1,∞ . (ii) There is a factorizable Banach limit θ on l∞ such that θ ≠ θγ for any state γ in (i). The above theorem has the consequences discussed in Chapters 7, 8, and 9. Let A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ . (a) The map A 󳨃→ Trω (AV), A ∈ ℒ(H), is a positive linear functional on the Calkin algebra ℒ(H) \ 𝒞0 (H). (b) Every Dixmier trace on ℒ1,∞ has the form of a “leading term in the asymptotic expansion of the heat trace”, that is, −1

Tr(Ae−(nV) ) = Trω (AV) ⋅ n + oω∘M (n),

n → ∞,

where oω∘M (n) means that the remainder term is a sequence in n such that (n + 1)−1 oω∘M (n) is a bounded sequence that vanishes under the extended limit ω ∘ M on l∞ . This copies the first-order expansion of Minakshisundaram’s heat trace Tr(etΔ ) as t → 0+ , where Δ is the Laplacian on a closed Riemannian manifold.

Introduction

| XXIII

(c) The “residue at the first simple pole” of a spectral ζ -function Tr(AV 1+s ), s > 0, is given by a Dixmier trace φθγ on ℒ1,∞ as Tr(AV 1+s ) =

φθγ (AV) s

+ oγ (s−1 ),

s → 0+ ,

where oγ (s−1 ) means that the remainder term is a function of s such that the function soγ (s−1 ) ∈ L∞ (0, 1) vanishes under the state γ. In general, Tr(AV 1+z ), ℜ(z) > 0, does not have a meromorphic extension to the half plane ℜz ≤ 0, but this firstorder expansion copies the form of the simple pole of the Minakshisundaram p and Pleijel ζ -function Tr((−Δ)− 2 (1+z) ) at z = 0 for the Laplacian Δ on a closed p-dimensional Riemannian manifold. Unlike the situation with the heat trace, not every Dixmier trace has this form. In the case of a closed Riemannian manifold, the Mellin transform is used to transfer between the asymptotic expansion of the heat trace Tr(etΔ ) as t → 0+ , and the merop morphicity of the zeta function Tr((−Δ)− 2 (1+s) ) on the complex plane. This method has limits for the general spectral behavior that is possible for weak trace class operators, and the approach in Chapter 8 is not based on the Mellin transform. Various statements for an operator A ∈ ℒ1,∞ to have a unique Dixmier trace in terms of heat trace and ζ -function residue formulas, and the kind of unique traceability on ℒ1,∞ implied by existence of an asymptotic expansion of the heat trace or meromorphicity of the spectral ζ -function, are given in Chapter 9. General theory of traces on two-sided ideals of the algebra ℒ(H) The general theory is based on two correspondences that bijectively associate a twosided ideal of the noncommutative algebra ℒ(H) to an ideal of the commutative algebra l∞ . Each correspondence lifts to a bijective association between traces on a twosided ideal of ℒ(H) and linear functionals of singular values or eigenvalue sequences in the ideal of l∞ . An subspace I of l∞ is called monotone if y ∈ l∞ and sup |y(k)| ≤ sup |x(k)|, k≥n

n ≥ 0,

k≥n

for some x ∈ I implies that y ∈ I. Let Π be the group of permutations of the semigroup ℤ+ . Define the operator π : l∞ → l∞ by (πx)(k) = x(π(k)),

x = {x(k)}k=0 ∈ l∞ , π ∈ Π. ∞

Definition. A symmetric sequence space J is a subspace of l∞ that is monotone and invariant under π : l∞ → l∞ for each π ∈ Π. A symmetric functional φ̂ : J → ℂ is a linear functional on J such that φ̂ = φ̂ ∘ π for each π ∈ Π.

XXIV | Introduction Define the right-shift operator T : l∞ → l∞ by T(x(0), x(1), x(2), . . .) = (0, x(0), x(1), . . .),

x = {x(k)}k=0 ∈ l∞ . ∞

Definition. A shift-invariant monotone ideal O is an ideal of l∞ that is monotone and invariant under T : l∞ → l∞ . A shift-invariant linear functional ℓ : O → ℂ is a linear functional on O such that ℓ = 21 ℓ ∘ T. The first correspondence in the next theorem is due to John Williams Calkin from 1941. The second correspondence is due to Albrecht Pietsch from the mid-1970s onwards. The theorem combines the results of Theorems 2.4.3 and 4.1.2. Theorem (Characterization of two-sided ideals of ℒ(H)). (a) Calkin correspondence. (i) If 𝒥 is a two-sided ideal of ℒ(H), then the commutative core J = {x ∈ l∞ : diag(x) ∈ 𝒥 } is a symmetric sequence space in l∞ . (ii) If J is a symmetric sequence space in l∞ , then 𝒥 = {A ∈ ℒ(H) : μ(A) ∈ J}

is a two-sided ideal of ℒ(H). (iii) The correspondences 𝒥 ?

μ diag

? J

between a two-sided ideal 𝒥 and a symmetric sequence space J are mutual inverses. (b) Pietsch correspondence. Define dyadic dilation L : l∞ → l∞ by . . . , x(n), . . .), Lx = (x(0), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(1), x(1), ⏟⏟ x(2), . .⏟,⏟⏟⏟⏟⏟⏟⏟⏟ x(2) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟.⏟⏟⏟ ⏟⏟, . . . , x(n), 2 times

2n times

4 times

and dyadic averaging L󸀠 : l∞ → l∞ by (L x)(n) = 2 󸀠

−n

2n+1 −2

∑ x(k),

k=2n −1

x ∈ l∞ , n ≥ 0.

(i) If 𝒥 is a two-sided ideal of ℒ(H), then the set O𝒥 = {x ∈ l∞ : diag(Lx) ∈ 𝒥 } is a shift-invariant monotone ideal of l∞ .

x ∈ l∞ ,

Introduction

| XXV

(ii) If O is a shift-invariant monotone ideal of l∞ , then the set 𝒥O = {A ∈ ℒ(H) : L μ(A) ∈ O} 󸀠

is a two-sided ideal of ℒ(H). (iii) The correspondences 𝒥 → O𝒥 and O → 𝒥O in the diagram Calkin correspondence

𝒥 ?

?

μ

diag

Pietsch correspondence

? J ? L

? ? O𝒥

L󸀠

are mutual inverses. Barry Simon, and Israel C. Gohberg and Mark G. Krein, studied quasi-Banach twosided ideals of ℒ(H) that are symmetrically quasinormed in the sense that ‖BAC‖𝒥 ≤ ‖B‖∞ ‖A‖𝒥 ‖C‖∞ ,

A ∈ 𝒥 , B, C ∈ ℒ(H),

for a quasinorm ‖ ⋅ ‖𝒥 on 𝒥 . Here ‖ ⋅ ‖∞ is the operator norm on ℒ(H). John von Neumann anticipated the Calkin correspondence with a paper in 1937 by associating with every n-dimensional symmetric sequence space an n2 -dimensional symmetric matrix space (coinciding as a linear space with the set of all n × n matrices), and posed the question whether quasinorms and completeness of symmetric sequences spaces lift to the symmetric matrix space. Theorems 2.4.9, 3.1.1, 3.1.2, and 4.1.6 show that the Calkin and Pietch correspondences provide bijective correspondences between quasiBanach two-sided ideals that are symmetrically quasinormed and quasi-Banach symmetric, respectively, quasi-Banach shift-invariant monotone, ideals of l∞ with quasinorms that are monotone and are invariant under permutation, and right shift, respectively. The next theorem characterizes traces on a two-sided ideal of ℒ(H) by invariant functionals on an ideal of l∞ . The Calkin correspondence reduces unitary invariance of traces to permutation invariance of symmetric functionals, which is then reduced further in the Pietsch correspondence to shift invariance. The theorem combines Theorems 3.1.3 and 4.1.4. Theorem (Characterization of traces). (a) Calkin correspondence. Let 𝒥 be a two-sided ideal of ℒ(H) with commutative core J. (i) If φ : 𝒥 → ℂ is a trace, then the mapping x 󳨃→ (φ ∘ diag)(x), is a symmetric functional φ̂ on J.

x ∈ J,

XXVI | Introduction (ii) If φ̂ : J → ℂ is a symmetric functional, then the mapping A 󳨃→ (φ̂ ∘ μ)(A),

0 ≤ A ∈ 𝒥,

is additive on the positive cone of 𝒥 and extends to a trace φ on 𝒥 . (iii) The correspondences φ?

∘diag ∘μ

? φ̂ .

between traces on 𝒥 and symmetric functionals on J are mutually inverse. (b) Pietsch correspondence. Let 𝒥 be a two-sided ideal of ℒ(H) and O𝒥 the corresponding monotone shift-invariant ideal of l∞ . (i) If φ : 𝒥 → ℂ is a trace, then the mapping x → (φ ∘ diag ∘ L)(x),

x ∈ O𝒥 ,

is a shift-invariant linear functional ℓφ on O𝒥 . (ii) If ℓ : O𝒥 → ℂ is a shift-invariant linear functional, then the mapping A → (ℓ ∘ L󸀠 ∘ μ)(A),

0 ≤ A ∈ 𝒥,

is additive on the positive cone of 𝒥 and extends to a trace φℓ on 𝒥 . (iii) The correspondences φ 󳨃→ ℓφ and ℓ 󳨃→ φℓ in the diagram Calkin correspondence

φ? ?

∘diag

Pietsch correspondence

∘μ

? φ̂ ? ∘L󸀠

? ? ℓ

∘L

are mutual inverses. If the ideal 𝒥 is quasi-Banach, then the triangle is bijective between continuous traces, continuous symmetric functionals, and continuous shift-invariant functionals, see Theorem 4.1.8. The reduction from a linear functional on a two-sided ideal that is invariant under the entire group of unitary operators on the Hilbert space H, to a linear functional on a symmetric sequence space invariant under the simpler group of permutations, then to a linear functional on a shift-invariant ideal that is invariant under a single operator T, is how we can determine the existence of traces and characterize them. The bijection of positive normalized traces on ℒ1,∞ with Banach limits on l∞ , and the bijection of Dixmier traces with factorizable Banach limits, follows from this reduction.

Introduction

| XXVII

The consequences of the previous theorem, discussed in Chapters 3 and 4, include the surprising result that, while Tr is the unique nonzero positive trace on ℒ1 , there exist other nonzero traces on ℒ1 , see Theorem 3.1.4. The correspondences of Calkin and Pietsch show that a trace on an arbitrary twosided ideal 𝒥 of ℒ(H) can be written as a linear function of singular values on the positive cone of 𝒥 , φ(A) = (φ ∘ diag)(μ(A)),

0 ≤ A ∈ 𝒥,

where φ ∘ diag is a symmetric functional on the commutative core J, and φ(A) = (ℓ ∘ L󸀠 )(μ(A)),

0 ≤ A ∈ 𝒥,

where ℓ is a shift-invariant functional on the shift-invariant monotone ideal O𝒥 associated to 𝒥 . Chapter 5 considers the extension of Victor Lidksii’s result on Tr, and whether traces can be written as linear functions of eigenvalue sequences. Definition. A two-sided ideal 𝒥 of ℒ(H) is logarithmic submajorization closed if B ∈ 𝒞0 (H) and n

n

k=0

k=0

∏ μ(k, B) ≤ ∏ μ(k, A),

∀n ≥ 0,

(3)

for some A ∈ 𝒥 implies that B ∈ 𝒥 . The condition in (3) was introduced by Weyl. It is not a restrictive condition since it is satisfied by any quasi-Banach ideal, including ℒ1,∞ . The next theorem is Theorem 5.1.2. Theorem (Spectral formula for traces). Let 𝒥 be a two-sided ideal of ℒ(H) with commutative core J and let φ be a trace on 𝒥 . (a) If 𝒥 is logarithmic submajorization closed, then (i) φ(A) = (φ ∘ diag)(λ(A)),

(ii)

A ∈ 𝒥,

where λ(A) ∈ J is any eigenvalue sequence of A and φ ∘ diag is a symmetric functional on J. φ(A) = (ℓ ∘ L󸀠 )(λ(A)),

A ∈ 𝒥,

where λ(A) ∈ J is any eigenvalue sequence of A and ℓ is a shift-invariant linear functional on the ideal O𝒥 .

XXVIII | Introduction (b) If 𝒥 is not closed with respect to logarithmic submajorization, then there exists an operator A ∈ 𝒥 such that λ(A) ∉ J for any eigenvalue sequence λ(A) of A. In particular, the spectral formulas in part (a) of the latter theorem cannot hold for each trace on the two-sided ideal if the ideal is not logarithmic submajorization closed. Structure of the text and subsequent volumes This text on the theory of singular traces on two-sided ideals of bounded operators of a separable Hilbert space to itself is Volume I of three planned volumes. Part I, of this Volume, motivates singular traces and ideals of bounded operators, and provides background in operator theory and singular values. Chapter 1 introduces bounded and compact operators of a separable Hilbert space to itself. It illustrates the trace on the ideal of trace class operators and Dixmier’s construction of traces on the ideal of weak trace class operators. Chapter 2 defines the singular value function of a bounded operator and partial orders on bounded operators derived from Hardy– Littlewood submajorization and logarithmic submajorization, which is Weyl’s product version of Hardy–Littlewood submajorization. It proves Calkin’s bijective correspondence between two-sided ideals of compact operators and monotone spaces of bounded sequences invariant under permutation. Part II, of this Volume, provides the general theory of traces on two-sided ideals of bounded operators. In Chapter 3 the Calkin correspondence is lifted to a bijective correspondence between traces and linear functionals invariant to permutations. The Figiel–Kalton theorem associating arithmetic means and the center of a sequence space invariant under permutation is proved in Chapter 3. Albrecht Pietch’s equivalent correspondence is described in Chapter 4. Chapter 5 proves the noncommutative version of the Figiel–Kalton theorem and that quasinilpotent operators in a logarithmic submajorization closed ideal are commutators. That every trace on a logarithmic submajorization closed ideal is a linear functional on eigenvalues follows. Part III, of this Volume, investigates formulas for traces on the ideal of weak trace class operators. Using Pietsch’s correspondence, Chapter 6 provides a formula for every positive trace on the set of weak trace class operators in terms of applying a Banach limit on l∞ to a dyadic average of eigenvalues. Dixmier’s formula for traces on the ideal of weak trace class operators is shown to be equivalent to every positive trace that is monotone for Hardy–Littlewood submajorization, which, in turn, relates to the Banach limit associated to the positive trace being factorizable. Chapter 7 shows that the trace on the ideal of the trace class operators is the only trace with a universal formula involving the diagonal matrix elements of an operator with respect to a given orthonormal basis of the Hilbert space. However, Chapter 7 shows that traces of some weak trace class operators can be written in terms of the diagonal matrix elements, provided they satisfy a decay condition called modulation. Chapter 8 examines the

Introduction

| XXIX

association between the Dixmier trace and the classical formula for the first residue of the ζ -function of the Laplace–Beltrami operator and the asymptotic expansion of the heat trace on a closed Riemannian manifold. There are infinitely many linearly independent traces on the ideal of weak trace class operators. Chapter 9 investigates when a weak trace class operator has a unique trace, meaning that it takes the same value for every trace on the ideal of weak trace class operators, and what the spectral, heat trace, and ζ -function residue formulas look like in this case. Volume II concerns applications of singular traces from Alain Connes’ noncommutative geometry. Part I of Volume II considers formulas for traces of Hilbert–Schmidt operators acting on square integrable functions of a p-dimensional Riemannian manifold. Connes’ trace formula linking the Dixmier trace and the noncommutative residue of a pseudodifferential operator of order −p on a closed Riemannian manifold is obtained as a corollary. Traces of Hilbert–Schmidt operators on isospectral deformations are also considered, with the noncommutative torus as an example. Part II of Volume II features a range of examples of applications of traces on the ideal of weak trace class operators, from recovery of the geometric measure on the limit set of a quasi-Fuschian group and integration of quantum differentials on the torus, to Connes’ character formula for the Hochschild class of a Fredholm module. Part III of Volume II extends the notion of a principal symbol to a pseudodifferential calculus of differential operators with coefficients in general C∗ -algebras, and trace formulas in this case. Volume III will consider the theory and applications of singular traces on twosided modules of semifinite von Neumann algebras. The singular value function, which is defined for bounded operators using the trace on the set of trace class operators in Volume I, is defined in Volume III using a semifinite faithful normal trace on the von Neumann algebra.

|

Part I: Preliminary material

1 What is a singular trace? The foundation of functional analysis has led to the study of the trace on the set of continuous operators from a separable Hilbert space to itself as an extension of the trace on the set of matrices, and led to the study of the ideal of trace class operators. In the late 1960s Jacques Dixmier constructed a trace on the ideal of weak trace class operators that vanished on the ideal of trace class operators contained within. Dixmier’s trace was not an extension of the matrix trace and began the study of singular traces. This chapter introduces bounded and compact operators from a separable Hilbert space to itself, and the eigenvalues and singular values of compact operators. The trace of a matrix and its invariance properties are also introduced. Traces are defined as linear functionals on two-sided ideals of compact operators which extend the invariance properties of the matrix trace. The trace on the ideal of trace class operators and the Dixmier trace on the ideal of weak trace class operators are introduced as examples. They illustrate the relationship of traces and ideals to singular values and eigenvalues that is the foundation of singular trace theory. This and later chapters assume a graduate-level knowledge of functional analysis. We refer to the monograph [219] for a graduate-level introduction to functional analysis. For other references, see the end notes to this chapter. We provide only those results, mostly without proof, needed for the foundation and motivation for the study of singular traces. Notation is also established.

1.1 Compact operators Let H be a complex Hilbert space, that is, an inner product vector space over the complex numbers that is complete in the associated vector norm. The inner product on H, always denoted by ⟨x, y⟩, x, y ∈ H, we take to be complex linear in the first variable. Also, ‖x‖ = √⟨x, x⟩, x ∈ H, always denotes the vector norm on H. Bounded linear operators on Hilbert spaces A linear operator A:H→H is bounded if there exists some constant C > 0 such that ‖Ax‖ ≤ C‖x‖ for all x ∈ H. The smallest constant is the operator norm ‖A‖∞ . We denote the set of bounded operators by ℒ(H). The bounded operators form the algebra of continuous https://doi.org/10.1515/9783110378054-001

4 | 1 What is a singular trace? linear operators of H to itself. In addition to the norm topology on the bounded operators given by the operator norm, the family of seminorms {‖Ax‖ : x ∈ H} forms a locally convex topology on the bounded operators called the strong operator topology. A Hilbert space is separable if it has a countable basis as a vector space. All Hilbert spaces we consider will be separable and infinite dimensional; the finite-dimensional theory is equivalent to linear algebra and can be obtained by restriction to a finitedimensional subspace. The Gram–Schmidt process familiar to linear algebra can be applied to infinite-dimensional Hilbert spaces, in fact, E. Schmidt applied the process initially to the Hilbert space l2 of square summable sequences [235]. Lemma 1.1.1 ([235], [219, Theorem II.6]). A separable infinite-dimensional Hilbert space H admits a countable orthonormal basis {en }∞ n=0 , i. e., ⟨en , em ⟩ = δmn and x = ∑∞ ⟨e , x⟩e for any x ∈ H where the infinite sum is understood as the vector n n=0 n norm limit of partial sums. If {en }∞ n=0 is an orthonormal basis of H, for fixed n, m ∈ ℤ+ , the linear operator en ⊗ em : H → H defined by (en ⊗ em )x := ⟨x, en ⟩em ,

x ∈ H,

is called a matrix element. An operator A is of finite rank if it is a finite linear combination of matrix elements in some basis. Equivalently, the subspace AH is finite dimensional. We denote the finite-rank operators by 𝒞00 (H) and square N × N matrices over the complex numbers by MN (ℂ). The maps, A ∈ 𝒞00 (H) 󳨃→ [⟨Aen , em ⟩]n,m=0,...,N N−1

A −1

,

for some {en }∞ n=0 and NA ∈ ℕ,

∑ anm en ⊗ em ←󳨂 [anm ]n,m=0,...,N−1 ∈ MN (ℂ),

n,m=0

for any {en }∞ n=0 ,

provide a representation of finite-rank operators as matrices, and matrices as linear transformations of finite-dimensional Hilbert subspaces of H. Lemma 1.1.2. If {en }∞ n=0 is an orthonormal basis of the separable Hilbert space H and A ∈ ℒ(H) is a bounded operator then ∞

A = ∑ ⟨Aen , em ⟩en ⊗ em , n,m=0

where the infinite sum is understood as a strong limit of partial sums. Proof. Since x = ∑∞ n=0 ⟨x, en ⟩en for any x ∈ H, we have ∞

∞

m=0

n,m=0

Ax = ∑ ⟨Ax, em ⟩em = ∑ ⟨Aen , em ⟩⟨x, en ⟩em .

1.1 Compact operators | 5

It follows that ℒ(H) is the strong closure of the finite-rank operators 𝒞00 (H) and the theory of operators on infinite-dimensional Hilbert spaces is an extension of linear algebra. Results and terminology from linear algebra transfer to operators, for example, the adjoint A∗ : H → H of a bounded operator A is the bounded operator with ⟨A∗ x, y⟩ = ⟨x, Ay⟩, x, y ∈ H, and A is self adjoint (or hermitian) if A = A∗ . A bounded self-adjoint operator A is positive, denoted A ≥ 0, if ⟨Ax, x⟩ ≥ 0 for every x ∈ H. Alternatively, A = B∗ B for some bounded operator B. Spectral theory The extension of the spectral theory of matrices to operators is the most important tool in operator theory. Definition 1.1.3. A complex number λ belongs to the resolvent ρ(A) of a bounded operator A ∈ ℒ(H) if A − λ1 : H → H is invertible, where 1 : H → H is the identity map. The resolvent is an open subset of the complex plane ℂ [88, Theorem 1.4]. The complement, the spectrum, is therefore closed. Definition 1.1.4. The spectrum σ(A) of a bounded operator A ∈ ℒ(H) is the set ℂ\ρ(A). Unlike matrices, for a bounded operator there is a distinction between being oneto-one and being invertible. Definition 1.1.5. The complex numbers λ ∈ σ(A) where A − λ1 is not a bijection are called the eigenvalues of A. Indeed, if A − λ1 is not injective this implies the existence of a nonzero x ∈ H such that Ax = λx. A bounded operator need not have any eigenvalues or eigenvectors. The shift operator U1 : l2 (ℤ) → l2 (ℤ) on the Hilbert space of double-sided square summable sequences, ∞

∞

U1 : {x(n)}n=−∞ 󳨃→ {x(n + 1)}n=−∞ ,

∞

∑ |x(n)|2 < ∞,

n=−∞

is the standard example of a bounded operator without eigenvalues and eigenvectors. Details can found in introductory texts on functional analysis, see [219, 228]. A bounded operator is compact if it is the operator norm limit of a sequence of finite-rank operators. We denote the set of compact operators by 𝒞0 (H). Compact operators possess eigenvalues and eigenvectors. Theorem 1.1.6 ([219, Theorem VI.15]). The spectrum of a compact operator consists of 0 and a countable set of eigenvalues whose only limit point, if it admits a limit point, is 0.

6 | 1 What is a singular trace? Definition 1.1.7. Let A ∈ 𝒞0 (H) be a compact operator and let λ be an eigenvalue of A. The eigenspace Eλ = {x ∈ H : (A − λ)x = 0} is a linear subspace of H. The dimension of the eigenspace Eλ is called the geometric multiplicity of λ. Definition 1.1.8. Let A ∈ 𝒞0 (H) be a compact operator and let λ be an eigenvalue of A. The set Rλ = {x ∈ H : ∃n ∈ ℕ such that (A − λ)n x = 0} is a linear subspace of H. The dimension of the linear space Rλ is called the algebraic multiplicity of λ. The geometric multiplicity of an eigenvalue is less than or equal to the algebraic multiplicity. Theorem 1.1.9 ([219, Theorem VI.15], [88, Theorem 2.21]). Let A ∈ 𝒞0 (H) be a compact operator and let λ be a nonzero eigenvalue of A. Then the algebraic multiplicity of λ is finite. The result is that, like for matrices, the nonzero spectrum of a compact operator consists entirely of eigenvalues of finite algebraic multiplicity, and can be ordered by nonincreasing absolute value and associated to a sequence in the space c0 of complexvalued sequences that vanish at infinity. Definition 1.1.10. An eigenvalue sequence ∞

λ(A) := {λ(n, A)}n=0 ∈ c0 of a compact operator A ∈ 𝒞0 (H) is the sequence of eigenvalues λ(n, A), n ≥ 0, of A, repeated according to algebraic multiplicity, such that the absolute values |λ(n, A)|, n ≥ 0, are nonincreasing. If A has a finite number N of eigenvalues, we embed the eigenvalue sequence in c0 as the sequence {λ(0, A), . . . , λ(N − 1, A), 0, 0, . . .} ∈ c00 . If A has no nonzero eigenvalues, the eigenvalue sequence of A is the zero sequence {0, 0, . . .}. The appearance of eigenvalues in the text assumes that, unless stated explicitly, they are the eigenvalues of a compact operator and are ordered in an eigenvalue sequence. The ordering of eigenvalues in this way is not necessarily unique. Example 1.1.11. The set of complex-valued bounded sequences l∞ can be embedded as bounded operators in any separable Hilbert space H. Let a ∈ l∞ act as an operator diag(a) on H by ∞

diag(a) := ∑ a(n)en ⊗ en , n=0

∞

a = {a(n)}n=0 ∈ l∞ ,

1.1 Compact operators | 7

where en ⊗ en are the rank-one projections relative to an orthonormal basis {en }∞ n=0 . Then diag(a) : H → H is a bounded linear operator with ‖diag(a)‖∞ = ‖a‖∞ := supn≥0 |a(n)|. Finite sequences map to finite-rank operators. If a = {a(n)}∞ n=0 ∈ c0 then diag(a) ∈ 𝒞0 (H) is compact and an eigenvalue sequence of diag(a) consists of the values a(n) ordered so that |a(n)| are nonincreasing (with zeros appended if a ∈ c00 ). The operators in Example 1.1.11 are called diagonal operators. A bounded operator A is normal if A = B + iC, where B = B∗ , C = C ∗ are self adjoint and BC = CB. The spectral, or diagonalization, theorem of normal matrices can be extended to compact normal operators [219, Theorem VI.17, Theorem VII.1]. We write it in the form of eigenvalue sequences. Theorem 1.1.12 (Spectral theorem). Let A ∈ 𝒞0 (H) be a compact normal operator. Then the algebraic and geometric multiplicities of each nonzero eigenvalue are equal and there exists an orthonormal basis {en }∞ n=0 of H such that ∞

A = ∑ λ(n, A)en ⊗ en , n=0

where λ(A) is an eigenvalue sequence and en ⊗ en are the rank-one projections relative to the basis {en }∞ n=0 . If f ∈ C(σ(A)) is a continuous function such that f (0) = 0 then there is a compact operator ∞

f (A) := ∑ f (λ(n, A))en ⊗ en n=0

on H such that σ(f (A)) = f (σ(A)). Singular values and decompositions If A is a compact operator, by the spectral theorem there is a positive compact operator |A|, which is unique and which is termed the absolute value of A, such that |A| = √A∗ A. The absolute value allows us to generalize singular values of matrices. Definition 1.1.13. The singular value sequence of a compact operator A ∈ 𝒞0 (H) is the ∞ eigenvalue sequence of |A| = √A∗ A, denoted μ(A) := {μ(n, A)}∞ n=0 = {λ(n, |A|)}n=0 . The singular value sequence can be used to extend the spectral decomposition of compact normal operators in Theorem 1.1.12. The Schmidt decomposition applies to arbitrary compact operators that may not be normal, and reverts to the spectral decomposition if the compact operator is positive. Theorem 1.1.14 (Schmidt decomposition). Let A ∈ 𝒞0 (H) be a compact operator. If {en }∞ n=0 is an orthonormal basis of H ordered such that |A|en = μ(n, A)en , n ≥ 0, then

8 | 1 What is a singular trace? there exists an orthonormal system {fn }∞ n=0 such that ∞

A = ∑ μ(n, A)fn ⊗ en , n=0

where μ(A) is the sequence of singular values of A. We describe the properties of the singular value sequence when considering the singular value function in Chapter 2, see also [247, Chapter 1]. Singular values generalize the commutative notion of nonincreasing rearrangement [165]. Example 1.1.15. If a = {a(n)}∞ n=0 ∈ c0 then diag(a) ∈ 𝒞0 (H) is compact and the sequence of singular values is the nonincreasing rearrangement of the sequence |a| = {|a(n)|}∞ n=0 , denoted μ(a) := μ(diag(a)) ∈ c0 . For a compact normal operator A, Theorem 1.1.12 implies that the sequence of singular values is the modulus of an eigenvalue sequence of A, μ(A) = |λ(A)|. The correspondence between singular values and eigenvalues breaks down for nonnormal operators. A nonnormal compact operator can have a zero eigenvalue sequence with nonzero singular values, as the following examples of quasinilpotent compact operators show. Definition 1.1.16. A compact operator A ∈ 𝒞0 (H) is said to be (a) nilpotent if An = 0 for some n ∈ ℕ. (b) quasinilpotent if σ(A) = {0}. Example 1.1.17. The operator A : ℂ2 → ℂ2 defined by setting A=(

0 0

1 ) 0

is nilpotent since A2 = 0. The eigenvalues are λ(A) = {0, 0}, yet the singular values are μ(A) = {1, 0}. Evidently every nilpotent operator is quasinilpotent, but the converse is not true for operators which are not of finite rank. Example 1.1.18. The Volterra operator A : L2 (0, 1) → L2 (0, 1) defined by setting t

(Ax)(t) = ∫ x(s) ds 0

is quasinilpotent but not nilpotent, see [158].

1.1 Compact operators |

9

Proof. By induction t

(An+1 x)(t) = ∫ 0

(t − s)n x(s) ds, n!

n ≥ 0.

For every 0 < t < 1 and λ ≠ 0, there is a convergent Neumann series representation −1

((A − λ1) x)(t) = −λ

−1

∞

n

−n

−1

∑ λ (A x)(t) = −λ x(t) − λ

−2

n=0

t

∫ e(t−s)/λ x(s) ds, 0

hence (A − λ1)−1 is bounded and σ(A) = {0}. To show that A is not nilpotent, it suffices to observe that (An χ(0,1) )(t) =

tn ≠ 0, n!

0 < t < 1.

For a normal operator A, or for a quasinilpotent operator, the singular value sequence of A dominates the modulus of its eigenvalue sequence. In general, the singular value sequence of a nonnormal operator need not dominate the modulus of the eigenvalue sequence. Example 1.1.19. The operator A : ℂ2 → ℂ2 defined by setting A=(

1 0

1 ) 1

has eigenvalues λ(0, A) = 1 and λ(1, A) = 1. Since |A|2 = A∗ A = (

1 1

0 1 )⋅( 1 0

1 1 )=( 1 1

1 ), 2

the operator |A| has the eigenvalues, listed in decreasing order, μ(0, A) = √

3 + √5 > |λ(0, A)| 2

μ(1, A) = √

3 − √5 < |λ(1, A)|. 2

and

However, the sum (or product) of the singular values of a compact operator dominates the sum (or product) of the absolute value of an eigenvalue sequence. The following lemmas for compact operators on an infinite-dimensional separable Hilbert space H are due originally to Hermann Weyl [291].

10 | 1 What is a singular trace? Lemma 1.1.20. Let A ∈ 𝒞0 (H) be a compact operator. For every n ≥ 0, we have n

n

k=0

k=0

∑ |λ(k, A)| ≤ ∑ μ(k, A).

Lemma 1.1.21. Let A ∈ 𝒞0 (H) be a compact operator. For every n ≥ 0, we have n

n

k=0

k=0

∏ |λ(k, A)| ≤ ∏ μ(k, A).

A nonnormal n×n matrix M of complex numbers can be decomposed into the sum of a normal matrix and a nilpotent matrix. The normal component is diagonalizable to a diagonal matrix that has the list of eigenvalues of M, with algebraic multiplicity, down the diagonal. Since the matrix trace vanishes for nilpotent matrices, this reduces the trace of the nonnormal matrix M to the trace of a normal matrix. The result that the trace of an arbitrary matrix can be calculated as the sum of its eigenvalues relies on this fact. Theorem 1.1.22. Let M = [Mkm ]n−1 k,m=0 ∈ Mn (ℂ) be an n × n matrix with complex entries. The matrix trace Tr of M (the sum of the diagonal entries of M) is equal to the sum of the eigenvalues of M repeated according to algebraic multiplicity, n−1

n−1

k=0

k=0

Tr(M) = ∑ Mkk = ∑ λ(k, M). Proof. Let M = N + R, where R is nilpotent and N is normal such that an orthogonal matrix U exists with U ∗ NU = diag{λ(0, M), . . . , λ(n − 1, M)}. The existence of U follows from the spectral theorem. Since Tr(R) = 0, we get Tr(M) = Tr(N) = Tr(U ∗ NU) = Tr(diag{λ(0, M), . . . , λ(n − 1, M)}). The last equality follows since the matrix trace is invariant under orthogonal transformations. Decomposing a matrix into normal and nilpotent parts is also central for spectral formulas for traces in the infinite-dimensional theory. The decomposition is in the form of a normal compact operator and a quasinilpotent compact operator. We will use it later in the book to identify when a singular trace of a compact operator is a function of its eigenvalue sequence, and what is the explicit form of that function. J. Ringrose extended the upper-triangular form of matrices to compact operators, [224], [225, Chapter 4]. The following decomposition was stated by T. West [289] in the context of Riesz operators, but for compact operators it follows directly from results on the upper-triangular form of a compact operator in [224]. Theorem 1.1.23. Let A ∈ 𝒞0 (H) be a compact operator. There exists a compact normal operator N and a compact quasinilpotent operator Q such that A = N + Q and λ(A) = λ(N).

1.2 Calkin correspondence

| 11

1.2 Calkin correspondence There is a remarkable correspondence between two-sided ideals of compact operators and sequence spaces generated by singular values due to J. W. Calkin (1909–1964) [36]. By equally remarkable results from the 1990s of Nigel Kalton and Kenneth Dykema [92] and Kenneth Dykema, Tadeusz Figiel, Gary Weiss, and Mariusz Wodzicki [94], traces on two-sided ideals can be identified with functionals on the corresponding sequence space. The functionals associated to traces are called symmetric functionals. Independently, Albrecht Pietsch developed an equivalent correspondence in the late 1980s involving dyadic averaging [200, 202, 274]. Pietsch’s approach has the advantage of providing an explicit formula for the symmetric functional for many ideals, including those that feature prominently in Alain Connes noncommutative geometry. Pietsch’s approach is described in Part II of this book, along with the equivalence with the Calkin correspondence. Statement of the correspondence The Calkin correspondence indicates a plethora of two-sided ideals of compact operators with traces. This is in a stark contrast to finite-dimensional linear algebra, where the matrix algebra has no proper ideals and admits only the matrix trace. Definition 1.2.1. A linear subspace 𝒥 of compact operators is a two-sided ideal of the algebra ℒ(H) if A ∈ 𝒥 and B ∈ ℒ(H) implies BA, AB ∈ 𝒥 . Definition 1.2.2. A symmetric sequence space J is a subspace of c0 such that a ∈ J and μ(b) ≤ μ(a) implies b ∈ J, where μ(a) = μ(diag(a)) is the nonincreasing rearrangement in Example 1.1.15. Symmetric sequence spaces can also be viewed as follows. Let π be a permutation of the set of nonnegative integers. Consider the permutation matrix associated to π, ∞

Π := ∑ eπ(n) ⊗ en , n=0

where {en }∞ n=0

is an orthonormal basis of H. If we identify the spaces l∞ and c0 with the corresponding subsets of diagonal matrices in ℒ(H) and 𝒞0 (H), respectively, as done in Example 1.1.11, a symmetric sequence space J is an arbitrary subset of c0 which is an ideal in l∞ and which is rearrangement invariant in the sense that Π∗ JΠ ⊂ J, for every permutation matrix Π. That is, for any element a ∈ J, we have Π∗ diag(a)Π = diag(b), where b ∈ J is given by b = {a(π(n))}∞ n=0 . The permutation matrices are examples of isometries of the Hilbert space, mapping an orthonormal basis to a rearrangement of the same orthonormal basis, Πen = eπ(n) .

12 | 1 What is a singular trace? A bounded operator U : H → H is unitary if U ∗ U = UU ∗ = 1 where 1 is the identity map 1 : H → H. Two operators A and B in ℒ(H) are unitarily equivalent if A = UBU ∗ ,

for some unitary U ∈ ℒ(H).

As in linear algebra, unitary operators are equivalent to transformations of orthonor∞ mal bases, that is, {Uen }∞ n=0 is an orthonormal basis if and only if {en }n=0 is, and the unitary operators form the full group of isometries of H. The embedding of a sequence a ∈ l∞ in ℒ(H) described in Example 1.1.11, a 󳨃→ diag(a),

a ∈ l∞ ,

is not unique and depends on the orthonormal basis. Each embedding of l∞ as diagonal operators is unitarily equivalent. For this reason, the orthonormal basis is not made explicit unless a certain result does not hold for unitarily equivalent operators. Similarly, the embedding of an eigenvalue sequence of a compact operator as a diagonal operator in ℒ(H), λ(A) 󳨃→ diag(λ(A)),

A ∈ 𝒞0 (H),

is unique up to permutation within eigenspaces whose eigenvalues have the same modulus. We often refer to an eigenvalue sequence without needing to specify which one, as results are either unitarily invariant, or invariant to the unitary operators representing permutation within eigenspaces whose eigenvalues have the same modulus. The Calkin correspondence will be written by us in the following way. If J is a symmetric sequence space then associate to it the subset 𝒥 of compact operators 𝒥 := {A ∈ 𝒞0 (H) : μ(A) ∈ J}.

Here μ(A) is the singular value sequence of a compact operator as in Definition 1.1.13. Conversely, if 𝒥 is a two-sided ideal then associate to it the sequence space J := {a ∈ c0 : diag(a) ∈ 𝒥 }. The sequence space J is called the commutative core of 𝒥 . The definition of the commutative core is independent of the basis chosen for diag since 𝒥 is a two-sided ideal. A proof of the following theorem due to Calkin is given in Chapter 2. Theorem 1.2.3 (Calkin correspondence). The correspondence J ↔ 𝒥 is a bijection between symmetric sequence spaces and two-sided ideals of compact operators. The Calkin correspondence is the statement that the diagonal of a two-sided ideal corresponds uniquely to a symmetric sequence space and that the diagonal determines the ideal.

1.2 Calkin correspondence

| 13

Given an ideal 𝒥 , we denote by 𝒥+ the cone of positive operators belonging to 𝒥 . Similarly, J+ denotes the positive cone of a sequence space J. The singular value sequence μ : 𝒥+ → J+ provides the mapping from a positive compact operator in the ideal 𝒥 to a positive sequence in the symmetric sequence space J. The Calkin correspondence dictates our notation, we always denote ideals in commutative algebras (like l∞ ) by straight letters (e. g., l∞ , c0 , J) and the corresponding noncommutative objects are denoted by curly letters (e. g., ℒ(H), 𝒞0 (H), 𝒥 ). We often consider the sequence spaces embedded as the diagonal inside their noncommutative counterpart. Examples of ideals The following sequence spaces and corresponding ideals are referenced repeatedly in the text. Example 1.2.4 (Finite-rank operators). The sequence space of finite nonzero sequences c00 ⊂ c0 corresponds to the ideal of finite rank operators 𝒞00 (H). The sequence space c00 is the minimal symmetric sequence space, hence the finite-rank operators form the minimal two-sided ideal of compact operators, that is, if 𝒥 is a two-sided ideal then 𝒞00 (H) ⊂ 𝒥 . Example 1.2.5 (Schatten ideals). The lp sequence spaces, p ≥ 1, ∞

lp := {a ∈ c0 : ‖a‖p := ( ∑ μ(n, a)p )

1/p

< ∞}

n=0

correspond to the Schatten ideals of compact operators ∞

p

1/p

ℒp := {A ∈ 𝒞0 (H) : ‖A‖p := ( ∑ μ(n, A) ) n=0

< ∞}.

Example 1.2.6 (Weak ideals). The weak-lp sequence spaces lp,∞ , p ≥ 1, lp,∞ := {a ∈ c0 : μ(n, a) = O((1 + n)−1/p ), n ≥ 0} correspond to the weak-lp ideals of compact operators ℒp,∞ := {A ∈ 𝒞0 (H) : μ(n, A) = O((1 + n)

−1/p

), n ≥ 0}.

Example 1.2.7 (Lorentz ideals). Let ψ : ℝ+ → ℝ+ be an increasing positive concave function. The Lorentz sequence space mψ is mψ := {a ∈ c0 : ‖a‖mψ := sup n≥0

n 1 ∑ μ(k, a) < ∞}. ψ(n + 1) k=0

14 | 1 What is a singular trace? The corresponding Lorentz ideals of compact operators are ℳψ := {A ∈ 𝒞0 (H) : ‖A‖ℳψ := sup n≥0

n 1 ∑ μ(k, A) < ∞}. ψ(n + 1) k=0

If ψp (t) := t 1−1/p , p > 1, then mψp = lp,∞ and so the weak-lp ideals are Lorentz ideals for p > 1. Let ψ(t) := log(1 + t), t > 0. Define the Lorentz sequence space m1,∞ := {a ∈ c0 : ‖a‖m1,∞ := sup n≥0

n 1 ∑ μ(k, a) < ∞}. log(2 + n) k=0

Lemma 1.2.8. The inclusions l1 ⊂ l1,∞ ⊂ m1,∞ are strict. Proof. To see that l1 ⊂ l1,∞ , it is sufficient to show that a = μ(a) ∈ l1 belongs to l1,∞ . Fix an arbitrary element a = μ(a) ∈ l1 and observe that n

n ⋅ a(n) ≤ ∑ a(k) ≤ ‖a‖1 , k=0

n ≥ 0.

Therefore, a(n) = O((1 + n)−1 ), n ≥ 0, and a ∈ l1,∞ . The inclusion is strict since the sequence {(1 + n)−1 }∞ n=0 does not belong to l1 and obviously belongs to l1,∞ . To see that l1,∞ ⊂ m1,∞ , it is sufficient to check that {(1 + n)−1 }∞ n=0 belongs to m1,∞ . The verification is immediate, since ∑nk=0 (1 + k)−1 ≤ 2 log(2 + n) for every n ≥ 0. To see that the latter inclusion is strict, consider the following sequence: a(0) = 1,

a(n) :=

k

2 2(k+1) −1

−

2 2k

,

2

2

2k ≤ n ≤ 2(k+1) −1 , k ≥ 0. 2

We claim that a ∈ m1,∞ . Indeed, for every natural number N such that N = 2(k+1) −1 for some k ≥ 0, we have N

N

n=0

n=0

∑ μ(n, a) = ∑ a(n) = 1 + 1 + 2 + 3 + ⋅ ⋅ ⋅ + k = 1 +

2 k(k + 1) ≤ log(2(k+1) ) 2

and this, together with the observation that the mapping N → ∑Nn=0 μ(n, a), N ≥ 1 is piecewise linear concave, yields the required estimate N

∑ μ(n, a) ≤ 2 log(N + 2),

n=0

∀N ≥ 1.

At the same time a ∉ l1,∞ . To see this, it is sufficient to verify that for every C > 1 the −1 ∞ inequality {a(n)}∞ n=0 ≤ C{(n + 1) }n=0 fails. Take an arbitrary C > 1 and choose k0 ≥ 1 2

so that C ≤ k0 . Take N = 2(k0 +1) −1 . By construction, we have a(N) = is strictly greater than

k0

2 −1

2(k0 +1)

≥

C . N

k0

2 −1

2(k0 +1)

2

−2k0

, which

1.2 Calkin correspondence

| 15

Example 1.2.9. In Alain Connes’ noncommutative geometry, the Lorentz ideal ℳ1,∞ := {A ∈ 𝒞0 (H) : ‖A‖ℳ1,∞ := sup n≥0

n 1 ∑ μ(k, A) < ∞} log(2 + n) k=0

associated to the sequence space m1,∞ has been labeled with similar notation to the weak ideal ℒ1,∞ . The ideal ℳ1,∞ has been variously called the Dixmier ideal, the Dixmier–Macaev ideal, and the dual of the Macaev ideal. The inclusions of ideals ℒ1 ⊂ ℒ1,∞ ⊂ ℳ1,∞

are strict by Lemma 1.2.8 and the Calkin correspondence. The distinction between traces (as defined below) on ℒ1 and traces on the weak trace class ideal ℒ1,∞ is fundamental, and is described in Part II. The distinction between traces on ℒ1,∞ and the Dixmier–Macaev ideal ℳ1,∞ can be observed in Part III. See the end notes for historical origins of the ideals ℒ1,∞ and ℳ1,∞ . Traces on ideals A trace on a two-sided ideal of compact operators is defined in analogy to the fact that the matrix trace is invariant under orthogonal transformations. Definition 1.2.10. A trace φ on a two-sided ideal 𝒥 of the algebra ℒ(H) is a unitarily invariant linear functional, that is, φ : 𝒥 → ℂ and φ(UAU ∗ ) = φ(A) for all A ∈ 𝒥 and all unitary operators U ∈ ℒ(H). Unitary invariance coincides with the property that a trace vanishes on commutators. Lemma 1.2.11. A linear functional φ : 𝒥 → ℂ is a trace if and only if φ(AB) = φ(BA),

A ∈ 𝒥 , B ∈ ℒ(H).

Alternatively, φ([A, B]) = 0, where [A, B] := AB − BA. Proof. If the condition holds, φ((UA)U ∗ ) = φ(U ∗ UA) = φ(A) for every unitary U ∈ ℒ(H). Hence φ is unitarily invariant. For the only if, notice that φ(UA) = φ(U ∗ (UA)U) = φ(AU) and φ([A, U]) = 0 for every unitary U ∈ ℒ(H). Every B ∈ ℒ(H) can be written as a linear combination of four unitary operators, [219, p. 209], B = ∑4i=1 αi Ui where Ui∗ Ui = I. Then [A, B] = ∑4i=1 αi [A, Ui ]. Hence φ([A, B]) = ∑4i=1 αi φ([A, Ui ]) = 0.

16 | 1 What is a singular trace? Singular values are unitarily invariant, μ(UAU ∗ ) = μ(A) for all unitaries U ∈ ℒ(H) and compact operators A ∈ 𝒞0 (H). Thus, using the Calkin correspondence μ : 𝒥+ → J+ , we can associate to every linear functional φ̂ : J → ℂ on the commutative core J of a two-sided ideal of compact operators 𝒥 the unitarily invariant functional φ : A 󳨃→ (φ̂ ∘ μ)(A),

A ≥ 0.

(1.1)

If, in addition, φ̂ vanishes on the subset K(J) := {μ(A + B) − μ(A) − μ(B) : 0 ≤ A, B ∈ 𝒥 } then (1.1) is additive on the positive cone of 𝒥 . Chapter 2 describes how every compact operator in the ideal 𝒥 can be written as a linear combination of four positive operators from 𝒥 . By using linear extension, that is, 4

φ(A) = ∑ αi φ(Ai ), i=1

4

A = ∑ αi Ai , i=1

αi ∈ ℂ, Ai ≥ 0, i = 1, 2, 3, 4,

formula (1.1) defines a trace on 𝒥 . Kalton showed that the subset K(J) belongs to the linear subspace Z(J) := Span{a − b : μ(a) = μ(b), 0 ≤ a, b ∈ J} called the center of the commutative core J. Definition 1.2.12. A linear functional φ̂ : J → ℂ on a symmetric sequence space J is called symmetric if φ̂ vanishes on the center Z(J). By the results of Kalton and Dykema, Figiel, Weiss, and Wodzicki on the form of commutators involving compact operators, every trace on a two-sided ideal arises from a symmetric functional. Part II of this book explains how traces of positive operators are completely determined by their action on the commutative core. The Calkin correspondence provides the mechanism, namely, if φ : 𝒥 → ℂ is a trace then the spectral theorem implies that φ(A) = (φ ∘ diag)(μ(A)),

0 ≤ A ∈ 𝒥.

The functional φ̂ := φ ∘ diag is linear on the symmetric space J, vanishes on Z(J), and (1.1) is satisfied. Therefore, the study of traces (linear functionals on a two-sided ideal that vanish on commutators) is equivalent to the study of linear functionals on the commutative core that vanish on the center.

1.3 Examples of traces |

17

The reduction to sequence spaces and symmetric functionals is why we can answer so many questions about existence, uniqueness, and other properties of traces. As mentioned above, Pietsch provided an equivalent description of the center of a symmetric sequence space and the linear functionals that vanish on the center. Pietsch’s approach is described in Chapter 4 and, in many cases of interest, it provides an explicit formula for the symmetric functional.

1.3 Examples of traces The matrix trace is a trace on the two-sided ideal of finite-rank operators. We describe two other examples of a trace on a two-sided ideal of compact operators. Their construction exemplifies (1.1). The first example is an extension of the matrix trace to the ideal of trace class operators [233], written in abstract form by John von Neumann [283], but precipitated by many others. Historical notes are at the end of this chapter. The second example is the Dixmier trace, which was constructed by Jacques Dixmier [76]. The restriction of the Dixmier trace to finite-rank operators vanishes, which answers in the negative whether every trace on an ideal of compact operators is an extension of the matrix trace.

1.3.1 The trace Let 0 ≤ A ∈ ℒ1 , where ℒ1 is the Schatten ideal in Example 1.2.5 for p = 1. The sequence of singular values of A is summable, μ(A) ∈ l1 . The sum is a linear functional on l1 , so define ∞

Tr(A) := ∑ μ(n, A), n=0

0 ≤ A ∈ ℒ1 .

(1.2)

The functional Tr is linear if the sum vanishes on Z(l1 ). We can avoid a direct computation by using two facts: (i) our knowledge from the spectral theorem that every positive compact operator has an orthonormal basis {fn }∞ n=0 of H with Afn = μ(n, A)fn , hence ∞

Tr(A) = ∑ ⟨Afn , fn ⟩, n=0

0 ≤ A ∈ ℒ1 ;

and (ii) under summation, for any orthonormal basis {en }∞ n=0 of H, ∞

∞

n=0

n=0

Tr(A) = ∑ ⟨Afn , fn ⟩ = ∑ ⟨Aen , en ⟩,

0 ≤ A ∈ ℒ1 .

18 | 1 What is a singular trace? By expressing Tr(A) using diagonal matrix entries with respect to a basis unrelated to A, it is immediate that Tr is linear. Its linear extension to all A ∈ ℒ1 coincides with the abstract formulation of the trace as the “sum of the diagonal” ∞

Tr(A) = ∑ ⟨Aen , en ⟩, n=0

A ∈ ℒ1 .

In this form Tr is an extension of the matrix trace. An operator A ∈ ℒ1 is called a trace class operator. 1.3.2 The Dixmier trace Let 0 ≤ A ∈ ℳ1,∞ , where ℳ1,∞ is the Lorentz ideal in Example 1.2.9. Generally, the partial sums n

∑ μ(j, A) = O(log(2 + n)),

j=0

0 ≤ A ∈ ℳ1,∞ , n ≥ 0

diverge, so it is not possible to use the sum, and the renormalization n 1 ∑ μ(j, A) = O(1), log(2 + n) j=0

0 ≤ A ∈ ℳ1,∞ , n ≥ 0

(1.3)

is not convergent. To obtain a scalar value, the renormalized sequence (1.3) can be ∗ combined with an element ω ∈ l∞ from the dual space of the bounded sequences, n 1 Trω (A) := ω({ ∑ μ(j, A)} log(2 + n) j=0

∞

), n=0

0 ≤ A ∈ ℳ1,∞ .

(1.4)

The positivity and linearity of Trω requires constraints on the element ω. In the course of this book, the discussion and the choice of particular linear functionals ω on l∞ will be examined due to the many formulas associated to the Dixmier trace. It is clear that if we would like Trω to be positive, we have to assume that ω ≥ 0. The difficulty in the formulation (1.4), and generally in the formulation (1.1), is its additivity. What is interesting is that the additivity of Trω implies that ω is a singular ∗ element of l∞ . We recall a few definitions. ∗ Denote by (l∞ )∗n and (l∞ )∗s the normal and singular parts of l∞ [146]. The normal ∗ ∗ part is equivalent to l1 , (l∞ )n ≅ l1 , and the singular part (l∞ )s is such that ∗ l∞ = (l∞ )∗n ⊕ (l∞ )∗s .

This formulation, and terminology, is the discrete analog of the decomposition of Borel measures on [0, ∞) into absolutely continuous and singular parts. The following the∗ orem shows, in particular, that if (1.4) is linear, then 0 ≤ ω ∈ l∞ must be singular.

1.3 Examples of traces |

19

∗ Conversely, if 0 ≤ ω ∈ l∞ is singular and of the particular form where it is invariant under the dilation operator σn : l∞ → l∞ , n ≥ 1:

σn (a(0), a(1), . . .) = (a(0), . . . , a(0), a(1), . .⏟,⏟⏟⏟⏟⏟⏟⏟⏟ a(1) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟.⏟⏟⏟ ⏟⏟, . . .), n times n times

a ∈ l∞ ,

then (1.4) defines a linear functional. Observe that ω ∘ σ2 = ω implies that ∞

n ∞ ω({a(n)}n=0 ) = ω({a(⌊ ⌋)} ), 2 n=0

a ∈ l∞ .

Indeed, we have ∞

n ∞ ω({a(⌊ ⌋)} ) = ω(a(0), a(0), a(1), a(1), a(2), a(2), . . .) = (ω ∘ σ2 )({a(n)}n=0 ). 2 n=0 ∗ Theorem 1.3.1. Let 0 ≤ ω ∈ l∞ be a positive linear functional on the algebra l∞ . (a) If Trω is additive on the positive cone of ℳ1,∞ , then ω is singular. (b) If ω is singular and if ω ∘ σ2 = ω, then Trω is additive on the positive cone of ℳ1,∞ and extends to a positive trace on ℳ1,∞ .

Proof. The first part is rather simple to prove, while the second part is less obvious. (a) Fix m ∈ ℕ. Let p, q ∈ ℒ(H) be mutually orthogonal projections, that is, p = p∗ = p2 , q = q∗ = q2 , and pq = qp = 0, such that Tr(p) = Tr(q) = m. Let χm denote the sequence {1, . . . , 1, 0, . . .} with m consecutive 1’s. We have μ(p) = μ(q) = χm and μ(p + q) = χ2m . It follows from (1.4) that Trω (p + q) = ω({

min{n + 1, 2m} } ), log(2 + n) n≥0

Trω (p) = Trω (q) = ω({

min{n + 1, m} } ). log(2 + n) n≥0

Since Trω (p + q) = Trω (p) + Trω (q), it follows that ω({

2 min{n + 1, m} − min{n + 1, 2m} } ) = 0. log(2 + n) n≥0

Since ω is a positive functional, it follows that 0 ≤ ω(χm ) ≤ ω({

2 min{n + 1, m} − min{n + 1, 2m} } ) = 0. log(2 + n) n≥0

Thus, ω vanishes on an arbitrary finite sequence. (b) Let A, B ∈ ℳ1,∞ be positive operators. We need the following fundamental inequalities for singular values. They are proved in Chapter 2 in Theorems 2.3.5 and 2.3.6. For every n ≥ 0, we have n

n

2n+1

k=0

k=0

k=0

∑ μ(k, A + B) ≤ ∑ μ(k, A) + μ(k, B) ≤ ∑ μ(k, A + B).

20 | 1 What is a singular trace? The first inequality immediately implies that Trω (A + B) ≤ Trω (A) + Trω (B). The second inequality implies that Trω (A) + Trω (B) ≤ ω({

2n+1 1 ∑ μ(k, A + B)} log(2 + n) k=0

).

n≥0

Since ω = ω ∘ σ2 , it follows that 2⌊ n ⌋+1

2 1 Trω (A) + Trω (B) ≤ ω({ ∑ μ(k, A + B)} ). n log(2 + ⌊ 2 ⌋) k=0 n≥0

Since ω is singular, it follows from the equality 2⌊ n ⌋+1

n 2 1 1 ∑ μ(k, A + B) = ∑ μ(k, A + B) + o(1) n log(2 + n) k=0 log(2 + ⌊ 2 ⌋) k=0

that Trω (A) + Trω (B) ≤ Trω (A + B). Thus, Trω is additive on the positive cone on ℳ1,∞ . Hence, Trω extends to a linear functional on ℳ1,∞ . The latter functional is unitarily invariant and is, therefore, a trace. The singular elements 0 ≤ ω ∈ (l∞ )∗s that are states, ω({1}∞ n=0 ) = 1, extend the ordinary limit ∞

ω({a(n)}n=0 ) = lim a(n), n→∞

∞

{a(n)}n=0 ∈ c,

from the space c of all convergent sequences to l∞ . The formulation (1.4) and the statement of Lemma 1.3.1(b) for dilation-invariant singular states are credited to Jacques Dixmier. The existence of dilation-invariant singular states is proved in Chapter 6. The trace Trω on ℳ1,∞ is called a Dixmier trace. Evidently, by restriction, a Dixmier trace is also a trace on the ideal ℒ1,∞ of weak trace class operators. Notice that ∞

n 1 1 1 Trω (diag({ } )) = ω({ } ) = 1, ∑ n + 1 n=0 log(2 + n) j=0 j + 1 n=0 ∞

so that Dixmier traces are not trivial. Notice also that, for a trace class operator 0 ≤ A ∈ ℒ1 , n 1 ∑ μ(j, A) = o(1), log(2 + n) j=0

n ≥ 0,

1.3 Examples of traces |

21

as n

∑ μ(j, A) = O(1),

j=0

n ≥ 0.

Then Trω (A) = 0,

0 ≤ A ∈ ℒ1 ,

since any singular state ω vanishes on o(1)-sequences. Thus Dixmier traces are not extensions of the trace Tr and vanish on ℒ1 . A Dixmier trace is an example of a singular trace on ℳ1,∞ . Definition 1.3.2. A trace φ : 𝒥 → ℂ on a two-sided ideal 𝒥 of the algebra ℒ(H) is a singular trace if it vanishes on finite-rank operators. Singular traces, since they vanish on finite-rank operators, have no analogue in finite-dimensional linear algebra. The identification in Part II of this book between traces on a two-sided ideal and symmetric functionals on the commutative core of the ideal will prove that Dixmier traces are not the only singular traces on ℳ1,∞ . Surprisingly, the same identification will show that the trace Tr is not the only trace on ℒ1 . The property that distinguishes Tr amongst other traces on ℒ1 is that it is monotone for submajorization, meaning that when n

n

k=0

k=0

∑ μ(k, A) ≤ ∑ μ(k, B),

0 ≤ A, B ∈ ℒ1 , n ≥ 0,

we have Tr(A) ≤ Tr(B). The formula (1.4) makes it clear that Dixmier traces are also monotone for submajorization. If n

n

k=0

k=0

∑ μ(k, A) ≤ ∑ μ(k, B),

0 ≤ A, B ∈ ℳ1,∞ , n ≥ 0,

then Trω (A) ≤ Trω (B). Part III proves that being monotone for submajorization characterizes Dixmier traces on ℳ1,∞ completely. The formulas from noncommutative geometry and mathematical physics associated to the Dixmier trace, including its association with the residue at the first simple pole of the zeta function z 󳨃→ Tr(A1+z ) as z → 0+ when 0 ≤ A ∈ ℒ1,∞ and −1

the asymptotics of the heat trace t 󳨃→ Tr(e−t(1+A) ) as t → 0+ when 0 ≤ A ∈ ℒ1,∞ , are derived in Part III from the identification of Dixmier traces with the traces monotone for submajorization.

22 | 1 What is a singular trace? 1.3.3 Spectral formulation of traces Theorem 1.1.22 stated the well-known result that the matrix trace in linear algebra can be written as the sum of the eigenvalues of the matrix. Is the same true for the extension of the matrix trace to trace class operators? Can the Dixmier trace be written as a formula involving eigenvalues? V. Lidskii showed in 1959 that if λ(A) = {λ(n, A)}∞ n=0 is an eigenvalue sequence of a trace class operator A ∈ ℒ1 then ∞

Tr(A) = ∑ λ(n, A), n=0

A ∈ ℒ1 .

This result does not follow trivially from (1.2), so much so that the formula is called the Lidskii formula. For a general trace φ on a two-sided ideal 𝒥 , the spectral formulation of φ is the demonstration that the linear extension of φ(A) = (φ ∘ diag)(μ(A)),

0 ≤ A ∈ 𝒥,

is, provided the eigenvalue sequence λ(A) belongs to the commutative core J, equivalent to φ(A) = (φ ∘ diag)(λ(A)),

A ∈ 𝒥.

The spectral formulation is not trivial because, as we recall from the examples on quasinilpotent operators in Section 1.1, eigenvalue sequences are not a linear extension of the singular value sequence μ : 𝒥+ → J+ . Kalton and Dykema [92], in their approach to the problem of whether traces on twosided ideals of compact operators are spectral, gave an example of a two-sided ideal 𝒥 with A ∈ 𝒥 such that λ(A) ∉ J where J is the commutative core of 𝒥 . That is, for any consistent choice of ordering of eigenvalue sequences, in general, λ : 𝒥 ↛ J. This indicates why the Calkin correspondence is based on singular values and not eigenvalue sequences. Kalton and Dykema identified a class of ideals 𝒥 for which λ:𝒥 →J does hold, and that, in the case of this class, the spectral formula holds for all traces on 𝒥 . Two of the authors extended the results of Kalton and Dykema and found the

1.4 Notes | 23

maximal class of ideals for which λ : 𝒥 → J. Specifically, an eigenvalue sequence λ(A) belongs to the commutative core J for every A ∈ 𝒥 if and only if n

n

k=0

k=0

∏ μ(k, B) ≤ ∏ μ(k, A),

n ≥ 0, B ∈ 𝒞0 (H)

implies that B ∈ 𝒥 . The condition is called logarithmic submajorization and the ideals that are solid for logarithmic submajorization form the maximal class of ideals for which traces on them are spectral. This result, described in Part II, provides an answer to the spectral formulation of traces on two-sided ideals of compact operators. Trace theorems for noncommutative geometry in Volume II rely on the spectral formulation of traces on ℒ1,∞ and estimates for eigenvalues of integral operators on ℝp “modulated” by the Laplacian operator.

1.4 Notes Compact operators For a history of the development of functional analysis, we refer the reader to J. Dieudonne [73], or the encyclopedic work of A. Pietsch [203]. An approachable introduction to the modern treatment of functional analysis is [219] by M. Reed and B. Simon. Innumerable works on C∗ -algebras and von Neumann algebras include the treatment of operators on Hilbert spaces, e. g., [77, 271]. Reference works are the tomes of N. Dunford and J. Schwartz [90, 91]. J. Retherford has written an introduction to functional analysis specifically aimed at an elementary proof of the Lidskii trace theorem [220]. Spectral theory The spectral theorem is probably the most important result in operator theory. Its milestones include D. Hilbert in 1906 [139] and F. Riesz in 1918 [221], who gave a complete treatment of what we now know as compact operators. Amazingly Hilbert conceived the continuous spectrum in 1906, and an associated spectral theorem in this case. Von Neumann in 1929 [280], and von Neumann [283] and M. Stone [252] in 1932, derived the modern version of the spectral theorem for bounded and unbounded operators and the abstract notion of a Hilbert space. L. Steen provides highlights in the history of spectral theory in [249]. H. R. Dowson provides a recommended exposition of the spectral theorem of Riesz operators and its historical development in Chapter 1 of [88]. The nonincreasing ordering of eigenvalues in a sequence with algebraic multiplicity goes back to Weyl [290, 291]. The term eigenvalue sequence is standard [247, p. 7].

24 | 1 What is a singular trace? Decomposition of compact operators Schmidt’s decomposition was published for integral operators by E. Schmidt in 1908 [235]. It appears in any introductory book on compact operators and singular values, e. g., [247, Chapter 1] and [201]. The orthonormal system {fn }∞ n=0 is defined by fn = UA en where UA is the partial isometry in the polar decomposition of A, see the next chapter. For historical comments see Pietsch [201], Steen [249], and Stewart [251]. The concept of a quasinilpotent operator originated from the theory of integral equations [278]. The decomposition of a compact operator into a normal compact operator (with the same eigenvalues with algebraic multiplicity) and a quasinilpotent compact operator follows from Theorems 2, 6, and 7 of J. Ringrose’s 1962 paper [224]. T. West gave the explicit statement in the context of Riesz operators in Section 7 of the 1966 paper [289]. Every compact operator on a separable Hilbert space is a Riesz operator. An exposition features in Chapter 3 of the monograph [88] by Dowson. For decomposition into normal and quasinilpotent parts see also [89, Theorem 8] in the context of spectral operators, but note that not all compact operators are spectral operators in the sense of Dunford. Calkin correspondence The fundamental 1941 paper of J. W. Calkin [36] was motivated by the “observation that the ring ℒ(H) of bounded everywhere defined operators in Hilbert space contains nontrivial two-sided ideals. This fact, which has escaped all but oblique notice in the development of the theory of operators, is of course fundamental from the point of view of algebra and at the same time differentiates ℒ(H) sharply from the ring of all linear operators over a unitary space with finite dimension number.”

The observation led Calkin to study and describe these ideals in terms of (commutative) ideals of sequence spaces and also led to the discovery of the Calkin algebra. Let H be a separable Hilbert space and let 𝒞0 (H) be the ideal of all compact operators on H. The ideal 𝒞0 (H) is norm closed in ℒ(H). The quotient C∗ -algebra 𝒞 = ℒ(H)/𝒞0 (H) is called the Calkin algebra. It is a noncommutative analogue of the quotient algebra l∞ /c0 . The sequence spaces lp , lp,∞ , p ≥ 1, and m1,∞ , and their corresponding ideals, ℒp , ℒp,∞ , and ℳ1,∞ , are extensively studied. M. Sh. Birman, M. Z. Solmjak, and others have derived a formidable collection of spectral and norm estimates [28, 29]. See also B. Simon’s book [247], where the proof is outlined for p > 1 that lp,∞ and mψp where

ψp (t) = t 1−1/p , t > 0, are the same subspace of c0 . The notations of ℒ1,∞ and l1,∞ came from the theory of weak-l1 spaces, see, e. g., [23]. The notations for symmetrically normed ideals used by Gohberg and Krein in the books [113, 114] are completely different, and inconsistent with the currently accepted notations of corresponding commutative objects like Lorentz spaces, see,

1.4 Notes | 25

e. g., [20, 159]. We also refer to the paper of Albrecht Pietsch [204], where he discussed the origin of the symmetrically normed ideal ℳ1,∞ and commutative space m1,∞ . The latter object was introduced by Sargent [230]. The term “Dixmier ideal” is often used in noncommutative geometry for ℳ1,∞ to refer to the dual of the Macaev ideal [179]. During the 1980s Albrecht Pietsch, while investigating the more general area of nuclear operators between Banach spaces [200], reworked the Calkin correspondence from the basis of dyadic decompositions [202]. Rearrangement-invariant sequence spaces A symmetric sequence space as introduced in Definition 1.2.2 is an ideal in the commutative algebra l∞ and is invariant with respect to the natural action of the symmetric group on that algebra (hence, the name). When E is equipped with a symmetric norm (or symmetric quasinorm), that is, with a norm ‖ ⋅ ‖ invariant with respect to the action of the symmetric group and such that ‖ab‖ ≤ ‖a‖‖b‖∞ for any a ∈ E and b ∈ l∞ , the symmetrically (quasi)normed space (E, ‖ ⋅ ‖E ) is one of the most fundamental objects in the field of Banach space geometry. We refer the reader to classic references such as [165] which records the development of the theory from the 1930s to 1977. The reader may also encounter the term “rearrangement-invariant (quasi)-Banach spaces” which is almost equivalent to our “symmetric (quasi)-Banach sequence space”, however, we caution the reader against identifying them. As a rule, “rearrangement-invariant sequence spaces” form a proper subclass of “symmetric sequence spaces.” Traces Traces of operators first appeared in E. I. Fredholm’s 1903 work [108] on integral operators. Solving an integral equation for continuous functions in a manner similar to solving linear equations, Fredholm developed an infinite-dimensional version of the determinant (see also H. von Koch [279]). The linear coefficient in the Fredholm series representing the infinite-dimensional characteristic equation, that is, the trace, took the form b

∫ K(x, x) dx a

for an integral operator SK with symmetric kernel K in C([a, b]2 ). D. Hilbert in 1904 [138] and E. Schmidt in 1907 [234] developed a spectral theorem for such integral operators, indicating the existence of an orthonormal system of ∞ continuous eigenvectors {un }∞ n=0 and eigenvalues {λn }n=0 (the terms were introduced by Hilbert in [138]) with ∞

K(x, y) = ∑ λn un (x)un (y), n=0

26 | 1 What is a singular trace? or in modern terms the spectral theorem ∞

SK = ∑ λn un ⊗ un . n=0

Hilbert’s work in 1906 on orthonormal systems [139] considered by Dieudonne [73, p. 110] as the paper establishing functional analysis as a new field in itself, allowed it to be shown when K(x, x) > 0 and ∑∞ n=0 ⟨SK en , en ⟩ < ∞ for a complete orthonormal ∞ system {en }n=0 , b

∞

∞

n=0

n=0

∫ K(x, x) dx = ∑ λn = ∑ ⟨SK en , en ⟩, a

see J. Mercer [184]. Thus the linear coefficient in the Fredholm series becomes recognizable to modern eyes as the canonical trace of a positive integral operator with continuous kernel. While it was clear that this linear coefficient was an extension of the matrix trace, see Koch [279], it was not named explicitly by these authors as a trace. Pietsch noted that Lalesco used the term trace in 1912 in a book on integral equations [208, p. 53], and traced the term “trace” in linear algebra to Dedekind [208, p. 52]. The general theory of bounded and compact operators developed over the period from 1906 to 1932, and, according to Pietsch [203, p. 137], J. von Neumann in 1932 [283] first explicitly wrote the trace (or “Spur”) ∞

Tr(S) = ∑ ⟨Sen , en ⟩ n=0

of a positive trace class operator S in an abstract Hilbert space with orthonormal basis {en }∞ n=0 , and explicitly showed independence of the value on the underlying orthonormal basis. In fact, von Neumann wrote it for all positive operators, allowing the value ∞, and foreshadowed the notion of a semifinite trace. From the spectral theorem, which had by then been established for all self-adjoint compact operators, it was clear that the trace of a positive trace class operator was equal to the sum of its eigenvalues. The same statement for every trace class operator would have to wait until V. B. Lidskii in 1959 [164], though there are indications that A. Grothendieck knew the result in 1955 [203, p. 404]. Schatten and von Neumann in 1946 [233] established the ideal of trace class operators and its duality with the compact operators, and preduality with the bounded operators. Inspired by Schatten and von Neumann’s class of trace class operators, the study of traces of nuclear operators on Banach spaces (that are not Hilbert spaces) was initiated by Ruston and Grothendieck, see [201]. This theory is more complicated and in general the matrix trace may not extend to the class of all nuclear operators [201, p. 213].

1.4 Notes | 27

From the origin of functional analysis, there has been a fundamental interplay between traces and determinants of linear operators, see [115] and [201]. The additivity of Dixmier’s trace was proved by Dixmier, under slightly different conditions [76]. Construction of traces Early in the 1950s Halmos proved that every bounded operator of the Hilbert space to itself is the sum of two commutators involving bounded operators [124, 125]. Brown, Pearcy, and Topping extended this in the 1970s to show that every compact operator is a sum of four commutators involving compact operators, and every Hilbert–Schmidt operator is a sum of four commutators involving Hilbert–Schmidt operators [35, 106, 126, 197]. From Lemma 1.2.11 it follows that there is no trace on the ideal of compact operators, and similarly there is no trace on the ideal of Hilbert–Schmidt operators. A fundamental question in functional analysis became whether the trace on the set of trace class operators was the only (nonzero) trace on a two-sided ideal of bounded operators. Jacques Dixmier’s construction of a nonzero trace on the ideal of compact operators whose series of singular values diverge as log(n) appeared in the 1960s [76]. From 1988 Alain Connes developed the theory of Dixmier’s trace [57, 58], and later with Henri Moscovici [71]. Alain Connes coined the name “Dixmier trace” in the 1988 publication [58] (and Dixmier’s trace was the topic of Connes’ course at College de France the year before). An extensive body of work followed with other authors generalizing the construction and applications initiated by Connes, e. g., [5, 6, 83, 121, 122, 168, 191– 193]. Other approaches to constructing singular traces developed independently. During the 1980s Albrecht Pietsch, working in the more general area of continuous operators between Banach spaces [200], identified a bijective correspondence between traces on two-sided ideals of continuous operators between Hilbert spaces and translation-invariant functionals [202]. Nigel Kalton answered Pietsch’s question from 1981 [199] concerning continuous traces different from “the trace” on proper quasiBanach ideals of the trace class ideal ℒ1 in 1987 by constructing singular traces [148, Theorem 6] on certain quasi-Banach ideals. Kalton also identified which quasi-Banach ideals within ℒ1 possessed no singular traces [148, Theorem 6]. Kalton in 1989 identified the commutator subspace of trace class operators studied by Gary Weiss [149, 288] and Anderson and Vaserstein [9, 10]. In doing so, Kalton proved the existence of traces different from “the trace” on ℒ1 [149]. An alternative unpublished proof of the same fact using shift invariant functionals on l1 is due to Tadeusz Figiel [149, p. 73]. Figiel learnt of Pietsch’s approach in 1981 (Srní, January 1981) [199, p. 89] and at a meeting in Georgenthal (East-Germany, April 1986). Pietsch’s paper [205] details a complete construction of traces on the Banach space equivalent

28 | 1 What is a singular trace? of the ideal ℳ1,∞ . It also notes the unpublished contributions of T. Figiel to the theory of constructing traces on ideals of compact operators. The commutator approach was subsequently developed in the 1990s for arbitrary ideals by Dykema, Figiel, Weiss, and Wodzicki [94]. The approach can be used to define traces and prove their existence [93, 94, 144, 294]. As described in the text, the correspondence of J. W. Calkin is used to transfer the construction and existence of symmetric functionals [83, 84, 107, 151, 152] on sequence or function spaces to the study of traces on operator ideals. The 1989 paper of Varga [274] contained an approach to the existence of traces on ideals generated by a single compact operator. With the exposition of Varga’s approach and its extension and generalization to general semifinite von Neumann algebras, we also refer to [83] and [120]. In the paper [294], M. Wodzicki introduced a concept for a Dixmier trace on an arbitrary operator ideal. Lidskii’s formula Lidskii in 1959 [164] used determinants to prove that the trace of every trace class operator is given by the sum of its eigenvalues. As mentioned above, Retherford’s text [220] is devoted to an elementary proof of Lidskii’s trace theorem. Also as mentioned, the result is not emulated in general Banach spaces. Determinant-free proofs of Lidskii’s theorem that are based on the Ringrose upper-diagonal form, such as those of Erdos [99] and Power [215], are those that lend themselves to extension to other operator ideals in the Hilbert space theory. Kalton in 1998 [150] developed the spectral version of the commutator characterization to investigate which traces are determined completely by eigenvalues [92, 150]. Spectrality of traces had been raised earlier in the Banach space setting by Pietsch [199]. Kalton and Dykema in 1998 [92] essentially solved the question of whether every trace on a two-sided ideal 𝒥 is spectral. By introducing logarithmic submajorization, Kalton and Dykema’s approach was completed by two of the authors [264]. This approach is described in Chapter 5, so we reserve further remarks until then.

2 Singular values and submajorization The previous chapter motivated the study of singular traces on ideals of the algebra of bounded operators on a separable Hilbert space ℒ(H). This chapter provides the prerequisites for their study. We introduce topologies on ℒ(H), the lattice of projections in ℒ(H), and the singular value function of a bounded operator. The singular value function is both an extension of the decreasing arrangement of sequences and an extension of the approximation numbers of matrices. We introduce the notion of a symmetric functional on an ideal of ℒ(H) as an invariant of singular values and show that it corresponds with a trace. The general theory of traces on ideals of ℒ(H) in Part II concerns identifying symmetric functionals on the ideal with symmetric functionals on its commutative core. This chapter introduces two preorders related to fundamental observations of H. Weyl (Lemmas 1.1.20 and 1.1.21 in Chapter 1). The first preorder, Hardy–Littlewood submajorization, is central in the classification of Dixmier traces and formulas for the Dixmier trace in terms of residues and asymptotics of a heat trace in Part III. The second preorder, logarithmic submajorization, is central to the spectral formulas for traces in Part II. Fundamental results on sublinearity of the singular value function and eigenvalue sequences are proven in this chapter. When paired with an identification of the common kernel of symmetric functionals, they prove the spectral formula for traces in Chapter 5. The background and context for this chapter is extensive, including classical sequence spaces and the operator theory of C ∗ - and von Neumann algebras. We reference here only those results required for later sections, or as motivation for later material. The background theories are comprehensively treated elsewhere, for example in, [20, 77, 159, 201, 214, 247, 271].

2.1 Projections in ℒ(H) In this section we introduce topologies on the algebra ℒ(H) and define the lattice of projections Proj(ℒ(H)). This is a special case of the theory of semifinite von Neumann algebras, which can be thought of as an extension of measure theory to operator algebras. Using spectral projections we describe the spectral theorem for bounded normal operators on H, extending Theorem 1.1.12 in Chapter 1. Topologies on ℒ(H) and von Neumann algebras Let H be a separable Hilbert space. We consider three basic topologies on ℒ(H). The norm (or uniform) topology on ℒ(H) is given by the operator norm ‖ ⋅ ‖∞ as mentioned in Chapter 1. The strong operator topology, also mentioned earlier, is the topology of pointwise convergence on ℒ(H): a basis of the neighborhoods of an element A ∈ ℒ(H) https://doi.org/10.1515/9783110378054-002

30 | 2 Singular values and submajorization in this topology is given by the sets Uϵ (A, {ξj }ni=1 ) := {B ∈ ℒ(H) : ∀j ‖(B − A)ξj ‖H < ϵ},

ϵ > 0, n ≥ 1,

where ξj ∈ H, i = 1, . . . , n. The weak operator topology on ℒ(H) is the topology in which the basis of the neighborhoods of an element A ∈ ℒ(H) is given by the sets Vϵ (A, {ξj }nj=1 , {ηj }nj=1 ) := {B ∈ ℒ(H) : |⟨ξj , (B − A)ηj ⟩| ≤ ϵ, ∀j},

ϵ > 0, n ≥ 1,

where ξj , ηj ∈ H, i = 1, . . . , n. Equivalently, a net {Ai }i∈𝕀 in ℒ(H) converges to A in the strong operator topology if Ai ξ → Aξ for all ξ ∈ H, and in the weak operator topology if ⟨Ai ξ , η⟩ → ⟨Aξ , η⟩ for all ξ , η ∈ H. Let ℳ be a subset of ℒ(H). The commutant ℳ󸀠 is the set 󸀠

ℳ := {B ∈ ℒ(H) : AB = BA, ∀A ∈ ℳ}.

The bicommutant ℳ󸀠󸀠 of ℳ is the commutant of ℳ󸀠 . The intersection of ℳ and its commutant ℳ󸀠 is called the center of ℳ. A subalgebra ℳ of ℒ(H) is unital if 1 ∈ ℳ and self-adjoint if A ∈ ℳ implies ∗ A ∈ ℳ. The famous von Neumann bicommutant theorem [281], [32, §2.4] says that a unital self-adjoint subalgebra ℳ ⊂ ℒ(H) coincides with its bicommutant ℳ󸀠󸀠 if and only if it is closed in the strong operator topology if and only if it is closed in the weak operator topology. An algebra ℳ satisfying one of these conditions is called a von Neumann algebra. The self-adjoint subalgebras 𝒩 ⊂ ℒ(H) closed in the uniform topology are called C∗ -algebras. The class of C∗ -algebras is significantly larger than that of von Neumann algebras. Definition 2.1.1. A projection p ∈ ℒ(H) is a self-adjoint idempotent p = p∗ = p2 . The Hilbert subspace pH is called the range of the projection p, also denoted Ran(p). The rank of the projection p is the dimension of pH. For each Hilbert subspace H0 of H, there exists a unique projection p such that pH = H0 . The bicommutant theorem implies that a von Neumann algebra is fully determined by its projections. This differs markedly from a C∗ -algebra which may not contain any nontrivial projections. A von Neumann algebra ℳ is called a factor if its center is isomorphic to ℂ. That B ∈ ℒ(H)󸀠 commutes with all rank-one projections identifies B as a scalar multiple of the identity 1 ∈ ℒ(H). Hence ℒ(H) is a factor. Recall that a self-adjoint operator A ∈ ℒ(H) is called positive if ⟨Aξ , ξ ⟩ ≥ 0 for all ξ ∈ H. The collection of all positive elements of ℒ(H) is denoted by ℒ(H)+ . This set is a proper closed cone in ℒ(H) and it induces a partial order in the set of all self-adjoint operators from ℒ(H) by setting A ≤ B if B − A ∈ ℒ(H)+ . Vigier’s theorem states that the von Neumann algebra ℒ(H) has the least upper bound property. Theorem 2.1.2 (Vigier’s theorem). If {Ai }i∈I is an increasing net in ℒ(H)+ , bounded from above by B in ℒ(H), then there exists A ∈ ℒ(H)+ such that Ai ↑ A strongly and A ≤ B.

2.1 Projections in ℒ(H)

| 31

The lattice of projections and measure theory Let Proj(ℒ(H)) denote the set of all projections in ℒ(H). The range Ran(p) of a projection p is a closed linear subspace of H, and every closed linear subspace is the range of a projection in Proj(ℒ(H)). The set Proj(ℒ(H)) is a lattice, with supremum and infimum determined by Ran(p ∨ q) := Ran(p) + Ran(q), Ran(p ∧ q) := Ran(p) ∩ Ran(q). Besides the von Neumann algebra ℒ(H), the only other von Neumann algebras relevant to this book are the algebra L∞ (X) of essentially bounded functions on a σ-finite measure space (X, Σ, ρ) and the special case of the bounded complex-valued sequences l∞ . Both are realized as von Neumann algebras on the separable Hilbert space of square integrable functions, or sequences, respectively. The lattice operations of Proj(ℒ(H)) generalize set union and intersection in the measure space. Example 2.1.3. Let (X, Σ, ρ) be a σ-finite measure space. Let L2 (X) denote the Hilbert space of all (equivalence classes of) square integrable functions on X, and let L∞ (X) denote the algebra of all (equivalence classes of) bounded functions on X. (a) The mapping f → Mf is an injective ∗-homomorphism from L∞ (X) into ℒ(L2 (X)). Here, the operator of pointwise multiplication Mf is given by the formula (Mf x)(t) = f (t)x(t),

x ∈ L2 (X).

(b) Every set A ∈ Σ generates a projection χA ∈ L∞ (X). Moreover, for every projection p ∈ L∞ (X), there exists a set A ∈ Σ such that p = χA . (c) If p = χA and q = χB , then p ∨ q = χA∪B ,

p ∧ q = χA∩B .

That is, the operations ∨, ∧ on the Boolean algebra of projections Proj(ℒ(H)) correspond to the operations ∪, ∩ on the σ-algebra Σ. (d) If X = (0, 1) or (0, ∞), Σ is the σ-algebra of Lebesgue measurable sets and ρ is Lebesgue measure, then L∞ (X) = L∞ (0, 1) or L∞ (X) = L∞ (0, ∞). (e) If X = ℤ+ , Σ = 2ℤ+ and ρ is counting measure, then L∞ (X) = l∞ . A commutative von Neumann algebra ℳ (that is, where AB = BA for all A, B ∈ ℳ) is called maximal if there are no commutative von Neumann algebras strictly containing ℳ. The von Neumann algebra L∞ (X) is a maximal commutative von Neumann subalgebra of ℒ(L2 (X)). Example 2.1.3 is especially important because of the following theorem. Theorem 2.1.4. A commutative von Neumann algebra on a separable Hilbert space H is isometrically isomorphic to L∞ (X) for some measure space (X, Σ, ρ).

32 | 2 Singular values and submajorization Example 2.1.3 indicates that the lattice of projections Proj(ℒ(H)) is an extension of the concept of a σ-algebra. If {pi }i∈I ⊂ Proj(ℒ(H)), then Vigier’s theorem (Theorem 2.1.2) implies that we can define ⋁ pi , ⋀ pi ∈ Proj(ℒ(H)) i∈I

i∈I

as the unique projections onto the closed linear span of {pi H}i∈I and onto the intersection ⋂i∈I pi H, respectively. If I is a countable set, then the operations ⋁i∈I and ⋀i∈I generalize the notions of countable union and countable intersection, respectively. The notion of “containment” A ⊆ B is generalized by the lattice of projections by stating that a projection p contains the projection q if q ≤ p (or equivalently that p∧q = q). An important difference between the lattice of measurable sets and the lattice of projections is that for the latter we do not have the distributive law. For general projections p, q, r ∈ Proj(H), it can happen that p ∨ (q ∧ r) ≠ (p ∨ q) ∧ (p ∨ r),

p ∧ (q ∨ r) ≠ (p ∧ q) ∨ (q ∧ r).

A projection p ∈ Proj(ℒ(H)) is said to be abelian if the reduced von Neumann algebra pℒ(H)p ⊆ ℒ(pH) is commutative. Example 2.1.5. Let {en }∞ n=0 be a complete orthonormal basis of a separable Hilbert space H. The rank-one projections pn := en ⊗ en : ξ → ⟨ξ , en ⟩en , n ≥ 0, introduced in Chapter 1, describe the minimal projections in ℒ(H). Since pn ℒ(H)pn is one-dimensional, each pn is an abelian projection. Spectral projections The σ-algebra of all Borel subsets of ℂ is denoted by ℬ(ℂ). A projection-valued measure E is a map E : ℬ(ℂ) → Proj(ℒ(H)) with E(ℂ) = 1 and such that the map F 󳨃→ ⟨E(F)ξ , η⟩,

F ∈ ℬ(ℂ)

is a complex-valued Borel measure for each ξ , η ∈ H. If A is a compact normal operator with eigenvalue λ ∈ σ(A), let EA (λ) be the projection onto the eigenspace of λ. Define EA as the projection valued counting measure assigning the projection EA (λ) to the points λ ∈ σ(A) \ {0} and the zero projection to all other points in ℂ. Then the spectral theorem for compact operators is equivalent to the statement A = ∫ λ dEA (λ), ℂ

2.1 Projections in ℒ(H)

| 33

which is shorthand notation for ⟨Aξ , η⟩ := ∫ λ d⟨EA (λ)ξ , η⟩,

ξ , η ∈ H.

ℂ

A similar result exists for bounded normal operators. Theorem 2.1.6 (Spectral theorem). If A ∈ ℒ(H) is a normal operator, then there exists a uniquely determined projection-valued measure EA : ℬ(ℂ) → Proj(ℒ(H)) supported on σ(A) such that A = ∫ λ dEA (λ). ℂ

If f is a Borel function bounded on σ(A), then f (A) = ∫ f (λ) dEA (λ) ℂ

defines a bounded operator f (A) ∈ ℒ(H). The projection-valued measure EA is called the spectral measure of A and the projections EA (F), F ∈ ℬ(ℂ), are called the spectral projections of A. If A is self-adjoint (respectively, positive), then the projection-valued measure EA is supported on ℝ (respectively, ℝ+ ). The projection EA (0, ∞) for a positive operator A ∈ ℒ(H) is frequently called the support projection of A. For a self-adjoint operator A ∈ ℒ(H), set A+ = AEA [0, ∞),

A− = −AEA (−∞, 0).

Since A = A+ − A− , it follows that every self-adjoint operator is a linear combination of two positive operators. The absolute value or modulus of a self-adjoint operator can be defined as |A| = A+ + A− . Every operator A ∈ ℒ(H) can be uniquely represented as a linear combination of two self-adjoint operators by the formula A = ℜA+iℑA. Here, the self-adjoint operators ℜA and ℑA are defined by setting 1 ℜA = (A + A∗ ), 2

ℑA =

1 (A − A∗ ). 2i

34 | 2 Singular values and submajorization Thus, every operator A ∈ ℒ(H) can be represented as a linear combination of four positive operators. The absolute value of an arbitrary operator A ∈ ℒ(H) is defined by Theorem 2.1.6 using the square root function applied to the positive operator A∗ A, |A| = √A∗ A. This is consistent with the definition |A| = A+ + A− when A is self-adjoint. A partial isometry is an operator U ∈ ℒ(H) such that U ∗ U = q and UU ∗ = p where p, q ∈ Proj(ℒ(H)). Equivalently, U : qH → pH is an isometry. For every operator A ∈ ℒ(H), there exists a partial isometry U ∈ ℒ(H) such that A = U|A|. This is called the polar decomposition of A.

2.2 Singular values This section introduces the singular value function of a bounded operator, which extends the singular value sequence of a compact operator. The singular value function is one of the central concepts in noncommutative measure theory. Conceptually, it is based on the classical distribution function, using instead the trace of projections in ℒ(H) as a measure of the size of sets. Volume III will introduce the singular value function associated to a semifinite faithful normal trace on a semifinite von Neumann algebra, and the affiliated spaces of measurable operators which correspond to unbounded measurable functions. The theory on ℒ(H) is simpler, and the material in this section can be found presented in such books as Gohberg and Krein [113, 114], and Simon [247]. Definition of the singular value function Definition 2.2.1. For every A ∈ ℒ(H) the singular value function of A, μ(A) : t → μ(t, A),

t ≥ 0,

is defined by the formula μ(t, A) = inf{‖A(1 − p)‖∞ : p ∈ Proj(ℒ(H)), Tr(p) ≤ t}. Since the trace of a projection is an integer, μ(A) is constant on intervals [n, n + 1) for n ∈ ℤ+ , and μ(A) is a step function ∞

μ(t, A) = ∑ μ(n, A)χ[n,n+1) (t), n=0

t ≥ 0.

2.2 Singular values | 35

Equivalently, the function μ(A) can be defined in terms of the spectral distribution function nA of the operator A ∈ ℒ(H). We set nA (s) := Tr(E|A| (s, ∞)),

s ≥ 0,

(2.1)

where E|A| denotes the spectral measure of the operator |A|. The singular value function can be expressed as follows: μ(t, A) = inf{s ≥ 0 : nA (s) ≤ t},

t ≥ 0.

(2.2)

Direction to the proof of this formula can be found in the end notes. The singular value function is an analogue of the classical notion of decreasing rearrangement [20]. Remark 2.2.2. Given a function x ∈ L∞ (0, ∞), the distribution function of x is defined by nx (s) = m({|x| > s}),

s≥0

where m is Lebesgue measure. The decreasing rearrangement t → μ(t, x) of the function x is μ(t, x) = inf{s ≥ 0 : nx (s) ≤ t},

t ≥ 0.

Volume III introduces the singular value function of operators affiliated to semifinite von Neumann algebras, which is jointly the generalization of the singular value function of operators A ∈ ℒ(H) and the decreasing rearrangement of functions x ∈ L∞ (0, ∞) and sequences x ∈ l∞ . Remark 2.2.3. If A ∈ ℒ(H), then the function t 󳨃→ μ(t, A) is the right inverse of the function nA , in the sense that for all t ≥ 0 we have nA (μ(t, A)) = t. However, it is not the left inverse, since μ(nA (t), A) = inf{s ≥ 0 : nA (s) ≤ nA (t)} ≤ t and the inequality can be strict. The singular value function extends the notion of the singular value sequence of a compact operator to all bounded operators. Lemma 2.2.4. For A ∈ ℒ(H) the function μ(A) is a step function with μ(t, A) = μ(n, A),

t ∈ [n, n + 1).

When A is a compact operator, μ(n, A), n ≥ 0, are the singular values of A.

36 | 2 Singular values and submajorization Proof. Let n ∈ ℤ+ . Note that Tr(p) is an integer for any projection p ∈ ℒ(H). Hence, if t ∈ [n, n + 1) then the condition Tr(p) ≤ t and Tr(p) ≤ n are equivalent. From the definition, μ(t, A) = inf{‖A(1 − p)‖∞ : p ∈ Proj(ℒ(H)), Tr(p) ≤ t}, we obtain μ(t, A) = μ(n, A),

t ∈ [n, n + 1).

Now, μ(n, A) = inf{s ≥ 0 : n|A| (s) ≤ n} = μ(n, |A|),

n ≥ 0.

If A is compact, then so is |A| and it follows from that the spectral decomposition of |A| in Theorem 1.1.12 that μ(n, |A|) is the nth largest eigenvalue of |A|. The following properties of the singular value function follow almost immediately from the definition. Lemma 2.2.5. Let A, B ∈ ℒ(H). (a) The function t → μ(t, A), t ≥ 0, is decreasing and right-continuous. (b) μ(0, A) = ‖A‖∞ . (c) The function μ(t, A) = μ(t, |A|) for all t ≥ 0. (d) If α ∈ ℂ, then μ(t, αA) = |α|μ(t, A) for all t ≥ 0. (e) If 0 ≤ B ≤ A, then μ(t, B) ≤ μ(t, A) for all t ≥ 0. The singular values of a finite rank operator A measure how well the operator can be approximated by operators of the rank strictly less than that of A, that is, μ(n, A) = inf{‖A − B‖∞ : rank(B) ≤ n},

n ≥ 0.

A similar formula, enhancing that of Definition 2.2.1, holds for an arbitrary operator A ∈ ℒ(H). Theorem 2.2.6. For every operator A ∈ ℒ(H), we have μ(t, A) = inf{‖A − B‖∞ : B ∈ ℒ(H), nB (0) ≤ t},

t ≥ 0.

For compact operators, optimal approximation by finite-rank operators defines the Schmidt decomposition in Theorem 1.1.14. The finite-rank operator n−1

Bn = ∑ μ(n, A)fn ⊗ en , k=0

n ≥ 0,

2.2 Singular values | 37

from the statement of Theorem 1.1.14 satisfies Tr(Bn ) ≤ n and attains the infimum μ(n, A) = ‖A − Bn ‖∞ which minimizes, in the uniform operator norm, the approximation of A by a finiterank operator of rank n. Properties of the singular value function We derive the inequalities of Fan. Fan’s inequalities describe how the singular values behave under addition and multiplication in the algebra ℒ(H). The usual triangle inequality for the absolute value of complex numbers is no longer valid for the absolute value of operators, that is, it may be that |A + B| ≰ |A| + |B|,

A, B ∈ ℒ(H).

Lemma 2.2.7 provides a very useful substitute. Lemma 2.2.7. Let Ak ∈ ℒ(H), k ≥ 0. There exist partial isometries Uk ∈ ℒ(H), k ≥ 0, such that 󵄨󵄨 n 󵄨󵄨 n 󵄨󵄨 󵄨 󵄨󵄨 ∑ Ak 󵄨󵄨󵄨 ≤ ∑ U ∗ |Ak |Uk , k 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 󵄨󵄨 k=0

n ≥ 0.

Lemma 2.2.8. Let 0 ≤ A, B ∈ ℒ(H). Then nA+B (0) ≤ nA (0) + nB (0). Proof. If either nA (0) or nB (0) is infinite then the result is trivial, and therefore we restrict attention to the case that nA (0) + nB (0) < ∞. Assume that nA (0) + nB (0) − nA+B (0) < 0. In this case, the dimension of the kernel of A + B exceeds the sum of the dimensions of the kernels of A and B, and hence there exists x ∈ H such that (A + B)x = 0, but x is orthogonal to the kernels of A and B. Thus, 0 = ⟨(A + B)x, x⟩ = ⟨Ax, x⟩ + ⟨Bx, x⟩. Since A and B are positive, it follows that ⟨Bx, x⟩ = ⟨Ax, x⟩ = 0 and thus Bx = 0 and Ax = 0, which contradicts the assumption that x is orthogonal to the kernels of A and B.

38 | 2 Singular values and submajorization The following corollary states the inequalities of Fan [105, 113]. They underlie the Calkin correspondence and many other results. In view of their importance, we provide short proofs. Corollary 2.2.9. Let A, B ∈ ℒ(H) and t, s ≥ 0. Then (a) μ(t + s, A + B) ≤ μ(t, A) + μ(s, B). (b) μ(t + s, AB) ≤ μ(t, A)μ(s, B). Proof. The proofs of the estimates are very similar. Fix ε > 0. By Theorem 2.2.6, there exists A1 such that nA1 (0) ≤ t,

‖A − A1 ‖∞ ≤ μ(t, A) + ε.

Similarly, there exists B1 such that nB1 (0) ≤ s,

‖B − B1 ‖∞ ≤ μ(s, B) + ε.

(a) We have ‖(A + B) − (A1 + B1 )‖∞ ≤ μ(t, A) + μ(s, B) + 2ε. By Lemma 2.2.7, there exist partial isometries U and V such that |A1 + B1 | ≤ U|A1 |U ∗ + V|B1 |V ∗ . Lemma 2.2.8 implies that nA1 +B1 (0) ≤ t + s. Since ε is arbitrarily small, the result follows by using Theorem 2.2.6 again. (b) Define an operator X by the formula X = AB − (A − A1 )(B − B1 ). It is clear that ‖AB − X‖∞ = ‖(A − A1 )(B − B1 )‖∞ ≤ (μ(t, A) + ε)(μ(s, B) + ε). Note that X = A1 (B − B1 ) + AB1 . It follows from Definition 2.2.1 and the assumption nB1 (0) ≤ s that μ(s, B1 ) = 0. Similarly μ(t, A1 ) = 0. By Definition 2.2.1 again, we obtain that μ(s, AB1 ) = 0 and nAB1 (0) ≤ s. Similarly arguing, we obtain that nA1 (B−B1 ) (0) ≤ t. From nAB1 (0) ≤ s,

nA1 (B−B1 ) (0) ≤ t,

we infer that nX (0) ≤ t + s. Since ε is arbitrarily small, the result follows by using Theorem 2.2.6 again.

2.2 Singular values | 39

Some additional properties of the singular value function are summarized in the following corollary. Corollary 2.2.10. Let A, B ∈ ℒ(H) and t ≥ 0. (a) Then μ(t, AB) ≤ ‖B‖∞ μ(t, A),

μ(t, BA) ≤ ‖B‖∞ μ(t, A),

μ(t, A∗ ) = μ(t, A).

(b) The function μ(A) is unitarily invariant, μ(t, A) = μ(t, U ∗ AU) for every unitary U ∈ ℒ(H) and, in fact, μ(t, UA) = μ(t, AU) = μ(t, A). (c) The operation A → μ(t, A) is continuous in the uniform norm on ℒ(H). More precisely, |μ(s, A) − μ(s, B)| ≤ ‖A − B‖∞ ,

∀s ≥ 0.

(d) If A ≥ 0 and if f : ℝ+ → ℝ+ is a continuous and increasing function, then μ(f (A)) = f (μ(A)). Proof. (a) The inequalities follow from Corollary 2.2.9 and the inequality μ(B) ≤ ‖B‖∞ . If A = U|A| is a polar decomposition of A, then |A∗ | = U|A|U ∗ . Therefore, μ(t, A∗ ) = μ(t, |A∗ |) = μ(t, U|A|U ∗ ) ≤ μ(t, |A|) = μ(t, A). Substituting A∗ in place of A, we obtain the reverse inequality μ(t, A) ≤ μ(t, A∗ ). (b) From (a) we have μ(t, U ∗ AU) ≤ μ(t, A) since ‖U‖∞ = ‖U ∗ ‖∞ = 1. Similarly, μ(t, A) = μ(t, U(U ∗ AU)U ∗ ) ≤ μ(t, U ∗ AU). (c) We apply Corollary 2.2.9 and the property μ(t, A) → ‖A‖∞ as t → 0+ for every A ∈ ℒ(H). (d) It is sufficient to establish the equality only for those values of t such that f is not a constant in the neighborhood of t. For such values of t, we have, by the spectral theorem, Ef (A) (f (t), ∞) = EA (t, ∞). Doubling under the singular value function and weak domination If A, B ∈ ℒ(H), then the direct sum A ⊕ B is conventionally defined as a linear operator on ℒ(H ⊕ H) given by (A ⊕ B)(x ⊕ y) = Ax ⊕ By. Dealing with the direct sum of Hilbert spaces will be inconvenient, and therefore we adopt the following notion of direct sum which will be more useful in the text. Definition 2.2.11. Let Ak ∈ ℒ(H), k ≥ 0 satisfy supk≥0 ‖Ak ‖∞ < ∞. If pk ∈ ℒ(H), k ≥ 0, are pairwise orthogonal projections, and if Bk ∈ pk ℒ(H)pk are such that μ(Bk ) = μ(Ak ), k ≥ 0, then we write ⨁ Ak = ∑ Bk . k≥0

k≥0

40 | 2 Singular values and submajorization Note that we have not strictly defined ⨁k≥0 Ak as an operator on H. There is freedom in the choice of the operators Bk . Since we will be concerned only with the singular value function μ(⨁k≥0 Ak ) rather than the direct sum operator itself, the specific choices of Bk will not be important. The key property of the direct sum operation relates to dilation. This is used repeatedly in the book to identify the scaling behavior of traces under the dilation operator. Define the dilation operator σ2 : L∞ (0, ∞) → L∞ (0, ∞) by the formula σ2 (x)(t) = x(t/2),

x ∈ L∞ (0, ∞), t > 0.

Lemma 2.2.12. If A ∈ ℒ(H), then σ2 ∘ μ(A) = μ(A ⊕ A). Proof. Let us fix an orthonormal basis {en }n≥0 in H. Set p1 (respectively, p2 ) to be the projection onto the linear span of {e2n }n≥0 (respectively, of {e2n+1 }n≥0 ). Set B1 e2n = μ(n, B)e2n , B2 e2n = 0,

B1 e2n+1 = 0,

B2 e2n+1 = μ(n, B)e2n+1 ,

n ≥ 0, n ≥ 0.

It is clear that μ(Bk ) = μ(A) and that Bk ∈ pk ℒ(H)pk , k = 1, 2. Thus, we may say that A ⊕ A = B1 + B2 . Recalling from Lemma 2.2.4 that μ(A) is a step function, we have B1 + B2 = diag(σ2 μ(A)). Then, μ(A ⊕ A) = μ(B1 + B2 ) = μ(diag(σ2 μ(A))) = σ2 μ(A). Part (c) of Corollary 2.2.10, that the function A → μ(t, A) is continuous in the uniform norm on ℒ(H), implies domination results of singular values for uniform convergence. In particular, if {Ai }i∈𝕀 ⊂ ℒ(H) for some index set 𝕀 are such that Ai → A in the uniform operator topology and μ(Ai ) ≤ μ(B) for all i ∈ 𝕀, then μ(A) ≤ μ(B). The next result shows domination of singular values holds for weak operator topology convergence. Lemma 2.2.13. Let {Ai }i∈𝕀 ⊂ ℒ(H) be a net of operators, and let B ∈ ℒ(H) be such that μ(Ai ) ≤ μ(B) for all i ∈ 𝕀. If Ai → A in the weak operator topology, then μ(A) ≤ μ(B). Proof. Let {pi }i∈𝕀 ⊂ ℒ(H) be a net of projections of rank k. We shall prove that every cluster point of {pi }i∈𝕀 in the weak operator topology has rank at most k. Let T ∈ ℒ(H) be a cluster point. Passing to a subnet if needed, we may assume without loss of generality that pi → T in the weak operator topology.

2.2 Singular values | 41

For each i ∈ 𝕀, write pi = ∑k−1 l=0 pl,i , where each pl,i is rank 1 projection. That is, there exists ξl,i ∈ H such that we have pl,i ζ = ⟨ζ , ξl,i ⟩ξl,i ,

ζ ∈ H.

By the Banach–Alaoglu theorem, the unit ball of the Hilbert space H is compact in the weak topology. Hence, again passing to a subnet if needed, we may assume that for every 0 ≤ l ≤ k − 1, there exists ξl ∈ H such that ξl,i → ξl weakly. Set k−1

Sζ = ∑ ⟨ζ , ξl ⟩ξl , l=0

ζ ∈ H.

We have that pi → S in weak operator topology. In particular, T = S. Thus rank(T) = rank(S) ≤ k. Now we prove the statement. Fix k ≥ 0, and for each i ∈ 𝕀 select pi ∈ Proj(H) such that ‖Ai (1 − pi )‖∞ ≤ μ(k, Ai ) ≤ μ(k, B) and such that rank(pi ) = k. Let T be a cluster point of the net {pi }i∈𝕀 ⊂ Proj(H) in the weak operator topology. Passing to a subnet if needed, we may assume that pi → T in the weak operator topology. Let n(T) denote the kernel projection of T, that is, n(T) is the maximal projection such that Tn(T) = 0. Since pi → T in the weak operator topology, it follows that n(T)pi n(T) → 0 in the weak operator topology. For every ζ ∈ H, we have ‖pi n(T)ζ ‖2 = ⟨pi n(T)ζ , pi n(T)ζ ⟩ = ⟨n(T)pi n(T)ζ , ζ ⟩ → 0. Thus, pi n(T) → 0 in the strong operator topology. We also have ‖Ai pi n(T)ζ ‖ ≤ ‖Ai ‖∞ ‖pi n(T)ζ ‖ ≤ ‖B‖∞ ‖pi n(T)ζ ‖ → 0. Hence, Ai pi n(T) → 0 in the strong operator topology. Consequently, Ai (1 − pi )n(T) = Ai n(T) − Ai pi n(T) → An(T) in the weak operator topology. Thus, ‖An(T)‖∞ ≤ lim inf ‖Ai (1 − pi )n(T)‖∞ ≤ lim inf ‖Ai (1 − pi )‖∞ ≤ μ(k, B). i∈𝕀

i∈𝕀

Since T is a cluster point of a net {pi }i∈𝕀 of projections of rank at most k, we have rank(T) = k. It follows that μ(k, A) ≤ ‖A − A(1 − n(T))‖∞ = ‖An(T)‖∞ ≤ μ(k, B).

42 | 2 Singular values and submajorization The singular value function and the singular value sequence The spectral theorem describes when the sequence space l∞ can be embedded in the diagonal of ℒ(H) according to an eigenbasis of a positive compact operator. The unitary invariance of the singular value function means that the singular value function associated to the embedding is independent of the eigenbasis. This identifies the singular value function as an extension of the decreasing rearrangement of sequences. Example 2.2.14. Let {en }n≥0 be an orthonormal basis of a separable Hilbert space H. Let diag(x) denote the embedding of x ∈ l∞ in ℒ(H) described in Example 1.1.11. For x ∈ l∞ , the function μ(diag(x)) is a step function independent of the orthonormal basis chosen, and μ(n, diag(x)), n ≥ 0, describes the decreasing rearrangement of the sequence x ∈ l∞ . For brevity, we denote μ(x) = μ(diag(x)). Strictly speaking, μ(x) for x ∈ l∞ is a step function in L∞ (0, ∞), as is μ(A) for any bounded operator A ∈ ℒ(H), with μ(t, A) = μ(n, A),

t ∈ [n, n + 1), n ≥ 0.

In the rest of the book, we shall use the same notation μ(A) when the context is clear that μ(A) refers either to the step function in L∞ (0, ∞), or the sequence of values {μ(n, A)}n≥0 in l∞ . For example, diag(μ(A)) refers to the operator associate to the sequence {μ(n, A)}n≥0 ∈ l∞ . If the context is not clear, then the maps P and E below can be used to lift sequences to step functions, and reduce step functions to sequences, respectively, where P : l∞ → L∞ (0, ∞),

(Px)(t) = ∑ x(n)χ[n,n+1) (t),

E : L∞ (0, ∞) → l∞ ,

(Ex)(n) = ∫ x(s) ds,

n≥0

x ∈ l∞ , t ≥ 0,

and n+1 n

x ∈ L∞ (0, ∞), n ≥ 0.

The maps E and P are continuous embeddings, and define isometries between l∞ and the algebra of step functions in L∞ (0, ∞) that are constant on intervals [n, n+1), n ≥ 0. If required, Eμ(A) will denote the sequence of values {μ(n, A)}n≥0 . A statement that is made in the text such as μ(A) = μ(diag(μ(A)))

2.3 Submajorization

| 43

is, more correctly, μ(A) = μ(diag(Eμ(A))), or P({μ(n, A)}n≥0 ) = μ(diag({μ(n, A)}n≥0 )). When it is required, the mappings between sequences and step functions will be made overt. The isometric embedding of l∞ as diagonal operators on ℒ(H) means that we often treat sequences implicitly as diagonal operators as mentioned in Chapter 1, occasionally dropping the reference to the embedding diag. Unitarily-invariant statements derived for operators and their singular values are then implicit statements on sequences and their decreasing rearrangements. We will often abbreviate the notation for operators and functionals on sequences, ∗ so for a functional ω ∈ l∞ , the value ω(x) = ω({x(n)}n≥0 ),

x = {x(n)}n≥0 ∈ l∞

will be denoted, when the context is clear, by ω(x(n)).

2.3 Submajorization The singular value function can be used to introduce preorders for operators A, B ∈

ℒ(H). Chapter 1 introduced the Calkin correspondence, which says that the commu-

tative core of an ideal is a symmetric sequence space. The Calkin correspondence is based on the preorder 0. By Lemma 2.3.4, there exist projections p, q ∈ ℒ(H) such that Tr(p), Tr(q) ≤ n and n

n

∫ μ(s, A) ds ≤ Tr(pAp) + ε,

∫ μ(s, B) ds ≤ Tr(qBq) + ε.

0

0

Define a projection r = p ∨ q. By definition, r is the projection onto the span of the images of p and q. Hence the dimension of the image of r is at most Tr(p) + Tr(q). Equivalently, Tr(r) ≤ Tr(p) + Tr(q) = 2n. Hence, 2n

Tr(pAp) + Tr(qBq) ≤ Tr(r(A + B)r) ≤ ∫ μ(s, A + B) ds. 0

Therefore, n

n

2n

∫ μ(s, A) ds + ∫ μ(s, B) ds ≤ ∫ μ(s, A + B) ds + 2ε. 0

0

0

Since ε > 0 is arbitrarily small, it follows that n

n

2n

∫ μ(s, A) ds + ∫ μ(s, B) ds ≤ ∫ μ(s, A + B) ds. 0

0

0

Now, 2n

2n−1

n−1

k=0

k=0

∫ μ(s, A + B) ds = ∑ μ(k, A + B) = ∑ μ(2k, A + B) + μ(2k + 1, A + B) 0

n

n−1

= ∑ (2σ 1 μ(A + B))(k) = ∫(2σ̂ 1 μ(A + B))(s) ds. k=0

2

0

2

The next theorem describes submajorization of the direct sum operation introduced in Definition 2.2.11. Theorem 2.3.7. If 0 ≤ A, B ∈ ℒ(H), then A ⊕ B ≺≺ A + B.

48 | 2 Singular values and submajorization Proof. Note that μ(A ⊕ B) = μ(μ(A) ⊕ μ(B)), where the right-hand side is understood as the decreasing rearrangement of the sequence μ(A) ⊕ μ(B). Thus, n

n

∫ μ(s, A ⊕ B) ds = ∫ μ(s, μ(A) ⊕ μ(B)) ds 0

0

n1

n2

= max ∫ μ(s, A) ds + ∫ μ(s, B) ds. n1 +n2 =n

0

0

Fix n ∈ ℤ+ . Fix n1 , n2 ∈ ℤ+ such that n1 + n2 = n. Fix ϵ > 0. Using Lemma 2.3.4, we can choose projections p1 and p2 such that n1

∫ μ(s, A) ds ≤ Tr(p1 Ap1 ) + ϵ,

Tr(p1 ) = n1 ,

0 n2

∫ μ(s, B) ds ≤ Tr(p2 Ap2 ) + ϵ,

Tr(p2 ) = n2 .

0

Let p = p1 ∨ p2 . We have Tr(p) ≤ Tr(p1 ) + Tr(p2 ) = n1 + n2 = n. It follows that n1

n2

∫ μ(s, A) ds + ∫ μ(s, B) ds ≤ Tr(p1 Ap1 ) + Tr(p2 Bp2 ) + 2ϵ 0

0

= Tr(Ap1 ) + Tr(Bp2 ) + 2ϵ ≤ Tr(Ap) + Tr(Bp) + 2ϵ n

= Tr(p(A + B)p) + 2ϵ ≤ ∫ μ(s, A + B) ds + 2ϵ. 0

Since ϵ > 0 is arbitrary, it follows that n2

n1

n

∫ μ(s, A) ds + ∫ μ(s, B) ds ≤ ∫ μ(s, A + B) ds. 0

0

0

Since n1 , n2 ∈ ℤ+ such that n1 + n2 = n are arbitrary, it follows that n

n

∫ μ(s, A ⊕ B) ds ≤ ∫ μ(s, A + B) ds. 0

0

Since n ∈ ℤ+ is arbitrary, the assertion follows.

2.3 Submajorization

| 49

There are several equivalent characterizations of submajorization. The following characterization of Hardy–Littlewood submajorization is known for ℒ(H) (see the proof in [113, Chapter II, Lemma 3.4]). Theorem 2.3.8. If 0 ≤ A, B ∈ ℒ(H) are compact operators, then B ≺≺ A if and only if Tr((B − t)+ ) ≤ Tr((A − t)+ ),

t > 0.

(2.3)

Proof. Fix t > 0. We have μ((A − t)+ ) = (μ(A) − t)+ , and (A − t)+ is of finite rank. It follows from (2.1) that nA (t)

∞

Tr((A − t)+ ) = ∫ (μ(s, A) − t)+ ds = ∫ (μ(s, A) − t) ds. 0

(2.4)

0

Recalling from Remark 2.2.2 that μ(t, A) = inf{s ≥ 0 : nA (s) ≤ t},

t ≥ 0,

we see that the function u

u → ∫(μ(s, A) − t) ds 0

is increasing when u < nA (t), and decreasing when u > nA (t). Hence this function attains its maximum at u = nA (t). If B ≺≺ A, then nB (t)

nB (t)

nA (t)

∫ (μ(s, B) − t) ds ≤ ∫ (μ(s, A) − t) ds ≤ ∫ (μ(s, A) − t) ds. 0

0

0

Inequality (2.3) follows now from (2.4). Suppose now that (2.3) holds. Fix u > 0 and set t = μ(u, A). It follows that u

nB (t)

∫(μ(s, B) − t) ds ≤ ∫ (μ(s, B) − t) ds = Tr((B − t)+ ) 0

0

u

≤ Tr((A − t)+ ) = ∫(μ(s, A) − t) ds. 0

50 | 2 Singular values and submajorization Hence, u

u

∫ μ(s, B) ds ≤ ∫ μ(s, A) ds. 0

0

Since u is arbitrary, we have B ≺≺ A. The next characterization is a special case of Weyl’s majorant theorem. Theorem 2.3.9. If 0 ≤ A, B ∈ ℒ(H) are compact operators, then the following conditions are equivalent: (a) B ≺≺ A. (b) For every convex function f : [0, ∞) → [0, ∞) such that f (0) = 0 and f (A) ∈ ℒ1 , we have f (B) ∈ ℒ1 and Tr(f (B)) ≤ Tr(f (A)). Proof. If B ≺≺ A, then also μ(B) ≺≺ μ(A). By Lemma II.3.4 in [113], we have Tr(f (B)) = ∑ f (μ(n, B)) ≤ ∑ f (μ(n, A)) = Tr(f (A)). n≥0

n≥0

Hence μ(f (B)) ∈ l1 and f (B) ∈ ℒ1 . To prove the converse assertion, consider the positive convex function f (s) = (s − t)+ for t > 0. Then (A − t)+ is of finite rank for every t > 0 and Tr((B − t)+ ) ≤ Tr((A − t)+ ),

t > 0,

by assumption. It follows from Theorem 2.3.8 that B ≺≺ A. Logarithmic submajorization Logarithmic submajorization will play a central role in the characterization of which traces are spectral, and when the commutative core of an ideal contains all the eigenvalue sequences of its operators. Definition 2.3.10. Let A, B ∈ ℒ(H). The operator B is said to be logarithmic submajorized by A and written B ≺≺log A if n

n

k=0

k=0

∏ μ(k, B) ≤ ∏ μ(k, A),

n ≥ 0.

Using the isometric embedding of l∞ as diagonal operators on ℒ(H), we write x ≺≺log y if diag(x) ≺≺log diag(y), x, y ∈ l∞ .

2.3 Submajorization |

51

The next characterization is essentially Weyl’s majorant theorem, see, e. g., Theorem 3.1 in [113], and it demonstrates the relationship between submajorization and logarithmic submajorization. Observe that B ≺≺log A is equivalent to n

n

k=0

k=0

∑ log(μ(k, B)) ≤ ∑ log(μ(k, A)),

n ≥ 0,

with the understanding that if μ(k, A) = 0, then log(μ(k, A)) = −∞. Theorem 2.3.11. If 0 ≤ A, B ∈ ℒ(H) are compact operators, then the following conditions are equivalent: (a) B ≺≺log A. (b) For every function f : [0, ∞) → [0, ∞) such that f ∘ exp is convex on ℝ, f (0) = 0, and f (A) ∈ ℒ1 , we have f (B) ∈ ℒ1 and Tr(f (B)) ≤ Tr(f (A)). In particular, B ≺≺log A implies that B ≺≺ A. Proof. Assume the statement in (a). We rewrite B ≺≺log A as n

n

k=0

k=0

∑ log(μ(k, B)) ≤ ∑ log(μ(k, A)),

n ≥ 0.

By Lemma II.3.4 in [113], since f ∘ exp is convex and f ∘ exp(x) → 0 as x → −∞ due to the condition f (0) = 0, this implies ∑ f ∘ exp ∘ log(μ(k, B)) ≤ ∑ f ∘ exp ∘ log(μ(k, B)),

k≥0

k≥0

n ≥ 0,

or ∑ f (μ(k, B)) ≤ ∑ f (μ(k, A)),

k≥0

k≥0

n ≥ 0.

The statement in (b) follows from the spectral theorem. Assume the statement in (b). Consider f (s) = (log(s) − t)+ , for fixed t ∈ ℝ. Then Tr((log(B) − t)+ ) ≤ Tr((log(A) − t)+ ),

t ∈ ℝ.

Fix n ≥ 0. Set t = log(μ(n, A)) and l = max{k ≥ 0 : μ(k, B) ≥ μ(n, A)}.

52 | 2 Singular values and submajorization We have l

∑ (log(μ(k, B)) − log(μ(n, A))) = Tr((log(B) − t)+ ) ≤ Tr((log(A) − t)+ )

k=0

n

= ∑ (log(μ(k, A)) − log(μ(n, A))). k=0

If l < n, then log(μ(k, B)) − log(μ(n, A)) ≤ 0,

l < k ≤ n.

Thus, n

∑ (log(μ(k, B)) − log(μ(n, A)))

k=0

l

n

k=0

k=l+1

= ∑ (log(μ(k, B)) − log(μ(n, A))) + ∑ (log(μ(k, B)) − log(μ(n, A))) l

≤ ∑ (log(μ(k, B)) − log(μ(n, A))). k=0

If l > n, then log(μ(k, B)) − log(μ(n, A)) ≥ 0,

n + 1 < k ≤ l.

Thus, n

∑ (log(μ(k, B)) − log(μ(n, A)))

k=0

l

l

k=0

k=n+1

= ∑ (log(μ(k, B)) − log(μ(n, A))) − ∑ (log(μ(k, B)) − log(μ(n, A))) l

≤ ∑ (log(μ(k, B)) − log(μ(n, A))). k=0

In all cases, whether l < n, l = n, or l > n, we have n

l

k=0

k=0

∑ (log(μ(k, B)) − log(μ(n, A))) ≤ ∑ (log(μ(k, B)) − log(μ(n, A))).

Consequently, n

n

k=0

k=0

∑ (log(μ(k, B)) − log(μ(n, A))) ≤ ∑ (log(μ(k, A)) − log(μ(n, A))).

2.3 Submajorization |

53

In other words, n

n

k=0

k=0

∑ log(μ(k, B)) ≤ ∑ log(μ(k, A)).

Since n ≥ 0 is arbitrary, the assertion in (a) follows. We now show that B ≺≺log A ⇒ B ≺≺ A. If the statement in (b) holds, then note that the statement of Theorem 2.3.9(b) is satisfied since f ∘ exp is convex for every convex function f . Logarithmic submajorization was introduced by H. Weyl, who proved Lemma 1.1.20 in Chapter 1. That is, if A ∈ ℒ(H) is a compact operator, then the singular value sequence logarithmically submajorizes an eigenvalue sequence λ(A) ≺≺log μ(A). From Lemma 1.1.2 in Chapter 1, A ∈ ℒ(H) has the matrix decomposition A = ∑ ⟨Aen , em ⟩en ⊗ em n,m≥0

according to an orthonormal basis {en }n≥0 of H. In this basis, A is called upper triangular if ⟨Aen , em ⟩ = 0,

n > m,

and the diagonal of A is ∑ ⟨Aen , en ⟩en ⊗ en = diag(⟨Aen , en ⟩).

n≥0

The following partial converse to Weyl’s lemma (Lemma 1.1.20) is a core result for the spectral formula of traces in Chapter 5. Theorem 2.3.12. Let x, y ∈ c0 be sequences convergent to zero such that y ≺≺log μ(x). Then there exists an upper-triangular compact operator A ∈ ℒ(H) such that μ(A) ≤ μ(x) and the diagonal of A is diag(y). In particular, λ(A) = λ(diag(y)), and λ(A) ≺≺log μ(x). The finite-dimensional version of Theorem 2.3.12 is known as Horn’s theorem, [140, Theorem 1]. It is the basis for proving Theorem 2.3.12.

54 | 2 Singular values and submajorization Theorem 2.3.13. Let x, y ∈ ℂn be such that x = μ(x), |y| = μ(y) and y ≺≺log x. If n−1

n−1

k=0

k=0

∏ |y(k)| = ∏ x(k),

then there exists an upper-triangular matrix A ∈ Mn (ℂ) such that μ(A) = x and such that the diagonal of A is y. Lemma 2.3.14. Let A ∈ Mn (ℂ) be upper-triangular and let y be the diagonal of A. For every permutation π of {0, . . . , n − 1} there exists a unitary U ∈ Mn (ℂ) such that U −1 AU is upper-triangular and such that the diagonal of U −1 AU = y ∘ π. Proof. We write π = π1 ∘⋅ ⋅ ⋅∘πm , where every πm is a transposition. Furthermore, we can assume that every πm permutes only two immediate neighbors. It, therefore, suffices to prove the assertion for the case when π is a transposition which permutes only two immediate neighbors (say, kth and (k + 1)th). We write A00 A=( 0 0

A01 A11 0

A02 A12 ) . A22

Here, A00 ∈ Mk (ℂ), A11 ∈ M2 (ℂ) and A22 ∈ Mn−k−2 (ℂ) are upper-triangular matrices, and A01 ∈ Mk2 (ℂ), A02 ∈ Mk,n−k−2 (ℂ), and A12 ∈ M2,n−k−2 (ℂ) are arbitrary matrices. Write yk 0

z ) yk+1

A11 = (

and choose a unitary matrix V ∈ M2 (ℂ) such that yk+1 0

w ). yk

V −1 A11 V = ( Set 1 U = (0 0

0 V 0

0 0) 1

and note that A00 U −1 AU = ( 0 0

A01 V V −1 A11 V 0

A02 V −1 A12 ) . A22

It is clear that U −1 AU is upper-triangular and that the diagonal of U −1 AU is y ∘ π.

2.4 Ideals and traces | 55

Corollary 2.3.15. Let x, y ∈ ℂn be such that x = μ(x) and y ≺≺log x. There exists an upper-triangular matrix A ∈ Mn (ℂ) such that the diagonal of A is y and μ(A) ≤ x. Proof. Let z = μ(z) be such that z(k) = x(k) for k < n − 1 and such that n−1

n−1

k=0

k=0

∏ z(k) = ∏ |y(k)|.

By Theorem 2.3.13 and Lemma 2.3.14, there exists an upper-triangular matrix A ∈ Mn (ℂ) such that μ(A) = z ≤ μ(x) and such that the diagonal of A is y. We now infer Theorem 2.3.12 from Corollary 2.3.15. Proof of Theorem 2.3.12. Let yn = yχ[0,n) ∈ l∞ ({0, 1, . . . , n − 1}). By Corollary 2.3.15, there exists An ∈ Mn (ℂ) such that (i) An is upper triangular; (ii) μ(An ) ≤ μ(x)χ[0,n) ; (iii) the diagonal of An is yn . Let us identify Mn (ℂ) with the subalgebra of operators on l2 supported in the “top left corner” in terms of the matrix basis {ej ⊗ ek }∞ j,k=0 where ej , j ≥ 0, is the standard basis vector of l2 with 1 in the jth position and 0 elsewhere. Then An is an upper-triangular element in ℒ(l2 ) for all n ≥ 0. By assumption, ‖An ‖∞ ≤ ‖x‖∞ , so that the sequence {An }n≥0 is bounded. Let A be a cluster point of {An }n≥0 in the weak-∗ topology on ℒ(H). Obviously, A is upper triangular and the diagonal of A is y. By Lemma 2.2.13, we have that μ(A) ≤ μ(x).

2.4 Ideals and traces This section proves the Calkin correspondence, which identifies the commutative core of an ideal with a symmetric sequence space. Extending the definitions in Chapter 1, we define ideals that are closed in a symmetric quasinorm, those that are solid for submajorizarion, and those that are solid for logarithmic submajorization. Linear functionals that respect these structures are similarly defined. Symmetric operator ideals and symmetric functionals Chapter 1 defined ideals of bounded operators in Definition 1.2.1 and a symmetric sequence space in Definition 1.2.2. This section proves the Calkin correspondence. The first step in the Calkin correspondence is to extend the notion of a symmetric sequence space in l∞ to ideals of ℒ(H). Recall that B ≤μ A denotes μ(B) ≤ μ(A). Lemma 2.4.1. Let 𝒥 be an ideal of ℒ(H) and let A ∈ 𝒥 . If B ∈ ℒ(H) is such that B ≤μ A, then B ∈ 𝒥 .

56 | 2 Singular values and submajorization Proof. By taking a polar decomposition, it follows from A ∈ 𝒥 that |A| ∈ 𝒥 , and that if |B| ∈ 𝒥 then B ∈ 𝒥 . Hence, replacing A and B with |A| or |B| if necessary, we assume that 0 ≤ B ∈ ℒ(H) and 0 ≤ A ∈ 𝒥 are such that μ(B) ≤ μ(A). Write A = ∑ μ(k, A)pk , k≥0

B = ∑ μ(k, B)qk , k≥0

where {pk }k≥0 (respectively, {qk }k≥0 ) are pairwise orthogonal rank-one spectral projections for A (respectively, for B). For each k ≥ 0, choose a partial isometry Uk such that Uk Uk∗ = pk and Uk∗ Uk = qk . We have Uk∗ pk Uk = Uk∗ Uk Uk∗ Uk = qk qk = qk . Since the isometries Uk have pairwise orthogonal images, we can define the strongly convergent sum U = ∑k≥0 Uk . We have U ∗ pk U = qk ,

k ≥ 0.

Therefore, U ∗ AU = ∑ μ(k, A)U ∗ pk U = ∑ μ(k, A)qk . k≥0

k≥0

Set μ(k, B) q . μ(k, A) k k≥0

C= ∑

Since μ(B) ≤ μ(A), C is a bounded operator on H. We have μ(k, B) qk ) ⋅ ( ∑ μ(k, A)qk ) = CU ∗ AU. μ(k, A) k≥0 k≥0

B = (∑

Since 𝒥 is an ideal, it follows that B ∈ 𝒥 . Definition 2.4.2. A subspace J of l∞ is called a symmetric sequence space if, for a ∈ J and for every b ∈ l∞ with μ(b) ≤ μ(a), we have b ∈ J. In Chapter 1 we defined the commutative core of a two-sided ideal 𝒥 of ℒ(H) by J = {x ∈ l∞ : diag(x) ∈ 𝒥 }. From Example 2.2.14 the definition of this sequence space and the association between decreasing rearrangements and the singular values of the corresponding diagonal operators are independent of the orthonormal basis chosen. The isometric embeddings of the sequence space J as diagonal operators in 𝒥 are unitarily equivalent. Theorem 2.4.3 below proves the Calkin correspondence stated in Theorem 1.2.3 in Chapter 1. Theorem 2.4.3 shows that the singular value function maps bijectively ideals to symmetric sequence spaces.

2.4 Ideals and traces | 57

Theorem 2.4.3. If 𝒥 is an ideal of ℒ(H), then the commutative core J = {x ∈ l∞ : diag(x) ∈ 𝒥 } is a symmetric sequence space. If J is a symmetric sequence space, then 𝒥 = {A ∈ ℒ(H) : μ(A) ∈ J}

is an ideal of ℒ(H) with commutative core J. This provides a canonical bijection between ideals and symmetric sequence spaces. Proof. Let 𝒥 be an ideal and let x ∈ J belong to the commutative core. Let y ∈ l∞ such that μ(y) ≤ μ(x). Then diag(y) ∈ ℒ(H) and diag(x) ∈ 𝒥 such that μ(diag(y)) = μ(y) ≤ μ(x) = μ(diag(x)). By Lemma 2.4.1, diag(y) ∈ 𝒥 . Hence, y ∈ J. It is clear that if 𝒥 is an ideal of ℒ(H) then J is a linear subspace since diag is linear. Hence J is a symmetric sequence space. Now, suppose J is a symmetric sequence space of l∞ . From Corollary 2.2.10(a) the set 𝒥 has the property that BAC ∈ 𝒥 when A ∈ 𝒥 and B, C ∈ ℒ(H). We prove that 𝒥 is a linear subspace of ℒ(H). If A, B ∈ 𝒥 , then μ(A) ∈ J and μ(B) ∈ J. Let z = μ(A) + μ(B) ∈ J. We have z ⊕ z ∈ J. It follows from the Corollary 2.2.9(a) and Lemma 2.2.12 that μ(A + B) ≤ σ2 (μ(A) + μ(B)) = σ2 (z) = μ(z ⊕ z) ∈ J. Then μ(A + B) ∈ J and A + B ∈ 𝒥 . Hence 𝒥 is a linear subspace of ℒ(H). The functor J → 𝒥 is the inverse to the functor 𝒥 → J, so the correspondence is bijective. Chapter 1 introduced a trace on a two-sided ideal as a unitarily-invariant linear functional in Definition 1.2.10 and Lemma 1.2.11. Given the symmetric structure of ideals and sequence spaces, it is natural to define a symmetric functional. Definition 2.4.4. Let 𝒥 be an ideal of ℒ(H). A linear functional φ : 𝒥 → ℂ is called symmetric if φ(A) = φ(B) whenever 0 ≤ A, B ∈ 𝒥 are such that μ(A) = μ(B). The symmetric functional φ is positive if φ : 𝒥+ → [0, ∞). Let J be a symmetric sequence space in l∞ . A linear functional φ : J → ℂ is called symmetric if φ(a) = φ(b) whenever 0 ≤ a, b ∈ J are such that μ(a) = μ(b). The symmetric functional φ is positive if φ : J+ → [0, ∞). The singular value sequence of a positive operator A ∈ 𝒥+ is its eigenvalue sequence. A symmetric functional is then a spectral invariant on the positive cone 𝒥+ . The following result shows that traces and symmetric functionals on 𝒥 coincide. Lemma 2.4.5. Let 𝒥 be a proper ideal of ℒ(H). Then every symmetric functional on 𝒥 is a trace, and every trace on 𝒥 is a symmetric functional.

58 | 2 Singular values and submajorization Proof. Suppose φ is a symmetric functional on 𝒥 . If 0 ≤ A ∈ 𝒥 and if U ∈ ℒ(H) is unitary, then μ(U ∗ AU) = μ(A). Since φ is symmetric, it follows that φ(U ∗ AU) = φ(A). By linearity, φ is a trace. Suppose φ is a trace. If 0 ≤ A, B ∈ 𝒥 are such that μ(A) = μ(B), then there exists (see, e. g., the proof of Lemma 2.4.1) a partial isometry U ∈ ℒ(H) such that U ∗ AU = B,

U ∗ U = EB (0, ∞),

UU ∗ = EA (0, ∞).

It follows that φ(B) = φ(U ∗ AU) = φ(UU ∗ A) = φ(EA (0, ∞)A) = φ(A). Traces and symmetric functionals coincide, and we refer to them interchangeably in the context of ideals of ℒ(H). Evidently, positive traces and positive symmetric functionals coincide on ideals of ℒ(H). The association between symmetric functionals and traces relies on the fact that ℒ(H) is a factor. On a commutative von Neumann algebra, every linear functional is a trace, but there may be no symmetric functionals. Example 2.4.6. On l∞ every continuous linear functional is a trace. However, there are no symmetric functionals on l∞ . Theorem 2.4.3 identifies the relationship between ideals of bounded operators and symmetric sequence spaces. Part II explains the relationship between traces on an ideal and symmetric functionals on the corresponding symmetric sequence space. Quasinormed ideals and continuous symmetric functionals This section defines a symmetric quasinorm and proves that every quasi-Banach twosided ideal of compact operators has an equivalent symmetric quasinorm. Definition 2.4.7. A quasinorm ‖ ⋅ ‖ : X → [0, ∞) on a linear space X has the same properties as a norm on X except for a modification of the triangle inequality: (a) ‖λ ⋅ x‖ = |λ|‖x‖, λ ∈ ℂ, x ∈ X, (b) ‖x‖ = 0 implies that x = 0, (c) ‖x + y‖ ≤ K(‖x‖ + ‖y‖), x, y ∈ X, for some K ≥ 1 called the concavity modulus. A quasinorm induces a topology on X with neighborhood basis of 0 given by {{x ∈ X : ‖x‖ < ε} : ε > 0}. A quasinorm on X is called p-convex for 0 < p ≤ 1 if we have the p-triangle inequality ‖x + y‖p ≤ ‖x‖p + ‖y‖p ,

x, y ∈ X.

2.4 Ideals and traces |

59

The Aoki–Rolewicz theorem states that every quasinormed vector space admits an equivalent p-convex norm for some 0 < p ≤ 1 [12, 226]. With a p-convex quasinorm, the topology of X is generated by the metric d(x, y) = ‖x − y‖p . A quasinormed linear space (X, ‖ ⋅ ‖) is called quasi-Banach if X is a complete metric space. A two-sided ideal 𝒥 of ℒ(H) is quasinormed if it admits a quasinorm ‖ ⋅ ‖𝒥 such that the quasinorm is jointly continuous for the left product map ℒ(H) × 𝒥 → 𝒥 . A two-sided ideal 𝒥 is symmetric by Lemma 2.4.1. A symmetric quasinorm preserves the symmetric property of an ideal or its commutative core. Definition 2.4.8. A symmetric quasinorm on an ideal 𝒥 is a quasinorm ‖⋅‖𝒥 such that ‖B‖𝒥 ≤ ‖A‖𝒥 when μ(B) ≤ μ(A), A, B ∈ 𝒥 . A symmetric quasinorm on an symmetric sequence space J is a quasinorm ‖ ⋅ ‖J such that ‖y‖J ≤ ‖x‖J when μ(y) ≤ μ(x), x, y ∈ J. Theorem 2.4.9(a) shows that a symmetric quasinorm in the terminology introduced above is jointly continuous, and that ideals with a symmetric quasinorm are exactly the symmetrically quasinormed ideals of compact operators studied by Gohberg and Krein, and Simon [113, 114, 247]. Theorem 2.4.9. Let ‖ ⋅ ‖𝒥 be a quasinorm on a two-sided ideal 𝒥 of ℒ(H). (a) The following are equivalent: i. 𝒥 is symmetrically quasinormed, in the sense that the quasinorm ‖ ⋅ ‖𝒥 satisfies ‖BAC‖𝒥 ≤ ‖B‖∞ ‖A‖𝒥 ‖C‖∞ ,

A ∈ 𝒥 , B, C ∈ ℒ(H).

(2.5)

ii. ‖ ⋅ ‖𝒥 is a symmetric quasinorm on 𝒥 . (b) If ‖ ⋅ ‖𝒥 is a symmetric quasinorm on 𝒥 , then ‖A‖𝒥 = ‖|A|‖𝒥 = ‖diag(μ(A))‖𝒥 ,

A ∈ 𝒥.

(c) If ‖ ⋅ ‖𝒥 is a symmetric quasinorm on 𝒥 , then ‖diag(σ2 ∘ μ(A))‖𝒥 ≤ 2K‖A‖𝒥 ,

A ∈ 𝒥,

where K is the concavity modulus of the quasinorm ‖ ⋅ ‖𝒥 . Proof. (b) From Definition 2.4.8, ‖A‖𝒥 = ‖B‖𝒥 if μ(A) = μ(B), A, B ∈ 𝒥 . The equalities now follow from μ(A) = μ(|A|) = μ(diag(μ(A)) for any A ∈ 𝒥 . (a) If the quasinorm ‖⋅‖𝒥 satisfies condition (2.5), then we can argue as in the proof of Lemma 2.4.1. That is, fix 0 ≤ B ∈ ℒ(H) and 0 ≤ A ∈ 𝒥 such that μ(B) ≤ μ(A) and

60 | 2 Singular values and submajorization select, for a given ε > 0, a partial isometry U such that |B| ≤ (1 + ε)U|A|U ∗ . Since ε > 0 is arbitrarily small, it follows that B ∈ 𝒥 and ‖B‖𝒥 ≤ ‖A‖𝒥 . Hence ‖ ⋅ ‖𝒥 is symmetric. If the quasinorm ‖ ⋅ ‖𝒥 is symmetric, then (2.5) follows from (b) and the inequality μ(BAC) ≤ ‖B‖∞ ‖C‖∞ μ(A), which in turn follows from part (a) of Corollary 2.2.10. (c) We have σ2 ∘μ(A) = μ(A⊕A) from Lemma 2.2.12. We also have μ(A⊕A) = μ(B1 +B2 ) for some B1 , B2 such that ‖B1 ‖𝒥 , ‖B2 ‖𝒥 ≤ ‖A‖𝒥 by part (a). Then, using part (b), ‖diag(σ2 ∘ μ(A))‖𝒥 ≤ ‖B1 + B2 ‖𝒥 ≤ 2K‖A‖𝒥 where K is the concavity modulus in the modified triangle inequality in Definition 2.4.7(c). In the definition of a quasinormed two-sided ideal (𝒥 , ‖ ⋅ ‖𝒥 ), the quasinorm ‖ ⋅ ‖𝒥 is not necessarily symmetric. It is only jointly continuous for the left product map by definition. However, an equivalent symmetric quasinorm exists. We always assume that an equivalent symmetric quasinorm is chosen. Lemma 2.4.10. Let (𝒥 , ‖ ⋅ ‖𝒥 ) be a quasinormed (resp., normed) ideal of ℒ(H). Then the formula |||A||| =

sup

‖B‖∞ ,‖C‖∞ ≤1

‖BAC‖𝒥 ,

A∈𝒥

defines an equivalent symmetric quasinorm (resp., norm) on 𝒥 . Proof. Since 𝒥 is a quasinormed ideal, the left multiplication map is jointly continuous from 𝒥 × ℒ(H) to 𝒥 . In particular, there exists ε > 0 such that ‖AC‖𝒥 ≤ 1 whenever ‖A‖𝒥 ≤ ε and ‖C‖∞ ≤ ε. Consequently, ‖AC‖𝒥 ≤ c1 ‖A‖𝒥 ‖C‖∞ ,

A ∈ 𝒥,

C ∈ ℒ(H),

A ∈ 𝒥,

B ∈ ℒ(H).

where we set c1 = ε−2 . Similarly, ‖BA‖𝒥 ≤ c2 ‖A‖𝒥 ‖B‖∞ , It follows that |||A||| ≤ c1 c2 ‖A‖𝒥 ,

A ∈ 𝒥.

On the other hand, we have (by setting B = C = 1) |||A||| ≥ ‖A‖𝒥 ,

A ∈ 𝒥.

Thus, |||⋅||| is an equivalent quasinorm on 𝒥 . It satisfies condition (2.5) in Theorem 2.4.9 and is, therefore, symmetric.

2.4 Ideals and traces | 61

A quasi-Banach ideal is a quasinormed two-sided ideal 𝒥 of ℒ(H) which is complete in the topology associated to the quasinorm. We often use the notation ℰ to distinguish a quasi-Banach ideal of ℒ(H) from an arbitrary two-sided ideal 𝒥 . If the quasinorm is a norm, then the quasi-Banach ideal is called a Banach ideal. Identical terminology, notation and results apply to a quasi-Banach or Banach symmetric sequence space. That the Calkin correspondence can be extended to a bijective correspondence between a symmetric quasinorm on a two-sided ideal and a symmetric quasinorm on a symmetric sequence space is discussed in Chapter 3. We define continuous traces and continuous symmetric functionals. Definition 2.4.11. Let (ℰ , ‖ ⋅ ‖ℰ ) be a quasi-Banach ideal of ℒ(H). A continuous trace is a symmetric functional φ ∈ ℰ ∗ . Let (E, ‖ ⋅ ‖E ) be a quasi-Banach symmetric sequence space in l∞ . A continuous symmetric functional is a symmetric functional φ ∈ ℰ ∗ . Positive symmetric functionals, as defined in Definition 2.4.4, are continuous traces. Lemma 2.4.12. A positive linear functional on a quasi-Banach ideal ℰ (resp., a quasiBanach symmetric sequence space E) is continuous. Proof. Assume the contrary. Choose a positive linear functional f : ℰ → ℂ which fails to be continuous, and is therefore unbounded. Let K be the concavity modulus of a symmetric quasinorm ‖ ⋅ ‖ℰ on ℰ . Since f is not bounded, it allows us to choose a sequence {Tk }k≥0 in ℰ such that ‖Tk ‖ℰ ≤ (2K)−k ,

|f (Tk )| ≥ 2k ,

k ≥ 0.

Since μ(Tk∗ ) = μ(Tk ), it follows that ‖ℜ(Tk )‖ℰ ≤

K (‖Tk ‖ℰ + ‖Tk∗ ‖ℰ ) = C‖Tk ‖ℰ ≤ 2−k K 1−k , 2

k ≥ 0.

Similarly, ‖ℑ(Tk )‖ℰ ≤ 2−k K 1−k ,

k ≥ 0.

On the other hand, we have 1󵄨 󵄨 2k−1 ≤ 󵄨󵄨󵄨f (Tk )󵄨󵄨󵄨 ≤ 2

1 󵄨󵄨 󵄨 󵄨f (ℜ(Tk ))󵄨󵄨󵄨 + 2󵄨

1 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨f (ℑ(Tk ))󵄨󵄨󵄨 ≤ max{󵄨󵄨󵄨f (ℜ(Tk ))󵄨󵄨󵄨, 󵄨󵄨󵄨f (ℑ(Tk ))󵄨󵄨󵄨}. 2󵄨

Set ℜ(Tk ),

Sk = {

ℑ(Tk ),

|f (ℜ(Tk ))| ≥ 2k−1 ,

|f (ℜ(Tk ))| < 2k−1 .

62 | 2 Singular values and submajorization We have |f (Sk )| ≥ 2k−1 ,

‖Sk ‖ℰ ≤ 2−k K 1−k ,

Sk = Sk∗ ,

k ≥ 0.

Set (Sk )+ , |f ((Sk )+ )| ≥ 2k−2 , Ak = { (Sk )− , |f ((Sk )+ )| < 2k−2 . We have f (Ak ) ≥ 2k−2 ,

‖Ak ‖ℰ ≤ 2−k K 1−k ,

Ak ≥ 0, k ≥ 0.

We claim that the series ∑k≥0 Ak converges in ℰ . Indeed, m

m

m

k=n

k=n

k=n m

m

k=n

k=n

‖ ∑ Ak ‖ℰ ≤ ∑ K k−n+1 ‖Ak ‖ℰ ≤ ∑ K k+1 ⋅ ‖Ak ‖ℰ ≤ ∑ K k+1 ⋅ 2−k K 1−k = K 2 ∑ 2−k ≤ K 2 21−n . Hence, the partial sums of the series ∑∞ k=0 Ak form a Cauchy sequence, and the series converges. Set A = ∑k≥0 Ak . Since Ak ≥ 0 for every k ≥ 0, it follows that A ≥ Ak for every k ≥ 0. Thus, f (A) ≥ f (Ak ) ≥ 2k−2 ,

k ≥ 0.

This contradicts that the domain of f is ℰ . This completes the proof. Fully symmetric ideals and fully symmetric functionals The study of Dixmier traces in Part III features ideals and functionals which are monotone with respect to Hardy–Littlewood submajorization. The operator spaces of interest to applications in mathematical physics and noncommutative geometry also generally possess a Fatou norm. In this section we introduce both. Definition 2.4.13. A quasi-Banach ideal ℰ of ℒ(H) is called: (a) strongly symmetric if, for A, B ∈ ℰ with μ(B) ≺≺ μ(A), we have ‖B‖ℰ ≤ ‖A‖ℰ ; (b) fully symmetric if, for A ∈ ℰ and B ∈ ℒ(H) with μ(B) ≺≺ μ(A), we have B ∈ ℰ and ‖B‖ℰ ≤ ‖A‖ℰ . Similar statements define strongly symmetric and fully symmetric quasi-Banach symmetric sequence spaces E.

2.4 Ideals and traces | 63

We note that ℰ is a strongly symmetric ideal if and only if it is a closed subspace of a fully symmetric ideal. Indeed, define ℱ by letting B ∈ ℱ if and only if there exists A ∈ ℰ such that μ(B) ≺≺ μ(A) and set ‖B‖ℱ := inf{‖A‖ℰ : A ∈ ℰ , μ(B) ≺≺ μ(A)}. It is easy to verify that ℱ is then a fully symmetric ideal containing ℰ as a closed subspace. The following property is widely used. In some standard references the requirement that the symmetrically normed ideal has a Fatou norm appears to be a part of the definition, e. g., Simon’s book [247]. Definition 2.4.14. The norm ‖ ⋅ ‖ℰ on a quasi-Banach ideal ℰ is a Fatou norm if the unit ball of ℰ is closed with respect to the convergence in the strong operator topology. All Banach ideals with a Fatou norm are strongly symmetric. Theorem 2.4.15. Every Banach ideal of ℒ(H) equipped with a Fatou norm is a closed subspace of a fully symmetric ideal. Proof. The assertion follows from the Calkin correspondence and classical results on Banach symmetric function spaces (see, e. g., [20, 159]). Definition 2.4.16. Let 𝒥 be a two-sided ideal of ℒ(H). A symmetric functional φ : 𝒥 → ℂ is called fully symmetric if φ(B) ≤ φ(A) whenever 0 ≤ A, B ∈ 𝒥 are such that μ(B) ≺≺ μ(A). Similar statements define a fully symmetric functional on a symmetric sequence space J. Every fully symmetric functional is positive. Definition 2.4.16 ensures that φ satisfies 0 ≤ φ(A) since 0 ≺≺ A when A ∈ 𝒥+ . Hence, if ℰ is a quasi-Banach ideal, then every fully symmetric functional on ℰ is continuous. Logarithmic submajorization closed ideals The discussion of the spectral formulation of traces in Chapter 5 will focus on those ideals which are monotone with respect to logarithmic submajorization. Definition 2.4.17. An ideal of compact operators 𝒥 is called logarithmic submajorization closed if A ∈ 𝒥 and B ∈ ℒ(H) such that B ≺≺log A implies that B ∈ 𝒥 . From the relation in Theorem 2.3.11, a fully symmetric ideal is logarithmic submajorization closed. Generally, a quasi-Banach ideal of ℒ(H) is logarithmic submajorization closed as the following proposition shows. Proposition 2.4.18. Every quasi-Banach ideal of ℒ(H) is closed with respect to logarithmic submajorization.

64 | 2 Singular values and submajorization Proof. Let ℰ be a quasi-Banach ideal and let A ∈ ℰ , B ∈ ℒ(H) be such that B ≺≺log A. By the Aoki–Rolewicz theorem, we may assume without loss of generality that the quasinorm of ℰ is p-convex for some 0 < p ≤ 1. Choose m such that 2K < m and set z = ∑ m−l σ2l μ(A). l≥0

Applying Theorem 2.4.9(c), we have ‖z‖pℰ ≤ ∑ m−lp ‖σ2l μ(A)‖pℰ ≤ ∑ m−lp ⋅ (2K)lp ‖A‖pℰ . l≥0

l≥0

In particular, z ∈ ℰ . Now, n

2−n

2n −1

μ(2 − 1, B) ≤ ( ∏ μ(k, A)) =m

2k

k

2−n

≤ (μ(2 − 1, A) ⋅ ∏ μ(2 − 1, A) )

k=0

2−21−n −n⋅2−n

n−1

n

k=0

−n

n

n−1

n

k

k

2k

2−n

⋅ m (m μ(2 − 1, A) ⋅ ∏(m μ(2 − 1, A)) ) n

k=0

≤ m2−n max mk μ(2k − 1, A) ≤ m2−n ∑ mk μ(2k − 1, A). 0≤k≤n

k=0

We have n

z(2n − 1) ≥ ∑ m−l μ(⌊ l=0 n

2n − 1 ⌋, A) 2l n

= ∑ m−l μ(2n−l − 1, A) = ∑ mk−n μ(2k − 1, A). l=0

k=0

Therefore, μ(2n − 1, B) ≤ m2 z(2n − 1). If 2n − 1 ≤ k < 2n+1 − 1, then μ(k, B) ≤ μ(2n − 1, B) ≤ m2 z(2n − 1) ≤ m2 ⋅ (σ2 z)(k). Since z ∈ ℰ , then B ∈ ℰ . Traces have stronger properties with respect to logarithmic submajorization than Hardy–Littlewood submajorization, which is demonstrated in Chapters 5 and 6. One notable property is that every positive trace on a logarithmic submajorization closed ideal respects logarithmic submajorization. Chapter 6 will show that the same is not true for fully symmetric traces and Hardy–Littlewood submajorization.

2.5 Examples of ideals | 65

Proposition 2.4.19. Let 𝒥 be an ideal of ℒ(H) closed with respect to logarithmic submajorization and let φ be a trace on 𝒥 . If φ is positive, then it is monotone with respect to logarithmic submajorization. Proposition 2.4.19 is proved in Section 5.6, since a short proof follows from the Ringrose decomposition of an operator in a logarithmic submajorization closed ideal into a normal and quasinilpotent part belonging to the ideal.

2.5 Examples of ideals We expand on the examples of two-sided ideals of the algebra ℒ(H) introduced in Chapter 1. Compact operators The ideal of compact operators 𝒞0 (H) is the norm closure of the linear span of projections in ℒ(H) with finite trace. It is a Banach ideal given the uniform norm on ℒ(H). The commutative core of 𝒞0 (H) is c0 , the space of complex-valued sequences which converge to zero at infinity. The sequence space c0 can be confirmed to be symmetric, verifying in this case Theorem 2.4.3, since μ(y) ≤ μ(x),

x ∈ c0 , y ∈ l∞ ,

implies that μ(y) ∈ c0 , hence y ∈ c0 . The sequence space c0 is fully symmetric, since y ≺≺ x,

x ∈ c0 , y ∈ l∞ ,

implies that y ∈ c0 . To see this, recall that z → 0 implies Cz → 0 where C : l∞ → l∞ is the Cesaro operator (Cz)(n) =

n 1 ∑ z(k), n + 1 k=0

z ∈ l∞ , n ≥ 0.

If y ≺≺ x and x ∈ c0 , then Cμ(y) ≤ Cμ(x), and Cμ(x) → 0. Hence, Cμ(y) → 0. Since μ(y) is monotonically nonincreasing, Cμ(y) → 0 implies that μ(n, y) → 0 as n → ∞, and hence y ∈ c0 . The full algebra of bounded linear operators on H, ℒ(H), admits no nontrivial traces. This follows from a theorem of Halmos which states that every bounded operator on H is a sum of finitely many commutators. We will consider a far-reaching generalization of Halmos’ theorem in Chapter 5. Therefore, only proper subspaces of ℒ(H) are of interest in the theory of traces. Since c0 is the largest proper symmetric sequence space contained in l∞ , from the Calkin correspondence of Theorem 2.4.3, all proper two-sided ideals of ℒ(H) are contained in the ideal of compact operators.

66 | 2 Singular values and submajorization Schatten ideals The lp sequence space, p > 0, ∞

p

1/p

lp := {x ∈ l∞ : ( ∑ μ(n, x) ) n=0

< ∞}

is a symmetric sequence space. The corresponding Schatten ideal (also called a Schatten–von Neumann ideal due to the fundamental contributions of John von Neumann) ℒp := {A ∈ ℒ(H) : μ(A) ∈ lp }

has commutative core lp . The sequence space lp is complete with the symmetric quasinorm ∞

1/p

p

‖x‖p := ( ∑ μ(n, x) ) n=0

,

x ∈ lp ,

which is a norm when p ≥ 1. That ℒp is a fully symmetric space when p ≥ 1 follows from Theorem 2.3.9 with f (t) = t p . The space ℒp is complete given the symmetric quasinorm, ∞

1/p

p

‖A‖p := ( ∑ μ(n, A) ) n=0

,

A ∈ ℒp .

It is a quasi-Banach ideal when 0 < p < 1 and a fully symmetric ideal when p ≥ 1. Using Corollary 2.2.10(d), the symmetric quasinorm can be written as 1/p

‖A‖p := Tr(|A|p )

,

A ∈ ℒp .

Chapter 3 provides a criterion for the existence of traces on two-sided ideals of terms of the action of the Cesaro operator on the commutative core of the ideal. The criterion makes it simple to verify that the Banach ideals ℒp , p > 1, admit no nonzero traces. As described after Definition 1.3.2 in Chapter 1, the trace Tr is a fully symmetric trace on the ideal ℒ1 . Duality and embedding According to the definition ℒ∞ := {A ∈ ℒ(H) : {μ(n, A)}n≥0 ∈ l∞ },

then we identify ℒ∞ = ℒ(H).

2.5 Examples of ideals | 67

This justifies the notation for the operator norm 󵄩 󵄩 ‖A‖∞ = 󵄩󵄩󵄩{μ(n, A)}n≥0 󵄩󵄩󵄩∞ = μ(0, A). The duality ∗

ℒ(H) → ℒ1

given by A 󳨃→ Tr(A⋅),

A ∈ ℒ(H)

is an isometry [75, 160, 190, 242]. This identifies the Banach dual ℒ∗1 of ℒ1 as ℒ(H). In other words, the ideal of trace class operators ℒ1 is identified as the predual of ℒ(H). These correspondences are a direct analogy of the duality l∞ → l1∗ given by ∞

b = {bn }∞ n=0 󳨃→ ϕb (a) := ∑ an bn ,

a = {an }∞ n=0 ∈ l1 ,

n=0

which can be recovered by restricting to diagonal operators. This is the Radon– Nikodym theorem for sequences, and the duality ℒ∗1 = ℒ∞ is a noncommutative analogy of the Radon–Nikodym theorem. Theorem 2.5.1. Up to a scalar multiple, Tr is the only continuous trace on the ideal ℒ1 . Proof. If φ ∈ ℒ∗1 , then there exists K ∈ ℒ(H) such that φ(A) = Tr(KA),

A ∈ ℒ1 .

If φ is a trace, then φ(AB) = φ(BA) implies that Tr(AKB) = Tr(BKA),

A, B ∈ ℒ1 .

In particular, taking B = [A, K]∗ yields Tr(|[A, K]|2 ) = 0, and hence [A, K] = 0. Let {en }∞ n=0 be an orthonormal basis for H. Taking Ax = ⟨x, en ⟩en and using [A, K]em = 0 yields Ken ⟨en , em ⟩ = en ⟨en , Kem ⟩,

n, m ≥ 0.

Thus if n ≠ m, we have ⟨en , Kem ⟩ = 0, and so K is diagonal in the basis {en }∞ n=0 . To see that K is a multiple of the identity, we apply Tr(AKB) = Tr(BKA) with Ax = en ⟨x, em ⟩ and B = em ⟨x, en ⟩. This yields ⟨en , Ken ⟩ = ⟨em , Kem ⟩,

n, m ≥ 0.

Hence K is a scalar multiple of the identity operator.

68 | 2 Singular values and submajorization From classical results on the interpolation of symmetric sequence spaces, every quasi-Banach ideal is continuously embedded in ℒ(H) and continuously contains a Schatten ideal. Theorem 2.5.2. If ℰ is a quasi-Banach ideal, then there exists p ∈ (0, 1] such that ℒp ⊂ ℰ ⊂ ℒ(H)

where the embeddings are continuous. If ℰ is a Banach ideal then p = 1. Proof. The assertion that ℰ ⊂ ℒ(H) follows from Theorem 2.4.9(b). Since diag(μ(0, A), 0, . . .) ≤ diag(μ(A)), and setting c = ‖diag(1, 0, . . .)‖ℰ , then 󵄩 󵄩 󵄩 󵄩 c‖A‖∞ = 󵄩󵄩󵄩diag(μ(0, A), 0, . . .)󵄩󵄩󵄩ℰ ≤ 󵄩󵄩󵄩diag(μ(A))󵄩󵄩󵄩ℰ = ‖A‖ℰ . The assertion that ℒp ⊂ ℰ follows directly from the Aoki–Rolewicz theorem [12, 226]. Without loss of generality, from the Aoki–Rolewicz theorem, n

‖A‖ℰ = inf{( ∑

k=0

‖Ak ‖pℰ )

1/p

n

: n ≥ 0, Ak ∈ ℰ , A = ∑ Ak } k=0

p

where p ∈ (0, 1] satisfies (2K) = 2. Here K is the modulus of concavity of the quasinorm. So p = 1 if the quasinorm is a norm. According to the Schmidt decomposition in Theorem 1.1.14, A = ∑ μ(k, A)fk ⊗ ek , k≥0

for an orthonormal system {fk }k≥0 and an orthonormal basis {ek }k≥0 . Setting Ak = μ(k, A)fk ⊗ ek ,

k ≥ 0,

we have that Ak is of finite rank, and hence Ak ∈ ℰ , k ≥ 0, and ‖Ak ‖ℰ ≤ μ(k, A)‖fk ⊗ ek ‖∞ ≤ μ(k, A),

k ≥ 0.

Hence, for all n ≥ 0, n

(∑ k=0

1/p p ‖Ak ‖ℰ )

n

p

1/p

≤ ( ∑ μ(k, A) ) k=0

Then, from the Aoki–Rolewicz theorem, ‖A‖ℰ ≤ ‖A‖ℒp .

≤ ‖A‖ℒp .

2.5 Examples of ideals | 69

Let (ℰ , ‖ ⋅ ‖ℰ ) be a quasi-Banach ideal of ℒ(H). Denote by ℰ0 the closure of the finite rank operators in the quasi-norm ‖ ⋅ ‖ℰ . Then ℰ0 is a quasi-Banach ideal called the separable part of ℰ (also called termed the regular part in Simon, [247]). In Simon’s terminology, the quasi-Banach ideal ℰ is regular if ℰ is equal to ℰ0 . Schatten ideals are regular. The condition of continuity and Theorem 2.5.2 makes the following statements evident. Lemma 2.5.3. If ℰ is a quasi-Banach ideal and φ ∈ ℰ ∗ is a continuous trace, then following statements are equivalent: (a) φ vanishes on finite rank operators (it is a singular trace). (b) φ vanishes on ℒp , p > 0, whenever ℒp ⊂ ℰ . (c) φ vanishes on ℰ0 . Lemma 2.5.3 indicates that regular quasi-Banach ideals admit no nonzero continuous singular traces. Weak ideals The weak-lp sequence spaces lp,∞ , p > 0, defined by lp,∞ := {x ∈ l∞ : μ(n, x) ≤ c ⋅ (1 + n)−1/p , c > 0, n ≥ 0}, are symmetric sequence spaces. The assignment ‖x‖p,∞ := inf{c > 0 : μ(n, x) ≤ c ⋅ (1 + n)−1/p , n ≥ 0},

x ∈ lp,∞ ,

or, equivalently, ‖x‖p,∞ := sup(1 + n)−1/p μ(n, x), n≥0

x ∈ lp,∞ ,

defines a symmetric quasinorm on lp,∞ . Under the Calkin correspondence of Theorem 2.4.3, the weak-lp ideal of operators ℒp,∞ := {A ∈ ℒ(H) : μ(A) ∈ lp,∞ }

has commutative core lp,∞ . The assignment ‖A‖p,∞ := sup(1 + n)−1/p μ(n, A), n≥0

A ∈ ℒp,∞ ,

defines a symmetric quasinorm on ℒp,∞ , in which ℒp,∞ is complete. Therefore ℒp,∞ is a quasi-Banach ideal. Operators from the quasi-Banach ideal ℒ1,∞ are called weak trace class operators. The weak ideals are not regular. If x(n) = (1 + n)−1/p , n ≥ 0, and xm (n) = x(n), n ≤ m, m ≥ 0, and 0 otherwise, then ‖diag(x) − diag(xm )‖p,∞ = 1,

m ≥ 0.

This is sufficient to prove that the weak ideals are not regular.

70 | 2 Singular values and submajorization Chapter 3 provides a criterion for the existence of traces on two-sided ideals in terms of the action of the Cesaro operator on the commutative core of the ideal. The criterion makes it simple to verify that the quasi-Banach ideals ℒp,∞ , p > 1, have no nonzero traces. The ideal ℒ1,∞ , in contrast, has many traces. Parts II and III describe the positive and fully symmetric traces on the ideal ℒ1,∞ . Volume II and the motivation for formulas for traces on ℒ1,∞ described in Part III show the primary role played by traces on the ideal ℒ1,∞ in Alain Connes’ noncommutative geometry. Lorentz ideals Lorentz ideals provide examples of ideals with fully symmetric traces and are central to the theory of Dixmier traces. Let ψ : ℝ+ → ℝ+ be an increasing positive concave function. The Lorentz sequence space mψ is the set of all x ∈ l∞ such that there exists c > 0 with μ(x) ≺≺ c ⋅ ψ󸀠 . That is, mψ := {x ∈ l∞ : μ(x) ≺≺ c ⋅ ψ󸀠 , c > 0}. Recall that we consider x embedded as a diagonal operator in ℒ(H), and therefore μ(x) is a step function in L∞ (0, ∞) such that μ(n, x), n ≥ 0, is the familiar decreasing rearrangement of the sequence |x|. It is clear that mψ is a closed fully symmetric sequence space when given the symmetric norm ‖x‖mψ := inf{c > 0 : μ(x) ≺≺ c ⋅ ψ󸀠 },

x ∈ mψ ,

or, equivalently, ‖x‖mψ := sup n≥0

n 1 ∑ μ(k, x), ψ(n + 1) k=0

x ∈ mψ .

The weak sequence space associated to ψ is lψ := {x ∈ l∞ : μ(x) ≤ c ⋅ ψ󸀠 , c > 0} provided that lim inf t→∞

ψ󸀠 (2t) > 0. ψ󸀠 (t)

In this case ‖x‖lψ := inf{c > 0 : μ(x) ≤ c ⋅ ψ󸀠 },

x ∈ lψ ,

defines a symmetric quasinorm. It is clear that mψ is the “submajorization” closure of lψ , in the sense that mψ is the smallest sequence space containing lψ which is closed under submajorization. In

2.5 Examples of ideals |

71

some cases, the sequence spaces lψ and mψ coincide. For example, if p > 1 and ψp (t) = t 1−1/p , t > 0, then lψp = lp,∞ is equivalent to the weak-lp sequence space, and it is known that lψp = mψp . In this case lp,∞ is closed under submajorization. Not every weak ideal is closed under submajorization, and the closure of lψ in the norm ‖ ⋅ ‖mψ need not be mψ . This provides an example of a Banach ideal that is not fully symmetric. Example 2.5.4. The closure of lψ in the norm ‖ ⋅ ‖mψ when the function ψ satisfies the condition lim

t→∞

ψ(2t) =1 ψ(t)

(2.6)

does not coincide with mψ and is, therefore, not fully symmetric. The details can be found in [159, Lemma II.5]. The corresponding Lorentz ideals of ℒ(H) are ℳψ := {A ∈ ℒ(H) : μ(A) ∈ mψ }

with symmetric norm ‖A‖ℳψ := sup n≥0

n 1 ∑ μ(k, A), ψ(n + 1) k=0

A ∈ ℳψ ,

and ℒψ := {A ∈ ℒ(H) : μ(A) ∈ lψ }

with symmetric quasinorm ‖A‖ℒψ := sup n≥0

μ(n, A) , ψ󸀠 (n + 1)

A ∈ ℒψ .

For the choice ψ(t) := log(1 + t),

t > 0,

which satisfies (2.6), the Lorentz sequence space is the Sargeant sequence space m1,∞ and the notation for the Lorentz ideal associated to ψ is ℳ1,∞ := {A ∈ ℒ(H) : μ(A) ∈ m1,∞ }.

Chapter 1 highlighted the difference between the weak-l1 sequence space l1,∞ and the sequence space m1,∞ , and hence the difference between the weak ideal ℒ1,∞ and the Lorentz ideal ℳ1,∞ .

72 | 2 Singular values and submajorization

2.6 Notes Von Neumann algebras and projections The operator norm topology, strong and weak operator topologies, and rings of bounded operators closed in the topologies were introduced by F. Murray and J. von Neumann [188]. Along with C∗ -algebras, von Neumann algebras form one of the pillars of operator algebra theory [77, 271]. The cited monographs, amongst many others, deal comprehensively with the theory of von Neumann algebras. As the manifestation of noncommutative measure theory, projections are central to the classification and structure of von Neumann algebras, a program initiated in 1937 by Murray and von Nuemann [187, 188] and completed in 1973 by Connes [55, 60]. A historical discussion is given in [295]. Theorem 2.1.3 is proved in [242], which shows that noncommutative von Neumann algebras are an extension of measure theory. That ℒ(H) is a factor is explained in [30, I.9.1.5]. The spectral theorem is a pillar of functional analysis, see, for example, the proof and descriptions in [219]. Singular value function Singular value sequences in functional analysis replicate the singular value decomposition of matrices. Schmidt in 1907 wrote down the singular value decomposition for integral operators with asymmetric kernels [234]. The term “singular value” appears to have originated from Schmidt’s work, and was in use by 1937, see the historical discussion in [251]. The term “s-number” originated later and appeared in Gohberg and Krein [113]. Gohberg and Krein [113, 114], Pietsch [201], and Simon [247] collect results on singular value sequences. Pietsch [201] has extensive historical notes. The singular value function can be traced to the fundamental paper by F. Murray and J. von Neumann [188]. The approximation formula in Theorem 2.2.6 is believed to have originated in [8]. Extensions to semifinite von Neumann algebras can be traced to A. Grothendieck’s announcement [118], and was almost simultaneously considered by M. Sonis [248] and V. I. Ovchinnikov [194, 195]. Later it was studied by F. J. Yeadon [298], T. Fack [101], and T. Fack and H. Kosaki in [104]. The proof of Lemma 2.2.5 can be found in [101, 104, 194, 298]. The proof of Theorem 2.2.6 may be found in [194, Theorem 1], [104, Proposition 2.4]. The result of Lemma 2.2.7 is well known (see [4] and [49]) and is frequently used. For further discussion on the inequalities of Fan, see [247]. Equation (2.2) is a noncommutative analogue of the classical formula linking the distribution function of a random variable and its decreasing rearrangement [20, 159]. The proof of the noncommutative version can be found in [194, Theorem 1], [101, Proposition 1.3], and [104, Proposition 2.2]. The formula given in Remark 2.2.2 is classical and still holds for rearrangements of measurable functions defined on an arbitrary σ-finite measure space [20, 159].

2.6 Notes | 73

Submajorization The notation 1, and an example of a Banach ideal with traces but no continuous traces, are given in Section 3.5. The behavior of the sequence spaces l1 and l1,∞ under the Cesaro operator C is of particular importance. Section 3.5 proves the following collection of statements. Recall that a trace is singular according to Definition 1.3.2 if it vanishes on the set of finite-rank operators. Theorem 3.1.4. (a) Let Tr be the trace on ℒ1 . i. There exist traces on ℒ1 that are not a scalar multiple of Tr. ii. Up to a scalar constant, Tr is the unique continuous trace on ℒ1 for the norm ‖ ⋅ ‖1 . (b) There exist non-zero traces on ℒ1,∞ . i. Every trace on ℒ1,∞ is singular and vanishes on ℒ1 . ii. There exist traces on ℒ1,∞ that are not continuous for the quasinorm ‖ ⋅ ‖1,∞ . (c) Let 𝒥 be a two-sided ideal of ℒ(H). i. If ℒ1,∞ ⊂ 𝒥 then every trace on 𝒥 is singular and vanishes on ℒ1 . ii. If ℒ1 ⊊ 𝒥 ⊊ ℒ1,∞ then there exists a trace φ on 𝒥 such that φ = Tr on the ideal of finite-rank operators. Every such trace φ is not positive on 𝒥 . iii. If ℒ1 ⊊ 𝒥 ⊊ ℒ1,∞ then there exists a trace φ on 𝒥 such that φ vanishes on ℒ1 and is positive on 𝒥 . iv. If ℒ1 ⊊ 𝒥 ⊊ ℒ1,∞ and 𝒥 is quasinormed then every continuous trace on 𝒥 is singular. Theorem 3.1.4 indicates that every ideal ℒ1 ⊊ 𝒥 ⊊ ℒ1,∞ admits an extension of the matrix trace, but it is not unique or positive, and it is not continuous if 𝒥 is quasinormed. Theorems 3.5.8 and 3.5.9 in Section 3.5 prove a decomposition of a continuous

3.2 Calkin correspondence for quasinorms | 87

trace on a quasinormed ideal 𝒥 into normal and singular parts. A trace φn on 𝒥 is normal if φn (A) = sup φn (Ai ) i∈I

for an increasing net {Ai }i∈I in ℒ(H)+ with least upper bound A, which is equivalent to φn respecting countable additivity for the lattice of projections in Section 2.1. Theorem 3.1.5. Suppose 𝒥 is a quasinormed ideal of ℒ(H) and φ is a continuous trace on 𝒥 . Then φ = φn + φs for a continuous normal trace φn on 𝒥 and a continuous singular trace φs on 𝒥 .

3.2 Calkin correspondence for quasinorms This section proves the statements of Theorems 3.1.1 and 3.1.2. That is, the Calkin correspondence provides a bijective correspondence between quasinorms on ideals and quasinorms on sequence spaces. Symmetric quasinorms were defined in Definition 2.4.8 in Section 2.4. The straightforward component of Theorem 3.1.1 is part (a): a symmetric quasinorm ‖ ⋅ ‖𝒥 on a two-sided ideal 𝒥 restricts to a symmetric quasinorm on the commutative core J by the assignment 󵄩 󵄩 x 󳨃→ 󵄩󵄩󵄩diag(x)󵄩󵄩󵄩𝒥 ,

x ∈ J.

Indeed, the linearity of x 󳨃→ diag(x) and the property μ(diag(x)) = diag(μ(x)), x ∈ J, yield that x 󳨃→ ‖diag(x)‖𝒥 is a symmetric quasinorm on J. It is also relatively straightforward to prove part (a) of Theorem 3.1.2, namely ℰ being quasi-Banach implies that the commutative core E is quasi-Banach in the assigned quasinorm. The limit of any Cauchy sequence of operators diag(xn ), n ≥ 0, belongs to ℰ by assumption and also belongs to the diagonal. This follows since the diagonal of ℒ(H) in any basis, being isomorphic with the von Neumann algebra l∞ , is closed in the operator norm and the strong operator topology. The difficult parts are Theorems 3.1.1(b) and 3.1.2(b). If ‖ ⋅ ‖J is a quasinorm on a symmetric sequence space J, it is not obvious that the assignment 󵄩 󵄩 A 󳨃→ 󵄩󵄩󵄩μ(A)󵄩󵄩󵄩J ,

A ∈ 𝒥,

defines a quasinorm on the corresponding ideal 𝒥 . The difficulty is that A 󳨃→ μ(A) is not linear. In general, the problem of lifting a quasinorm on a sequence space to a quasinorm on an ideal goes back to the work of J. von Neumann and R. Schatten.

88 | 3 Calkin correspondence for norms and traces Although A 󳨃→ μ(A) is not linear, from Corollary 2.2.9 it is subadditive “up to dilation”, and we complete the proofs of Theorems 3.1.1 and 3.1.2 by considering the behavior of the symmetric quasinorm under dilation. The dilation operator σn : l∞ → l∞ , n ∈ ℕ, was introduced in Section 1.3.2 as σn (x(0), x(1), . . .) = (x(0), . . . , x(0), x(1), . . . , x(1), . . .), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n times

n times

x ∈ l∞ .

Lemma 3.2.1. Let (J, ‖ ⋅ ‖J ) be a quasinormed symmetric sequence space. We have 󵄩󵄩 󵄩 n 󵄩󵄩σn (x)󵄩󵄩󵄩J ≤ nK ‖x‖J ,

x ∈ J, n ≥ 1.

Here, K is the concavity modulus of J. Proof. For x ∈ J we have n−1

σn (x) = ∑ xk , k=0

where x( m−k ), xk (m) = { n 0,

m = k mod n, m ≠ k mod n.

By the quasitriangle inequality, we have n−1

n−1

k=0

k=0

󵄩󵄩 󵄩 k+1 n 󵄩󵄩σn (x)󵄩󵄩󵄩J ≤ ∑ K ‖xk ‖J ≤ K ∑ ‖xk ‖J . Since μ(xk ) = μ(x) for 0 ≤ k < n, we have ‖xk ‖J = ‖x‖J ,

0 ≤ k < n.

Combining these estimates, we complete the proof. Proof of Theorem 3.1.1. Part (a) is straightforward as discussed. Let us prove part (b). By Corollary 2.2.9, for A, B ∈ 𝒥 we have μ(A + B) ≤ σ2 μ(A) + σ2 μ(B). Thus, 󵄩 󵄩 󵄩 󵄩 ‖A + B‖𝒥 = 󵄩󵄩󵄩μ(A + B)󵄩󵄩󵄩J ≤ 󵄩󵄩󵄩σ2 μ(A) + σ2 μ(B)󵄩󵄩󵄩J . By the quasitriangle inequality and Lemma 3.2.1, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩σ2 μ(A) + σ2 μ(B)󵄩󵄩󵄩J ≤ K 󵄩󵄩󵄩σ2 μ(A)󵄩󵄩󵄩J + K 󵄩󵄩󵄩σ2 μ(B)󵄩󵄩󵄩J 󵄩 󵄩 󵄩 󵄩 ≤ 2K 3 󵄩󵄩󵄩μ(A)󵄩󵄩󵄩J + 2K 3 󵄩󵄩󵄩μ(B)󵄩󵄩󵄩J ,

3.2 Calkin correspondence for quasinorms | 89

where K is the concavity modulus of J. It follows that ‖A + B‖𝒥 ≤ 2K 3 ‖A‖𝒥 + 2K 3 ‖B‖𝒥 ,

A, B ∈ 𝒥 .

In other words, (𝒥 , ‖ ⋅ ‖𝒥 ) is a quasinormed ideal (with concavity modulus at most 2K 3 ). This gives us part (b). Part (c) is clear from 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩μ(diag(x))󵄩󵄩󵄩J = 󵄩󵄩󵄩μ(x)󵄩󵄩󵄩J = ‖x‖J ,

x ∈ J,

and 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩diag(μ(A))󵄩󵄩󵄩𝒥 = 󵄩󵄩󵄩μ(diag(μ(A)))󵄩󵄩󵄩𝒥 = 󵄩󵄩󵄩μ(A)󵄩󵄩󵄩𝒥 = ‖A‖𝒥 ,

A ∈ 𝒥.

We now prove closure in the corresponding quasinorms is preserved by the Calkin correspondence. Lemma 3.2.2. Let {Tk }k≥1 ∈ ℒ(H). We have ∞

∞

k=1

k=1

μ( ∑ Tk ) ≤ ∑ σ2k μ(Tk ) where it is assumed the series on the right converges in l∞ . Proof. We aim to show the convergence of the series in the left-hand side in l∞ together with the required estimate. Since ∑∞ k=n σ2k μ(Tk ) → 0 in l∞ as n → ∞, we get ∞

∞

∞

k=n

k=n

k=n

∑ (σ2k μ(Tk ))(0) = ∑ μ(0, Tk ) = ∑ ‖Tk ‖∞ → 0,

n → ∞.

Fix ϵ > 0 and choose N(ϵ) so large that ∞

∑ ‖Tk ‖∞ < ϵ,

k=n

n ≥ N(ϵ).

We have 󵄩󵄩 m 󵄩󵄩 m ∞ 󵄩󵄩 󵄩 󵄩󵄩 ∑ Tk 󵄩󵄩󵄩 ≤ ∑ ‖Tk ‖∞ ≤ ∑ ‖Tk ‖∞ , 󵄩󵄩 󵄩󵄩 󵄩󵄩k=n 󵄩󵄩∞ k=n k=n

m ≥ n ≥ N(ϵ).

So, the partial sums form a Cauchy sequence. We now have n

∞

k=1

k=1

∑ Tk → ∑ Tk ,

n → ∞,

90 | 3 Calkin correspondence for norms and traces in ℒ(H) and, therefore, n

∞

k=1

k=1

μ( ∑ Tk ) → μ( ∑ Tk ),

n → ∞,

(3.1)

in l∞ . Repeatedly applying Corollary 2.2.9, we obtain an estimate l

l

k=1

k=1

μ(t, ∑ Ak ) ≤ ∑ μ(tk , Ak ),

l

t ≥ ∑ tk . k=1

Thus, n

n

k=1

k=1

μ(t, ∑ Tk ) ≤ ∑ μ(

∞ t t , T ) ≤ μ( k , Tk ), ∑ k 2k 2 k=1

t > 0.

In other words, we have n

∞

k=1

k=1

μ( ∑ Tk ) ≤ ∑ σ2k μ(Tk ).

(3.2)

Combining (3.2) and (3.1), we complete the proof. Lemma 3.2.3. Let (E, ‖ ⋅ ‖E ) be a quasi-Banach sequence space with concavity modulus K. Let {xk }k≥1 be a sequence in E. If ∞

∑ K k ‖xk ‖E < ∞,

k=1

then the series ∑∞ k=1 xk converges in E. Proof. Let Xn = ∑nk=1 xk . We claim that {Xn }n≥1 is a Cauchy sequence in E. Fix ϵ > 0 and choose N such that ∞

∑ K k ‖xk ‖E < ϵ.

k=N

Let m > n ≥ N. By the quasitriangle inequality 󵄩󵄩 m 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ‖Xn − Xm ‖E = 󵄩󵄩 ∑ xk 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩k=n+1 󵄩󵄩E m

≤ ∑ K k−n ‖xk ‖E k=n+1

3.2 Calkin correspondence for quasinorms | 91 ∞

≤ ∑ K k−n ‖xk ‖E k=n+1 ∞

≤ ∑ K k ‖xk ‖E < ϵ. k=n+1

Since ϵ > 0 is arbitrary, it follows that (Xn )n≥1 is a Cauchy sequence in E. Since E is complete in the quasinorm ‖ ⋅ ‖E , it follows that (Xn )n≥1 converges in E. Proof of Theorem 3.1.2. Part (a) is straightforward as discussed. Let us prove part (b). Choose a Cauchy sequence {Sm }m≥1 in (ℰ , ‖ ⋅ ‖ℰ ) and let us establish its convergence. Firstly, the series is Cauchy in ℒ(H) and, hence, converges uniformly to S. Choose a strictly increasing sequence {mk }k≥1 such that k

‖Smk+1 − Smk ‖ℰ ≤ (4K)−k K −2 ,

k ≥ 1,

where K is the concavity modulus of E. By Lemma 3.2.1 with xk = μ(Smk+1 − Smk ), we have k 󵄩 󵄩 ∑ K k 󵄩󵄩󵄩σ2k μ(Smk+1 − Smk )󵄩󵄩󵄩E ≤ ∑ K k ⋅ 2k K 2 ‖Smk+1 − Smk ‖ℰ ≤ ∑ 2−k < ∞.

k≥1

k≥1

k≥1

By Lemma 3.2.3, the series ∞

∑ σ2k μ(Smk+1 − Smk )

k=1

converges in E. Thus, 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ σ k μ(Sm − Sm )󵄩󵄩󵄩 → 0, 2 k+1 k 󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩k=n 󵄩E

n → ∞.

(3.3)

We have ∞

S − Sm1 = ∑ Smk+1 − Smk . k=1

By Lemma 3.2.2, we have ∞

μ(S − Sm1 ) ≤ ∑ σ2k μ(Smk+1 − Smk ). k=1

By the preceding paragraph, the right-hand side is in E. Hence, so is the left-hand side. In other words, S − Sm1 ∈ ℰ . Since Sm1 ∈ ℰ , it follows that S ∈ ℰ .

92 | 3 Calkin correspondence for norms and traces We have ∞

S − Smn = ∑ Smk+1 − Smk . k=n

By Lemma 3.2.2, we have ∞

μ(S − Smn ) ≤ ∑ σ2k μ(Smk+1 − Smk ). k=n

By (3.3), the right-hand side tends to 0 in E as n → ∞. Hence, so does the left-hand side. In other words, S − Smn → 0 in ℰ as n → ∞. Since {Sm }m≥1 is Cauchy sequence, it follows that S − Sm → 0 in ℰ as m → ∞.

3.3 Discrete Figiel–Kalton theorem Let J be a symmetric sequence space. Denote by D(J) the set of all differences of decreasing rearrangements of sequences in J. That is, D(J) = {x ∈ J : x = μ(x1 ) − μ(x2 ),

x1 , x2 ∈ J}.

The space D(J) is a real linear subspace of J, which follows from the property μ(x1 ) + μ(y1 ) = μ(μ(x1 ) + μ(y1 )), x1 , y1 ∈ J. We also introduce the set Z(J) of differences of equimeasurable sequences in J. That is, Z(J) = Span{x1 − x2 : x1 , x2 ∈ J are self-adjoint and equimeasurable}. The space Z(J) is a real linear subspace in E. We remind the reader that equimeasurable means that the positive parts of x1 and x2 have the same singular value sequence and so do the negative parts. The Figiel–Kalton theorem for symmetric sequence spaces, initially stated by Figiel and Kalton for symmetric function spaces on L∞ (0, ∞), relates equimeasurable differences to Cesaro averages. Denote by C : l∞ → l∞ the Cesaro operator (Cx)(n) =

n 1 ∑ x(k), n + 1 k=0

x ∈ l∞ , n ≥ 0.

Theorem 3.3.1 (Figiel–Kalton theorem). Let J be a symmetric sequence space and let x ∈ D(J). Then Cx ∈ J if and only if x ∈ Z(J). The Figiel–Kalton theorem is proved by a combination of Theorems 3.3.2 and 3.3.7 below.

3.3 Discrete Figiel–Kalton theorem

| 93

Sufficiency of Cesaro averaging This section proves the following statement. Theorem 3.3.2. Let J be a symmetric sequence space and let x ∈ D(J). If Cx ∈ J, then x ∈ Z(J). The proof is based on proving the statement first for sequences x that have zero mean on dyadic intervals 2n ≤ k < 2n+1 and second for sequences which are constant on dyadic intervals. The following elementary lemma was stated by Pearcy and Topping in [196]. Lemma 3.3.3. Let λ ∈ ℝn be such that ∑n−1 k=0 λ(k) = 0. There exists a permutation π of the set {0, 1, . . . , n − 1} such that 󵄨󵄨 k 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨 ∑ λ(π(i))󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨λ(k)󵄨󵄨󵄨. 󵄨󵄨 0≤k≤n−1 0≤k≤n−1󵄨󵄨󵄨i=0 󵄨 Proof. Without loss of generality, all values λk , 0 ≤ k ≤ n − 1, are distinct. We may assume that μ(0, λ+ ) ≥ μ(0, λ− ). There exists a unique number π(0) such that λ(π(0)) = μ(0, λ+ ) and set n0 = 1. We use induction to define nk and π(k), 0 ≤ k ≤ n − 1. Let ni and π(i), 0 ≤ i ≤ k are already defined. If ∑ki=1 λ(π(i)) < 0, then set λ(π(k+1)) = μ(nk , λ+ ) and nk+1 = nk + 1. If ∑ki=1 λ(π(i)) ≥ 0, then set λπ(k+1) = −μ(k + 1 − nk , λ+ ) and nk+1 = nk . The next lemma will allow a sequence x ∈ D(J) whose dyadic averages vanish to be represented as a difference of two equimeasurable sequences. Lemma 3.3.4. Let λ ∈ ℝn be such that ∑n−1 k=0 λ(k) = 0. There exist equimeasurable elements λ1 , λ2 ∈ ℝn such that λ = λ1 − λ2 and such that ‖λ1 ‖∞ ≤ ‖λ‖∞ ,

‖λ2 ‖∞ ≤ ‖λ‖∞ .

Proof. Let π be the permutation constructed in Lemma 3.3.3. Set θ = λ ∘ π. Let k

λ3 (k) = ∑ θ(l), l=0

0 ≤ k < n.

Set λ4 (k) = λ3 (k − 1),

1 ≤ k < n,

λ4 (0) = 0.

It is immediate that λ3 − λ4 = θ. By construction, λ3 is a permutation of λ4 , so that they are equimeasurable. By Lemma 3.3.3, we have ‖λ3 ‖∞ ≤ ‖θ‖∞ ,

‖λ4 ‖∞ ≤ ‖θ‖∞ .

Setting λ1 = λ3 ∘ π −1 ,

λ2 = λ4 ∘ π −1 ,

94 | 3 Calkin correspondence for norms and traces we obtain λ1 − λ2 = (λ3 − λ4 ) ∘ π −1 = θ ∘ π −1 = θ. This completes the proof. Define the bounded linear operator E 󸀠 : l∞ → l∞ by, here x ∈ l∞ , E 󸀠 (x) = (x(0), x(1),

x(2) + x(3) x(4) + x(5) + x(6) + x(7) ,..., , . . . , . . .). ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 4 2 times

4 times

That is, E 󸀠 (x)(0) = x(0) and, for n ≥ 0, 󸀠

−n

E (x)(k) = 2

2n+1 −1

2n ≤ k < 2n+1 .

∑ x(j),

j=2n

It follows that E 󸀠 (x) is constant on dyadic intervals 2n ≤ k < 2n+1 , n ≥ 0, and therefore E 󸀠 is an idempotent. The next result proves Theorem 3.3.2 for those sequences x such that E 󸀠 (x) vanishes. Lemma 3.3.5. Let J be a symmetric sequence space and let x ∈ D(J). If E 󸀠 (x) = 0, then x ∈ Z(J). Proof. Let n ≥ 0. Let xn (k) = x(k + 2n ),

0 ≤ k < 2n .

By the condition E 󸀠 (x) = 0 and Lemma 3.3.4, there exist equimeasurable yn and zn n from ℝ2 such that xn = yn − zn and ‖yn ‖∞ ≤ ‖xn ‖∞ ,

‖zn ‖∞ ≤ ‖xn ‖∞ .

Set y(k) = yn (k − 2n ),

z(k) = zn (k − 2n ),

2n ≤ k < 2n+1 .

Since E 󸀠 (x) = 0, we have x(0) = 0 and set y(0) = z(0) = 0. It is immediate that y is equimeasurable with z and x = y − z. It remains to show that y, z ∈ J. For this purpose, recall that x ∈ D(J) and choose a = μ(a) ∈ J such that |x| ≤ a. We now have 󵄨 󵄨 μ(2n , y) ≤ ‖yχ[2n ,∞) ‖∞ = sup ‖yl ‖∞ ≤ sup ‖xl ‖∞ = sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨 ≤ μ(2n , a). l≥n

l≥n

k≥2n

3.3 Discrete Figiel–Kalton theorem

| 95

If 2n ≤ m < 2n+1 , then μ(m, y) ≤ μ(2n , y) ≤ μ(2n , a) ≤ μ(⌊

m ⌋, a) = μ(m, a ⊕ a). 2

Since J is a symmetric sequence space, it follows that y ∈ J. Since y and z are equimeasurable, it follows that also z ∈ J. We prove Theorem 3.3.2 if the sequence x is constant on dyadic intervals. Lemma 3.3.6. Let J be a symmetric sequence space and let x ∈ D(J). If Cx ∈ J and if x = E 󸀠 (x), then x ∈ Z(J). Proof. Define a sequence z by setting z(k) = (Cx)(2⌈log2 (k+1)⌉ − 1),

k ≥ 0.

For 2n ≤ k < 2n+1 , it follows from the definition of z that 󵄨󵄨2n+1 −1 󵄨󵄨 󵄨󵄨 k 󵄨󵄨 󵄨 n+1 󵄨󵄨 󵄨󵄨 󵄨󵄨 −n−1 󵄨󵄨󵄨2 −1 󵄨󵄨 󵄨󵄨 󵄨󵄨 −n−1 󵄨󵄨󵄨 −n−1 󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨z(k)󵄨󵄨 = 2 󵄨󵄨 ∑ x(l)󵄨󵄨 ≤ 2 󵄨󵄨 ∑ x(l)󵄨󵄨 + 2 󵄨󵄨 ∑ x(l)󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 l=0 󵄨 󵄨l=0 󵄨 󵄨 l=k+1 󵄨 Recall that x ∈ D(J) and choose a = μ(a) ∈ J such that |x| ≤ a. We now have 󵄨󵄨2n+1 −1 󵄨󵄨 2n+1 −1 󵄨󵄨 󵄨 󵄨󵄨 ∑ x(l)󵄨󵄨󵄨 ≤ ∑ μ(l, a) ≤ 2n+1 μ(k, a). 󵄨󵄨 󵄨󵄨 󵄨󵄨 l=k+1 󵄨󵄨 l=k+1 It follows that |z| ≤ |Cx| + μ(a). In particular, z ∈ J. On the other hand, we have, for 2n ≤ k < 2n+1 and using x = E 󸀠 (x), 2n+1 −1

2n −1

l=0

l=0

(2z − σ2 z)(k) = 2−n ∑ x(l) − 2−n ∑ x(l) 2n+1 −1

= 2−n ∑ x(l) = x(2n ) = x(k), l=2n

where we have used the assumption that x = E 󸀠 (x). In other words, 2z − σ2 z = x. Since z ∈ J and 2z is equimeasurable with σ2 z, we have 2z − σ2 z ∈ Z(J). Thus, x ∈ Z(J).

96 | 3 Calkin correspondence for norms and traces We can now prove Theorem 3.3.2 in full generality. Proof of Theorem 3.3.2. Let J be a symmetric sequence space and let x ∈ D(J). Let y = E 󸀠 (x),

z = x − E 󸀠 (x).

Since x ∈ D(J), it follows that |x| ≤ μ(a) for some a ∈ J. Indeed, if 2n ≤ k < 2n+1 , then 󵄨󵄨2n+1 −1 󵄨󵄨 2n+1 −1 󵄨󵄨 k 󵄨󵄨 󵄨 −n 󵄨󵄨 −n ∑ μ(l, a) ≤ μ(2n , a) ≤ μ(⌊ ⌋, a). 󵄨󵄨y(k)󵄨󵄨󵄨 = 2 󵄨󵄨󵄨 ∑ x(l)󵄨󵄨󵄨 ≤ 2 󵄨󵄨 2 n 󵄨󵄨󵄨 l=2n l=2 󵄨 Hence y, z ∈ J. As E 󸀠 is linear and E 󸀠 (μ(a)) = μ(E 󸀠 (μ(a))), we have that E 󸀠 : D(J) → D(J) since E 󸀠 (μ(a) − μ(b)) = μ(E 󸀠 (μ(a))) − μ(E 󸀠 (μ(b))),

a, b ∈ J.

It follows that y, z ∈ D(J) with y = E 󸀠 (y),

E 󸀠 (z) = 0,

since E 󸀠 is an idempotent. If, in addition, Cx ∈ J, then also Cy ∈ J. Indeed, if 2n ≤ k < 2n+1 , then 󵄨󵄨 k 󵄨󵄨 󵄨󵄨 k 󵄨󵄨 k k 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 (k + 1)󵄨󵄨(Cy)(k) − (Cx)(k)󵄨󵄨 = 󵄨󵄨 ∑ y(l) − ∑ x(l)󵄨󵄨 ≤ 󵄨󵄨 ∑ y(l) − ∑ x(l)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 l=0 l=2n 󵄨l=0 󵄨 󵄨l=2 󵄨 󵄨󵄨 k 󵄨󵄨 󵄨󵄨 k 󵄨󵄨 k k 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ 󵄨󵄨󵄨 ∑ y(l)󵄨󵄨󵄨 + 󵄨󵄨󵄨 ∑ x(l)󵄨󵄨󵄨 ≤ ∑ μ(l, a ⊕ a) + ∑ μ(l, a) 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 l=2n 󵄨l=2 󵄨 󵄨l=2 󵄨 l=2n n n ≤ (k + 1)μ(2 , a) + (k + 1)μ(2 , a ⊕ a). Note that μ(2n , a) = σ2 μ(2n+1 , a) = μ(2n+1 , a ⊕ a) ≤ μ(k, a ⊕ a). Hence (k + 1)μ(2n , a) + (k + 1)μ(2n , a ⊕ a) ≤ (k + 1)μ(k, a⊕2 ) + (k + 1)μ(k, a⊕4 ), and we have |Cy − Cx| ≤ μ(a⊕2 ) + μ(a⊕4 ). By Lemma 3.3.6, we have y ∈ Z(J). By Lemma 3.3.5, we have z ∈ Z(J). It follows that x = y + z ∈ Z(J).

3.3 Discrete Figiel–Kalton theorem

| 97

Necessity of Cesaro averaging This section proves the following statement. Theorem 3.3.7. Let J be a symmetric sequence space and let x ∈ D(J). If x ∈ Z(J), then Cx ∈ J. The first step is proving that the center Z(J) is the linear span of differences of positive equimeasurable elements. Lemma 3.3.8. Let J be a symmetric sequence space. We have Z(J) = Span{x1 − x2 : x1 , x2 ∈ J are positive and equimeasurable}. Proof. We write K−1

x = ∑ x1k − x2k , k=0

where, for each 0 ≤ k < K, the elements x1k , x2k ∈ J are self-adjoint and equimeasurable. We write 2K−1

x = ∑ y1k − y2k , k=0

where (we denote ⌊ k2 ⌋ by l to lighten the notations) (x1l )+ ,

y1k = {

(x2l )− ,

k = 0 mod 2, k = 1 mod 2,

(x2l )+ , k = 0 mod 2, y2k = { (x1l )− , k = 1 mod 2.

For every 0 ≤ k < 2K, the elements y1k , y2k ∈ J are positive and equimeasurable. The next lemma is a fundamental observation. It will also prove the additivity of a symmetric functional lifted from the commutative core to the corresponding ideal. Lemma 3.3.9. If 0 ≤ x, y ∈ l∞ , then 0 ≤ C(μ(x) + μ(y) − μ(x + y)) ≤ μ(x + y). Proof. Let n ∈ ℤ+ . By Theorem 2.3.5, we have n

n

n

k=0

k=0

k=0

∑ μ(k, x + y) ≤ ∑ μ(k, x) + ∑ μ(k, y).

By Theorem 2.3.6, we have n

n

2n+1

n

k=0

k=0

k=0

k=0

∑ μ(k, x) + ∑ μ(k, y) ≤ ∑ μ(k, x + y) ≤ ∑ μ(k, x + y) + (n + 1)μ(n, x + y).

98 | 3 Calkin correspondence for norms and traces Combining those inequalities and dividing them by n + 1, we write 0≤

n 1 ∑ μ(k, x) + μ(k, y) − μ(k, x + y) ≤ μ(n, x + y). n + 1 k=0

Since n ∈ ℤ+ is arbitrary, the assertion follows. Corollary 3.3.10. Let J be a symmetric sequence space and let {xk }Kk=1 ⊂ J be a positive sequence. We have K

K

k=1

k=1

C( ∑ μ(xk )) − C(μ( ∑ xk )) ∈ J. Proof. For K = 2, the assertion follows from Lemma 3.3.9, since if z0 ∈ l∞ and 0 < z0 ≤ z from some z = μ(z) ∈ J then z0 ∈ J. Since x, y ∈ J, we get x + y ∈ J, and 0 ≤ C(μ(x) + μ(y) − μ(x + y)) ≤ μ(x + y). The general case follows by induction on K. We can now prove Theorem 3.3.7. Proof of Theorem 3.3.7. By Lemma 3.3.8, we can write K−1

x = ∑ x1k − x2k , k=0

where, for each 0 ≤ k < K, the elements x1k , x2k ∈ J are positive and equimeasurable. By assumption, x = μ(a) − μ(b), a, b ∈ J. In other words, we have K−1

K−1

k=0

k=0

μ(a) + ∑ x2k = μ(b) + ∑ x1k . By Corollary 3.3.10, we have K

K

k=1

k=1

K

K

k=1

k=1

C(μ(a) + ∑ μ(x2k )) − Cμ(μ(a) + ∑ x2k ) ∈ J, C(μ(b) + ∑ μ(x1k )) − Cμ(μ(b) + ∑ x1k ) ∈ J. Taking the difference of these equations, we arrive at K

K

k=1

k=1

C(μ(a) + ∑ μ(x2k )) − C(μ(b) + ∑ μ(x1k )) ∈ J. Since μ(x1k ) = μ(x2k ), 1 ≤ k ≤ K, it follows that Cx = C(μ(a) − μ(b)) ∈ J.

3.4 Calkin correspondence for traces | 99

3.4 Calkin correspondence for traces This section proves the statement of Theorem 3.1.3. That is, the Calkin correspondence provides a bijective correspondence between a trace φ on a two-sided ideal and a symmetric functional φ̂ on the commutative core. Symmetric functionals were defined in Definition 2.4.4. It is evident from the restriction φ 󳨃→ φ ∘ diag of a trace to a symmetric functional and the lift of a symmetric functional to a trace on the positive cone φ̂ 󳨃→ φ̂ ∘ μ that the bijective correspondence preserves positivity. This section also proves that continuous traces are in bijective correspondence with continuous symmetric functionals. That a trace φ : 𝒥 → ℂ on a two-sided ideal 𝒥 restricts to a symmetric functional φ̂ on the commutative core J by the assignment ̂ φ(x) = φ(diag(x)),

x ∈ J,

is straightforward since diag is linear and μ(diag(x)) = diag(μ(x)),

x ∈ J.

That φ̂ is continuous in the induced quasinorm on J from Theorem 3.1.1 if φ is continuous for a quasinorm on 𝒥 is also straightforward, since 󵄨󵄨 ̂ 󵄨󵄨 󵄩 󵄩 󵄨󵄨φ(x)󵄨󵄨 ≤ ‖φ‖ ⋅ 󵄩󵄩󵄩diag(x)󵄩󵄩󵄩𝒥 = ‖φ‖ ⋅ ‖x‖J ,

x ∈ J.

Lifting a symmetric functional φ̂ on a symmetric sequence space J to a trace φ : 𝒥 → ℂ on the corresponding two-sided ideal 𝒥 by linear extension of the assignment ̂ φ(A) = φ(μ(A)),

A ∈ 𝒥+ ,

is harder since μ is not additive. The same stumbling block was faced in Section 3.2. That φ is continuous in the induced quasinorm on 𝒥 from Theorem 3.1.1 if φ̂ is continuous for a quasinorm on J is straightforward since 󵄨󵄨 󵄨 ̂ ⋅ 󵄩󵄩󵄩󵄩μ(A)󵄩󵄩󵄩󵄩J = ‖φ‖ ̂ ⋅ ‖A‖𝒥 , 󵄨󵄨φ(A)󵄨󵄨󵄨 ≤ ‖φ‖

A ∈ 𝒥+ .

Therefore to complete the proof of Theorem 3.1.3 we only require the following result.

100 | 3 Calkin correspondence for norms and traces Theorem 3.4.1. Let J be a symmetric sequence space and let 𝒥 be the corresponding ideal of ℒ(H). If φ : J → ℂ is a symmetric functional, then the mapping A → φ(μ(A)),

0 ≤ A ∈ 𝒥,

is additive on the positive cone of 𝒥 . In particular, it extends to a trace on 𝒥 . The approach below uses the Figiel–Kalton theorem and Lemma 3.3.9 to identify that the difference μ(A) + μ(B) − μ(A + B) ∈ Z(J) for any A, B ∈ 𝒥+ and, immediately, the difference vanishes for any symmetric functional. The bijective correspondence between traces and symmetric functionals given by Theorem 3.1.3 means that traces exist when symmetric functionals exist, and this can be determined from the Figiel–Kalton theorem by invariance of the commutative core under Cesaro averaging. Theorem 3.4.2. Let 𝒥 be an ideal of ℒ(H) with commutative core J. The following conditions are equivalent: (a) Every trace on 𝒥 is trivial. (b) Every symmetric functional on J is trivial. (c) C : J → J. We now prove Theorems 3.4.1 and 3.4.2. The following lemma is stated as a separate assertion for convenience. Lemma 3.4.3. If φ is an additive and positively homogeneous mapping on the positive cone 𝒥+ , then φ admits a unique extension to a linear functional on 𝒥 . Furthermore, if φ is unitarily invariant on 𝒥+ then the unique extension to 𝒥 is a trace. Proof of Theorem 3.4.1. Let φ be a symmetric functional on J. Let 0 ≤ A, B ∈ 𝒥 . Define x ∈ J by setting x = μ(A) + μ(B) − μ(A + B). By the definition of D(J), we have x ∈ D(J). From Lemma 3.3.9, Cx ∈ J. By Theorem 3.3.2, we have x ∈ Z(J). In particular, φ(x) = 0. By the linearity of φ, we have φ(μ(A + B)) = −φ(x) + φ(μ(A)) + φ(μ(B)) = φ(μ(A)) + φ(μ(B)). This proves additivity on the positive cone. Besides being additive, φ ∘ μ is positive homogeneous and unitarily invariant. By Lemma 3.4.3, φ ∘ μ uniquely extends to a trace.

3.5 Existence and structure of traces | 101

The next lemma proves that Z(J) is the common kernel for all symmetric functionals on a symmetric sequence space. Lemma 3.4.4. Let J be a symmetric sequence space and let x ∈ J. The following conditions are equivalent: (a) x ∈ Z(J). (b) φ(x) = 0 for every symmetric linear functional φ on J. Proof. The implication (a) ⇒ (b) is trivial. Let us prove the implication (b) ⇒ (a). Let π : J → J/Z(J) be the quotient mapping. Suppose π(x) ≠ 0. Let f be a linear functional on J/Z(J) such that f (π(x)) ≠ 0. Set φ = f ∘ π. Let x1 , x2 ∈ J be positive and equimeasurable. Clearly, x1 − x2 ∈ Z(J) so that π(x1 − x2 ) = 0 and, therefore, φ(x1 − x2 ) = 0. Hence, φ is symmetric. By construction, φ(x) ≠ 0. This contradicts (b). Hence, the assumption π(x) ≠ 0 is incorrect and, therefore, x ∈ Z(J). Proof of Theorem 3.4.2. The conditions (a) and (b) are equivalent by Theorem 3.1.3. Let us prove the equivalence of conditions (b) and (c). Suppose (c) holds. Fix a symmetric functional φ : J → ℂ. Let 0 ≤ x ∈ J, so that μ(x) ∈ D(J). By Theorem 3.3.2, we have μ(x) ∈ Z(J). Thus, φ(x) = φ(μ(x)) = 0 for every symmetric functional φ on J. Since x ∈ J is arbitrary, it follows that φ = 0. This proves (b). Suppose now that (b) holds. Let x ∈ J and note that φ(μ(x)) = 0 for every symmetric linear functional φ on J. By Lemma 3.4.4, we have μ(x) ∈ Z(J). By Theorem 3.3.7, we have that Cμ(x) ∈ J. Since |Cx| ≤ Cμ(x), it follows that Cx ∈ J. Since x ∈ J is arbitrary, it follows that C : J → J. This proves (c).

3.5 Existence and structure of traces Theorem 3.4.2 proves that no nonzero traces exist on a two-sided ideal 𝒥 of ℒ(H) if and only if the commutative core J is invariant under the Cesaro operator C : l∞ → l∞ defined by (Cx)(n) =

n 1 ∑ x(k), n + 1 k=0

x ∈ l∞ , n ≥ 0.

The invariance of the commutative core J under the Cesaro operator can be tested for many ideals. Examples of existence results For 1 ≤ p ≤ ∞, Chapter 1 and Section 2.5 introduced the Schatten ideals ℒp = {A ∈ ℒ(H) : μ(A) ∈ lp }

102 | 3 Calkin correspondence for norms and traces closed with respect to the symmetric norm 󵄩 󵄩 ‖A‖p = 󵄩󵄩󵄩μ(A)󵄩󵄩󵄩p ,

A ∈ ℒp ,

and the weak ideals ℒp,∞ = {A ∈ ℒ(H) : μ(A) ∈ lp,∞ }

closed with respect to the symmetric quasinorm 󵄩 󵄩 ‖A‖p,∞ = 󵄩󵄩󵄩μ(A)󵄩󵄩󵄩p,∞ ,

A ∈ ℒp,∞ .

The Schatten ideals and the weak ideals, p > 1, have no traces as a consequence of the classical Hardy inequality. Corollary 3.5.1. For 1 < p ≤ ∞, the Schatten ideals ℒp and the weak ideals ℒp,∞ admit no nontrivial traces. Proof. For p > 1, the operator C : lp → lp is bounded by the Hardy inequality [19]. For 1 < q < p, lp,∞ is an interpolation space between lq and l∞ . By interpolation, C : lp,∞ → lp,∞ is bounded for p > 1. The assertion follows from Theorem 3.4.2. For the trace class operators ℒ1 , the Figiel–Kalton theorem implies that the trace Tr is not the only trace on ℒ1 . Corollary 3.5.2. There exists a trace φ : ℒ1 → ℂ such that φ ≠ α ⋅ Tr for a constant α. However, up to scalar constant, Tr is the unique continuous trace on ℒ1 . Proof. Take a = μ(a) ∈ l1 such that a(n) =

1

,

n ≥ 0,

b(0) = ∑ a(k) , b(n) = 0,

n ≥ 1.

(n + 1) log2 (2 + n)

and b = μ(b) ∈ l1 such that k≥0

Set x = a − b ∈ Dl1 . Clearly, ∑k≥0 x(k) = 0, hence Tr vanishes on diag(x). However, n+1

n 1 1 1 1 (Cx)(n) = dx ∼ − , n ≥ 1, ∑ x(k) ∼ ∫ 2 n + 1 k=0 n+1 n log(1 + n) x log (x) 1

3.5 Existence and structure of traces | 103

and, therefore, Cx ∉ l1 . Since x ∈ Dl1 , Theorem 3.3.7 implies that x ∉ Zl1 . By Lemma 3.4.4, we have φ(x) ≠ 0 for some symmetric linear functional φ on l1 . By Theorem 3.1.3, φ lifts to a trace on ℒ1 which cannot be scalar multiple of Tr. Theorem 2.5.1 proved that, up to scalar constant, Tr is the unique continuous trace on ℒ1 . The next example demonstrates that there is a Banach ideal 𝒥 which admits nontrivial traces but does not admit a nontrivial continuous trace. The example involves Orlicz ideals. Let Φ be a convex function on [0, ∞) such that Φ(t) > 0 for all t > 0 and such that 0 = Φ(0) = lim

t→0

Φ(t) t = lim . t→∞ Φ(t) t

Denote by lΦ the Orlicz sequence space (see, e. g., [159, 165]) lΦ := {x ∈ l∞ : ∃s > 0 such that Φ(

|x| ) ∈ l1 }. s

The Orlicz norm ‖ ⋅ ‖lΦ is defined by 󵄩󵄩 |x| 󵄩󵄩󵄩 󵄩 ‖x‖lΦ := inf{s > 0 : 󵄩󵄩󵄩Φ( )󵄩󵄩󵄩 ≤ 1}, 󵄩󵄩 s 󵄩󵄩1

x ∈ lΦ .

Equipped with the norm ‖x‖lΦ , lΦ becomes a symmetric Banach sequence space. Let ℒΦ be the corresponding Banach ideal of ℒ(H) with commutative core lΦ . Corollary 3.5.3. There exists an Orlicz function Φ such that the Banach ideal ℒΦ admits a nontrivial trace but does not admit a nontrivial continuous trace. Proof. By Proposition 2.b.5 in [166] there exists an Orlicz sequence space lΦ which is not an interpolation space between lp and l∞ for any p > 1. For such a space, we have C : lΦ ↛ lΦ , through a combination of [159, Theorem II.6.4] and [159, Theorem II.6.6]. By Theorem 3.4.2, ℒΦ admits a nontrivial trace. Let φ : lϕ → ℂ be a continuous symmetric functional. For every n ∈ ℕ, we have 󵄨󵄨 󵄨 1 󵄨 󵄨󵄨 󵄨 󵄨󵄨 1 󵄨󵄨φ(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨φ( σn x)󵄨󵄨󵄨 ≤ ‖φ‖lΦ →ℂ ⋅ ‖σn x‖lΦ , 󵄨󵄨 n 󵄨󵄨 n

x ∈ lΦ .

The first equality follows directly from Lemma 2.2.12. For every Orlicz space lΦ with lΦ ≠ l1 , we have that lim

n→∞

1 ‖σ x‖ = 0, n n lΦ

x ∈ lϕ .

This fact can be observed in the proof of Proposition 4.8 in [153]. It follows that φ(x) = 0. Since x ∈ lΦ is arbitrary, it follows that φ = 0. Hence, every continuous trace on ℒΦ is trivial.

104 | 3 Calkin correspondence for norms and traces Structure of traces The behavior of the symmetric sequence space l1 of summable sequences and the symmetric sequence space l1,∞ of weakly-summable sequences under the Cesaro operator characterizes the traces admitted by ideals. Definition 1.3.2 called a trace φ : 𝒥 → ℂ singular if it vanishes on the set of finite-rank operators. For continuous traces on quasinormed ideals, or all traces on any ideal containing ℒ1,∞ , extensions of the matrix trace Tr and singular traces characterize all traces. Corollary 3.5.4. Traces exist on ℒ1,∞ . Every trace on ℒ1,∞ is singular and vanishes on ℒ1 . Proof. Let x = μ(x) ∈ l1,∞ . Note that Cx ∈ l1,∞ if and only if x ∈ l1 . By Theorem 3.3.2, we have that x ∈ Zl1,∞ if and only if x ∈ l1 . Hence there exists traces by Lemma 3.4.4 and φ(x) = 0 when x ∈ l1 for every symmetric linear functional φ on l1,∞ . Since x = μ(x) ∈ l1 is arbitrary, it follows that φ|l1 = 0 for every symmetric linear functional φ on l1,∞ . In fact, ℒ1,∞ is the minimal ideal of ℒ(H) in the partial order of set inclusion such that all traces are singular. Theorem 3.5.5. If 𝒥 is an ideal of ℒ(H), then the following statements are equivalent. (a) Every trace φ : 𝒥 → ℂ is singular. (b) ℒ1,∞ ⊂ 𝒥 . Proof. Suppose ℒ1,∞ ⊂ 𝒥 . If φ is a trace on 𝒥 , then φ|ℒ1,∞ is a trace on ℒ1,∞ . By Corollary 3.5.4, φ is singular. Suppose ℒ1,∞ ⊄ 𝒥 . In particular, we have Ce0 ∉ J, where J is the commutative core of the ideal 𝒥 and e0 = (1, 0, . . .) ∈ c00 . By Theorem 3.3.7, e0 ∉ Z(J). By Lemma 3.4.4, there exists a symmetric linear functional φ on J such that φ(e0 ) ≠ 0. By Theorem 3.1.3, φ lifts to a trace on 𝒥 . This trace is not singular. The proof of Theorem 3.5.5 showed the statement that any ideal 𝒥 that does not contain ℒ1,∞ admits an extension of the matrix trace. Corollary 3.5.6. If 𝒥 is an ideal of ℒ(H), then the following statements are equivalent. (a) There exists a trace φ : 𝒥 → ℂ such that φ = Tr on the ideal of finite-rank operators 𝒞00 . (b) ℒ1,∞ ⊄ 𝒥 . Proof. By the proof of Theorem 3.5.5, the statements are equivalent, without loss of generality, to the existence of a trace φ such that φ(p) = 1 where p = diag(e0 ). If φ(p) = 1 for some rank 1 projection p, then, since an arbitrary rank 1 projection q is unitarily equivalent to p, φ(q) = 1 for all rank 1 projections. Hence φ = Tr on the ideal of finite-rank operators. Suppose ℒ1 ⊊ 𝒥 ⊊ ℒ1,∞ .

3.5 Existence and structure of traces | 105

By Corollary 3.5.6, the two-sided ideal 𝒥 admits an extension of the matrix trace. The next result shows that the extension of the matrix trace from 𝒞00 to 𝒥 is not unique. It also demonstrates that ℒ1,∞ admits traces that are not continuous. Corollary 3.5.7. There are traces on ℒ1,∞ that do not vanish on any ideal 𝒥 where ℒ1 ⊊ 𝒥 ⊊ ℒ1,∞ . Hence, there exist traces on ℒ1,∞ that are not continuous. Proof. Let x = μ(x) ∈ J be such that x ∉ l1 . Here J is the commutative core of 𝒥 . Since μ(x) ∉ l1 , Cx ∉ l1,∞ . Since x = μ(x) ∈ D(J) we have from Theorem 3.3.1 that x ∉ Zl1,∞ . By Lemma 3.4.4, there exists a symmetric linear functional φ : l1,∞ → ℂ such that φ(x) ≠ 0. By Theorem 3.4.1, there exists a trace φ on ℒ1,∞ such that φ(diag(x)) ≠ 0. This proves the first statement. Let 𝒥 = (ℒ1,∞ )0 be the closure of 𝒞00 in the quasinorm of ℒ1,∞ . By Lemma 2.5.3, all continuous traces on ℒ1,∞ vanish on (ℒ1,∞ )0 . The statement that all traces on ℒ1,∞ are continuous contradicts the first statement. Every trace on an ideal 𝒥 such that ℒ1,∞ ⊂ 𝒥 is singular. Every continuous trace on an quasinormed ideal 𝒥 such that ℒ1 ⊊ 𝒥 ⊊ ℒ1,∞

is also singular by the next result. Hence an extension of Tr from 𝒞00 to 𝒥 given by Corollary 3.5.6 is not continuous. Theorem 3.5.8. If 𝒥 is a quasinormed ideal of ℒ(H), then the following statements are equivalent. (a) Every continuous trace φ : 𝒥 → ℂ is singular. (b) 𝒥 ⊄ ℒ1 . Proof. Let φ be a nonsingular continuous trace on 𝒥 . Let A be a finite-rank operator such that φ(A) ≠ 0. It follows that either φ(ℜA) ≠ 0 or φ(ℑA) ≠ 0. Hence, without loss of generality, A = A∗ . Let A = ∑ λ(k, A)pk k≥0

be the Schmidt decomposition of A. Since rank(A) < ∞, it follows that the sum is actually finite. Clearly, φ(A) = ∑ λ(k, A)φ(pk ). k≥0

Since pk and p0 are unitarily equivalent, it follows that φ(pk ) = φ(p0 ). Thus, φ(A) = ∑ λ(k, A) ⋅ φ(p0 ). k≥0

106 | 3 Calkin correspondence for norms and traces Since φ(A) ≠ 0, it follows that φ(p0 ) ≠ 0. Without loss of generality, φ(p0 ) = 1. Let 0 ≤ B ∈ 𝒥 and let B = ∑ μ(k, B)qk k≥0

be its Schmidt decomposition. By linearity, n

n

k=0

k=0

φ( ∑ μ(k, B)qk ) = ∑ μ(k, B)φ(qk ). Since p0 and qk are unitarily equivalent, it follows that φ(qk ) = φ(p0 ) = 1. Thus, n

n

k=0

k=0

φ( ∑ μ(k, B)qk ) = ∑ μ(k, B). Since φ is continuous, it follows that 󵄨󵄨 󵄨󵄨 󵄩󵄩 n 󵄩󵄩 n 󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨φ( ∑ μ(k, B)qk )󵄨󵄨󵄨 ≤ ‖φ‖𝒥 →ℂ 󵄩󵄩󵄩 ∑ μ(k, B)qk 󵄩󵄩󵄩 ≤ ‖φ‖𝒥 →ℂ ‖B‖𝒥 . 󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄨󵄨󵄨 󵄨󵄨 󵄩󵄩k=0 󵄩󵄩𝒥 󵄨 k=0 Thus, n

∑ μ(k, B) ≤ ‖φ‖𝒥 →ℂ ‖B‖𝒥 .

k=0

Letting n → ∞, we obtain ‖B‖1 ≤ ‖φ‖𝒥 →ℂ ⋅ ‖B‖𝒥 . Since 0 ≤ B ∈ 𝒥 is arbitrary, it follows that 𝒥 ⊂ ℒ1 . Conversely, if 𝒥 ⊂ ℒ1 , then Tr|𝒥 is continuous and nonsingular. Combined, the above results prove Theorem 3.1.4. Theorem 3.5.8 proves the decomposition statement in Theorem 3.1.5 when 𝒥 ⊄ ℒ1 , since φ = φs where φs is continuous and singular. The next result completes the proof of Theorem 3.1.5. Theorem 3.5.9. Let 𝒥 ⊂ ℒ1 be a quasinormed ideal of ℒ(H). If φ is a continuous trace on 𝒥 , then there exists a unique decomposition φ = α ⋅ Tr + φs where α is a scalar constant and φs is a continuous singular trace on 𝒥 .

3.6 Notes | 107

Proof. Let p be a rank-one projection. Set φn = φ(p) ⋅ Tr|𝒥 and φs = φ − φ(p) ⋅ Tr|𝒥 . That φ(p) ⋅ Tr|𝒥 is continuous on 𝒥 is demonstrated in the proof Theorem 3.5.8. That φn and hence φs are traces is obvious. Suppose φs is not singular. Repeating the argument in the proof of Theorem 3.5.8, we conclude that φs (q) ≠ 0 for some rank-one projection q. Since p and q are unitarily equivalent, it follows that φs (p) = φs (q). However, φs (p) = 0 by definition. This contradiction shows that φ is singular. We have established the existence of the decomposition. For uniqueness, suppose φ = α󸀠 ⋅ Tr|𝒥 + φ󸀠s and φ = α ⋅ Tr|𝒥 + φs . Then φ(p) = α = α󸀠 . Hence φs = φ󸀠s .

3.6 Notes Correspondence of symmetric quasinorms That a symmetric norm on an ideal restricts to a symmetric norm on its diagonal, and hence commutative core, is straightforward. Lifting a symmetric norm from the commutative core to the ideal and proving closure is a problem from John von Neumann’s paper [282] published in 1937. Solving it has been a long standing goal of the second named author. The notes to Chapter 2 discussed that von Neumann’s [282] paper introduced symmetric norms in the finite-dimensional setting, in particular, for the matrix ideals ℒp . The paper implied the question whether every symmetric norm on the symmetric sequence space J defines a symmetric norm on 𝒥 by the assignment 󵄩 󵄩 ‖A‖𝒥 := 󵄩󵄩󵄩μ(A)󵄩󵄩󵄩J ,

A ∈ 𝒥.

This was solved in 2008 [152]. If J is complete, then 𝒥 is complete. Hence all Banach ideals are in bijective correspondence with symmetric Banach sequence spaces. The question was resolved for symmetric quasinorms and quasi-Banach ideals in 2013 [163, 258]. For a Banach ideal, the statements in Theorems 3.1.1 and 3.1.2 appeared in the paper [152] as a combination of [152, Theorem 8.7] and [152, Theorem 8.11]. These results generalize a number of earlier results in the literature, in particular, where the completeness of symmetric operator spaces was established under certain additional conditions on the (quasi)norm (see, e. g., [300], [299, Proposition 2.8(ii)], [257, Corollary 1], [259, Theorem 1.2.4], [296, Lemma 4.1], [79, Theorem 4.5], [82, Corollary 2.4]). The results in [152] were achieved by the introduction of uniform submajorization. Uniform submajorization is discussed in Volume III. For the historical background to the paper [152], we quote from a memorial written by the third named author, reprinted with permission from the Nigel Kalton Memorial website:1 1 http://kaltonmemorial.missouri.edu/

108 | 3 Calkin correspondence for norms and traces

“…main achievement is an infinite-dimensional analogue of a finite-dimensional result of John von Neumann from 1937 and the new notion which was invented to obtain this analogue. In that remarkable paper, von Neumann laid the foundation of what was later to become the theory of symmetrically normed (or unitarily-invariant) operator ideals. It is also the first paper where the so-called “noncommutative Lp -spaces” made their appearance. I read this paper of von Neumann as a very young man and realized the beauty of its ideas and noted that it suggested immediately a number of infinite-dimensional questions which were to occupy me for the next 25 years. One of these problems which I tried to resolve in my PhD thesis (1988) was whether a positive unitarilyinvariant functional on a given unitarily-invariant ideal ℰ in the algebra ℒ(H) of all bounded linear operators on a Hilbert space H is a (Banach) norm provided this is the case for its restriction to the diagonal subspace of ℰ (that is, on the set of all operators from ℰ which are diagonal with respect to a given orthonormal basis in H). In my PhD thesis, I answered this question under the additional assumption that the norm is monotone with respect to Hardy–Littlewood [sub]majorization. That was a frustrating restriction and, for many years, I returned again and again to that problem. It was only natural that at some stage in 2006 while [Nigel Kalton and I] were finalizing our work (begun earlier in Oberwolfach) on singular traces which are not monotone with respect to the Hardy–Littlewood [sub]majorization, I again looked at this infinite-dimensional analogue of von Neumann’s result and explained it to Nigel together with a rather long account of various approaches I had tried in the past. Basically, it took Nigel a couple of weeks and a long flight from Adelaide to the US to come up with an outstanding new idea which we later termed ‘uniform Hardy–Littlewood [sub]majorization’.”

The method of proof in this chapter does not use uniform submajorization. Lemmas 3.2.1–3.2.3, which are the key results, use the properties of dilation and symmetry. Essentially, it follows from the application of the doubling trick Lemma 2.2.12. Correspondence of symmetric functionals In the Banach lattices setting, the papers by G. Ya. Lozanovskii [176–178] studied the “singular” part (Mψ∗ )s of the dual to the commutative analogs Mψ of Lorentz operator ideals. This work did not directly lead to the notion of “singular traces” but was nevertheless helpful to recognize the connections between commutative and noncommutative theories, see the comment in [83]. However, it should be noted that in the literature devoted to the study of singular functionals on Lorentz function spaces, the concept of symmetric functionals was not introduced. The second-named author’s first contact with such an object came from noncommutative geometry, from Alain Connes’ use of Dixmier traces and the subsequent exposure to a wide audience. The concept of symmetric functionals on ideals of operators and sequence spaces, as positive continuous functionals which take the same value on positive elements with the same distribution function, was introduced in 1998 [83]. Symmetric functionals in this sense and their connection to traces was developed further in [84, 84, 85, 107, 151, 152, 154, 168]. The use of singular traces in noncommutative geometry inspired similar approaches from 1996, [6, 120–122]. The term “symmetric functional” was used by Figiel and Kalton [107], who studied a similar object without reference to positivity. The terminology of Figiel and Kalton is used in this volume. The previ-

3.6 Notes | 109

ous book [170] used the term symmetric functional in the sense of [83], a continuous symmetric functional on a Banach ideal. We refer to [107] for additional historical comments, but also for potential connections between symmetric functionals and analytic functions. Given the Calkin correspondence between ideals of compact operators and symmetric sequence spaces, it was natural also to ask whether all linear symmetric functionals on the sequence space lifted to the ideal. Theorem 3.1.3 provides a natural bijection between the set of all symmetric functionals on the ideal 𝒥 and the commutative core J. It was observed first for the case of fully symmetric functionals for fully symmetric ideals and sequence spaces in 1998 in [83]. The case of continuous symmetric functionals for Banach ideals and sequence spaces was treated in [151, Theorem 5.2]. Most of the cited work proved the analogous correspondence between symmetric bimodules of semifinite von Neumann algebras and symmetric function spaces simultaneously. For quasi-Banach ideals and the algebra ℒ(H) specifically, Pietsch’s work predates the above. Independently, the bijection between traces and symmetric functionals had been observed by Pietsch for Banach and quasi-Banach ideals as early as 1990 [202]. Pietsch later showed the bijection on all ideals of compact operators [210, Theorem 4]. Pietsch’s approach factors through dyadic decompositions of compact operators and is discussed in the next chapter. The equivalence between Pietsch’s approach and the symmetric functional approach was established in [207, Theorem 6.3]. The connection between the symmetric functional approach in [83] and that of Figiel–Kalton resulted from a collaboration between the second named author and Nigel Kalton; reprinted with permission from the Nigel Kalton Memorial website: “In 1998, together with B. de Pagter, P. Dodds and E. Semenov, we approached the notion of a Dixmier trace, which we called, at that time, a symmetric functional, as a part of a general theory of singular traces on symmetrically normed operator ideals of ℒ(H) (and, more generally, on symmetrically normed operator bimodules on semifinite von Neumann algebras). At that time, I was vaguely aware of earlier work of Nigel with Figiel and with Dykema. However, their framework was subtly different and I could not immediately see its implications for our own work. [A] meeting in Oberwolfach [in 2004] was a perfect chance to tell to Nigel about our results and also state a number of problems some of which we tried ourselves and some of which were fairly new and which I thought of as important from the viewpoint of noncommutative geometry. Somewhat incredibly, Nigel was interested and the very next day he approached me with a tentative solution to the problem which we thought was very hard. Nigel suggested a way how one can attempt to construct symmetric functionals (and unitarily invariant linear functionals on a special Lorentz ideal of importance in noncommutative geometry) which were not monotone with respect to the Hardy–Littlewood [sub]majorization. The idea was so nice that I liked it straight away. The problem was with its technical implementation, or rather with my understanding of the latter. It was some time before I could present a strict and complete technical record of Nigel’s idea, which was initially stated in two or three lines on a table napkin. Indeed, many times during my subsequent collaboration with Nigel, I noted that he didn’t seem to experience the technical difficulties which would set back anybody else. Back in 2004, we parted with the agreement that I would try to continue working with the argument and see for what class of Lorentz spaces it is applicable.

110 | 3 Calkin correspondence for norms and traces

That took a few years (the paper was only published in 2008) and a few meetings in Adelaide and Missouri which still are (and forever will be) a treasured part of my memory. Our further collaboration was firmly centered on the theory of singular traces. I was able to contribute to our work my knowledge of (sometimes obscure) works on the theory of symmetric spaces from the former Soviet Union (Braverman, Mekler, Russu, Sedaev); however, the earlier ideas and techniques of Nigel (from his papers with Figiel and Dykema, which I mentioned earlier) have come to play an essential role in our approach. It was an absolute pleasure for me to see how ideas and techniques born from completely different perspectives (and motivations) became central to the study of singular traces and their applications in noncommutative geometry. Eventually, it had become clear to us that there exists a single thread which permeates works on commutators, unitary orbits, various geometrical questions in the theory of symmetric operator spaces and noncommutative geometry which was both fascinating and fruitful. Of course, this realization would never have happened if Nigel was not the mathematical giant that he was.”

The lack of linearity of μ is the hurdle to lifting a symmetric functional φ on the diagonal to an additive functional φ ∘ μ on the positive cone of the ideal. The method of proving additivity originated in the proof of [151, Theorem 5.2]. In [151] the difference A + B − (A + B) is lifted to a direct sum to be treated as a commutator, and the noncommutative version of the Figiel–Kalton theorem combining [92, 94, 107], is applied. This chapter proves the discrete version of the Figiel–Kalton theorem from [107] and uses the observation that C(μ(A) + μ(B) − μ(A + B)) ∈ J. Interestingly, this observation is originally from Dixmier [76] using inequalities of Hersch [133, 134] to prove additivity of the Dixmier trace (see the historical notes to Chapter 6). Figiel–Kalton theorem The technical operator E 󸀠 used in the proof of the Figiel–Kalton theorem is similar to the dyadic operations used in Chapter 4. In terms of the dyadic dilation operator L and dyadic averaging operator L󸀠 defined in Chapter 4, the operator E 󸀠 is E 󸀠 (x) − e0 ⋅ x = (TLL󸀠 T ∗ )(x),

x ∈ l∞ ,

where e0 = (1, 0, . . .) and T is the right shift operator. The components TLL󸀠 T ∗ and e0 ⋅ are idempotents on l∞ that annul each other, hence E 󸀠 is an idempotent. The statement of the Figiel–Kalton theorem presented here is original to this book. The Figiel–Kalton theorem was originally stated for symmetric function spaces involving the continuous Cesaro mean operator [107]. Existence of traces The notes to Chapter 1, under “Construction of traces”, provided a historical account on the existence results.

3.6 Notes | 111

Kalton in 1989 identified the commutator subspace of trace class operators studied by Gary Weiss [149, 288] and Anderson and Vaserstein [9, 10]. In doing so, Kalton proved the existence of traces different from “the trace” on ℒ1 [149]. An alternative unpublished proof of the same fact using shift invariant functionals on l1 is due to Tadeusz Figiel [149, p. 73]. Corollary 3.5.1 is the Hardy inequality. Bennet in [19] described the result C : lp → lp , p > 1, as the “crudest form of the Hardy inequality”, in his work on multipliers z ∈ l∞ such that C(y ⋅ z) ∈ lp for y ∈ lp . Some examples of two-sided ideals, which are not Lorentz ideals, possessing traces were presented in [84]. The commutator approach was subsequently developed in the 1990s for arbitrary ideals by Dykema, Figiel, Weiss, and Wodzicki [94]. See [94, Section 5] for other examples of ideals that do or do not support nontrivial traces. Many existence results for ideals are given in [93, 94, 144, 294]. The ideal ℒ1,∞ plays a special role in the commutator subspace theory as explained in [94]. See also [144]. That the ideal (ℒ1,∞ )0 supports an extension of the trace Tr which was not unique and not continuous was proven in [93]. The equivalent conditions in Theorem 3.5.6 were noted in [93]. Existence of continuous traces Pietsch in [207, 210] considers those ideals that admit traces, admit traces but no continuous trace, and admit continuous traces, according to the spectrum of the operator ( 21 T − 1) introduced in the next chapter. Exact conditions for the existence of continuous traces on Banach ideals [170, Theorem 4.1.3] and quasi-Banach ideals [212, Theorem 2.1] are known. Corollary 3.5.3 proves that there are Banach ideals that admit traces, but no continuous traces at all. Are there Banach ideals where all traces are continuous? Pietsch has shown that all traces being continuous is equivalent to a finite number of linearly independent singular traces on a quasi-Banach ideal [209, Theorem 8.14]. Banach ideals admit either an infinite number of linearly independent singular traces, or none [209, Proposition 8.17]. Therefore, Banach ideals that admit a singular trace, must also admit a discontinuous trace. This dichotomy, that generally an ideal admits either an infinite number of linearly independent singular traces or none, is an open conjecture [144], [209, Problem 4.6]. Kalton identified which quasi-Banach ideals within ℒ1 possessed no singular traces [148, Theorem 6], or were “uniquely-traceable”. Yoshida–Hewitt decomposition for traces Kalton proved Theorems 3.5.8 and 3.5.9 in [148]. Theorem 3.1.5 states that continuous symmetric functionals decompose into normal and singular parts. In the framework of the theory of Banach lattices, it is customary to consider a decomposition of an arbitrary continuous functional on a given Banach lattice into a direct sum of “normal” (that is, continuous with respect to order convergence) and

112 | 3 Calkin correspondence for norms and traces “singular” parts. This decomposition is usually linked with the classical theorem of K. Yosida and E. Hewitt [301], which states that any bounded additive measure can be uniquely represented as the sum of a countably additive measure and a purely finitely additive measure, the so-called singular part, which is characterized by the fact that its absolute value does not dominate any nonzero positive countably additive measure. For Banach ideals of compact operators, a complete analogy to the Yosida and Hewitt result was achieved in [78]. Proposition 2.7 in [78] extended Dixmier’s earlier results on ℒ(H) [74].

4 Pietsch correspondence 4.1 Introduction The Calkin correspondence in Chapter 3 associated two-sided ideals to symmetric sequence spaces, quasi-Banach ideals to quasi-Banach symmetric sequence spaces, and traces on ideals to symmetric functionals. The restriction to the diagonal of the ideal provides the bijective associations. Under restriction we observe that the notion of a trace, that is a linear functional invariant to the full group of unitary operators in ℒ(H), is reduced to the notion of a linear functional invariant to the permutation, or symmetry, group Π of unitary operators which preserves the diagonal. The reduction to the simpler group enables the existence of functionals invariant to the permutation group to be characterized by the familiar and concrete Cesaro operator C : l∞ → l∞ in the Figiel–Kalton theorem. From this Chapter 3 derived results on the existence of traces. However, looking beyond the existence of symmetric functionals to constructions and questions such as the cardinality of the set of continuous traces, the set of linear functionals invariant under the permutation group is still difficult to concretely identify. Pietsch correspondence on ideals and traces Albrecht Pietsch, working in the more general area of continuous operators between Banach spaces, developed an alternative to the Calkin correspondence. It is based, in the case of Hilbert spaces, on the dyadic dilation of the diagonal. Pietsch’s approach provides a bijective correspondence between two-sided ideals 𝒥 of ℒ(H) and shiftinvariant monotone ideals O of l∞ , see Definition 4.1.1 and Theorem 4.1.2 below. The correspondence is an isometric bijection between quasi-Banach ideals of ℒ(H) and quasi-Banach shift-monotone invariant ideals of l∞ . We obtain, with the maps defined below and Theorem 4.1.6, therefore the triangle Calkin correspondence

𝒥 ?

?

μ

diag

Pietsch correspondence

? J ? L

? ? O𝒥

L󸀠

between the ideal 𝒥 , the commutative core J, and a shift-invariant monotone ideal O𝒥 , that respects quasinorms and closure with respect to quasinorms. Define the right-shift operator T : l∞ → l∞ by T(x(0), x(1), . . .) = (0, x(0), x(1), . . .),

x ∈ l∞ .

The operator T is a right inverse of the left shift operator T ∗ : l∞ → l∞ defined by T ∗ (x(0), x(1), . . .) = (x(1), x(2), x(3), . . .), https://doi.org/10.1515/9783110378054-005

x ∈ l∞ .

114 | 4 Pietsch correspondence If e0 denotes the standard basis vector e0 = (1, 0, 0, . . .) ∈ l∞ then TT ∗ e0 = 0 and T is not the left inverse of T ∗ . Definition 4.1.1. Let O be an ideal in l∞ . Then O is (a) shift-invariant if Tx ∈ O for every x ∈ O; (b) monotone if for every x ∈ O and every y ∈ l∞ such that 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨y(k)󵄨󵄨󵄨 ≤ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨, k≥n

k≥n

n ≥ 0,

we have y ∈ O. Define the dyadic dilation operator L : l∞ → l∞ by Lx = (x(0), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(1), x(1), ⏟⏟ x(2), . .⏟,⏟⏟⏟⏟⏟⏟⏟⏟ x(2) . . . , x(n), . . .), ⏟⏟⏟⏟⏟⏟⏟⏟⏟.⏟⏟⏟ ⏟⏟, . . . , x(n), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 times

2n times

4 times

x ∈ l∞ ,

and the dyadic averaging operator L󸀠 : l∞ → l∞ as 2n+1 −2

(L󸀠 x)(n) = 2−n ∑ x(k),

x ∈ l∞ , n ≥ 0.

k=2n −1

That is, (L󸀠 x)(n) is the average value of x(k) in the interval 2n − 1 ≤ k < 2n+1 − 1. Despite also being a form of dyadic averaging, this is not the same as the operator E 󸀠 from Section 3.3. The operator L is a right inverse of L󸀠 , that is, (L󸀠 ∘ L)(x) = x,

x ∈ l∞ .

(4.1)

That dyadic dilation and averaging map shift-invariant monotone ideals of l∞ to symmetric sequence spaces bijectively is the core of the following result. Theorem 4.1.2 (Pietsch correspondence). (a) If 𝒥 is a two-sided ideal of ℒ(H), then the set O𝒥 = {x ∈ l∞ : diag(Lx) ∈ 𝒥 } is a shift-invariant monotone ideal of l∞ . (b) If O is a shift-invariant ideal of l∞ such that e0 ∈ O, the set 󸀠

𝒥O = {A ∈ ℒ(H) : L μ(A) ∈ O}

is a two-sided ideal of ℒ(H). (c) The correspondences 𝒥 → O𝒥 and O → 𝒥O are mutual inverses:

4.1 Introduction | 115

(i) if 𝒥 is a two-sided ideal of ℒ(H) and O = O𝒥 , then 𝒥O = 𝒥 ; (ii) if O is a shift-invariant monotone ideal of l∞ and 𝒥 = 𝒥O , then O𝒥 = O. Note that in Theorem 4.1.2(b) we do not require O to be monotone. However, we do so in Theorem 4.1.2(c). In terms of the commutative core J of an ideal 𝒥 , the corresponding shift-invariant monotone ideal in l∞ is O𝒥 = L󸀠 J,

J = LO𝒥 .

Definition 4.1.3. Let O be a shift-invariant monotone ideal of l∞ . We say that a linear functional ℓ : O → ℂ is shift-invariant if ℓ ∘ 21 T = ℓ. Pietsch’s correspondence provides a bijective correspondence between traces on two-sided ideals and shift-invariant linear functionals on shift-invariant monotone ideals. In transferring to a shift-invariant monotone ideal from a symmetric sequence space, the reduction of linear functionals invariant under permutations to linear functionals invariant under the right shift operator is an advantage. The permutation group is generated by an uncountable set, while the shift semigroup has a single generator. The factor 21 is required to identify shift invariant functionals with symmetric functionals – the right shift on a shift-invariant monotone ideal will relate to doubling and hence dilation by 2 in the corresponding symmetric sequence space. We observe in Chapter 6 when shift-invariant linear functionals can be explicitly identified. We obtain in this chapter, with the maps defined below and Theorem 4.1.9, the triangle Calkin correspondence

φ? ?

∘diag ∘μ

? φ̂ ? ∘L󸀠

Pietsch correspondence

? ? ℓ

∘L

between traces, symmetric functionals, and shift-invariant functionals, that respects continuity of traces and linear functionals. Theorem 4.1.4. Let 𝒥 be an ideal of ℒ(H) and O𝒥 the corresponding shift-invariant monotone ideal of l∞ . (a) If φ : 𝒥 → ℂ is a trace, then the mapping x → (φ ∘ diag ∘ L)(x),

x ∈ O𝒥 ,

is a shift-invariant linear functional ℓφ on O𝒥 . (b) If ℓ : O𝒥 → ℂ is a shift-invariant linear functional, then the mapping A → ℓ ∘ L󸀠 ∘ μ(A),

0 ≤ A ∈ 𝒥,

is additive on the positive cone of 𝒥 and extends to a trace φℓ on 𝒥 . (c) The correspondences φ 󳨃→ ℓφ and ℓ 󳨃→ φℓ are mutually inverse.

116 | 4 Pietsch correspondence Understanding, as in Chapter 3, that a trace φ on 𝒥 restricts to a symmetrical functional on the diagonal, the formula in Theorem 4.1.4(a) describing every shift-invariant linear functional can be abbreviated to ℓ(x) = φ(Lx),

x ∈ O𝒥 .

The formula in Theorem 4.1.4(b) indicates that every trace φ on a two-sided ideal 𝒥 has the explicit form on the positive cone φ(A) = ℓ(L󸀠 μ(A)),

0 ≤ A ∈ 𝒥,

(4.2)

in terms of a corresponding shift-invariant functional ℓ. The shift-invariant monotone ideal O that corresponds to the quasi-Banach symmetric sequence space l1,∞ , O = L󸀠 l1,∞ , is the principal ideal of l∞ generated by the sequence x0 (n) = 2−n ,

n ≥ 0.

Chapter 6 shows that O is in bijective correspondence with l∞ . The linear functionals on l∞ that are the image of the shift-invariant functionals on O under this bijection can be identified, and (4.2) will provide an explicit formula for all traces on the weak ideal ℒ1,∞ in Chapter 6. Continuous Pietsch correspondence and regular traces The Pietsch correspondence maps between quasinormed ideals and quasinormed shift-invariant monotone ideals, provided the appropriate notion of a shift-invariant quasinorm is defined. Definition 4.1.5. A quasinorm ‖⋅‖O on a shift-invariant monotone ideal O in l∞ is shiftinvariant monotone if (a) T : O → O is continuous, (b) if x, y ∈ O are such that 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨y(k)󵄨󵄨󵄨 ≤ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨, k≥n

k≥n

n ≥ 0,

then ‖y‖O ≤ cO ‖x‖O for some constant cO > 0. A shift-invariant monotone ideal is quasinormed if it admits a shift-invariant monotone quasinorm. If the quasinormed ideal (O, ‖⋅‖O ) is complete, then the shift-invariant monotone ideal O is called quasi-Banach. Theorem 4.1.6. Let 𝒥 be an ideal of ℒ(H) with corresponding shift-invariant monotone ideal O𝒥 .

4.1 Introduction | 117

(a) If ‖ ⋅ ‖𝒥 is a symmetric quasinorm on 𝒥 , then 󵄩 󵄩 x 󳨃→ 󵄩󵄩󵄩diag(Lx)󵄩󵄩󵄩𝒥 ,

x ∈ O𝒥

is a shift-invariant monotone quasinorm on O𝒥 . (b) If ‖ ⋅ ‖O𝒥 is a shift-invariant monotone quasinorm on O𝒥 , then 󵄩 󵄩 A 󳨃→ 󵄩󵄩󵄩L󸀠 μ(A)󵄩󵄩󵄩O , 𝒥

A∈𝒥

is a symmetric quasinorm on 𝒥 . (c) The correspondences ‖ ⋅ ‖𝒥 → ‖ ⋅ ‖O𝒥 and ‖ ⋅ ‖O𝒥 → ‖ ⋅ ‖𝒥 are mutual inverses up to equivalence of quasinorms. Moreover, the statements that 𝒥 is quasi-Banach and that O𝒥 is quasi-Banach are equivalent. Nigel Kalton, when answering Albrecht Pietsch’s question of whether continuous traces other than the matrix trace exist on principal ideals within ℒ1 , introduced the notion of a regular trace on a two-sided ideal of compact operators. Definition 4.1.7. Let 𝒥 be a two-sided ideal of ℒ(H) and let φ : 𝒥 → ℂ be a trace. We say that φ is regular if, for every A ∈ 𝒥 , the mapping B 󳨃→ φ(AB),

B ∈ ℒ(H),

is continuous in the uniform norm of ℒ(H). Similarly, let O be a shift-invariant monotone ideal of l∞ and let ℓ : O → ℂ be a shift-invariant linear functional. We say that ℓ is regular if, for every a ∈ O, the mapping b → ℓ(ab),

b ∈ l∞ ,

is continuous in the norm of l∞ . Section 4.5 shows that on an arbitrary two-sided ideal 𝒥 the set of regular traces is spanned by the set of positive traces. Recall from Definition 2.4.4 that a trace φ : 𝒥 → ℂ is positive if φ(A) ≥ 0 for all A ∈ 𝒥+ , and, by Lemma 2.4.12, positive traces are continuous when 𝒥 is quasi-Banach. Regular traces therefore play the role of continuous traces on arbitrary ideals. A trace φ on 𝒥 is hermitian if φ(A∗ ) = φ(A),

A ∈ 𝒥,

where the bar denotes the complex conjugate of the complex number φ(A). A shiftinvariant linear functional ℓ on a shift-invariant monotone ideal O in l∞ is hermitian if ℓ(x∗ ) = ℓ(x),

x ∈ O.

118 | 4 Pietsch correspondence Proposition 4.5.2 proves that the set of all hermitian regular shift-invariant linear functionals on O is a lattice with respect to operations ∨ and ∧ defined by (ℓ1 ∨ ℓ2 )(x) = sup{ℓ1 (u) + ℓ2 (v) : 0 ≤ u, v ∈ O, x = u + v},

0 ≤ x ∈ O,

(ℓ1 ∧ ℓ2 )(x) = inf{ℓ1 (u) + ℓ2 (v) : 0 ≤ u, v ∈ O, x = u + v},

0 ≤ x ∈ O.

and

Similarly, a lattice on the set of hermitian traces on a two-sided ideal 𝒥 is defined by the operations ∨ and ∧, (φ1 ∨ φ2 )(A) = sup{φ1 (B) + φ2 (C) : 0 ≤ B, C ∈ 𝒥 , A = B + C}, (φ1 ∧ φ2 )(A) = inf{φ1 (B) + φ2 (C) : 0 ≤ B, C ∈ 𝒥 , A = B + C},

0 ≤ A ∈ 𝒥, 0 ≤ A ∈ 𝒥.

The correspondence φ → ℓφ and ℓ → φℓ in Theorem 4.1.4 between traces and shift-invariant linear functionals preserves positivity by construction. The following results show that the correspondence preserves regularity, order, and continuity. Theorem 4.1.8. Let 𝒥 be an ideal of ℒ(H). (a) If φ : 𝒥 → ℂ is a regular trace, then ℓφ : O𝒥 → ℂ is a regular shift-invariant linear functional. (b) If ℓ : O𝒥 → ℂ is a regular shift-invariant linear functional, then the trace φℓ : 𝒥 → ℂ is regular. (c) The correspondences φ → ℓφ and ℓ → φℓ preserve the lattice operations ∨ and ∧ on hermitian regular shift-invariant functionals and traces defined above. Theorem 4.1.9. Let 𝒥 be a quasi-Banach ideal of ℒ(H). (a) If φ : 𝒥 → ℂ is a continuous trace, then ℓφ : O𝒥 → ℂ is a continuous shift-invariant linear functional. (b) If ℓ : O𝒥 → ℂ is a continuous shift-invariant linear functional, then the trace φℓ : 𝒥 → ℂ is continuous. From the preservation of lattice operations, the hermitian regular traces form a lattice. On a quasi-Banach ideal the continuous hermitian traces also form a lattice. The following decomposition of regular and continuous traces into positive parts is a consequence of the preservation of lattice operations. Theorem 4.1.10 (Jordan decomposition). Every regular trace φ on a two-sided 𝒥 of ℒ(H) is the linear combination of four positive traces on 𝒥 . Conversely, every linear combination of positive traces on 𝒥 is a regular trace. If 𝒥 is quasi-Banach, then every continuous trace φ on 𝒥 is the linear combination of four positive traces on 𝒥 .

4.2 Pietsch correspondence for ideals | 119

4.2 Pietsch correspondence for ideals This section proves the statement of Theorem 4.1.2. That is, the Pietsch correspondence between ideals of ℒ(H) and shift-invariant monotone ideals in l∞ . The following sequence of results from Lemma 4.2.1 to Lemma 4.2.3 link, in Lemma 4.2.4, the monotone property in a shift-invariant monotone ideal to the symmetric property of a symmetric sequence space. Lemma 4.2.1. For every x ∈ l∞ and n ≥ 0, we have 󵄨 󵄨 μ(2n − 1, Lx) = sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨, k≥n

n ≥ 0.

Proof. Fix n ∈ ℕ and write Lx = Lx ⋅ χ[0,2n −1) + Lx ⋅ χ[2n −1,∞) . Using Corollary 2.2.9, we have μ(2n − 1, Lx) ≤ μ(2n − 1, Lx ⋅ χ[0,2n −1) ) + μ(0, Lx ⋅ χ[2n −1,∞) ). Clearly, μ(2n − 1, Lx ⋅ χ[0,2n −1) ) = 0,

μ(0, Lx ⋅ χ[2n −1,∞) ) = ‖Lx ⋅ χ[2n −1,∞) ‖∞ .

It follows that 󵄨 󵄨 μ(2n − 1, Lx) ≤ ‖Lx ⋅ χ[2n −1,∞) ‖∞ = sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨. k≥n

In order to prove the converse inequality, choose a set A ⊂ ℤ+ such that |A| = 2n −1. If k ≥ n, then Lx contains 2k ≥ 2n instances of x(k). Hence, Lx ⋅ χAc contains at least 2k − |A| = 2k − (2n − 1) ≥ 2n − (2n − 1) = 1 instance of x(k). Thus, 󵄨 󵄨 ‖Lx ⋅ χAc ‖∞ ≥ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨. k≥n

By the definition of the decreasing rearrangement of a sequence (see Example 2.2.14), we have μ(2n − 1, Lx) = inf{‖Lx ⋅ χAc ‖∞ : |A| ≤ 2n − 1}. It follows that 󵄨 󵄨 μ(2n − 1, Lx) ≥ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨. k≥n

120 | 4 Pietsch correspondence Corollary 4.2.2. For every x ∈ l∞ and n ≥ 0, we have 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨x(k)󵄨󵄨󵄨 ≤ (L󸀠 μ(Lx))(n) ≤ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨. k≥n

k≥n+1

Proof. By the definition of L󸀠 , we have 2n+1 −2

(L󸀠 μ(Lx))(n) = 2−n ∑ μ(k, Lx) ≤ μ(2n − 1, Lx). k=2n −1

Applying Lemma 4.2.1 to the right-hand side above, we obtain 󵄨 󵄨 (L󸀠 μ(Lx))(n) ≤ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨. k≥n

Conversely, 󸀠

(L μ(Lx))(n) = 2

−n

2n+1 −2

∑ μ(k, Lx) ≥ μ(2n+1 − 1, Lx).

k=2n −1

Again using Lemma 4.2.1, we obtain the bound 󵄨 󵄨 (L󸀠 μ(Lx))(n) ≥ sup 󵄨󵄨󵄨x(k)󵄨󵄨󵄨. k≥n+1

The next lemma relates the monotone condition 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨y(k)󵄨󵄨󵄨 ≤ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨, k≥n

k≥n

n ≥ 0,

to the dilation operator σ2 from Lemma 2.2.12. Lemma 4.2.3. If x, y ∈ l∞ are such that 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨y(k)󵄨󵄨󵄨 ≤ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨, k≥n

k≥n

n ≥ 0,

then μ(Ly) ≤ σ2 μ(Lx). Proof. It follows from Lemma 4.2.1 that μ(2n − 1, Ly) ≤ μ(2n − 1, Lx),

n ≥ 0.

Consequently, μ(m, Ly) ≤ μ(2n − 1, Ly) ≤ μ(2n − 1, Lx) ≤ μ(⌊

m ⌋, Lx) 2

4.2 Pietsch correspondence for ideals | 121

whenever 2n − 1 ≤ m < 2n+1 − 1. Consequently, for such m we obtain μ(m, Ly) ≤ μ(⌊

m ⌋, Lx). 2

Since n is arbitrary, the result follows. Recall that L󸀠 ∘ L : l∞ → l∞ is the identity operator on sequences. The next lemma shows that L ∘ L󸀠 : l∞ → l∞ maps a symmetric sequence space to itself. Lemma 4.2.4. We have LL󸀠 μ(A) ≤ σ2 μ(A),

μ(A) ≤ σ2 LL󸀠 μ(A).

Proof. Consider n ≥ 0 and let m ∈ [2n − 1, 2n+1 − 2]. We have 󸀠

−n

(LL μ(A))(m) = 2

2n+1 −2

∑ μ(k, A) ≤ μ(2n − 1, A) ≤ μ(⌊

k=2n −1

m ⌋, A) ≤ σ2 μ(A). 2

This proves the first estimate. Let m ∈ [2n+1 − 2, 2n+2 − 3], so that ⌊ m2 ⌋ ∈ [2n − 1, 2n+1 − 2]. We have n+1

(LL󸀠 μ(A))(⌊

2 −2 m ⌋) = 2−n ∑ μ(k, A) ≥ μ(2n+1 − 2, A) ≥ μ(m, A). 2 k=2n −1

This proves the second estimate. We can now prove the first part of Theorem 4.1.2. Proof of Theorem 4.1.2(a). Let x, y ∈ O𝒥 . We have Lx, Ly ∈ J and, therefore, L(x + y) ∈ J. Hence, x + y ∈ O𝒥 . Similarly, λx ∈ O𝒥 for every λ ∈ ℂ. Thus, O𝒥 is a linear subspace in l∞ . Let x ∈ O𝒥 and y ∈ l∞ . If |y| ≤ |x|, then |Ly| = L(|y|) ≤ L(|x|) = |Lx|. Since Lx ∈ J, it follows that Ly ∈ J. Hence, y ∈ O𝒥 . Thus, O𝒥 is an ideal of l∞ . Let x ∈ O𝒥 and note that L(Tx) = (0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(0), x(0), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(1), . . . , x(1), . . . , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(n − 1), . . . , x(n − 1), . . .). 2 times

4 times

Thus, μ(L(Tx)) = μ(Lx ⊕ Lx).

2n times

122 | 4 Pietsch correspondence Since Lx ∈ J, it follows that Lx ⊕ Lx ∈ J. Hence, L(Tx) ∈ J and, therefore, Tx ∈ O𝒥 . Thus, O𝒥 is shift-invariant. Let x ∈ O𝒥 and y ∈ l∞ be such that 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨y(k)󵄨󵄨󵄨 ≤ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨. k≥n

k≥n

Since Lx ∈ J, it follows from Lemma 4.2.3 that Ly ∈ J. Hence, y ∈ O𝒥 . Thus, O𝒥 is monotone. The harder calculations in Chapter 3 on the Calkin correspondence were due to the singular value map μ : 𝒥+ → J+ not being additive. Additivity followed from the behavior of the dilation operator σ2 : l∞ → l∞ under symmetry and the doubling trick of Lemma 2.2.12. The next lemma is the equivalent in the Pietsch correspondence and links additivity under dyadic averaging to the right-shift operator. Lemma 4.2.5. If A, B ∈ ℒ(H), then L󸀠 μ(A + B) ≤ T 2 (L󸀠 μ(A) + L󸀠 μ(B)) + ‖A + B‖∞ (e0 + e1 ). Here e0 = (1, 0, 0, . . .) and e1 = (0, 1, 0, . . .). Proof. By the definition of L󸀠 , we have 󸀠

2n+1 −2

−n

(L μ(A + B))(n) = 2

∑ μ(k, A + B) ≤ μ(2n − 1, A + B),

k=2n −1

n ≥ 0.

Next, μ(2n − 1, A + B) ≤ μ(2n − 2, A + B) ≤ μ(2n−1 − 1, A) + μ(2n−1 − 1, B), and μ(2n−1 − 1, A) ≤ 22−n μ(2n−1 − 1, B) ≤ 22−n

2n−1 −2

∑ k=2n−2 −1 2n−1 −2

∑ k=2n−2 −1

μ(k, A) = (L󸀠 μ(A))(n − 2),

n ≥ 2,

μ(k, B) = (L󸀠 μ(B))(n − 2),

n ≥ 2.

Combining these inequalities, we obtain (L󸀠 μ(A + B))(n) ≤ (L󸀠 μ(A))(n − 2) + (L󸀠 μ(B))(n − 2),

n ≥ 2.

Since (L󸀠 μ(A + B))(1) ≤ (L󸀠 μ(A + B))(0) ≤ ‖A + B‖∞ , we have L󸀠 μ(A + B) ≤ T 2 (L󸀠 μ(A) + L󸀠 μ(B)) + ‖A + B‖∞ (e0 + e1 ), and this completes the proof.

n ≥ 1,

4.3 Pietsch correspondence for traces | 123

We can now prove the second part of Theorem 4.1.2. Proof of Theorem 4.1.2(b). Let O be a shift-invariant ideal of l∞ and let A, B ∈ 𝒥O . By the definition of 𝒥O , we have L󸀠 μ(A) ∈ O and L󸀠 μ(B) ∈ O. By Lemma 4.2.5, we have L󸀠 μ(A + B) ≤ T 2 (L󸀠 μ(A) + L󸀠 μ(B)) + ‖A + B‖∞ (e0 + e1 ). Since O is shift-invariant and e0 ∈ O, it follows that the above right-hand side is in O. Since O is an ideal of l∞ , it follows that the left-hand side is also in O. That is, L󸀠 μ(A + B) ∈ O and, therefore, A + B ∈ 𝒥O . If A ∈ 𝒥O and if C ∈ ℒ(H), then μ(AC) ≤ ‖C‖∞ μ(A). It follows that L󸀠 μ(AC) ≤ ‖C‖∞ L󸀠 μ(A), and hence L󸀠 μ(CA) ∈ O and by definition CA ∈ 𝒥O . Similarly, AC ∈ 𝒥O . Combining the assertions in these paragraphs, we infer that 𝒥O is an ideal of ℒ(H). Proof of Theorem 4.1.2(c). Let O be a shift-invariant monotone ideal of l∞ and let 𝒥 =

𝒥O . We claim that O𝒥 = O.

Let x ∈ l∞ and let

󵄨 󵄨 z(n) = sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨, k≥n

n ≥ 0.

By Corollary 4.2.2, we have T ∗ z ≤ L󸀠 μ(Lx) ≤ z,

n ≥ 0.

If x ∈ O, then also z ∈ O (since O is monotone). Thus, L󸀠 μ(Lx) ∈ O for every x ∈ O. In other words, Lx ∈ 𝒥O = 𝒥 . Thus, x ∈ O𝒥 . Since x ∈ O is arbitrary, it follows that O ⊂ O𝒥 . If x ∈ O𝒥 , then Lx ∈ 𝒥 = 𝒥O . Hence, L󸀠 μ(Lx) ∈ O and, therefore, T ∗ z ∈ O. Since O is shift-invariant, it follows that z = TT ∗ z + z(0)e0 ∈ O. Since |x| ≤ z, it follows that x ∈ O. Since x ∈ O𝒥 is arbitrary, it follows that O𝒥 ⊂ O. Conversely, let 𝒥 be an ideal of ℒ(H) and let O = O𝒥 . We claim that 𝒥O = 𝒥 . Let A ∈ 𝒥 . By Lemma 4.2.4, we have LL󸀠 μ(A) ≤ σ2 μ(A). Hence, LL󸀠 μ(A) ∈ 𝒥 and 󸀠 L μ(A) ∈ O𝒥 = O. Thus, A ∈ 𝒥O . Since A ∈ 𝒥 is arbitrary, it follows that 𝒥 ⊂ 𝒥O . If A ∈ 𝒥O , then L󸀠 μ(A) ∈ O = O𝒥 . This means LL󸀠 μ(A) ∈ 𝒥 . By Lemma 4.2.4, we have μ(A) ≤ σ2 LL󸀠 μ(A). It follows that A ∈ 𝒥 . Since A ∈ 𝒥O is arbitrary, it follows that 𝒥O ⊂ 𝒥 .

4.3 Pietsch correspondence for traces This section proves the statement of Theorem 4.1.4. That is, the Pietsch correspondence associates bijectively a trace on an ideal of ℒ(H) to a shift-invariant linear functional on a shift-invariant monotone ideal of l∞ . A shift-invariant monotone linear functional ℓ : O → ℂ was defined in Definition 4.1.3 by the property ℓ ∘ 21 T = ℓ.

124 | 4 Pietsch correspondence If φ : 𝒥 → ℂ is a trace, then proving that ℓφ (x) = (φ ∘ diag ∘ L)(x),

x ∈ O𝒥 ,

is a shift-invariant linear functional ℓφ on O𝒥 is relatively straightforward. The proof below shows that the 21 in the invariance statement arises from dilation and doubling. If ℓ : O → ℂ is a shift-invariant monotone linear functional, then proving that φℓ (A) = (ℓ ∘ L󸀠 )(μ(A)),

0 ≤ A ∈ 𝒥O ,

is additive on the positive cone of 𝒥O is more involved. The following lemma identifies the equivalent of the center of a symmetric sequence space (Definition 1.2.2) for a shiftinvariant monotone ideals. Lemma 4.3.1. Let O be a translation-invariant ideal of l∞ and let x ∈ O. If n

{2−n ∑ 2k x(k)} k=0

n≥0

∈ O,

then ℓ(x) = 0 for every translation-invariant linear functional ℓ : O → ℂ. Proof. Set n

z(n) = 2−n ∑ 2k x(k), k=0

n ≥ 0.

For n ≥ 1, we have n−1

n

k=0

k=0

(Tz)(n) = z(n − 1) = 21−n ∑ 2k x(k) = 21−n ∑ 2k x(k) − 2x(n) = 2z(n) − 2x(n). For n = 0, we have z(n) = x(n) and (Tz)(n) = 0. Thus, 2x = 2z − Tz. Hence, 2ℓ(x) = 2ℓ(z) − ℓ(Tz) = 0. Proof of Theorem 4.1.4(a). The functional ℓ = φ ∘ diag ∘ L on O𝒥 is well defined and linear. Let x ∈ O𝒥 . We now have L(Tx) = (0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(0), x(0), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(1), . . . , x(1), . . . , x(n − 1), . . . , x(n − 1), . . .) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 times

2n times

4 times

= (T ∘ σ2 )((x(0), x(1), x(1), x(2), . .⏟,⏟⏟⏟⏟⏟⏟⏟⏟ x(2) . . . , x(n), . . .)) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟.⏟⏟⏟ ⏟⏟, . . . , x(n), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 times

= (T ∘ σ2 )(Lx).

4 times

2n times

4.3 Pietsch correspondence for traces | 125

Thus, ℓ(Tx) = φ(L(Tx)) = (φ ∘ T)(σ2 (Lx)) = φ(σ2 (Lx)) = φ(Lx ⊕ Lx) = 2φ(Lx) = 2ℓ(x).

Hence ℓ is translation invariant. Proof of Theorem 4.1.4(b). Let 0 ≤ A, B ∈ 𝒥 . Set x = L󸀠 (μ(A) + μ(B) − μ(A + B)). By definition, x ∈ O𝒥 . Note that n

2n+1 −2

k

∑ 2 x(k) = ∑ μ(m, A) + μ(m, B) − μ(m, A + B). m=0

k=0

By Theorems 2.3.5 and 2.3.6, we have n

0 ≤ ∑ 2k x(k) ≤ 2n+1 μ(2n+1 − 1, A + B). k=0

Hence, n

0 ≤ 2−n ∑ 2k x(k) ≤ 2(Lμ(A + B))(n), k=0

n ≥ 0.

Hence, x ∈ O𝒥 satisfies the conditions of Lemma 4.3.1. In particular, ℓ(x) = 0. The functional φ(A) = ℓ(L󸀠 μ(A)) is well defined for 0 ≤ A ∈ 𝒥 . We have φ(A) + φ(B) − φ(A + B) = ℓ(x) = 0. So, φ : 𝒥+ → ℂ is additive (and, obviously, homogeneous). Every additive and homogeneous functional on 𝒥+ uniquely extends to a linear functional on 𝒥 . The extension φ : 𝒥 → ℂ is linear and unitarily invariant. The next lemma will prove that the correspondence between φℓ and ℓφ is bijective, by the fact that the map LL󸀠 : J → J preserves the center of the commutative core J of an ideal 𝒥 . Lemma 4.3.2. Let J be a symmetric sequence space. For every x ∈ J such that x = μ(x), we have x − LL󸀠 x ∈ Z(J). Proof. We use the Figiel–Kalton theorem. Assume x = μ(x). Since μ(LL󸀠 x) = LL󸀠 μ(x), it is clear that x − LL󸀠 x ∈ D(J). For 2n − 1 ≤ k < 2n+1 − 1, n+1

󵄨󵄨 󵄨 󸀠 󵄨󵄨C(x − LL x)(k)󵄨󵄨󵄨 ≤

1 2 −2 ∑ x(k) ≤ x(2n − 1) ≤ (σ2 x)(k). k + 1 j=2n −1

Hence C(x − LL󸀠 x) ∈ J. The assertion follows from Theorem 3.3.2.

126 | 4 Pietsch correspondence Proof of Theorem 4.1.4(c). Let φ : 𝒥 → ℂ be a trace and let ℓ = ℓφ . We claim that φℓ = φ. Let 0 ≤ A ∈ 𝒥 . We have φℓ (A) = ℓ(L󸀠 μ(A)) = φ(LL󸀠 μ(A)). By Lemma 4.3.2, we have φℓ (A) = φ(μ(A)) = φ(A). Since 0 ≤ A ∈ 𝒥 is arbitrary, it follows that φ = φℓ . Conversely, let ℓ : O𝒥 → ℂ be a shift-invariant linear functional and let φ = φℓ . We have ℓφ = φℓ ∘ L = ℓ.

4.4 Pietsch correspondence for quasinorms This section proves the statement of Theorem 4.1.6. That is, the Pietsch correspondence associates bijectively a symmetric quasinorm on an ideal of ℒ(H) to a shiftinvariant monotone quasinorm on a shift-invariant monotone ideal of l∞ . Recall that shift-invariant monotone quasinorms were defined in Definition 4.1.5. Symmetric quasinorms were defined in Definition 2.4.8. That a symmetric quasinorm ‖ ⋅ ‖𝒥 on a two-sided ideal 𝒥 restricts to a quasinorm ‖ ⋅ ‖O𝒥 on O𝒥 by the assignment 󵄩 󵄩 x 󳨃→ 󵄩󵄩󵄩diag(Lx)󵄩󵄩󵄩𝒥 ,

x ∈ O𝒥 ,

is straightforward since diag and L are linear. The shift-invariance and monotonicity properties are checked below. Lifting a shift-invariant monotone quasinorm ‖ ⋅ ‖O on a shift-invariant monotone ideal O to a quasinorm ‖ ⋅ ‖O𝒥 on the corresponding ideal 𝒥O by the assignment 󵄩 󵄩 A 󳨃→ 󵄩󵄩󵄩L󸀠 μ(A)󵄩󵄩󵄩O ,

A ∈ 𝒥O ,

is more direct than in the Calkin correspondence case since, from Lemma 4.2.5, L󸀠 μ satisfies a triangle inequality up to shift invariance and monotonicity. The shiftinvariance and monotonicity conditions on the quasinorm on O then provide the triangle inequality. Lemma 4.4.1. For a quasinormed shift-invariant ideal (O, ‖ ⋅ ‖O ) in l∞ such that e0 ∈ O, the ideal (𝒥O , ‖ ⋅ ‖𝒥O ) is quasinormed. Proof. Let A, B ∈ O𝒥 . Clearly, e1 = Te0 ∈ O. It follows from Lemma 4.2.5 that L󸀠 μ(A + B) ≤ T 2 (L󸀠 μ(A) + L󸀠 μ(B)) + ‖A + B‖∞ (e0 + e1 ).

4.4 Pietsch correspondence for quasinorms | 127

By the quasitriangle inequality, we have 󵄩 󵄩 2 󸀠 󵄩 󵄩󵄩 󸀠 󸀠 󵄩󵄩L μ(A + B)󵄩󵄩󵄩O ≤ KO (󵄩󵄩󵄩T (L μ(A) + L μ(B))󵄩󵄩󵄩O + ‖A + B‖∞ ‖e0 + e1 ‖O ), where KO is the concavity modulus of the quasinorm ‖ ⋅ ‖O . Since O is a shift-invariant quasi-Banach ideal, it follows that 󵄩 󵄩 󵄩 󸀠 󵄩󵄩 󸀠 2 󸀠 󵄩󵄩L μ(A + B)󵄩󵄩󵄩O ≤ KO (‖T‖O→O 󵄩󵄩󵄩L μ(A) + L μ(B)󵄩󵄩󵄩O + ‖A + B‖∞ ‖e0 + e1 ‖O ). Again using the quasitriangle inequality, we obtain 󵄩󵄩 󸀠 󵄩 󵄩󵄩L μ(A + B)󵄩󵄩󵄩O 󵄩 󵄩 󵄩 󵄩 ≤ KO2 (‖T‖2O→O (󵄩󵄩󵄩L󸀠 μ(A)󵄩󵄩󵄩O + 󵄩󵄩󵄩L󸀠 μ(B)󵄩󵄩󵄩O ) + (‖A‖∞ + ‖B‖∞ )(‖e0 ‖O + ‖e1 ‖O )). Since 󵄩 󵄩 ‖X‖∞ ≤ KO󸀠 󵄩󵄩󵄩L󸀠 μ(X)󵄩󵄩󵄩O ,

X ∈ 𝒥O ,

it follows that 󵄩󵄩 󸀠 󵄩 󵄩 󵄩 󸀠 󵄩 󸀠󸀠 󵄩 󸀠 󵄩󵄩L μ(A + B)󵄩󵄩󵄩O ≤ KO (󵄩󵄩󵄩L μ(A)󵄩󵄩󵄩O + 󵄩󵄩󵄩L μ(B)󵄩󵄩󵄩O ). The assertion follows from the definition of the quasinorm in 𝒥O . We can now prove Theorem 4.1.6. Proof of Theorem 4.1.6(a). We prove that the quasinorm ‖ ⋅ ‖O𝒥 is shift-invariant and monotone. We have L(Tx) = (0, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(0), x(0), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(1), . . . , x(1), . . . , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(n − 1), . . . , x(n − 1), . . .) 2 times

2n times

4 times

= (T ∘ σ2 )((x(0), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(1), x(1), ⏟⏟ x(2), . .⏟,⏟⏟⏟⏟⏟⏟⏟⏟ x(2) . . . , x(n), . . .)) ⏟⏟⏟⏟⏟⏟⏟⏟⏟.⏟⏟⏟ ⏟⏟, . . . , x(n), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 times

2n times

4 times

= (T ∘ σ2 )(Lx). Hence, 󵄩 󵄩 󵄩 󵄩 ‖Tx‖O𝒥 = 󵄩󵄩󵄩L(Tx)󵄩󵄩󵄩𝒥 = 󵄩󵄩󵄩(T ∘ σ2 )(Lx)󵄩󵄩󵄩𝒥 ≤ 2K𝒥 ‖Lx‖𝒥 = 2K𝒥 ‖x‖O𝒥 . Here, K𝒥 is the concavity modulus for the quasi-Banach space (𝒥 , ‖ ⋅ ‖𝒥 ). Thus, T is a bounded mapping on O𝒥 . Let x, y ∈ O𝒥 be such that 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨y(k)󵄨󵄨󵄨 ≤ sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨, k≥n

k≥n

n ≥ 0.

By Lemma 4.2.3, we have ‖y‖O𝒥 = ‖Ly‖𝒥 ≤ ‖Lx ⊕ Lx‖𝒥 ≤ 2K𝒥 ‖Lx‖𝒥 = 2K𝒥 ‖x‖O𝒥 .

128 | 4 Pietsch correspondence Proof of Theorem 4.1.6(b). By Lemma 4.4.1, the assignment ‖A‖𝒥O = ‖L󸀠 μ(A)‖O , A ∈ 𝒥O , is a quasinorm on 𝒥O . Since ‖A‖𝒥O = ‖μ(A)‖𝒥O , the quasinorm is symmetric. Proof of Theorem 4.1.6(c). Let (O, ‖ ⋅ ‖O ) be a quasi-Banach shift-invariant monotone ideal and let (𝒥 , ‖ ⋅ ‖𝒥 ) = (𝒥O , ‖ ⋅ ‖𝒥O ). We prove that ‖x‖O ≈ ‖Lx‖𝒥O ,

x ∈ O.

Set 󵄨 󵄨 z(n) = sup󵄨󵄨󵄨x(k)󵄨󵄨󵄨. k≥n

We have z ∈ O and ‖z‖O = ‖x‖O . By Corollary 4.2.2, we have T ∗ z ≤ L󸀠 μ(Lx) ≤ z. Thus, 󵄩 󵄩 ‖Lx‖𝒥O = 󵄩󵄩󵄩L󸀠 μ(Lx)󵄩󵄩󵄩O ≤ ‖z‖O = ‖x‖O and 󵄩 󵄩 󵄩 󵄩 ‖Lx‖𝒥O = 󵄩󵄩󵄩L󸀠 μ(Lx)󵄩󵄩󵄩O ≥ 󵄩󵄩󵄩T ∗ z 󵄩󵄩󵄩O . Clearly, z = z(0)e0 + TT ∗ z. Thus, 󵄨 󵄨 󵄩 󵄩 ‖z‖O ≤ KO (󵄨󵄨󵄨z(0)󵄨󵄨󵄨 ⋅ ‖e0 ‖O + 󵄩󵄩󵄩TT ∗ z 󵄩󵄩󵄩O ) 󵄩 󵄩 ≤ KO (‖z‖∞ ⋅ ‖e0 ‖O + ‖T‖O→O 󵄩󵄩󵄩T ∗ z 󵄩󵄩󵄩O ) ≤ KO (‖x‖∞ ⋅ ‖e0 ‖O + ‖T‖O→O ‖Lx‖𝒥O ). Since ‖x‖∞ ≤ KO󸀠 ‖Lx‖𝒥O , we have that ‖ ⋅ ‖O𝒥 ≈ ‖ ⋅ ‖O . Conversely, let (𝒥 , ‖ ⋅ ‖𝒥 ) be a quasi-Banach ideal and (O, ‖ ⋅ ‖O ) = (O𝒥 , ‖ ⋅ ‖O𝒥 ). We have 󵄩 󵄩 󵄩 󵄩 ‖A‖𝒥O = 󵄩󵄩󵄩L󸀠 μ(A)󵄩󵄩󵄩O = 󵄩󵄩󵄩LL󸀠 μ(A)󵄩󵄩󵄩𝒥 .

4.4 Pietsch correspondence for quasinorms | 129

By Lemma 4.2.4, 󵄩 󵄩 󵄩 󵄩󵄩 󸀠 󵄩󵄩LL μ(A)󵄩󵄩󵄩𝒥 ≤ 󵄩󵄩󵄩σ2 μ(A)󵄩󵄩󵄩𝒥 ≤ 2K𝒥 ‖A‖𝒥 , 󵄩 󵄩 󵄩 󵄩 ‖A‖𝒥 ≤ 󵄩󵄩󵄩σ2 LL󸀠 μ(A)󵄩󵄩󵄩𝒥 ≤ 2K𝒥 󵄩󵄩󵄩LL󸀠 μ(A)󵄩󵄩󵄩𝒥 . Hence, ‖A‖𝒥O ≤ 2K𝒥 ‖A‖𝒥 ,

‖A‖𝒥 ≤ 2K𝒥 ‖A‖𝒥O .

The following lemmas prove the final statement of Theorem 4.1.6. The Pietsch correspondence bijectively associates quasi-Banach ideals of ℒ(H) to quasi-Banach shiftinvariant monotone ideals of l∞ . Lemma 4.4.2. For every quasi-Banach ideal (𝒥 , ‖⋅‖𝒥 ) in ℒ(H), the corresponding quasinormed shift-invariant monotone ideal (O𝒥 , ‖ ⋅ ‖O𝒥 ) is complete. Proof. Let (xn )n≥0 be a Cauchy sequence in O𝒥 . It follows that (Lxn )n≥0 is a Cauchy sequence in 𝒥 . Since (𝒥 , ‖ ⋅ ‖𝒥 ) is a quasi-Banach space, it follows that Lxn → y in 𝒥 . In particular, Lxn → y in ℒ(H) and, therefore, in weak operator topology. Let m ≥ 0 and consider 2m − 1 ≤ k1 , k2 < 2m+1 − 1. We have 0 = ⟨(Lxn )ek1 , ek1 ⟩ − ⟨(Lxn )ek2 , ek2 ⟩ → ⟨yek1 , ek1 ⟩ − ⟨yek2 , ek2 ⟩. Hence, ⟨yek1 , ek1 ⟩ = ⟨yek2 , ek2 ⟩. Since each Lxn is diagonal operator, then so is y. Consequently, y = Lx for some x ∈ l∞ . Since Lx ∈ 𝒥 , it follows that x ∈ O𝒥 . We now have ‖xn − x‖O𝒥 = ‖Lxn − Lx‖𝒥 → 0,

n → ∞.

This shows the completeness of the quasi-normed space (O𝒥 , ‖ ⋅ ‖O𝒥 ). Lemma 4.4.3. For every quasi-Banach shift-invariant monotone ideal (O, ‖ ⋅ ‖O ) in ℒ(H), the quasinormed ideal (𝒥O , ‖ ⋅ ‖𝒥O ) is complete. Proof. Let {An }n≥0 be a Cauchy sequence in 𝒥O . It is also a Cauchy sequence in ℒ(H). Denote its limit in ℒ(H) by A. Choose nk → ∞ such that ‖Ank+1 − Ank ‖𝒥O ≤ ϵk+1 , Here, ε is chosen in such a way that 2K𝒥O KO ε < 1. Denote, for brevity, Vk = Ank+1 − Ank . We have A = An1 + ∑ Vk , k≥1

where the series converges in ℒ(H).

k ≥ 1.

130 | 4 Pietsch correspondence By Lemma 3.2.2, we have μ(∑ Vk ) ≤ ∑ σ2k μ(Vk ). k≥1

k≥1

Set xk = L󸀠 σ2k μ(Vk ). By Lemma 4.2.4, we have σ2k μ(Vk ) ≤ σ2 Lxk ,

Lxk ≤ σ2k+1 μ(Vk ).

Obviously, 󵄩 󵄩 ‖xk ‖O = ‖Lxk ‖𝒥O ≤ 󵄩󵄩󵄩σ2k+1 μ(Vk )󵄩󵄩󵄩𝒥 ≤ (2K𝒥O ϵ)k+1 , O

k ≥ 1.

Hence, μ(∑ Vk ) ≤ ∑ σ2 Lxk = σ2 L(∑ xk ). k≥1

k≥1

k≥1

Note that 󵄩󵄩 m 󵄩󵄩 m m (2K𝒥O KO ϵ)n+1 󵄩󵄩 󵄩 󵄩󵄩 ∑ xk 󵄩󵄩󵄩 ≤ ∑ K k ‖xk ‖O ≤ ∑ (2K𝒥 KO ϵ)k+1 ≤ . O 󵄩󵄩 󵄩󵄩 O 1 − 2K𝒥O KO ϵ 󵄩󵄩k=n 󵄩󵄩O k=n k=n Hence, the sequence {∑nk=1 xk }n≥1 is Cauchy in O. However, (O, ‖ ⋅ ‖O ) is complete and, hence, the series ∑k≥1 xk converges in O. Hence, L(∑k≥1 xk ) ∈ 𝒥O and, therefore, ∑k≥1 Vk ∈ 𝒥O . Consequently, A ∈ 𝒥O . We now have μ( ∑ Vk ) ≤ ∑ σ2k μ(Vk ) ≤ ∑ Lxk = L( ∑ xk ). k≥m

k≥m

k≥m

k≥m

Since ∑k≥m xk is a decreasing sequence, it follows that 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ‖A − Anm ‖𝒥O ≤ 󵄩󵄩󵄩L( ∑ xk )󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ xk 󵄩󵄩󵄩 → 0, 󵄩󵄩 󵄩󵄩𝒥O 󵄩󵄩 󵄩 k≥m k≥m 󵄩O

m → ∞.

Thus, Anm → A in 𝒥O . Since {An }n≥0 is Cauchy in 𝒥O , it follows that An → A in 𝒥O .

4.5 Lattice of regular traces | 131

4.5 Lattice of regular traces Kalton’s notion of a regular trace on a two-sided ideal 𝒥 of ℒ(H) was given in Definition 4.1.7. A trace φ : 𝒥 → ℂ is regular if B 󳨃→ φ(AB),

B ∈ ℒ(H),

is continuous in the uniform norm of ℒ(H) for each A ∈ 𝒥 . Similarly, a shift-invariant functional ℓ : O → ℂ on a shift-invariant monotone ideal O of l∞ is regular if b 󳨃→ φ(ab),

b ∈ l∞ ,

is continuous in the norm of l∞ for each a ∈ O. Pietsch correspondence of regular traces This section proves the initial statements of Theorem 4.1.8, namely that the bijective correspondence in Theorem 4.1.4 maps regular traces to regular shift-invariant linear functionals. Proof of Theorem 4.1.8 (a)–(b). Let φ : 𝒥 → ℂ be a regular trace. We claim that ℓφ = φ∘ L : O𝒥 → ℂ is also regular. Indeed, for a ∈ O𝒥 and b ∈ l∞ , we have that L(ab) = La⋅Lb, where La ∈ 𝒥 and Lb ∈ ℒ(H). Thus, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨ℓφ (ab)󵄨󵄨󵄨 = sup 󵄨󵄨󵄨φ(La ⋅ Lb)󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨φ(La ⋅ B)󵄨󵄨󵄨 < ∞.

‖b‖∞ ≤1

‖B‖∞ ≤1

‖b‖∞ ≤1

This proves the claim. Let O be a shift-invariant monotone ideal of l∞ and let ℓ : O → ℂ be a regular shift-invariant linear functional. We claim that φℓ : 𝒥O → ℂ is a regular trace. Let A ∈ 𝒥 and B ∈ ℒ(H) be positive operators. We have 1

1

1

1

φℓ (AB) = φℓ (B 2 AB 2 ) = ℓ(L󸀠 μ(B 2 AB 2 )). It is immediate that 1

1

1

1

μ(B 2 AB 2 ) ≤ ‖B‖∞ μ(A). Thus, L󸀠 μ(B 2 AB 2 ) ≤ ‖B‖∞ L󸀠 μ(A). Let b ∈ l∞ be such that 1

1

L󸀠 μ(B 2 AB 2 ) = L󸀠 μ(A) ⋅ b,

‖b‖∞ ≤ ‖B‖∞ .

132 | 4 Pietsch correspondence Clearly, φℓ (AB) = ℓ(L󸀠 μ(A) ⋅ b). Therefore, 󵄨 󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨φℓ (AB)󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨ℓ(L󸀠 μ(A) ⋅ b)󵄨󵄨󵄨 < ∞.

‖B‖∞ ≤1 B≥0

‖b‖∞ ≤1

The claim follows now by linearity in A and B. Lattice of hermitian regular traces A lattice on hermitian regular traces is introduced in this section and the Pietsch correspondence is proved to be a lattice isomorphism. This section proves the Jordan decomposition in Theorem 4.1.10, namely that every regular trace is the span of four positive traces. Therefore the set of regular traces is the span of all positive traces since the proof that positive traces are regular is straightforward. Assume without loss of generality that 0 ≤ A ∈ 𝒥 and 0 ≤ B ∈ ℒ(H). If φ : 𝒥+ → ℂ is a positive trace then φ(AB) = φ(B1/2 AB1/2 ) = φ(μ(B1/2 AB1/2 )) ≤ ‖B‖∞ φ(μ(A)). Hence φ is regular. We introduce the notion of a positive part of a shift-invariant linear function ℓ : O → ℂ. Define ℓ+ (x) := sup{ℓ(u) : 0 ≤ u ≤ x},

0 ≤ x ∈ O.

Lemma 4.5.1. For every hermitian regular shift-invariant functional ℓ : O → ℂ, ℓ+ extends by linearity from the positive cone of O to a shift invariant positive linear functional on O. Proof. First, let us show that ℓ+ (a) is well defined for every 0 ≤ a ∈ O. Indeed, if 0 ≤ u ≤ a, then u = ab with ‖b‖∞ ≤ 1. Hence 󵄨 󵄨 sup{ℓ(u) : 0 ≤ u ≤ a} ≤ sup 󵄨󵄨󵄨ℓ(ab)󵄨󵄨󵄨 < ∞, ‖b‖∞ ≤1

since ℓ is assumed to be regular. Next, let 0 ≤ a1 , a2 ∈ O and let 0 ≤ b ≤ a1 +a2 . Set b1 = min{b, a1 } and b2 = (b−a1 )+ . It is immediate that b = b1 + b2 , 0 ≤ b1 ≤ a1 and 0 ≤ b2 ≤ a2 . Hence, ℓ(b) = ℓ(b1 ) + ℓ(b2 ) ≤ ℓ+ (a1 ) + ℓ+ (a2 ). Taking the supremum over b, we obtain ℓ+ (a1 + a2 ) ≤ ℓ+ (a1 ) + ℓ+ (a2 ).

4.5 Lattice of regular traces | 133

Next, let 0 ≤ a1 , a2 ∈ O and fix ε > 0. Choose 0 ≤ b1 ≤ a1 and 0 ≤ b2 ≤ a2 such that ℓ+ (a1 ) ≤ ℓ(b1 ) + ε,

ℓ+ (a2 ) ≤ ℓ(b2 ) + ε.

Let b = b1 + b2 . Clearly, 0 ≤ b ≤ a1 + a2 . Thus, ℓ+ (a1 ) + ℓ+ (a2 ) ≤ 2ε + ℓ(b1 ) + ℓ(b2 ) = 2ε + ℓ(b) ≤ 2ε + ℓ+ (a1 + a2 ). Since ε > 0 is arbitrary, we conclude that ℓ+ (a1 ) + ℓ+ (a2 ) ≤ ℓ+ (a1 + a2 ). Combining the assertions, we infer that ℓ+ (a1 ) + ℓ+ (a2 ) = ℓ+ (a1 + a2 ),

0 ≤ a1 , a2 ∈ O.

Thus, ℓ+ is additive (and positive homogeneous) on O. Every such functional extends to a linear functional on O. We denote the extension of ℓ+ by linearity again by ℓ+ . Obviously, ℓ+ is a positive functional. It remains to show that ℓ+ is shift-invariant. If 0 ≤ u ≤ a ∈ O, then 0 ≤ Tu ≤ Ta. Conversely, if 0 ≤ v ≤ Ta, then v(0) = 0 and, therefore, v = Tu with 0 ≤ u ≤ a. Hence, ℓ+ (Ta) = sup{ℓ(v) : 0 ≤ v ≤ Ta} = sup{ℓ(Tu) : 0 ≤ u ≤ a} = 2 sup{ℓ(u) : 0 ≤ u ≤ a} = 2ℓ+ (a). Thus, ℓ+ is shift-invariant. The next result introduces the lattice operations on hermitian regular shiftinvariant linear functionals. Observe that the positive part defined above is ℓ+ = ℓ ∨ 0. Proposition 4.5.2. The set of all hermitian regular shift-invariant linear functionals on O is a lattice with respect to the operations ∨ and ∧ defined by (ℓ1 ∨ ℓ2 )(x) = sup{ℓ1 (u) + ℓ2 (v) : 0 ≤ u, v ∈ O, x = u + v}, (ℓ1 ∧ ℓ2 )(x) = inf{ℓ1 (u) + ℓ2 (v) : 0 ≤ u, v ∈ O, x = u + v},

0 ≤ x ∈ O, 0 ≤ x ∈ O.

Proof. The assertion follows from Lemma 4.5.1 and the equalities ℓ1 ∨ ℓ2 = ℓ2 + (ℓ1 − ℓ2 )+ ,

ℓ1 ∧ ℓ2 = ℓ1 − (ℓ1 − ℓ2 )+ .

We now prove that the lattice operations lift from hermitian regular shift-invariant monotone linear functionals to hermitian regular traces. Theorem 4.5.3. Let 𝒥 be an ideal of ℒ(H). The set of all hermitian regular traces on 𝒥 is a lattice with respect to the operations ∨ and ∧ defined by (φ1 ∨ φ2 )(A) = sup{φ1 (B) + φ2 (C) : 0 ≤ B, C ∈ 𝒥 , A = B + C}, (φ1 ∧ φ2 )(A) = inf{φ1 (B) + φ2 (C) : 0 ≤ B, C ∈ 𝒥 , A = B + C},

0 ≤ A ∈ 𝒥, 0 ≤ A ∈ 𝒥.

134 | 4 Pietsch correspondence To prove Theorem 4.5.3 and Theorem 4.1.8(c), introduce the positive part of a regular hermitian trace by φ+ (A) := sup{φ(B) : 0 ≤ B ≤ A},

0 ≤ A ∈ 𝒥.

Observe that the positive part of a regular trace is well defined. If 0 ≤ B ≤ A, then μ(B) = μ(A) ⋅ diag(b) for some b ∈ l∞ . Hence φ(B) = φ(μ(B)) = φ(μ(A)diag(b)) and 󵄨 󵄨 φ+ (A) ≤ sup 󵄨󵄨󵄨φ(μ(A)B)󵄨󵄨󵄨 < ∞. ‖B‖∞ ≤1

The next sequence of lemmas from Lemma 4.5.4 to Lemma 4.5.6 prove that, in Lemma 4.5.7, the positive part φ+ can be written in terms of the positive part (ℓφ )+ of the corresponding shift-invariant linear functional. Lemma 4.5.4. Let 𝒥 be an ideal of ℒ(H) and let φ : 𝒥 → ℂ be a regular hermitian trace. We have φ+ (A) = φ+ (LL󸀠 μ(A)),

0 ≤ A ∈ 𝒥.

Proof. Let 0 ≤ A ∈ 𝒥 . We have φ+ (A) ≤ sup{φ(b) : b = μ(b) ∈ l∞ , 0 ≤ b ≤ μ(A)}. If b = μ(b) is such that 0 ≤ b ≤ μ(A), then it follows from Lemma 4.3.2 that b − LL󸀠 b ∈ Z(J). Thus, φ+ (A) ≤ sup{φ(LL󸀠 b) : b = μ(b) ∈ l∞ , 0 ≤ b ≤ μ(A)} ≤ sup{φ(LL󸀠 b) : b ∈ l∞ , 0 ≤ LL󸀠 b ≤ LL󸀠 μ(A)} ≤ sup{φ(c) : c ∈ l∞ , 0 ≤ c ≤ LL󸀠 μ(A)} = φ+ (LL󸀠 μ(A)). Conversely, φ+ (LL󸀠 μ(A)) ≤ sup{φ(b) : b = μ(b) ∈ l∞ , 0 ≤ b ≤ LL󸀠 μ(A)}. If b = μ(b) is such that 0 ≤ b ≤ LL󸀠 μ(A), then it follows from Lemma 4.3.2 that b−LL󸀠 b ∈ Z(J). Thus, φ+ (LL󸀠 μ(A)) ≤ sup{φ(LL󸀠 b) : b = μ(b) ∈ l∞ , 0 ≤ b ≤ LL󸀠 μ(A)} ≤ sup{φ(LL󸀠 b) : b ∈ l∞ , 0 ≤ LL󸀠 b ≤ LL󸀠 μ(A)}.

4.5 Lattice of regular traces | 135

Since LL󸀠 b ≤ LL󸀠 μ(A), it follows that LL󸀠 b = LL󸀠 c for some 0 ≤ c ≤ μ(A). Since b = μ(b), it follows that we may also assume c = μ(c). Thus, φ(LL󸀠 μ(A)) ≤ sup{φ(LL󸀠 c) : c ∈ l∞ , 0 ≤ c ≤ μ(A)}. If c = μ(c) is such that 0 ≤ c ≤ μ(A), then it follows from Lemma 4.3.2 that c − LL󸀠 c ∈ Z(J). Thus, φ+ (LL󸀠 μ(A)) ≤ sup{φ(c) : c ∈ l∞ , 0 ≤ c ≤ μ(A)} = φ+ (A). Lemma 4.5.5. Let O be a shift-invariant monotone ideal of l∞ . If 0 ≤ a ∈ O, then Cμ(b) − Cb ∈ 𝒥O for all 0 ≤ b ≤ La. Proof. Since O is monotone, it follows that h ∈ O, where h(n) = sup a(n). m≥n

It is evident that, 0 ≤ b ≤ Lh. We may, therefore, assume without loss of generality that h = a and a is decreasing. Let k ≥ 0 and let A ⊂ ℤ+ be such that |A| = k + 1 with k

∑ μ(l, b) = ∑ b(l).

l=0

l∈A

We have, k

k

∑ b(l) = ∑ b(l) + ∑ b(l) ≤ ∑ b(l) + ∑ (La)(l) ≤ ∑ b(l) + (k + 1)(La)(k + 1).

l∈A

l∈A l≤k

l∈A l>k

l=0

l∈A l>k

l=0

Dividing by k, we obtain 1 k 1 k ∑ μ(l, b) ≤ ∑ b(l) + (La)(k), k + 1 l=0 k + 1 l=0

k ≥ 0.

In other words, Cμ(b) ≤ Cb + La. On the other hand, we have Cb ≤ Cμ(b) since b ≥ 0. Combining these estimates yields 0 ≤ Cμ(b) − Cb ≤ La. Since La ∈ 𝒥O , we complete the proof.

136 | 4 Pietsch correspondence Lemma 4.5.6. Let 𝒥 be an ideal of ℒ(H) with commutative core J. Let 0 ≤ a ∈ O𝒥 . For all x ∈ J such that 0 ≤ x ≤ La, there exists 0 ≤ b ≤ a such that x − Lb ∈ Z(J). Proof. Let b = L󸀠 x. Note that 0 ≤ L󸀠 x ≤ L󸀠 La = a. Thus, 0 ≤ b ≤ a. We claim that Cx − CLb ∈ J. Indeed, if 2k − 1 ≤ n < 2k+1 − 1, then n

n

n

n

m=0

m=0

m=2k −1

m=2k −1

∑ x(m) − ∑ Lb(m) = ∑ x(m) − ∑ (Lb)(m).

Thus, 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨󵄨 n 1 󵄨󵄨󵄨󵄨 n 󵄨󵄨 󵄨 󵄨󵄨 ∑ x(m)󵄨󵄨󵄨 + 󵄨󵄨 ∑ Lb(m)󵄨󵄨󵄨 󵄨󵄨(Cx)(n) − (CLb)(n)󵄨󵄨󵄨 ≤ 󵄨󵄨 n + 1 󵄨󵄨 k 󵄨󵄨 n + 1 󵄨󵄨󵄨 k 󵄨m=2 −1 󵄨 󵄨 m=2 −1 ≤

2 n+1

n

∑ (La)(m) = m=2k −1

2(n − 2k + 2) ⋅ (La)(n) ≤ 2(La)(n). n+1

By Lemma 4.5.5, we have Cμ(x) − Cx ∈ J and Cμ(Lb) − CLb ∈ J. Thus, Cμ(x) − Cμ(Lb) = (Cμ(x) − Cx) + (Cx − CLb) − (Cμ(Lb) − CLb) ∈ J. By Theorem 3.3.7, μ(x) − μ(Lb) ∈ Z(J). Consequently, x − Lb = (x − μ(x)) + (μ(x) − μ(Lb)) + (μ(Lb) − Lb) ∈ Z(J). The next lemma shows that the positive part φ+ of a trace φ can be written in terms of the positive part (ℓφ )+ of the corresponding shift-invariant linear functional ℓ. Lemma 4.5.7. Let 𝒥 be an ideal of ℒ(H) and let φ : 𝒥 → ℂ be a regular hermitian trace. We have φ+ (A) = (ℓφ )+ (L󸀠 μ(A)),

0 ≤ A ∈ 𝒥.

Proof. Let 0 ≤ A ∈ 𝒥 . By Lemma 4.5.4, we have φ+ (A) = sup{φ(x) : x ∈ l∞ , 0 ≤ x ≤ LL󸀠 μ(A)}. We have, φ+ (A) ≥ sup{φ(Lb) : b ∈ l∞ , 0 ≤ Lb ≤ LL󸀠 μ(A)} = sup{ℓφ (b) : b ∈ l∞ , 0 ≤ b ≤ L󸀠 μ(A)} = (ℓφ )+ (L󸀠 μ(A)). If 0 ≤ x ≤ L(L󸀠 μ(A)), then, using Lemma 4.5.6, we find 0 ≤ b ≤ L󸀠 μ(A) such that x − Lb ∈ Z(J). Thus, φ+ (A) ≤ sup{φ(Lb) : b ∈ l∞ , 0 ≤ Lb ≤ LL󸀠 μ(A)} = sup{ℓφ (b) : b ∈ l∞ , 0 ≤ b ≤ L󸀠 μ(A)} = (ℓφ )+ (L󸀠 μ(A)).

4.5 Lattice of regular traces | 137

We can now prove Theorem 4.5.3 and Theorem 4.1.8(c). Proof of Theorem 4.5.3 and Theorem 4.1.8(c). Let 𝒥 be an ideal of ℒ(H) and let φ : 𝒥 → ℂ be a regular trace. By Lemma 4.5.7, we have φ+ (A) = (ℓφ )+ (L󸀠 μ(A)),

0 ≤ A ∈ 𝒥.

Since ℓφ is a regular shift-invariant linear functional on O𝒥 , it follows by Lemma 4.5.1 that (ℓφ )+ is also a regular shift-invariant linear functional on O𝒥 . Thus, φ+ is a regular trace on 𝒥 and ℓφ+ = (ℓφ )+ . It follows immediately that φ1 ∨ φ2 and φ1 ∧ φ2 are regular traces on 𝒥 and that ℓφ1 ∨φ2 = ℓφ1 ∨ ℓφ2 ,

ℓφ1 ∧φ2 = ℓφ1 ∧ ℓφ2

for all regular traces φ1 and φ2 on 𝒥 . On a quasi-Banach ideal 𝒥 , each positive trace φ is continuous by Lemma 2.4.12. Hence every regular trace is continuous. That every continuous trace is regular is straightforward, if A ∈ 𝒥 and B ∈ ℒ(H) then 󵄨󵄨 󵄨 󵄨󵄨φ(BA)󵄨󵄨󵄨 ≤ ‖φ‖𝒥 →ℂ ⋅ ‖BA‖𝒥 ≤ ‖φ‖𝒥 →ℂ ⋅ ‖B‖∞ ⋅ ‖A‖𝒥 . Hence the set of regular traces and continuous traces coincide for a quasi-Banach ideal. The following proof shows that the bijection between continuous traces and continuous shift-invariant functionals is an isometry. Proof of Theorem 4.1.9. Let φ : 𝒥 → ℂ be a continuous trace. Then 󵄨󵄨 󵄨 󵄨󵄨φ(Lx)󵄨󵄨󵄨 ≤ ‖φ‖𝒥 →ℂ ⋅ ‖Lx‖𝒥 = ‖φ‖𝒥 →ℂ ⋅ ‖x‖O𝒥 by Theorem 4.1.6. Hence, ℓφ = φ ∘ L : O𝒥 → ℂ is continuous on O𝒥 . Let ℓ : O → ℂ be a shift-invariant linear functional. For every positive A ∈ 𝒥 , we have φℓ (A) = ℓ(L󸀠 μ(A)). Since ℓ is continuous, it follows that 󵄨󵄨 󵄨 󵄩 󸀠 󵄩 󵄨󵄨φℓ (A)󵄨󵄨󵄨 ≤ ‖ℓ‖O→ℂ ⋅ 󵄩󵄩󵄩L μ(A)󵄩󵄩󵄩O . By Theorem 4.1.6, we have 󵄨󵄨 󵄨 󵄨󵄨φℓ (A)󵄨󵄨󵄨 ≤ ‖ℓ‖O→ℂ ⋅ ‖A‖𝒥O . Hence, φℓ is continuous on the positive cone of 𝒥O . By linearity, φℓ is continuous on 𝒥O .

138 | 4 Pietsch correspondence Proof of Theorem 4.1.10. Let φ be a hermitian regular trace on the ideal 𝒥 . By Theorem 4.5.3, φ decomposes into a linear combination of two positive traces, φ = φ ∨ 0 − 0 ∨ (−φ). Let φ be a regular trace on the ideal 𝒥 . Then 1 1 φ = (φ + φ) + i(φ − φ), 2 2i where φ+φ and i(φ−φ) are hermitian regular traces. The statement is therefore proved for regular traces on the ideal 𝒥 . The statement for a quasi-Banach ideal follows from the fact that the regular traces and continuous traces coincide for a quasi-Banach ideal.

4.6 Notes The notes to Chapter 1 provide some history to the development of the Pietsch correspondence alongside the constructions of Dixmier, and the development of the Figiel– Kalton theorem and its noncommutative analogue for commutators. Pietsch considered the more general area of continuous operators between Banach spaces. The notions of symmetric functionals (see [83, 84, 107, 151, 152]) developed during the 1990s independently of Pietsch’s notion of shift-invariant functionals. Pietsch correspondence Albrecht Pietsch’s bijective identification of traces using invariance under right shifts rather than the permutation group [205, 207, 210, 211, 244] has become as important as the Calkin correspondence, as noted in the survey [172]. It is more constructive than the bijective identification between traces and symmetric functionals on the commutative core. Pietsch introduced the dyadic partial sums in the late 1980s on quasi-Banach ideals, summarized in the 1990 publication [202]. Pietsch later showed that the dyadic approach produces the bijection between traces on ideals of compact operators and shift-invariant linear functionals on shift-invariant monotone ideals of l∞ for all two-sided ideals of compact operators [210, Theorem 4], [211, Theorem 4.6]. See also Pietsch’s surveys [213] and [208], and historical comments in [210]. The bijection obtained by Pietsch is a consequence of his approach to dyadic decompositions [198, 202, 207], which provides more direct proofs of the bijection than using the Schmidt decomposition, especially for the proof of additivity. Shift-invariant functionals and shift-invariant monotone sequence spaces introduced by Pietsch are largely unstudied compared to rearrangement invariance, symmetric functionals, and symmetric sequence spaces. Their equivalence was established in [207, Theorem 6.3], see

4.6 Notes | 139

also [213], which identified the triangle of relations between the Pietsch correspondence, the Calkin correspondence, and the dyadic maps L and L󸀠 . The advantages of reducing from the permutation group to the semigroup generated by right shifts were noted by Pietsch [213, p. 365]. There is, as of yet, no neat classification for general shift-invariant monotone states such as their identification with Banach limits in the case of l1,∞ discussed in Chapter 6. The 1995 doctoral thesis of J. Varga [275] also contains an approach based n ∞ on dyadic partial sums. The bijection {an }∞ n=1 → {2 an }n=1 transfers between Pietsch’s shift-invariant monotone ideals and those introduced by Varga, and the bijection between traces on operator ideals and shift-invariant functionals on associated unbounded shift-invariant monotone sequence space is proved as [275, Theorem 4] in Varga’s thesis. Varga assumes that the operator ideals are full, in his terms, a condition that eventually is satisfied by all proper ideals of compact operators (private communication A. Pietsch). The paper [244] repeats many of Pietsch’s existing results for the ideal ℒ1,∞ . Regular traces on arbitrary ideals Kalton introduced the notion of a regular, or separably continuous, trace in [148]. The μ(n,B) principal ideals 𝒥A := {B ∈ ℒ(H) : supn≥0 μ(n,A) < ∞} for a trace class operator A are ideals within the trace class operators that can be Banach, quasi-Banach, or neither. As a substitute for continuous traces on principal ideals, Kalton linked traces to the uniform topology of ℒ(H). Kalton called a trace φ on 𝒥A separately continuous if the functional S 󳨃→ φ(SB),

S ∈ ℒ(H)

is uniformly continuous in S for each B ∈ 𝒥A . A necessary and sufficient condition was found on the singular values sequence μ(A) for when the set of separately continuous traces was spanned by the trace Tr [148, Theorem 6]. The condition does not exhaust the principal ideals within the trace class operators that admit traces. From this the existence of “unusual” traces inequivalent to Tr was proven by Kalton. For the existence and uniqueness of traces on principal ideals, we refer further to [93, 121, 148, 274, 294]. Our previous work on singular traces [170] and Volume III use uniform submajorization to prove the Jordan decomposition of continuous traces on a Banach ideal. Here, for quasi-Banach ideals Pietsch’s dyadic decomposition is used. Jordan decomposition of continuous traces on the quasi-Banach ideal ℒ1,∞ appears in [72, Corollary 2.2].

5 Spectrality of traces 5.1 Introduction A foundational result in linear algebra is that the matrix trace Tr is spectral, meaning that Tr(A) depends only on the spectrum λ(A) of A. Theorem 1.1.22 showed that the trace of a matrix is the sum of its eigenvalues repeated with algebraic multiplicity. This remains true for the operator trace Tr on ℒ1 extending the matrix trace, a nontrivial result known as Lidskii’s theorem. This chapter considers when a trace φ on a two-sided ideal 𝒥 of the algebra ℒ(H) of bounded operators on a separable Hilbert space H is spectral, meaning that the value φ(A) depends only on the eigenvalues of an arbitrary compact operator A ∈ 𝒥 listed with multiplicity. Definition 5.1.1. A trace φ on a two-sided ideal 𝒥 is spectral if, for all A, B ∈ 𝒥 , φ(A) = φ(B)

whenever λ(A) = λ(B).

Here λ(A) = λ(B) indicates an eigenvalue sequence of the compact operator A, counted with algebraic multiplicities as defined in Definition 1.1.10, is equal to some eigenvalue sequence of the compact operator B. Equivalently, the operators diag(λ(A)) and diag(λ(B)) are unitarily equivalent. Chapter 2, in Lemma 2.4.5, identified traces on a two-sided ideal with symmetric linear functionals on that ideal of operators. Every trace is a spectral invariant on the positive cone of the ideal, in the sense that if 0 ≤ A ∈ 𝒥 and φ is a trace on 𝒥 , then φ depends only on μ(A) = λ(A). The conditions under which this remains true for arbitrary nonpositive or non-self-adjoint operators is the focus of this chapter. Spectral formula for traces According to the Calkin correspondence between traces on an ideal 𝒥 and symmetric functionals on the commutative core J, described in Theorem 3.1.3 in Chapter 3, for every trace φ on 𝒥 there exists a unique symmetric functional φ̂ on J such that ̂ φ(A) = (φ̂ ∘ μ)(A) = φ(μ(A)),

A ∈ 𝒥+ ,

on the positive cone 𝒥+ of 𝒥 . Here, μ(A) is the sequence of singular values of the compact operator A. This chapter proves that, when 𝒥 is a logarithmic submajorization closed ideal in the sense of Definition 2.4.17, the singular values μ(A) can be replaced by an eigenvalue sequence λ(A), and the formula ̂ φ(A) = (φ̂ ∘ λ)(A) = φ(λ(A)), https://doi.org/10.1515/9783110378054-006

A ∈ 𝒥,

(5.1)

142 | 5 Spectrality of traces holds. That is, the trace φ(A) of A ∈ 𝒥 is equal to the corresponding symmetric functional φ̂ on the commutative core J applied to λ(A). The linear functional φ̂ ∘ λ is well defined – it is independent of which eigenvalue sequence is chosen, and is the linear extension to the whole space J of the additive functional φ̂ ∘ μ on the positive cone of J. Consequently, all traces on logarithmic submajorization closed ideals are spectral. Additionally, formula (5.1) using the symmetric functional φ̂ on the commutative core J is only possible when 𝒥 is logarithmic submajorization closed. When 𝒥 is not logarithmic submajorization closed, there exists an operator A ∈ 𝒥 such that λ(A) ∈ ̸ J. These statements, and equivalent ones involving the Pietsch correspondence in Chapter 4, are summarized in the following extension of Lidskii’s theorem. Theorem 5.1.2 (Spectral formula for traces). Let 𝒥 be an ideal of ℒ(H) and let φ be a trace on 𝒥 . (a) If 𝒥 is closed with respect to logarithmic submajorization, then (i) φ(A) = (φ̂ ∘ λ)(A)

(ii)

for all A ∈ 𝒥 , where φ̂ is the corresponding symmetric functional on the commutative core J. φ(A) = (ℓφ ∘ L󸀠 )(λ(A))

for all A ∈ 𝒥 , where ℓφ is the corresponding shift-invariant linear functional on the ideal O𝒥 and L󸀠 is dyadic averaging as described in Chapter 4. (b) If 𝒥 is not closed with respect to logarithmic submajorization, then there exists an operator A ∈ 𝒥 such that λ(A) ∉ J for any eigenvalue sequence λ(A) of A. Banach and quasi-Banach ideals are logarithmic submajorization closed. Therefore the ideal of trace class operators ℒ1 is logarithmic submajorization closed. Theorem 5.1.2 implies Lidskii’s formula, described in Chapter 1, for the trace Tr on ℒ1 . The weak trace class operators ℒ1,∞ and the Dixmier–Macaev ideal ℳ1,∞ are also logarithmic submajorization closed, as is every Lorentz ideal from Section 2.5. The Figiel– Kalton theorem in Chapter 3 proves that there are other traces on ℒ1 besides Tr, and traces on ℒ1,∞ and ℳ1,∞ , including the Dixmier traces described in Chapter 1. They are all spectral traces and can be written as a formula involving an eigenvalue sequence and the corresponding symmetric functional. Any sequence of the eigenvalues of a compact operator A listed by multiplicity can be substituted into the formula, even if they are not in order of nonincreasing absolute value. If x is such as a sequence then diag(x) = Udiag(λ(A))U ∗

5.1 Introduction | 143

for some unitary U ∈ ℒ(H) and eigenvalue sequence λ(A), and ̂ φ(A) = φ(diag(λ(A)) = φ(Udiag(λ(A)U ∗ ) = φ(diag(x)) = φ(x). Equivalently, symmetric functionals are invariant under the permutation group. Theorem 5.1.2 is not a complete answer to the spectral formulation of traces on two-sided ideals of ℒ(H). Theorem 5.1.2 does not exclude that a trace on an ideal 𝒥 that is not logarithmic submajorization closed may have an extension to a trace on a logarithmic submajorization closed ideal containing 𝒥 . Examples exist of a nonpositive trace φ on an ideal 𝒥 that is not logarithmic submajorization closed such that φ(Q) = 1 for a quasinilpotent operator Q ∈ 𝒥 . Hence φ cannot be spectral and cannot have an extension to a logarithmic submajorization closed ideal contained 𝒥 . Every positive trace on the same ideal 𝒥 extends to a positive trace on a logarithmic submajorization closed ideal containing 𝒥 , we refer to the notes at the end of this chapter. Hence, traces on an ideal 𝒥 that is not closed with respect to logarithmic submajorization can be spectral. To obtain Theorem 5.1.2, we study the commutator subspace [𝒥 , ℒ(H)] of an ideal 𝒥 of ℒ(H). Spectral characterization of commutators Definition 5.1.3. Let 𝒥 be a two-sided ideal of ℒ(H). The subspace [𝒥 , ℒ(H)] = Span({[A, B] : A ∈ 𝒥 , B ∈ ℒ(H)}) is called the commutator subspace of 𝒥 . Clearly, the commutator subspace [𝒥 , ℒ(H)] is the common kernel of all traces on 𝒥 . The spectrality of traces in Theorem 5.1.2 is not trivial; for an operator A ∈ 𝒥 written as the sum of four positive operators A = A1 − A2 + iA3 − iA4 , it is not automatic that the difference (μ(A1 ) − μ(A2 ) + iμ(A3 ) − iμ(A4 )) − λ(A) should belong to the center Z(J) of the commutative core J. Or, more succinctly written as operators, it is not clear in general why we should have A − diag(λ(A)) ∈ [𝒥 , ℒ(H)]. The spectral formulation of traces in Theorem 5.1.2 would follow from (5.2). Recall that the Cesaro operator C : l∞ → l∞

(5.2)

144 | 5 Spectrality of traces is defined by (Cx)(n) :=

n 1 ∑ x(k), 1 + n k=0

x ∈ l∞ , n ≥ 0.

The invariance of the commutative core to the Cesaro operator and the commutator subspace are related by the following noncommutative version of the Figiel–Kalton theorem (Theorem 3.3.1), which also completes the Calkin correspondence between the center Z(J) of the commutative core J and the commutator subspace [𝒥 , ℒ(H)] of the ideal 𝒥 , [𝒥 , ℒ(H)] ?

λ diag

? Z(J) .

Theorem 5.1.4. Let 𝒥 be an ideal of ℒ(H) with commutative core J and one of the following holds: (i) A ∈ 𝒥 is normal; or (ii) 𝒥 is logarithmic submajorization closed and A ∈ 𝒥 is arbitrary. Then the following statements are equivalent: (a) A ∈ [𝒥 , ℒ(H)]. (b) For any eigenvalue sequence λ(A) of A, Cλ(A) ∈ J. (c) For any eigenvalue sequence λ(A) of A there exists x ∈ J such that 󵄨󵄨 󵄨 󵄨󵄨Cλ(A)󵄨󵄨󵄨 ≤ μ(x). When the ideal 𝒥 is logarithmic submajorization closed with commutative core J, as a consequence of Theorem 5.1.4, then A ∈ [𝒥 , ℒ(H)] if and only if λ(A) ∈ Z(J). A refinement of Theorem 5.1.4 identifies when the difference of two operators belong to the commutator subspace in terms of the difference in their eigenvalue sequences. Theorem 5.1.5. Suppose 𝒥 is an ideal of ℒ(H) with commutative core J and either: (i) A, B ∈ 𝒥 are normal; or (ii) 𝒥 is logarithmic submajorization closed and A, B ∈ 𝒥 are arbitrary. Then the following statements are equivalent: (a) A − B ∈ [𝒥 , ℒ(H)]. (b) For any eigenvalue sequences λ(A) of A and λ(B) of B, C(λ(A) − λ(B)) ∈ J.

5.2 Normal operators in the commutator subspace

| 145

(c) For any eigenvalue sequences λ(A) of A and λ(B) of B there exists x ∈ J such that 󵄨 󵄨󵄨 󵄨󵄨C(λ(A) − λ(B))󵄨󵄨󵄨 ≤ μ(x). Once it is known that λ(A) ∈ J, then equation (5.2) follows from Theorem 5.1.5 since A and diag(λ(A)) have the same eigenvalues with the same multiplicities.

5.2 Normal operators in the commutator subspace This section proves the statement of Theorem 5.1.4 for normal operators in a two-sided ideal 𝒥 of ℒ(H). It is the original theorem of Dykema, Figiel, Weiss, and Wodzicki, lifting the discrete Figiel–Kalton theorem from a statement about the center of the commutative core J to a statement about Cesaro averages of eigenvalue sequences of normal operators and the commutator subspace. Theorem 5.2.1. Let 𝒥 be an ideal of ℒ(H). Let A ∈ 𝒥 be a normal operator. We have A ∈ [𝒥 , ℒ(H)] if and only if Cλ(A) ∈ J. We prove Theorem 5.2.1 using the following lemmas. Recall that if A = A∗ ∈ 𝒥 is self-adjoint then A = A+ − A− for positive operators A+ , A− ∈ 𝒥 . Self-adjoint operators A, B ∈ ℒ(H) are equimeasurable if μ(A+ ) = μ(B+ ) and μ(A− ) = μ(B− ) where A± and B± are the positive and negative parts of A and B, respectively. The doubling operator A ⊕ A was defined in Definition 2.2.11. Lemma 5.2.2. Let 𝒥 be an ideal of ℒ(H). If self-adjoint operators A1 , A2 ∈ 𝒥 are equimeasurable, then A1 − A2 ∈ [𝒥 , ℒ(H)]. In particular, 2A − A ⊕ A ∈ [𝒥 , ℒ(H)]. Proof. Let {ek }k≥0 (resp., {fk }k≥0 ) be an orthonormal sequence such that A1 ek = λ(k, A1 )ek (resp., A2 fk = λ(k, A2 )fk ). Let U be a partial isometry such that Uek = fk and such that Ue = 0 if e is orthogonal to every ek , k ≥ 0. We have that U ∗ U = EA1 (0, ∞), UU ∗ = EA2 (0, ∞) and UA1 U ∗ = A2 . It follows that A1 − A2 = A1 − UA1 U ∗ = U ∗ UA1 − UA1 U ∗ = [U ∗ , UA1 ] ∈ [𝒥 , ℒ(H)]. In particular, 2A − A ⊕ A = (A − A ⊕ 0) + (A − 0 ⊕ A) ∈ [𝒥 , ℒ(H)] + [𝒥 , ℒ(H)]. The next lemma proves that the center of the commutative core lifts to the commutator subspace.

146 | 5 Spectrality of traces Lemma 5.2.3. Let 𝒥 be an ideal of ℒ(H). If x ∈ Z(J), then diag(x) ∈ [𝒥 , ℒ(H)]. Proof. Since x ∈ Z(J), it follows by definition that x is a linear combination of differences of equimeasurable elements. That is, one can write n

x = ∑ αk (x2k−1 − x2k ), k=1

∗ ∗ where αk ∈ ℂ, x2k−1 = x2k−1 ∈ J, x2k = x2k ∈ J, and x2k−1 and x2k are equimeasurable. Thus, n

diag(x) = ∑ αk (diag(x2k−1 ) − diag(x2k )). k=1

Since the operators diag(x2k−1 ) and diag(x2k ) are equimeasurable, the assertion follows from Lemma 5.2.2. Lemma 5.2.5 below uses the Figiel–Kalton theorem (Theorem 3.3.1 from Section 3.3) to show that Cλ(A) ∈ J for A = A∗ ∈ 𝒥 implies that every trace on 𝒥 vanishes on A. To employ the Figiel–Kalton theorem, we need to understand the difference between the Cesaro average of eigenvalues and the decomposition of singular values. Lemma 5.2.4. Let A be a self-adjoint compact operator. We have 󵄨󵄨 󵄨 󵄨󵄨C(λ(A) − μ(A+ ) + μ(A− ))󵄨󵄨󵄨 ≤ 2μ(A). Proof. Since A is self-adjoint and compact, there exists an eigenvalue sequence λ(A) = {λ(k, A)}∞ k=0 . It is clear that n

n

n

{λ(k, A)}k=0 ⊂ {μ(k, A+ )}k=0 ∪ {−μ(k, A− )}k=0 . On the other hand, the set n n n 󵄨 󵄨 ({μ(k, A+ )}k=0 ∪ {−μ(k, A− )}k=0 )\{λ(k, A)}k=0 ⊂ {λ : |λ| ≤ 󵄨󵄨󵄨λ(n, A)󵄨󵄨󵄨}

has cardinality less than 2(n + 1). It follows that 󵄨󵄨 n 󵄨󵄨 n 󵄨󵄨 󵄨 󵄨󵄨 ∑ μ(k, A+ ) − μ(k, A− ) − ∑ λ(k, A)󵄨󵄨󵄨 ≤ 2(n + 1)󵄨󵄨󵄨λ(n, A)󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨k=0 󵄨󵄨 k=0 Dividing both sides by n + 1 and using the definition of C, we conclude the proof. We can now use the Figiel–Kalton theorem. Lemma 5.2.5. Let 𝒥 be an ideal of ℒ(H). If A = A∗ ∈ 𝒥 is such that Cλ(A) ∈ J, then A ∈ [𝒥 , ℒ(H)].

5.2 Normal operators in the commutator subspace

| 147

Proof. Let x = μ(A+ ) − μ(A− ) ∈ D(J) where the linear subspace D(J) of J was defined in Section 3.3. Since Cλ(A) ∈ J, it follows from Lemma 5.2.4 that Cx ∈ J. By Theorem 3.3.2, we have x ∈ Z(J). By Lemma 5.2.3, diag(x) ∈ [𝒥 , ℒ(H)]. By Lemma 5.2.2, we have A+ − diag(μ(A+ )) ∈ [𝒥 , ℒ(H)],

A− − diag(μ(A− )) ∈ [𝒥 , ℒ(H)].

Thus, A − diag(x) ∈ [𝒥 , ℒ(H)]. The assertion follows now from the preceding paragraph. The following lemma is a simplified version of Proposition 5.7 from [94]. It demonstrates that the Cesaro average of eigenvalues of commutators belong to the commutative core of an ideal and is the main technical component in the reverse direction to Lemma 5.2.5. Lemma 5.2.6. Let A, B ∈ ℒ(H) be self-adjoint operators. If A is compact, then 󵄨󵄨 󵄨 󵄨󵄨C(λ([A, B]))󵄨󵄨󵄨 ≤ 4‖B‖∞ μ(A) + μ([A, B]). Proof. By hypothesis, i[A, B] is self-adjoint. Let {ek }k≥0 and {fk }k≥0 be orthonormal bases in H such that [A, B]ek = λ(k, [A, B])ek ,

Afk = λ(k, A)fk ,

k ≥ 0.

For a given n ∈ ℕ, let p (resp., q) be the projection onto the span of {ek }n−1 k=0 (resp., of {fk }n−1 ). Clearly, Tr(p) = Tr(q) = n and k=0 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 ∗󵄩 󵄩󵄩[A, B](1 − p)󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ λ(k, [A, B])ek ek 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩k=n 󵄩 󵄨󵄨 󵄨󵄨 = max󵄨󵄨λ(k, [A, B])󵄨󵄨 k≥n 󵄨󵄨 󵄨 = 󵄨󵄨λ(n, [A, B])󵄨󵄨󵄨 = μ(n, [A, B]), where the last line following from the self-adjointness of i[A, B]. Similarly, 󵄩󵄩 󵄩 󵄩󵄩A(1 − q)󵄩󵄩󵄩 = μ(n, A). Setting r = q ∨ p, we obtain r[A, B]r = [rAr, rBr] + rA(1 − r)Br − rB(1 − r)Ar

148 | 5 Spectrality of traces and, therefore, Tr(r[A, B]r) = Tr(rA(1 − r)Br − rB(1 − r)Ar). Hence, 󵄩 󵄩 󵄨 󵄨󵄨 󵄨󵄨Tr(r[A, B]r)󵄨󵄨󵄨 ≤ 2󵄩󵄩󵄩A(1 − r)󵄩󵄩󵄩 ⋅ ‖B‖ ⋅ Tr(r) ≤ 4nμ(n, A)‖B‖. On the other hand, we have r[A, B]r = p[A, B]p + (r − p)[A, B](r − p) + p[A, B](r − p) + (r − p)[A, B]p and, therefore, Tr(p[A, B]p) = Tr(r[A, B]r) − Tr((r − p)[A, B](r − p)). We have 1 − p ≥ r − p and, therefore, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Tr((r − p)[A, B](r − p))󵄨󵄨󵄨 = 󵄨󵄨󵄨Tr((r − p)(1 − p)[A, B](1 − p)(r − p))󵄨󵄨󵄨 󵄩 󵄩 ≤ 󵄩󵄩󵄩(1 − p)[A, B](1 − p)󵄩󵄩󵄩 ⋅ Tr(r − p) ≤ nμ(n, [A, B]). It follows that 󵄨󵄨 n−1 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ λ(k, [A, B])󵄨󵄨󵄨 = 󵄨󵄨󵄨Tr(p[A, B]p)󵄨󵄨󵄨 ≤ 4nμ(n, A)‖B‖ + nμ(n, [A, B]), 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨k=0 󵄨󵄨 which is the claim. Recall that for any two sided ideal 𝒥 , if A ∈ 𝒥 then A = ℜA + iℑA for self-adjoint operators ℜA, ℑA ∈ 𝒥 . To employ Lemma 5.2.6, we need Lemma 5.2.9 which is a statement on the difference between the Cesaro average of eigenvalues of a compact operator and of eigenvalues of the real and imaginary parts. The following lemmas are used in the proof of Lemma 5.2.9. Lemma 5.2.7 is a restatement of Lemma 3.3.9, so we omit the proof. Lemma 5.2.7. If A1 , A2 ∈ ℒ(H) are positive operators, then 0 ≤ Cμ(A1 ) + Cμ(A2 ) − Cμ(A1 + A2 ) ≤ μ(A1 + A2 ). Lemma 5.2.8. If A1 , A2 ∈ ℒ(H) are compact self-adjoint operators, then 󵄨󵄨 󵄨 󵄨󵄨Cλ(A1 ) + Cλ(A2 ) − Cλ(A1 + A2 )󵄨󵄨󵄨 ≤ 6σ4 μ(A1 ) + 6σ4 μ(A2 ).

5.2 Normal operators in the commutator subspace

| 149

Proof. Denote for brevity, (A1 )+ = X1 ,

(A1 )− = Y1 ,

(A1 + A2 )+ = X3 ,

(A2 )+ = X2 ,

(A1 + A2 )− = Y3 .

(A2 )− = Y2 ,

Let A = X1 + X2 + Y3 = Y1 + Y2 + X3 . By Lemma 5.2.7, we have 0 ≤ Cμ(X1 ) + Cμ(X2 ) + Cμ(Y3 ) − Cμ(A) ≤ 2μ(A), 0 ≤ Cμ(Y1 ) + Cμ(Y2 ) + Cμ(X3 ) − Cμ(A) ≤ 2μ(A).

Therefore, 󵄨󵄨 󵄨 󵄨󵄨C(μ(X1 ) − μ(Y1 ) + μ(X2 ) − μ(Y2 ) + μ(Y3 ) − μ(X3 ))󵄨󵄨󵄨 ≤ 4μ(A). By Lemma 5.2.4, we have C(λ(A1 ) − μ(X1 ) + μ(Y1 )) ≤ 2μ(A1 ),

C(λ(A2 ) − μ(X2 ) + μ(Y2 )) ≤ 2μ(A2 ),

C(λ(A1 + A2 ) − μ(X3 ) + μ(Y3 )) ≤ 2μ(A1 + A2 ). Therefore, 󵄨󵄨 󵄨 󵄨󵄨C(λ(A1 ) + λ(A2 ) − λ(A1 + A2 ))󵄨󵄨󵄨 ≤ 2μ(A1 ) + 2μ(A2 ) + 2μ(A1 + A2 ) + 4μ(A).

(5.3)

Note from Fan’s inequalities, Lemma 2.2.9, that t t t μ(t, A) ≤ μ( , X1 ) + μ( , X2 ) + μ( , Y3 ), 4 4 2

t > 0,

or equivalently, μ(A) ≤ σ4 μ(X1 ) + σ4 μ(X2 ) + σ2 μ(Y3 ). Thus, μ(A) ≤ σ4 μ(A1 ) + σ4 μ(A2 ) + σ2 μ(A1 + A2 ) ≤ 2σ4 μ(A1 ) + 2σ4 μ(A2 ),

(5.4)

and similarly, μ(A1 + A2 ) ≤ σ2 μ(A1 ) + σ2 μ(A2 ) ≤ σ4 μ(A1 ) + σ4 μ(A2 ). The assertion follows by combining (5.3), (5.4), and (5.5). Lemma 5.2.9. For every compact normal operator A, we have 󵄨󵄨 󵄨 󵄨󵄨C(λ(A) − λ(ℜA) − iλ(ℑA))󵄨󵄨󵄨 ≤ 5μ(A).

(5.5)

150 | 5 Spectrality of traces Proof. Fix n ≥ 0. Since A is a normal operator, it follows that n

σ(A) ∩ {λ : |λ| > μ(n, A)} ⊂ {λ(k, A)}k=0 . On the other hand, the set n

{λ(k, A)}k=0 \(σ(A) ∩ {λ : |λ| > μ(n, A)}) has a cardinality less then n + 1 and is contained in the disk {|λ| ≤ μ(n, A)}. Therefore, 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∑ λ(k, A) − 󵄨󵄨 ≤ (n + 1)μ(n, A). λ ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 λ∈σ(A),|λ|>μ(n,A) 󵄨󵄨

(5.6)

Arguing similarly and using the normality of A, we have 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∑ λ(k, ℜA) − 󵄨󵄨 ≤ (n + 1)μ(n, A). λ ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 λ∈σ(ℜA),|λ|>μ(n,A) 󵄨󵄨 Since A is normal operator, it follows that ℜσ(A) = σ(ℜA). Therefore, 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∑ λ(k, ℜA) − ℜλ󵄨󵄨󵄨 ≤ (n + 1)μ(n, A). ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 λ∈σ(A),|ℜλ|>μ(n,A) 󵄨

(5.7)

We have 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ℜλ − ℜλ󵄨󵄨󵄨 ≤ |ℜλ| ∑ ∑ ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨 λ∈σ(A),|λ|>μ(n,A) λ∈σ(A),|ℜλ|≤μ(n,A)μ(n,A) ≤

∑ λ∈σ(A),|λ|>μ(n,A)

μ(n, A) ≤ (n + 1)μ(n, A).

It follows now from (5.7) that 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∑ λ(k, ℜA) − ℜλ󵄨󵄨󵄨 ≤ 2(n + 1)μ(n, A). ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 λ∈σ(A),|λ|>μ(n,A) 󵄨

(5.8)

󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∑ λ(k, ℑA) − 󵄨󵄨 ≤ 2(n + 1)μ(n, A). ℑλ ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 󵄨󵄨 λ∈σ(A),|λ|>μ(n,A)

(5.9)

Similarly,

Combining (5.8) and (5.9), we obtain 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∑ λ(k, ℜA) + iλ(k, ℑA) − 󵄨󵄨 ≤ 4(n + 1)μ(n, A). λ ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 λ∈σ(A),|λ|>μ(n,A) 󵄨󵄨

5.3 Subharmonic functions on matrix algebras | 151

It follows now from (5.6) that 󵄨󵄨 󵄨󵄨 n 󵄨 󵄨󵄨 󵄨󵄨 ∑ λ(k, A) − λ(k, ℜA) − iλ(k, ℑA)󵄨󵄨󵄨 ≤ 5(n + 1)μ(n, A). 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 Dividing both sides by n + 1 and using the definition of C, we conclude the argument. Proof of Theorem 5.2.1. Let A ∈ [𝒥 , ℒ(H)]. That is, A is a linear combination of commutators, and admits a decomposition n

n

k=1

k=1

A = ∑ [Ak , Bk ] = ∑ [ℜAk , ℜBk ] − [ℑAk , ℑBk ] + i[ℜAk , ℑBk ] + i[ℑAk , ℜBk ] where each Ak ∈ 𝒥 and Bk ∈ ℒ(H). Taking real parts, we obtain n

ℜA = i ∑ [ℜAk , ℑBk ] + [ℑAk , ℜBk ]. k=1

Therefore, ℜA ∈ [𝒥 , ℒ(H)] and, similarly, ℑA ∈ [𝒥 , ℒ(H)]. It follows from Lemma 5.2.6 that Cλ([ℜAk , ℑBk ]), Cλ([ℑAk , ℜBk ]) ∈ J,

1 ≤ k ≤ n.

It follows from Lemma 5.2.8 that Cλ(ℜA) ∈ J. Similarly, Cλ(ℑA) ∈ J. Since μ(A) ∈ J, it follows from Lemma 5.2.9 that Cλ(A) ∈ J. Conversely, let Cλ(A) ∈ J. It follows from Lemma 5.2.9 that Cλ(ℜA) ∈ J. It follows from Lemma 5.2.5 that ℜA ∈ [𝒥 , ℒ(H)]. Similarly, ℑA ∈ [𝒥 , ℒ(H)]. Therefore, A ∈ [𝒥 , ℒ(H)].

5.3 Subharmonic functions on matrix algebras The preceding section proved Theorem 5.1.4 for normal operators in an arbitrary ideal 𝒥 of compact operators. To extend this to arbitrary nonnormal operators, we use Kalton’s approach involving subharmonic functions of matrices. If A is a compact operator then there exist a compact normal operator N and a compact quasinilpotent operator Q such that A=N +Q where A and N have a same eigenvalue sequence, λ(A) = λ(N). This is Ringrose’s decomposition of a compact operator, stated in Theorem 1.1.23 in Chapter 1. From Weyl’s lemma, Lemma 1.1.21, 󵄨 󵄨 󵄨 󵄨 μ(N) = 󵄨󵄨󵄨λ(N)󵄨󵄨󵄨 = 󵄨󵄨󵄨λ(A)󵄨󵄨󵄨 ≺≺log μ(A).

152 | 5 Spectrality of traces If 𝒥 is a logarithmic submajorization closed ideal as in Definition 2.4.17, then A ∈ 𝒥 and μ(N) ≺≺log μ(A) implies that N ∈ 𝒥 . Hence Q = A−N ∈ 𝒥. Proving Theorem 5.1.4 for an arbitrary A ∈ 𝒥 in a logarithmic submajorization closed ideal 𝒥 reduces to showing that Q ∈ [ℒ(H), 𝒥 ] for a quasinilpotent operator Q ∈ 𝒥 . This latter result is technically quite difficult. We follow the technique of Kalton and Dykema, based upon Kalton’s lifting of subharmonic functions to matrix algebras, and the method of the authors’ that extends some of Kalton and Dykema’s geometric estimates to a logarithmic submajorization estimate. Lifting subharmonic functions to matrix algebras We begin by recalling the definition of a subharmonic function. A smooth function u : ℂ → ℝ is called harmonic if Δu = 0 where Δ is the Laplace operator on ℝ2 identified with ℂ. Equivalently, u is a smooth function such that for every z ∈ ℂ and r ∈ ℝ we have 2π

u(z) =

1 ∫ u(z + eiθ r) dθ. 2π 0

A function u : ℂ → ℝ is said to be subharmonic if 2π

u(z) ≤

1 ∫ u(z + eiθ r) dθ 2π 0

for every z ∈ ℂ and every r ∈ ℝ. Equivalently, u ∈ C 2 (ℂ) is subharmonic if and only if Δu ≥ 0 (see p. 41 in [131]). We quantize the notion of a subharmonic function. Definition 5.3.1. A function u : Mn (ℂ) → ℝ is said to be subharmonic if 2π

1 u(A) ≤ ∫ u(A + eiθ B) dθ 2π 0

for every A, B ∈ Mn (ℂ). If equality holds for all A, B ∈ Mn (ℂ) then u is said to be harmonic. Fix n ≥ 1. For a given subharmonic function u : ℂ → ℝ, we define a function û : Mn (ℂ) → ℝ by setting ̂ u(A) = ∑ u(λ). λ∈σ(A)

5.3 Subharmonic functions on matrix algebras | 153

Here, σ(A) means the spectrum of A counted with algebraic multiplicity. Observe that ̂ if A is a normal matrix, then u(A) is well-defined by the functional calculus and u(A) = Tr(u(A)). For a general matrix A with Jordan decomposition A = N + Q where N is ̂ normal and Q is nilpotent, we have u(A) = Tr(u(N)). The main result on lifting subharmonic functions on ℂ to matrix algebras is the following theorem. Theorem 5.3.2. For every subharmonic function u ∈ C 2 (ℂ), the function û : Mn (ℂ) → ℝ is subharmonic in the sense of Definition 5.3.1. The rest of this section proves Theorem 5.3.2. Proving Theorem 5.3.2 for harmonic functions is relatively straightforward, as shown in the next lemma. To prove the same result for subharmonic functions we will use Riesz’ representation of a subharmonic function in terms of a harmonic function and a potential. Lemma 5.3.3. If u is harmonic as a function on ℂ then û is harmonic as a function on Mn (ℂ) in the sense of Definition 5.3.1. In particular, let A, B ∈ Mn (ℂ) and let D = {z ∈ ℂ : |z| < R} with R > ‖A‖ + ‖B‖. For every harmonic function u : D → ℝ, we have 2π

1 ̂ + eiθ B) dθ. ̂ u(A) = ∫ u(A 2π 0

Proof. Let f : D → ℂ be the mapping defined by setting f (z) = z n . Writing m

m

(A + eiθ B) = Am + ∑ eikθ Ck ,

Ck ∈ Mn (ℂ),

k=1

we obtain m

m

Tr((A + eiθ B) ) = Tr(Am ) + ∑ Tr(Ck )eikθ . k=1

Thus, 2π

2π

0

0

1 1 n f ̂(A) = Tr(An ) = ∫ Tr((A + eiθ B) ) dθ = ∫ f ̂(A + eiθ B) dθ. 2π 2π By linearity, the equality 2π

1 f ̂(A) = ∫ f ̂(A + eiθ B) dθ 2π

(5.10)

0

holds for every polynomial f : D → ℂ. Recall that for every analytic function f : D → ℂ, its Taylor polynomials converge to f uniformly on the compact subsets of D. Hence,

154 | 5 Spectrality of traces equality (5.10) holds for an arbitrary analytic function f : D → ℂ. Taking real parts, we obtain equality (5.10) for the real part of an arbitrary analytic function f : D → ℂ. Since every harmonic function u : D → ℂ is the real part of an analytic function, the assertion follows. Before introducing Riesz’ representation of subharmonic functions, we will need a special form of Fubini’s theorem. Theorem 5.3.4. Let u be a measurable function on [0, 1]×[0, 1] such that u+ ∈ L1 ([0, 1]× [0, 1]). It follows that 1 1

1 1

∫ ∫ u(s, t) ds dt = ∫ ∫ u(s, t) dt ds = 0 0

0 0

u(s, t) ds dt.

∫ [0,1]×[0,1]

Proof. Fubini theorem states that the assertion is true when u ∈ L1 ([0, 1] × [0, 1]), and Tonelli’s theorem states that the assertion is true when u is only positive and measurable. We combine these using the assumption that u+ ∈ L1 ([0, 1] × [0, 1]). We have 1 1

1 1

∫ ∫ u+ (s, t) ds dt = ∫ ∫ u+ (s, t) dt ds = 0 0

0 0

u+ (s, t) ds dt < ∞

∫ [0,1]×[0,1]

by Fubini’s theorem. We have 1 1

1 1

∫ ∫ u− (s, t) ds dt = ∫ ∫ u− (s, t) dt ds = 0 0

0 0

u− (s, t) ds dt

∫ [0,1]×[0,1]

by Tonelli’s theorem. Subtracting those equalities, we obtain the assertion. Riesz representation of a subharmonic function The following lemma is a well-known special case of Riesz’ representation theorem, providing a representation of a subharmonic function on a domain in terms of a harmonic function and a finite Borel measure on the closure of the domain. Lemma 5.3.5. For every subharmonic function u ∈ C ∞ (ℂ) and for every bounded domain D ⊂ ℂ, there exists a harmonic function u0 : D → ℂ such that u(z) = u0 (z) +

1 ∫ log(|z − ζ |) ⋅ (Δu)(ζ ) dζ , 2π

z ∈ D.

D

Proof. According to [100, Chapter 2, Theorem 1], the function v(z) =

1 ∫ log(|z − ζ |)(Δu)(ζ ) dζ , 2π D

z ∈ D.

(5.11)

5.3 Subharmonic functions on matrix algebras | 155

(i. e., the second term in (5.11)) solves the differential equation (Δv)(z) = (Δu)(z),

z ∈ D.

In other words, Δ(u − v) = 0 in D and, thus, u0 = u − v is a harmonic function in D. The function z 󳨃→

1 log(|z|), 2π

z ∈ ℤ,

is sometimes called a Riesz potential. Informally speaking, the next lemma proves subharmonicity of the function A → log(|det(A)|) for matrices. We recall that, for a matrix A ∈ Mn (ℂ), we have det(A) = ∏ λ. λ∈σ(A)

Lemma 5.3.6. For all matrices A, B ∈ Mn (ℂ), we have 2π

1 󵄨 󵄨 󵄨 󵄨 log󵄨󵄨󵄨det(A)󵄨󵄨󵄨 ≤ ∫ log(󵄨󵄨󵄨det(A + eiθ B)󵄨󵄨󵄨) dθ. 2π

(5.12)

0

Proof. The roots of the polynomial (−λ)m − 1 are −e2πki/m , 0 ≤ k ≤ m. Hence, for every λ ∈ ℂ and every m ∈ ℕ, we have 󵄨󵄨m−1 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∏ (1 + e2πki/m λ)󵄨󵄨󵄨 = 󵄨󵄨󵄨(−λ)m − 1󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 k=0 󵄨󵄨 If λ ∈ ℂ and |λ| ≠ 1, then the set {1+eiθ λ, θ ∈ (0, 2π)} is separated from 0 and, therefore, the mapping θ → log(|1 + eiθ λ|) is continuous. Hence, for every λ ∈ ℂ with |λ| ≠ 1, we have 2π

2π m−1 󵄨 󵄨 󵄨 󵄨 ∑ log(󵄨󵄨󵄨1 + e2πki/m λ󵄨󵄨󵄨) ∫ log(󵄨󵄨󵄨1 + eiθ λ󵄨󵄨󵄨) dθ = lim m→∞ m k=0 0

󵄨󵄨m−1 󵄨󵄨1/m 󵄨󵄨 󵄨󵄨 2πki/m = 2π lim log(󵄨󵄨󵄨 ∏ (1 + e λ)󵄨󵄨󵄨 ) m→∞ 󵄨󵄨 󵄨󵄨 󵄨 k=0 󵄨 󵄨󵄨 󵄨󵄨1/m m = 2π log( lim 󵄨󵄨(−λ) − 1󵄨󵄨 ) = 2π max{log(|λ|), 0}. m→∞

For every λ with |λ| = 1, we have λ = eiθ0 and, therefore, 2π

2π

2π

0

0

0

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫ log(󵄨󵄨󵄨1 + eiθ λ󵄨󵄨󵄨) dθ = ∫ log(󵄨󵄨󵄨1 + ei(θ+θ0 ) 󵄨󵄨󵄨) dθ = ∫ log(󵄨󵄨󵄨1 + eiθ 󵄨󵄨󵄨) dθ = 0. The latter equality can be found in [1, p. 78].

156 | 5 Spectrality of traces For every C ∈ Mn (ℂ), we have 󵄨 󵄨 log(󵄨󵄨󵄨det(1 + eiθ C)󵄨󵄨󵄨) =

∑ μ∈σ(1+eiθ C)

󵄨 󵄨 log(|μ|) = ∑ log(󵄨󵄨󵄨1 + eiθ λ󵄨󵄨󵄨). λ∈σ(C)

Therefore, 2π

2π

󵄨 󵄨 󵄨 󵄨 ∫ log(󵄨󵄨󵄨det(1 + eiθ C)󵄨󵄨󵄨) dθ = ∑ ∫ log(󵄨󵄨󵄨1 + eiθ λ󵄨󵄨󵄨) dθ ≥ 0. λ∈σ(C) 0

0

(5.13)

If det(A) = 0, then the left-hand side of (5.12) is −∞ and the assertion is trivial. If det(A) ≠ 0, then A is an invertible matrix. Setting C = A−1 B, we have A + eiθ B = A(1 + eiθ C). Therefore, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 log(󵄨󵄨󵄨det(A + eiθ B)󵄨󵄨󵄨) = log(󵄨󵄨󵄨det(A)󵄨󵄨󵄨) + log(󵄨󵄨󵄨det(1 + eiθ C)󵄨󵄨󵄨). The estimate (5.12) follows from the above inequality and (5.13). 1 2π

The next lemma proves that the matrix equivalent of Riesz’ potential z 󳨃→ log(|z|) is subharmonic on a bounded domain.

Lemma 5.3.7. Let u ∈ C ∞ (ℂ) be a subharmonic function. For every bounded domain D ⊂ ℂ, the function 󵄨 󵄨 A → ∫ log(󵄨󵄨󵄨det(A − ζ )󵄨󵄨󵄨) ⋅ (Δu)(ζ ) dζ ,

A ∈ Mn (ℂ),

D

is subharmonic. Proof. For every C ∈ Mn (ℂ), we have |det(C)| ≤ ‖C‖n . For every ζ ∈ D, setting C = A + eiθ B − ζ , we obtain 󵄨 󵄨 󵄩 󵄩 log+ (󵄨󵄨󵄨det(A + eiθ B − ζ )󵄨󵄨󵄨) ≤ n log+ (󵄩󵄩󵄩A + eiθ B − ζ 󵄩󵄩󵄩) ≤ n log+ (2R), where R = max{‖A‖ + ‖B‖, sup |ξ |}. ξ ∈D

Since Δu ≥ 0, it follows that 󵄨 󵄨 log+ (󵄨󵄨󵄨det(A + eiθ B − ζ )󵄨󵄨󵄨) ⋅ (Δu)(ζ ) ≤ n log+ (2R) ⋅ (Δu)(ζ ),

ζ ∈ D, θ ∈ [0, 2π].

Integrating the above inequality over ζ ∈ D and θ ∈ [0, 2π], we obtain ∫ D×[0,2π]

󵄨 󵄨 log+ (󵄨󵄨󵄨det(A + eiθ B − ζ )󵄨󵄨󵄨) ⋅ (Δu)(ζ ) dζ dθ < ∞.

(5.14)

5.3 Subharmonic functions on matrix algebras | 157

Using (5.14) and Theorem 5.3.4, it follows that 2π

󵄨 󵄨 ∫ (∫ log(󵄨󵄨󵄨det(A + eiθ B − ζ )󵄨󵄨󵄨) ⋅ (Δu)(ζ ) dζ ) dθ 0

D

2π

󵄨 󵄨 = ∫( ∫ log(󵄨󵄨󵄨det(A + eiθ B − ζ )󵄨󵄨󵄨) dθ) ⋅ (Δu)(ζ ) dζ D

0

󵄨 󵄨 ≥ 2π ∫ log(󵄨󵄨󵄨det(A − ζ )󵄨󵄨󵄨) ⋅ (Δu)(ζ ) dζ , D

where the last inequality uses Lemma 5.3.6. With a matrix harmonic function from Lemma 5.3.3 and the subharmonic matrix potential in Lemma 5.3.7, we can now combine them using the Riesz representation in Lemma 5.3.5 to prove Theorem 5.3.2 for smooth functions. Lemma 5.3.8. For every subharmonic function u ∈ C ∞ (ℂ), the function û : Mn (ℂ) → ℝ is subharmonic. Proof. Simply writing the definition of det, we obtain ∑ log(|λ − ζ |) =

λ∈σ(A)

󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 log(|μ|) = log(󵄨󵄨󵄨 ∏ μ󵄨󵄨󵄨) = log(󵄨󵄨󵄨det(A − ζ )󵄨󵄨󵄨). 󵄨󵄨 󵄨 μ∈σ(A−ζ ) μ∈σ(A−ζ ) 󵄨 ∑

Fix a domain D = {z ∈ ℂ : |z| < R} with R > ‖A‖ + ‖B‖. Using Lemma 5.3.5, select functions u0 and v such that u = u0 + v, u0 is harmonic in D and v is a potential (the second term in (5.11)). Therefore, ̂ v(A) =

1 ∫ ∑ log(|λ − ζ |) ⋅ (Δu)(ζ ) dζ 2π λ∈σ(A) D

1 󵄨 󵄨 = ∫ log(󵄨󵄨󵄨det(A − ζ )󵄨󵄨󵄨) ⋅ (Δu)(ζ ) dζ . 2π D

It follows from Lemma 5.3.7 that 2π

1 ̂ + eiθ B) dθ ≥ v(A). ̂ ∫ v(A 2π

(5.15)

0

It follows from Lemma 5.3.3 that 2π

1 ∫ û 0 (A + eiθ B) dθ = û 0 (A). 2π 0

Adding (5.15) and (5.16), we arrive at the assertion.

(5.16)

158 | 5 Spectrality of traces The final step in proving Theorem 5.3.2 is to approximate twice differentiable subharmonic functions by smooth subharmonic functions. Lemma 5.3.9. If u : ℂ → ℝ is subharmonic and continuous, then there exists a sequence {un }n≥1 ⊂ C ∞ (ℂ) of subharmonic functions such that un ↓ u pointwise. Proof. Let ϕ ∈ C ∞ (ℂ) be positive, compactly supported, and rotation invariant. Assume also that ‖ϕ‖1 = 1. Let un = u ∗ n2 σ 1 ϕ where σ 1 denotes dilation (σ 1 ϕ)(z) = n n n ϕ(nz), n ≥ 0, z ∈ ℂ. Since u is continuous, it is immediate that un → u pointwise. Since ϕ is positive, it follows that each un is subharmonic. We now aim to show that the sequence of functions {un }n≥1 is monotonically decreasing. The crucial fact we are using (see Theorem 2.12 in [131]) is that the function 2π

gx : r → ∫ u(x + reiθ ) dθ,

r>0

0

is monotone increasing. We now have un (x) = n2 ∫ u(y)ϕ(n(x − y)) dm(y) = n2 ∫ u(x + y)ϕ(ny) dm(y) ℂ

ℂ

∞ 2π

= n2 ∫ ∫ u(x + reiθ )ϕ(nr)r dr dθ 0 0 ∞

∞

0

0

r = n2 ∫ gx (r)ϕ(nr)r dr = ∫ gx ( )ϕ(r)r dr. n Obviously, the right-hand side is decreasing in n. Now, we can prove Theorem 5.3.2. Proof of Theorem 5.3.2. By Lemma 5.3.9, there exists a sequence {un }n≥0 ⊂ C ∞ (ℂ) of subharmonic functions such that un ↓ u pointwise. By Lemma 5.3.8, we have 2π

û n (A) ≤

1 ∫ û n (A + eiθ B) dθ. 2π 0

̂ It is immediate from the definition of û that û n (T) ↓ u(T) for every T ∈ Mn (ℂ). By the monotone convergence theorem, we have 2π

2π

̂ + eiθ B). lim ∫ û n (A + e B) dθ = ∫ u(A

n→∞

0

iθ

0

5.4 Kalton’s estimate for quasinilpotent operators | 159

Thus, 2π

̂ u(A) ≤

1 ̂ + eiθ B) dθ. ∫ u(A 2π 0

5.4 Kalton’s estimate for quasinilpotent operators A quasinilpotent compact operator Q decomposes into real and imaginary parts, Q = ℜQ + iℑQ, where ℜQ and ℑQ are self-adjoint compact operators. This section and the next prove estimates for the Cesaro averages of the eigenvalues of ℜQ and ℑQ. Let 𝒥 be a logarithmic submajorization closed ideal of ℒ(H) with commutative core J. Then ℜQ, ℑQ ∈ 𝒥 , and the estimates will show that Cλ(ℜQ), Cλ(ℑQ) ∈ J. Theorem 5.2.1 for normal compact operators can then be used to show that ℜQ and ℑQ belong to the commutator subspace [ℒ(H), 𝒥 ]. Set log+ (x) := max{log(|x|), 0},

x ∈ ℝ.

Theorem 5.4.1 follows Kalton, and links sums of eigenvalues of ℜQ on a punctured plane with sums of eigenvalues of log+ (2e|Q|). Theorem 5.4.1. For every compact quasinilpotent operator Q ∈ ℒ(H), we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ λ󵄨󵄨󵄨 ≤ 40eTr(log+ (2e|Q|)). 󵄨󵄨 󵄨 λ∈σ(ℜQ) 󵄨 |λ|>1

To prove the statement of Theorem 5.4.1 we first consider nilpotent matrices, and approximate a compact quasinilpotent operator uniformly by nilpotent matrices. Kalton’s estimate for nilpotent matrices Suppose Q is a nilpotent matrix. We prove the matrix version of Theorem 5.4.1. Theorem 5.4.2. Let Q ∈ Mn (ℂ) be a nilpotent matrix. We have ∑ λ∈σ(ℜQ) |λ|>1

λ ≤ 40eTr(log+ (2e|Q|)).

160 | 5 Spectrality of traces Note that T + eiθ T ∗ is normal for every T ∈ Mn (ℂ). Theorem 5.3.2 on lifting subharmonic function to matrices is the central component of proving the next lemma. Lemma 5.4.3. Let Q ∈ Mn (ℂ) be nilpotent. If u ∈ C 2 (ℂ) is such that Δu ≥ 0, then 2π

∫ Tr(u(Q + eiθ Q∗ )) dθ ≥ 0. 0

Proof. By Theorem 5.3.2, we have that û is subharmonic. In particular, we have 2π

2π

0

0

1 1 ̂ + eiθ Q∗ ) dθ = ̂ 0 = u(Q) ≤ ∫ u(Q ∫ Tr(u(Q + eiθ Q∗ )) dθ. 2π 2π Set v(z) := ℜ(z)χ(1,∞) (|z|),

w(z) := log+ (e|z|),

z ∈ ℂ.

First, we need a subharmonic function which approximates −v. Lemma 5.4.4. There exists a subharmonic function u ∈ C 2 (ℂ) such that |u + v| ≤ 6ew. Proof. Define an increasing function ϕ ∈ C 2 (−∞, ∞) by setting 0, t < 0, { { { 5 4 3 ϕ(t) = {6t − 15t + 10t , 0 ≤ t ≤ 1, { { t > 1. {1, Define an increasing convex function ψ ∈ C 2 (−∞, ∞) by setting ψ󸀠󸀠 = 2ϕ󸀠 + |ϕ󸀠󸀠 | and and ψ󸀠 (1) = 23 . In ψ󸀠 (0) = ψ(0) = 0. An elementary computation shows that ψ(1) = 23 8 4 1 particular, we have ψ(t) = 0 for t < 0 and ψ(t) = 23 (t − ) for t > 1. 4 2 Let the function u : ℂ → ℝ be defined by the formula u(z) = eψ(log(|z|)) − ℜz ⋅ ϕ(log(|z|)). Clearly, u ∈ C 2 (ℂ). We claim that Δu ≥ 0. Indeed, we have Δ(ψ(log(|z|))) = (ψ ∘ log)󸀠󸀠 (|z|) + |z|−1 (ψ ∘ log)󸀠 (|z|) = |z|−2 ψ󸀠󸀠 (log(|z|)) and Δ(ℜz ⋅ ϕ(log(|z|))) = ℜz ⋅ Δ(ϕ(log(|z|))) + 2∇(ϕ ∘ log)(|z|) ⋅ ∇ℜz ℜz = ℜz ⋅ |z|−2 ϕ󸀠󸀠 (log(|z|)) + 2(ϕ ∘ log)󸀠 (|z|) |z| ℜz 󸀠 󸀠󸀠 = 2 (2ϕ (log(|z|)) + ϕ (log(|z|))). |z|

5.4 Kalton’s estimate for quasinilpotent operators | 161

In particular, 1 󸀠󸀠 e 󵄨 |ℜz| 󸀠󸀠 󵄨󵄨 ψ (log(|z|)) ≤ ψ (log(|z|)) ≤ 2 ψ󸀠󸀠 (log(|z|)). 󵄨󵄨Δ(ℜz ⋅ ϕ(log(|z|)))󵄨󵄨󵄨 ≤ |z| |z|2 |z| It follows that Δu ≥ 0 and, therefore, the function u is subharmonic. By the construction of u and v, we have |u + v| ≤ 6ew. We can now prove Theorem 5.4.2 for a nilpotent matrix Q. Proof of Theorem 5.4.2. Let A(θ) = Q + eiθ Q∗ ,

B(θ) = ℜ(Q + eiθ Q∗ ).

By Lemma 5.4.4, we have 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Tr(u(A(θ))) + Tr(v(A(θ)))󵄨󵄨󵄨 ≤ 6eTr(log+ (e󵄨󵄨󵄨A(θ)󵄨󵄨󵄨)). We have, μ(A(θ)) ≤ 2σ2 μ(Q), and, therefore, 󵄨 󵄨 Tr(log+ (e󵄨󵄨󵄨A(θ)󵄨󵄨󵄨)) ≤ 2Tr(log+ (2e|Q|)). It is immediate that 󵄨󵄨 󵄨 󵄨󵄨v(z) − v(ℜz)󵄨󵄨󵄨 ≤ χ(1,∞) (|z|) ≤ log+ (e|z|),

z ∈ ℂ.

Hence, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Tr(v(A(θ))) − Tr(v(B(θ)))󵄨󵄨󵄨 ≤ Tr(log+ (e󵄨󵄨󵄨A(θ)󵄨󵄨󵄨)) ≤ 2Tr(log+ (2e|Q|)). It follows that 󵄨󵄨 󵄨 󵄨󵄨Tr(u(A(θ))) + Tr(v(B(θ)))󵄨󵄨󵄨 ≤ 14eTr(log+ (2e|Q|)). Since, 1 1 B(θ) = (1 + cos(θ)) ⋅ ℜQ + sin(θ) ⋅ ℑ(Q), 2 2 by results on matrices in Appendix A.3, Lemmas A.3.3 and A.3.2, we have 󵄨󵄨 󵄨󵄨 1 1 󵄨󵄨 󵄨 󵄨󵄨Tr(v(B(θ))) − (1 + cos(θ)) ⋅ Tr(v(ℜQ)) − sin(θ) ⋅ Tr(v(ℑQ))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 ≤ 2n|B(θ)| (1) + 3n|ℜQ| (1) + 3n|ℑQ| (1).

(5.17)

162 | 5 Spectrality of traces We have μ(B(θ)) ≤ 2σ2 μ(Q), where σ2 is the dilation operator and, therefore, n|B(θ)| (1) ≤ 2n2|Q| (1) ≤ 2Tr(log+ (2e|Q|)). Similarly, μ(ℜQ) ≤ σ2 μ(Q), therefore, n|ℜQ| (1) ≤ 2n|Q| (1) ≤ 2Tr(log+ (2e|Q|)), and μ(ℑQ) ≤ 2σ2 μ(Q), thus, n|ℑQ| (1) ≤ 2n|Q| (1) ≤ 2Tr(log+ (2e|Q|)). Hence, 󵄨󵄨 󵄨󵄨 1 1 󵄨 󵄨󵄨 󵄨󵄨Tr(v(B(θ))) − (1 + cos(θ)) ⋅ Tr(v(ℜQ)) − sin(θ) ⋅ Tr(v(ℑQ))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 ≤ 16Tr(log+ (2e|Q|)) ≤ 6eTr(log+ (2e|Q|)). Substituting this expression into (5.17), we obtain 󵄨󵄨 󵄨󵄨 1 1 󵄨󵄨 󵄨 󵄨󵄨Tr(u(A(θ))) + (1 + cos(θ)) ⋅ Tr(v(ℜQ)) + sin(θ) ⋅ Tr(v(ℑQ))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 ≤ 20eTr(log+ (2e|Q|)). Integrating over θ ∈ (0, 2π) and taking Lemma 5.4.3 into account, we obtain 2π

1 1 1 ∫ ( (1 + cos(θ)) ⋅ Tr(v(ℜQ)) + sin(θ) ⋅ Tr(v(ℑQ))) dθ 2π 2 2 0

≤ 20eTr(log+ (2e|Q|)). Computing the integral, we complete the proof.

5.4 Kalton’s estimate for quasinilpotent operators | 163

Approximating compact quasinilpotent operators by nilpotent matrices We now want to lift Theorem 5.4.2 from nilpotent matrices to quasinilpotent compact operators. Recall every compact operator is a uniform limit of matrices with respect to some selected orthonormal basis of the Hilbert space H. The first component of lifting is showing that a quasinilpotent compact operator is a uniform limit of nilpotent matrices. Lemma 5.4.5. Every compact quasinilpotent operator in ℒ(H) is a uniform limit of nilpotent operators of finite rank. In selected orthonormal bases, these operators are finitedimensional matrices. Proof. Let Q ∈ ℒ(H) be a compact quasinilpotent operator. Using polar decomposition, Q = U|Q| where U is a partial isometry and |Q| is a positive compact operator. We have the uniform convergence 1 Tn = U|Q|E|Q| ( , ∞) → Q. n Let Hn = ker(Tn )⊥ ∩ Tn (H). By the assumption, we have codim(ker(Tn )) < ∞ and dim(Tn (H)) < ∞. Hence, dim(Hn ) < ∞. It is clear that Tn (H) ⊂ Hn and Hn⊥ ⊂ ker(Tn ). It follows that Tn : Hn → Hn and Tn : Hn⊥ → 0. Fix ε > 0 and consider the closed set Fε = {z ∈ ℂ : |z| ≥ ε}. Clearly, σ(Q) ∩ Fε = 0. It follows from [113, 1.2.3 in Chapter 1] that there exists N such that for every n > N we have σ(Tn ) ∩ Fε = 0. Since ε can be taken arbitrarily small, we obtain sup{|λ| : λ ∈ σ(Tn )} → 0.

(5.18)

By the Schur theorem (see p. 64 in [304]), every matrix is upper-triangular with respect to some orthonormal basis. Choose a basis in Hn such that Tn : Hn → Hn is upper triangular. Let Dn : Hn → Hn be the diagonal of Qn and let Qn = Tn − Dn : Hn → Hn . It is clear that Qn : Hn → Hn is a strictly upper-triangular matrix. Hence, Qn : Hn → Hn is nilpotent. Extend Qn : Hn → Hn to the operator Qn : H → H by setting Qn x = 0 for x ∈ Hn⊥ . It is immediate that Qn : H → H is a nilpotent operator represented as a finite-dimensional matrix. The diagonal coefficients of Tn : Hn → Hn are precisely the eigenvalues of Tn . By (5.18), we have that Dn → 0 in the uniform norm. Hence, Qn − Q = (Tn − Q) − Dn → 0,

n → ∞,

in the uniform norm. The second component of lifting Theorem 5.4.2 is that, conditionally, sums of eigenvalues of a compact operator over a punctured plane are also approximated by the same sums for matrices.

164 | 5 Spectrality of traces Lemma 5.4.6. Let A ∈ ℒ(H) be a compact self-adjoint operator such that 1, −1 ∉ σ(A). If An , n ∈ ℕ is a sequence of compact self-adjoint operators such that An → A uniformly, then ∑ λ∈σ(An ),|λ|>1

λ→

∑ λ∈σ(A),|λ|>1

λ.

Proof. Let f (t) = (t − 1)+ ,

t ∈ ℝ.

It is immediate that ∑ λ∈σ(An ),|λ|>1

λ = Tr(f (An )) − Tr(f (−An )) + nAn (1) − n−An (1).

The condition 1 ∉ σ(A) (resp., −1 ∉ σ(A)) guarantees that nAn (1) → nA (1) (resp., that n−An (1) → n−A (1)) as n → ∞. Hence, for all sufficiently large n, we have that nAn (1) = nA (1) and n−An (1) = n−A (1). We now have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩f (An ) − f (A)󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩f (An ) − f (A)󵄩󵄩󵄩∞ ⋅ (rank(f (An )) + rank(f (A))) 󵄩 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩f (An ) − f (A)󵄩󵄩󵄩∞ ⋅ (nAn (1) + nA (1)) = 2nA (1) ⋅ 󵄩󵄩󵄩f (An ) − f (A)󵄩󵄩󵄩∞ . Since An → A in the uniform norm, it follows that f (An ) → f (A) in the uniform norm. The argument above shows that f (An ) → f (A) in ℒ1 . Similarly, f (−An ) → f (−A) in ℒ1 . Combining the results of the preceding paragraphs, we arrive at ∑ λ∈σ(An ),|λ|>1

λ → Tr(f (A)) − Tr(f (−A)) + nA (1) − n−A (1) =

∑ λ∈σ(A),|λ|>1

λ.

We are now in a position to lift Theorem 5.4.2. Proof of Theorem 5.4.1. By Lemma 5.4.5, there exists a sequence of finite-dimensional nilpotent matrices Qn , n ∈ ℕ, such that Qn → Q uniformly. Hence, Q∗n → Q∗ and, therefore, ℜQn → ℜQ. We have |Qn |2 = Q∗n Qn → Q∗ Q = |Q|2 . Since the operation A → A1/2 is continuous on the positive cone of ℒ(H), it follows that |Qn | → |Q| uniformly. If 1, −1 ∉ σ(ℜQ), it follows from Lemma 5.4.6 that ∑ |λ|>1,λ∈σ(ℜQn )

λ→

∑

λ,

n → ∞.

|λ|>1,λ∈σ(ℜQ)

We also have that Tr(log+ (2e|Qn |)) → Tr(log+ (2e|Q|)),

n → ∞.

5.5 Logarithmic submajorization estimates for quasinilpotent operators | 165

By Theorem 5.4.2, we have λ ≤ 40eTr(log+ (2e|Qn |)).

∑

(5.19)

|λ|>1,λ∈σ(ℜQn )

Passing n → ∞, we obtain the inequality λ ≤ 40eTr(log+ (2e|Q|))

∑ |λ|>1,λ∈σ(ℜQ)

in the case 1, −1 ∉ σ(ℜQ). In general, since ℜQ is compact, we have (1, 1 + ε] ∩ σ(ℜQ) = 0 and [−1 − ε, −1) ∩ σ(ℜQ) = 0 for every sufficiently small ε > 0. It follows that λ = (1 + ε)

∑

λ.

∑ |λ|>1,λ∈σ((1+ε)−1 ℜQ)

|λ|>1,λ∈σ(ℜQ)

Applying the preceding paragraph to the operator (1 + ϵ)−1 Q, we infer that λ ≤ 40e(1 + ε)Tr(log+ (

∑ |λ|>1,λ∈σ(ℜQ)

2e |Q|)). 1+ϵ

Passing ϵ → 0, we arrive at ∑

λ ≤ 40eTr(log+ (2e|Q|))

|λ|>1,λ∈σ(ℜQ)

in the general case. Substituting Q → −Q, we obtain the assertion of the theorem.

5.5 Logarithmic submajorization estimates for quasinilpotent operators Let Q be a compact quasinilpotent operator. This section proves that the Cesaro averages of the self-adjoint compact operators ℜQ and ℑQ are logarithmically submajorized by a quadrupling of Q, or, equivalently, the dilation σ4 μ(Q). Theorem 5.5.1. For every compact quasinilpotent operator Q, we have Cλ(ℜQ) ≺≺log 2400Q⊕4 ,

Cλ(ℑQ) ≺≺log 2400Q⊕4 .

The estimate in Theorem 5.5.1, combined with Theorem 5.2.1, will prove the main result that Q ∈ [ℒ(H), 𝒥 ] for any quasinilpotent operator Q belonging to a logarithmically closed ideal 𝒥 . The sequence Cλ(ℜQ) is not quite dominated by the singular values of the operator Q ⊕ Q. To overcome this we introduce the technical device of an operator S : l∞ → l∞

166 | 5 Spectrality of traces that inflates the singular values of the operator Q ⊕ Q enough so that essentially, up to scalar factors, Cλ(ℜQ) ≤ Sμ(Q ⊕ Q). However, S moderates the inflation enough so that essentially, up to scalar factors, Sμ(Q) ≺≺log μ(Q ⊕ Q). The combination of the inflation and moderation estimates is our approach to proving the statement of Theorem 5.5.1 in this section, and is the origin of the quadrupling of Q appearing in the statement. Logarithmic submajorization estimates for the operator S Define a nonlinear homogeneous operator S : l∞ → l∞ by setting (Sx)(k) = μ(k, x)(1 +

∏k μ(m, x) 1 log( m=0 k+1 )), k+1 μ(k, x)

k ≥ 0.

(5.20)

This section proves the moderating estimate for this operator. Theorem 5.5.2. If x = μ(x) ∈ l∞ , then Sx ≺≺log 4(x ⊕ x). The following sequence of technical lemmas are used to prove Theorem 5.5.2. The first lemma shows that Sx is a decreasing rearrangement. Lemma 5.5.3. For every x ∈ l∞ , we have Sx = μ(Sx). Proof. Without loss of generality, we may assume that x = μ(x). We have to prove (Sx)(k + 1) ≤ (Sx)(k), k ≥ 0. First, note that, for every constant C > 0, the function x → x(1 +

k+1 C log( )), k+2 x

x ∈ [0, C]

is increasing. Fixing the values x(0), . . . , x(k), and setting C = (∏km=0 x(m))1/(k+1) , we infer that the function x(k + 1) → (Sx)(k + 1) = x(k + 1)(1 +

k+1 C log( )) k+2 x(k + 1)

is also increasing. So, for given x(0), . . . , x(k), the function x(k + 1) → (Sx)(k + 1) attains its maximal value when x(k + 1) takes its maximal value, which is x(k). Therefore, (Sx)(k + 1) ≤ x(k)(1 +

k

x(m) ∏ k+1 C 1 log( )) = x(k)(1 + log( m=0 k+1 )). k+2 x(k) k+2 x(k)

It is immediate that the right-hand side of the latter inequality does not exceed (Sx)(k). This proves the assertion.

5.5 Logarithmic submajorization estimates for quasinilpotent operators | 167

Lemma 5.5.4. If x = μ(x) ∈ l∞ , then (a) for every n ≥ 0 and for every k ≥ n, we have (Sx)(k) ≤ x(n)(1 +

∏n x(m) 1 log( m=0 n+1 )); k+1 x(n)

(b) for every n ≥ 0 and for every k ≤ n, we have (Sx)(k) ≤ x(k)(1 +

∏n x(m) 1 log( m=0 n+1 )). k+1 x(n)

Proof. Despite the similarity of both estimates, they require essentially different proofs. (a) Fixing the values x(0), . . . , x(k − 1), and arguing as in the proof of Lemma 5.5.3, we infer that the function x(k) → (Sx)(k) is increasing. Hence, (Sx)(k) ≤ x(k − 1)(1 +

∏k−1 x(m) 1 log( m=0 k )). k+1 x(k − 1)

Fixing the values x(0), . . . , x(k − 2), and repeating the argument above, we infer that the function of the variable x(k − 1) ∈ [0, x(k − 2)] standing at the right-hand side of the preceding inequality is increasing. Hence, (Sx)(k) ≤ x(k − 2)(1 +

∏k−2 x(m) 1 log( m=0 k−1 )). k+1 x(k − 2)

Repeating the argument for k − 2, k − 3, . . . , n + 1, we conclude the proof. (b) Since both sides of the inequality are homogeneous with respect to x, we may assume without loss of generality that x(n) = 1. It follows that x(k) ≥ 1, 1 ≤ k ≤ n, and, therefore, ∏km=0 x(m) x(k)k+1

k

n

m=0

m=0

≤ ∏ x(m) ≤ ∏ x(m) =

∏nm=0 x(m) . x(n)n+1

Combining the preceding estimate with the Definition of S given in (5.20) yields the assertion. Lemma 5.5.5. For every u > 0 and for every n ∈ ℕ, we have 2n

∏(1 + k=0

u ) ≤ 22n+u+2 . k+1

Proof. Let u ∈ [m − 1, m] with m ∈ ℕ. We have 2n

∏(1 + k=0

2n ∏2n (m + k + 1) u m ) ≤ ∏(1 + ) = k=02n k+1 k+1 ∏k=0 (k + 1) k=0 (2n + m + 1)! = ≤ 22n+m+1 ≤ 22n+u+2 . m!(2n + 1)!

168 | 5 Spectrality of traces Proof of Theorem 5.5.2. Fix n ≥ 0 and denote s=

∏nm=0 x(m) . x(n)n+1

By Lemma 5.5.4, (a) and (b), we have 2n+1

n

k=0

k=0

2n+1 1 1 log(s)) ⋅ ∏ x(n)(1 + log(s)) k+1 k + 1 k=n+1

∏ (Sx)(k) ≤ ∏ x(k)(1 + 2

n

2n+1

= (∏ x(k)) ⋅ s−1 ∏ (1 + k=0

k=0

1 log(s)) k+1

and 2n

n

k=0

k=0

∏(Sx)(k) ≤ ∏ x(k)(1 + 2

n−1

2n 1 1 log(s)) ⋅ ∏ x(n)(1 + log(s)) k+1 k + 1 k=n+1 2n

= (∏ x(k)) x(n) ⋅ s−1 ∏(1 + k=0

k=0

1 log(s)). k+1

Since s ≥ 1, it follows from Lemma 5.5.5 that 2n+2 1 1 log(s)) ≤ ∏ (1 + log(s)) ≤ 22n+4+log(s) ≤ 42n+2 s k+1 k + 1 k=0

2n+1

∏ (1 + k=0

and 2n

1 log(s)) ≤ 22n+2+log(s) ≤ 42n+1 s. k+1

∏(1 + k=0

Therefore, 2n+1

∏ (Sx)(k) ≤ 4

2n+2

k=0

2

n

2n+1

(∏ x(k)) = 42n+2 ∏ μ(k, x ⊕ x) k=0

k=0

(5.21)

and 2n

2n+1

∏(Sx)(k) ≤ 4

k=0

n−1

2

2n

(∏ x(k)) x(n) = 42n+1 ∏ μ(k, x ⊕ x). k=0

k=0

Since n ≥ 0 is arbitrary, the assertion follows by combining (5.21) and (5.22).

(5.22)

5.5 Logarithmic submajorization estimates for quasinilpotent operators | 169

Domination of Cesaro averages This section proves the inflation estimate for S when applied to a compact quasinilpotent operator Q. Kalton’s estimate in Theorem 5.4.1 is required in the following lemma. Lemma 5.5.6. For every compact quasinilpotent operator Q, we have 󵄨 󵄨󵄨 ⊕2 󵄨󵄨Cλ(ℜQ)󵄨󵄨󵄨 ≤ 40eSμ((2eQ) ). A similar assertion holds for ℑQ. Proof. Since ℜQ is self-adjoint, we infer from Lemma 2.2.9 that, for every n ≥ 0, 1 󵄨󵄨 󵄨 ∗ ⊕2 ⊕2 󵄨󵄨λ(n, ℜQ)󵄨󵄨󵄨 = μ(n, ℜQ) = μ(n, (Q + Q )) ≤ μ(n, Q ) ≤ μ(n, (2eQ) ). 2 Fix n ≥ 0. If μ(n, (2eQ)⊕2 ) = 0, then μ(n, ℜQ) = 0 and, hence, rank(Q) ≤ n. Thus, (Cλ(ℜQ))(n) = =

n 1 ∑ λ(k, ℜ(Q)) n + 1 k=0

1 ∞ 1 Tr(ℜQ) = 0. ∑ λ(k, ℜQ) = n + 1 k=0 n+1

In this case, it is evident that |(Cλ(ℜQ))(n)| ≤ 40e(Sμ((2eQ)⊕2 ))(n). Assume that μ(n, (2eQ)⊕2 ) ≠ 0. Set 󵄨 󵄨 m(n) = max{m ≥ 0 : 󵄨󵄨󵄨λ(m, ℜQ)󵄨󵄨󵄨 > μ(n, 2e(Q⊕2 ))},

n ≥ 0.

By the inequality above, we have m(n) ≤ n. Clearly, m(n)

σ(ℜQ) ∩ {λ : |λ| > μ(n, 2e(Q⊕2 ))} = {λ(k, ℜQ)}k=0 and

󵄨󵄨m(n) 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 n 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ λ(k, ℜQ)󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨λ(k, ℜQ)󵄨󵄨󵄨 + 󵄨󵄨󵄨 ∑ λ(k, ℜQ)󵄨󵄨󵄨. 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 k=0 󵄨󵄨 󵄨k=0 󵄨 k=m(n)+1 Therefore, 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 ∑ λ(k, ℜQ)󵄨󵄨󵄨 ≤ (n + 1)μ(n, (2eQ)⊕2 ) + 󵄨󵄨󵄨 λ󵄨󵄨󵄨. ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 󵄨󵄨 󵄨λ∈σ(ℜQ),|λ|>μ(n,(2eQ)⊕2 ) 󵄨󵄨

(5.23)

In order to estimate the second summand at the right-hand side of (5.23), we define an operator Q0 by setting Q0 =

1 Q⊕2 . μ(n, (2eQ)⊕2 )

170 | 5 Spectrality of traces It is clear that λ(Q⊕2 ) = σ2 λ(Q) = 0 and, therefore, Q0 is a quasinilpotent operator. We write 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ λ(k, ℜQ)󵄨󵄨󵄨 ≤ (n + 1)μ(n, (2eQ)⊕2 ) + 1 μ(n, (2eQ)⊕2 )󵄨󵄨󵄨 λ󵄨󵄨󵄨. ∑ 󵄨󵄨 󵄨 󵄨󵄨 󵄨 2 󵄨󵄨 󵄨λ∈σ(ℜQ0 ),|λ|>1 󵄨󵄨 󵄨󵄨k=0 Applying Theorem 5.4.1, we obtain n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 λ󵄨󵄨󵄨 ≤ 80eTr(log+ (2e|Q0 |)) = 80e ∑ log(μ(k, 2eQ0 )). ∑ 󵄨󵄨 󵄨󵄨 󵄨 k=0 λ∈σ(ℜQ0 ),|λ|>1 󵄨

Therefore, 󵄨󵄨 n 󵄨󵄨 n ⊕2 󵄨󵄨 󵄨 󵄨󵄨 ∑ λ(k, ℜQ)󵄨󵄨󵄨 ≤ (n + 1)μ(n, (2eQ)⊕2 ) + 40eμ(n, (2eQ)⊕2 ) ∑ log( μ(k, (2eQ) ) ). 󵄨󵄨󵄨 󵄨󵄨󵄨 μ(n, (2eQ)⊕2 ) k=0 󵄨k=0 󵄨 Dividing both sides by (n + 1) and appealing to the definition of S given in (5.20) yields the assertion. Proof of Theorem 5.5.1. Let Q be a quasinilpotent compact operator with real part ℜQ and imaginary part ℑQ. We have μ(Cλ(ℜQ)) ≤ 40eSμ((2eQ)⊕2 ) by Lemma 5.5.6. By Lemma 5.5.3 and Theorem 5.5.2, Sμ((2eQ)⊕2 ) ≺≺log 4μ((2eQ)⊕2 ⊕ (2eQ)⊕2 ). Hence μ(Cλ(ℜQ)) ≺≺log 160e ⋅ 2e ⋅ μ(Q⊕2 ⊕ Q⊕2 ) ≤ 2400μ(Q⊕4 ). Similarly for the imaginary part ℑQ.

5.6 Quasi-nilpotent operators are commutators Ringrose’s decomposition of a compact operator on a separable Hilbert space H from Theorem 1.1.23 in Chapter 1 has the following refinement when the compact operator belongs to a logarithmic submajorization closed ideal of the algebra ℒ(H). Theorem 5.6.1 (Ringrose’s decomposition). Let 𝒥 be a logarithmic submajorization closed ideal of ℒ(H) with commutative core J. If A ∈ 𝒥 is a compact operator, then A = N + Q for a compact normal operator N ∈ 𝒥 with λ(A) = λ(N) ∈ J and a quasinilpotent compact operator Q ∈ [ℒ(H), 𝒥 ].

5.6 Quasi-nilpotent operators are commutators |

171

Proof. Let A ∈ 𝒥 . By Theorem 1.1.23, we have A = N + Q, where N is a normal compact operator such that λ(A) = λ(N) and Q is a quasinilpotent operator. By Lemma 1.1.21, we have that N ≺≺log A. Since 𝒥 is closed with respect to logarithmic submajorization, it follows that N ∈ 𝒥 and, therefore, Q ∈ 𝒥 . Since N is normal, then λ(N) ∈ J. Hence λ(A) ∈ J. Since Q ∈ 𝒥 , it follows that ℜQ, ℑQ ∈ 𝒥 . Since Q⊕4 ∈ 𝒥 and since 𝒥 is closed with respect to the logarithmic submajorization, it follows that Cλ(ℜQ) ∈ J and Cλ(ℑQ) ∈ J by Theorem 5.5.1. By Theorem 5.2.1, we have that ℜQ ∈ [𝒥 , ℒ(H)] and ℑQ ∈ [𝒥 , ℒ(H)]. Hence, Q ∈ [𝒥 , ℒ(H)]. With Theorem 5.6.1 we can prove the spectral formulation of traces on logarithmic submajorization closed ideals, Theorem 5.1.2, and the noncommutative version of the Figiel–Kalton theorem, Theorems 5.1.4 and 5.1.5. The next lemma extends Lemma 4.3.2. To translate an eigenvalue sequence formula from symmetric functionals to shift-invariant monotone linear functionals, we need to prove λ(A) − LL󸀠 λ(A) ∈ Z(J). Lemma 5.6.2. Let J be a symmetric sequence space. For every x ∈ J such that |x| = μ(x), we have x − LL󸀠 x ∈ Z(J). Proof. If y is such that |y| ≤ μ(x), then |C(y − LL󸀠 y)| ≤ 2σ2 μ(x). Indeed, if 2n − 1 ≤ k < 2n+1 − 1, then C(y − LL󸀠 y)(k) =

k 1 1 k ∑ y(l) − (LL󸀠 y)(l) = ∑ y(l) − (LL󸀠 y)(l). k + 1 l=0 k + 1 l=2n −1

Thus, k

1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󸀠 ∑ 󵄨󵄨y(l)󵄨󵄨 + 󵄨󵄨(LL󸀠 y)(l)󵄨󵄨󵄨 󵄨󵄨C(y − LL y)(k)󵄨󵄨󵄨 ≤ k + 1 l=2n −1󵄨 󵄨 󵄨 ≤

k k 1 ∑ μ(l, x) + (LL󸀠 μ(x))(l) ≤ 2μ(2n − 1, x) ≤ 2μ(⌊ ⌋, x). k + 1 l=2n −1 2

Let y = ℜx and let z = LL󸀠 y. We have |y| ≤ μ(x) and |z| ≤ LL󸀠 μ(x). By Lemma 4.5.5, we have Cμ(y+ ) − Cy+ ∈ J,

Cμ(y− ) − Cy− ∈ J,

Cμ(z+ ) − Cz+ ∈ J,

Cμ(z− ) − Cz− ∈ J.

By the preceding paragraph, we have C(y+ − y− − z+ + z− ) = C(y − z) = C(y − LL󸀠 y) ∈ J.

172 | 5 Spectrality of traces Combining these inclusions, we obtain C(μ(y+ ) − μ(y− ) − μ(z+ ) + μ(z− )) ∈ J. By Theorem 3.3.7, we obtain μ(y+ ) − μ(y− ) − μ(z+ ) + μ(z− ) ∈ Z(J). Thus, y − LL󸀠 y = y − z = y+ − y− − z+ + z− ∈ Z(J). In other words, ℜx − LL󸀠 (ℜx) ∈ Z(J). Similarly, ℑx − LL󸀠 (ℑx) ∈ Z(J). Proof of Theorem 5.1.2. (a) Suppose 𝒥 is closed with respect to logarithmic submajorization. From Theorem 5.6.1, A = N + Q, where N ∈ 𝒥 is a normal compact operator such that λ(A) = λ(N) and Q is a quasinilpotent operator such that Q ∈ [𝒥 , ℒ(H)]. Since φ vanishes on [𝒥 , ℒ(H)], φ(A) = φ(N) + φ(Q) = φ(N) = φ(λ(N)) = φ(λ(A)). This proves the formula in (i). Denote for brevity x = λ(A) ∈ J. By Lemma 5.6.2, we have x − LL󸀠 x ∈ Z(J). Thus, φ(A) = φ(x) = φ(LL󸀠 x) = (φ ∘ L)(L󸀠 x) = ℓφ (L󸀠 x) = ℓφ (L󸀠 λ(A)). This proves the formula in (ii). (b) Since 𝒥 is not closed with respect to logarithmic submajorization, it follows that there exist x ∈ J and y ∈ l∞ such that y ≺≺log x but y ∉ J. By Theorem 2.3.12, there exists an upper-triangular operator A ∈ ℒ(H) such that μ(A) ≤ μ(x) and such that the diagonal of A is y. Since μ(A) ≤ μ(x), it follows that A ∈ 𝒥 . Also, λ(A) = λ(diag(y)) and, therefore, λ(A) ∉ J. Proposition 2.4.19 indicated the stronger properties of positive traces with respect to logarithmic submajorization. The proposition is a corollary of Theorem 5.1.2, so it is proved here.

5.6 Quasi-nilpotent operators are commutators |

173

Proof of Proposition 2.4.19. Let x = μ(x) and y = μ(y) be such that y ≺≺log x. By Theorem 2.3.12, there exists an upper-triangular operator A ∈ ℒ(H) such that μ(A) ≤ μ(x) and such that diagonal of A is y. If x ∈ J, then the inequality μ(A) ≤ μ(x) implies that A ∈ 𝒥 . By Theorem 5.1.2, we have φ(y) = φ(λ(A)) = φ(A). Recall that φ is hermitian. Taking real parts, we obtain φ(y) = ℜ(φ(A)) = φ(ℜA). Since φ is positive, it follows that φ(y) ≤ φ(|ℜA|). By Lemma 2.2.7, there exist partial isometries U and V such that 󵄨 󵄨 2|ℜA| ≤ U ∗ |A|U + V ∗ 󵄨󵄨󵄨A∗ 󵄨󵄨󵄨V. Therefore, 󵄨 󵄨 2φ(|ℜA|) ≤ φ(UU ∗ |A|) + φ(VV ∗ 󵄨󵄨󵄨A∗ 󵄨󵄨󵄨) ≤ φ(|A|) + φ(|A|∗ ) = 2φ(|A|). Since μ(A) ≤ μ(x), it follows that φ(y) ≤ φ(|A|) = φ(μ(A)) ≤ φ(x). This completes the proof. The next lemma is used to prove equivalence of statement (c) in Theorem 5.1.4 with the other statements of Theorem 5.1.4. Lemma 5.6.3. Let J be a symmetric sequence space. If x, y ∈ J are such that |x| ≤ μ(y) and Cx ∈ J, then there exists z ∈ J such that |Cx| ≤ μ(z). Proof. Let m ≥ k. We have (Cx)(m) =

k m 1 1 ∑ x(l) + ∑ x(l). m + 1 l=0 m + 1 l=k+1

Thus, m 1 󵄨󵄨 󵄨 k + 1 󵄨󵄨 󵄨 󵄨 󵄨 ∑ μ(l, x) ≤ 󵄨󵄨󵄨(Cx)(k)󵄨󵄨󵄨 + μ(k, y). 󵄨󵄨(Cx)(m)󵄨󵄨󵄨 ≤ 󵄨󵄨(Cx)(k)󵄨󵄨󵄨 + m+1 m + 1 l=k+1

174 | 5 Spectrality of traces Set 󵄨 󵄨 z(k) = sup󵄨󵄨󵄨(Cx)(m)󵄨󵄨󵄨. m≥k

It follows that, 0 ≤ z ≤ |Cx| + μ(y). In particular, z ∈ J. Evidently, z = μ(z) and |Cx| ≤ z. Proof of Theorem 5.1.4. From Theorem 5.6.1, A = N + Q, where N ∈ 𝒥 is a normal compact operator such that λ(A) = λ(N) and Q is a quasinilpotent operator such that Q ∈ [𝒥 , ℒ(H)]. Hence, A ∈ [𝒥 , ℒ(H)] if and only if N ∈ [𝒥 , ℒ(H)]. By Theorem 5.2.1, N ∈ [𝒥 , ℒ(H)] if and only if Cλ(N) ∈ I. Since λ(N) = λ(A), the equivalence of the statements (a) and (b) for the case when A is not normal follows. If |Cλ(A)| ≤ μ(z) for some z ∈ J, it is evident that Cλ(A) ∈ J. If Cλ(A) ∈ J, then setting x = λ(A) and y = |λ(A)| in Lemma 5.6.3 proves that |Cλ(A)| ≤ μ(z) for some z ∈ J. Hence the statements (b) and (c) are equivalent. The following lemma extends Lemma 5.2.8 to arbitrary operators in 𝒥 . It is used to prove Theorem 5.1.5 that extends Theorem 5.1.4 to differences of operators and differences of eigenvalue sequences. Lemma 5.6.4. Let 𝒥 be an ideal of ℒ(H) closed with respect to logarithmic submajorization. If A1 , A2 ∈ 𝒥 , then Cλ(A1 + A2 ) ∈ Cλ(A1 ) + Cλ(A2 ) + J where J is the commutative core of 𝒥 . In particular, there exists z ∈ J such that 󵄨󵄨 󵄨 󵄨󵄨Cλ(A1 + A2 ) − Cλ(A1 ) − Cλ(A2 )󵄨󵄨󵄨 ≤ μ(z). Proof. Let A = N + Q with N, Q as in Theorem 5.6.1. We prove the first display. Taking real and imaginary parts, we write ℜN = ℜA − ℜQ,

ℑN = ℑA − ℑQ.

By Lemma 5.2.8, we have Cλ(ℜN) ∈ Cλ(ℜA) − Cλ(ℜQ) + J,

Cλ(ℑN) ∈ Cλ(ℑA) − Cλ(ℑQ) + J.

By Theorem 5.1.4, we have Cλ(ℜQ) ∈ J,

Cλ(ℑQ) ∈ J.

5.7 Notes | 175

Thus, Cλ(ℜN) ∈ Cλ(ℜA) + J,

Cλ(ℑN) ∈ Cλ(ℑA) + J.

By Lemma 5.2.9, we have Cλ(N) ∈ Cλ(ℜN) + iCλ(ℑN) + J. Consequently, Cλ(A) = Cλ(N) ∈ Cλ(ℜN) + iCλ(ℑN) + J = Cλ(ℜA) + iCλ(ℑA) + J. that

For operators A1 , A2 ∈ 𝒥 , we infer from Lemma 5.2.8 and the preceding paragraph C(λ(A1 + A2 )) ∈ C(λ(ℜ(A1 + A2 ))) + iC(λ(ℑ(A1 + A2 ))) + J

⊂ C(λ(ℜA1 )) + C(λ(ℜA2 )) + iC(λ(ℑA1 )) + iC(λ(ℑA2 )) + J ⊂ C(λ(A1 )) + C(λ(A2 )) + J.

Let J be the commutative core of 𝒥 . To prove the second display, set 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 x = λ(A1 + A2 ) − λ(A1 ) − λ(A2 ), y = 󵄨󵄨󵄨λ(A1 + A2 )󵄨󵄨󵄨 + 󵄨󵄨󵄨λ(A1 )󵄨󵄨󵄨 + 󵄨󵄨󵄨λ(A2 )󵄨󵄨󵄨. By the first display, Cx ∈ J. Evidently, |x| ≤ y = μ(y). Since J is closed with respect to logarithmic submajorization, we get y ∈ J by Theorem 5.6.1. The assertion follows from Lemma 5.6.3. Proof of Theorem 5.1.5. It follows from Theorem 5.1.4 that A1 − A2 ∈ [𝒥 , ℒ(H)] if and only if Cλ(A1 − A2 ) ∈ J. By the first display in Lemma 5.6.4, Cλ(A1 − A2 ) ∈ J if and only if C(λ(A1 ) − λ(A2 ))) ∈ J. Hence the statements (a) and (b) are equivalent. Lemma 5.6.4 applied to the operators A1 = A − B ∈ 𝒥 and A2 = B proves the equivalence of statements (b) and (c).

5.7 Notes Commutator subspaces Paul Halmos in 1952 and 1954 [124, 125] observed that every bounded operator on a Hilbert space is the sum of two commutators, and also likely made the first use of a commutator subspace as a result for traces, i. e., the consequence is that ℒ(H) admits no nontrivial traces when H is infinite dimensional. In 1971 Carl Pearcy and David Topping [197] initiated the study of commutator subspaces of Schatten ideals of compact operators. We refer to the survey article of Gary Weiss [287] for additional background. Halmos [125, p. 198] questioned what the spectrum of commutators look like. The modern statement, involving eigenvalues and the Cesàro operator, originated with

176 | 5 Spectrality of traces Nigel Kalton in 1989 [149]. Precursors to this result can be seen in Weiss’ papers [285] and [286]. Kalton proved Theorem 5.1.4 for the ideal of trace class operators ℒ1 . Incidentally, this showed that the commutator subspace [ℒ1 , ℒ∞ ] was smaller than the kernel of the trace Tr. Hence there exist traces (but not continuous traces) other than Tr on ℒ1 (Corollary 3.5.2). Section 5.2 follows Dykema, Figiel, Weiss, and Wodzicki [94]. In [94, Theorem 5.1] the statement of Theorem 5.2.1 was proved for every two-sided ideal of compact operators. The result of [94] combined with Kalton’s observation in [150] on quasinilpotent operators, proves Theorem 5.1.4 for geometrically stable ideals. Many other corollaries of commutator results, and answers to questions of Halmos, Pearcy, and Topping, are shown in [94]. The correspondence between the common kernel of traces [𝒥 , ℒ(H)] and the common kernel of symmetric functionals Z(J) is implicit in the bijective correspondence between traces and symmetric functionals and not usually stated. Pietsch identified a subspace of shift-invariant monotone ideal of l∞ that vanishes for all shiftinvariant linear functionals and proved the bijective identification with the center Z(J) of the commutative core J using the dyadic maps L and L󸀠 of Chapter 4 [207, p. 347]. The refinement of [94] that identifies when the difference of two operators belongs to the commutator subspace in terms of the difference in their eigenvalue sequences originated with [150, Theorem 3.1] and [155, Theorem 3.2], updated with the result of [264]. Theorem 5.1.4 is essentially a combination of results of Kalton [150], Dykema, Figiel, Weiss, Wodzicki [94] and Sukochev and Zanin [264]. Spectral formula Background to Lidskii’s original formula has already been mentioned in Chapter 1. Lidskii in 1959 [164] proved that the trace of every trace class operator is given by the sum of its eigenvalues. Nigel Kalton, using the identification of the commutator subspace for normal operators due to Dykema, Figiel, Weiss, and Wodzicki (which appeared chronologically later, [94]), proved Theorem 5.1.2 in 1998 with the condition of geometric stability [150]. With Dykema, also in 1998, results were extended to the semifinite case [92]. See also [102]. Fack in 2004 noted that Dixmier’s trace specifically was spectral [102]. If J is the commutative core of 𝒥 , geometric stability is the condition that {(∏nk=0 x(k))1/(n+1) } ∈ J for every x ∈ J (cf. Lemma 2.4.18). Sections 5.3 and 5.4 follow Kalton and Dykema. The representation lemma, Lemma 5.3.5, is due to Riesz from 1926 [222, 223]. A proof can be found in [277, Section 3.10]. Lemma 5.3.9 was proved in [131], see Theorem 3.8 on p. 103. Sections 5.5 and 5.6 follow Sukochev and Zanin [264]. The main technical difference between the proof in this volume and the first edition [170] is that Corollary 5.5.8 assuming geometric stability in the first edition is replaced by Theorem 5.5.1 assuming logarithmic submajorization. The formulation in Theorem 5.1.2, with closure under logarithmic submajorization as the condition for an ideal 𝒥 , appeared in 2014 in [264,

5.7 Notes | 177

Theorem 8]. That there is a logarithmic submajorization closed ideal that is not geometrically stable is proved in [264, Theorem 36]. The way we have stated the Lidskii theorem appeared first in 2013 in [155]. That all Banach and quasi-Banach ideals are logarithmic submajorization closed, as stated in Proposition 2.4.18, was proved in [264, Lemma 35] following [150]. An arbitrary ideal 𝒥 has a logarithmic submajorization closure, n

n

j=0

j=0

LE(𝒥 ) := {B ∈ ℒ(H) : ∏ μ(j, B) ≤ ∏ μ(j, A), ∀n ≥ 0, for some A ∈ 𝒥 }. Then LE(𝒥 ) is the smallest logarithmic submajorization closed ideal containing 𝒥 [264, Lemma 33]. Consequently, LE(𝒥 ) is the smallest ideal whose diagonal contains the eigenvalue sequences of 𝒥 , i. e., A ∈ 𝒥 implies diag(λ(A)) ∈ LE(𝒥 ). While a native spectral formulation is denied to traces on an arbitrary ideal 𝒥 , it is natural to ask which traces on 𝒥 extend to traces on LE(𝒥 ) and thereby obtain a spectral formulation in LE(𝒥 ). Theorem 36 in [264] presents a principal ideal 𝒥A which is not logarithmic submajorization closed, but every positive trace on the principal ideal is monotone with respect to logarithmic submajorization. By Theorem 10 in [264], such a positive trace extends to a positive trace on the logarithmic submajorization closure LE(𝒥A ) and is spectral. Example 1.5 of [92] proves that there is a nonpositive trace on 𝒥A that does not extend to the submajorization closure. Identifying ideals with positive traces that do not extend to the logarithmic submajorization closure is still an open problem. Ringrose’s decomposition Theorem 5.6.1 is a refinement of the Ringrose’s decomposition (Theorem 1.1.23) and, to the best of our knowledge, appeared in this form first in the proof of [264, Theorem 7].

|

Part III: Formulas for traces on ℒ1,∞

This part of the book discusses formulas for Dixmier traces and positive traces on the ideal of weak trace class operators ℒ1,∞ . The ideal of weak trace class operators was defined in Section 2.5 using the Calkin correspondence with the space of weakly summable sequences l1,∞ , ℒ1,∞ = {A ∈ ℒ(H) : μ(A) ∈ l1,∞ }.

The formulas for traces are motivated by spectral results from classical differential geometry, including Weyl’s asymptotic formula for the eigenvalues of the Dirichlet Laplacian on a bounded open domain, Mariusz Wodzicki’s noncommutative residue which relates traces on pseudodifferential operators to the zeta function of the Beltrami–Laplace operator on a compact smooth manifold with no boundary, and the first term in the asymptotic expansion of the Minakshisundaram–Pleijel heat kernel. Motivations are provided in the introductions to each chapter of Part III. The motivations bridge the theoretical results of Part II to the application of singular traces in Alain Connes’ noncommutative extension of differential geometry in Volume II. Also discussed in this part is the distinction between Dixmier traces on the quasi-Banach ideal ℒ1,∞ and the dual of the Macaev ideal ℳ1,∞ . Dixmier traces and positive traces Chapter 6 uses the Ringrose decomposition from Chapter 5 to define a Dixmier trace Trω on ℒ1,∞ by the formula Trω (A) = ω(

n 1 ∑ λ(k, A)), log(2 + n) k=0

A ∈ ℒ1,∞ ,

where ω is an extended limit on l∞ applied to the sequence {

n 1 ∈ l∞ , ∑ λ(k, A)} log(2 + n) k=0 n≥0

and λ(A) is an eigenvalue sequence of A. An extended limit is a state on l∞ that extends the usual limit on convergent sequences to all bounded sequences. Part III gives three characterizations of Dixmier traces. A positive trace is fully symmetric if it is monotone for Hardy–Littlewood submajorization. Chapter 6 proves that every fully symmetric trace on ℒ1,∞ is, up to scalar multiplication, a Dixmier trace. This is the first characterization of a Dixmier trace. This characterization also shows that a Dixmier trace defined on ℒ1,∞ has a unique extension to the two-sided ideal ℳ1,∞ , the smallest ideal that contains ℒ1,∞ and is monotone for Hardy–Littlewood submajorization, provided that the extended limit is invariant to the dilation operator. Chapter 6 also proves that the set of normalized positive traces on ℒ1,∞ is in bijective correspondence with the set of Banach limits on l∞ . The set of Dixmier traces is https://doi.org/10.1515/9783110378054-007

182 | Formulas for traces on ℒ1,∞ in bijective correspondence with the smaller set of factorizable Banach limits on l∞ . This is the second characterization of a Dixmier trace. The third characterization, described below, is that the Dixmier trace generalizes the leading term in an asymptotic expansion of a heat trace. Diagonal formulas Chapter 7 investigates diagonal formulas for Dixmier traces and positive traces on ℒ1,∞ . Dixmier traces are spectral by results from Chapter 5, so Trω (A) = ω(

n 1 ∑ λ(k, A)), log(2 + n) k=0

A ∈ ℒ1,∞ ,

where ω is an extended limit on l∞ and λ(A) is an eigenvalue sequence of A ∈ ℒ1,∞ , in the same manner as ∞

Tr(A) = ∑ λ(k, A), k=0

A ∈ ℒ1 .

However, it is shown that there exists no orthonormal basis {en }n≥0 of the separable Hilbert space H such that Trω (A) = ω(

n 1 ∑ ⟨Aek , ek ⟩) log(2 + n) k=0

for every A ∈ ℒ1,∞ , while, given any orthonormal basis {en }n≥0 of the separable Hilbert space H, ∞

Tr(A) = ∑ ⟨Aek , ek ⟩ k=0

for every A ∈ ℒ1 . Dixmier traces are not a “divergent sum” of the diagonal of operators in ℒ1,∞ for any orthonormal basis. In fact, a diagonal formula fails for all traces on an ideal 𝒥 except for a scalar multiple of Tr. Chapter 7 introduces the idea of a modulated operator with respect to a fixed positive operator 0 ≤ V ∈ ℒ1,∞ , and it is proved that a diagonal formula holds on the subspace of modulated operators in the ideal ℒ1,∞ for an eigenbasis of V ordered such that Ven = μ(n, V)en , n ≥ 0. Of particular interest in Volume II will be operators that are modulated by an inverse power of the Laplacian on ℝp or a p-dimensional closed Riemannian manifold. Heat trace and ζ-function formulas Chapter 8 considers formulas for a Dixmier trace as the first term in an asymptotic expansion of a heat trace and a ζ -function residue, as used in noncommutative geometry and mathematical physics. The heat trace formulation involves the behavior of

Formulas for traces on ℒ1,∞

| 183

the function −1

n 󳨃→ Tr(e−(nA) ),

n ≥ 0,

as n → ∞, where 0 ≤ A ∈ ℒ1,∞ . The associated heat trace formula is −1 1 ξω (A) := ω( Tr(e−(nA) )), n

0 ≤ A ∈ ℒ1,∞ ,

where ω is an extended limit on l∞ . The ζ -function residue formulation involves the behavior of the function s 󳨃→ Tr(A1+s ),

s ∈ (0, 1),

as s → 0+ . The associated ζ -function residue formula is ζγ (A) := γ(sTr(A1+s )),

0 ≤ A ∈ ℒ1,∞ ,

where γ is a state on L∞ (0, 1) such that γ(f ) = a for any function f ∈ L∞ (0, 1) with lims→0+ f (s) = a. Chapter 8 studies the conditions on the states γ and ω under which the ζ -residue and heat traces functionals coincide with a Dixmier trace on the positive cone of ℒ1,∞ . Using the characterization in Chapter 6 of Dixmier traces as the set of all fully symmetric functionals, Chapter 8 obtains the third characterization that every Dixmier trace Trω on ℒ1,∞ for an extended limit ω coincides with the heat trace functional ξω∘M , Trω (A) = ξω∘M (A),

0 ≤ A ∈ ℒ1,∞ ,

where ω ∘ M is the extended limit on l∞ obtained by composing ω with the logarithmic mean operator M : l∞ → l∞ , (Mx)(n) =

n x(k) 1 , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

Surprisingly, a similar result between ζ -function residues and Dixmier traces fails. The set of traces arising from ζ -function residues is strictly smaller than the set of Dixmier traces on ℒ1,∞ . Measurability criteria The cardinality of the set of Dixmier traces on ℒ1,∞ is infinite. This follows from the second characterization that Dixmier traces are in bijective correspondence with factorizable Banach limits. This is in stark contrast with the uniqueness of the positive trace Tr on ℒ1 . For an operator A ∈ ℒ1,∞ with oscillatory asymptotic spectral behavior λ(k, A) ∼

(log(k))i , k

k → ∞,

184 | Formulas for traces on ℒ1,∞ the value Trω (A) depends on the extended limit ω on l∞ . For an operator A ∈ ℒ1,∞ , such as an inverse power of the Dirichlet Laplacian on a bounded smooth domain, satisfying asymptotic spectral behavior similar to Weyl’s law λ(k, A) ∼

1 , k

k → ∞,

then it is clear from the formula of the Dixmier trace that Trω (A) = 1 for every extended limit ω on l∞ . Alain Connes introduced the operators that have the same value for every Dixmier trace in the book “Noncommutative Geometry”, calling them measurable operators. From examples of measurable operators, the function s → Tr(A1+s ), s > 0, generally possesses a meromorphic continuation and a simple pole at s = 0, and an −1 expansion of the heat trace n → Tr(e−(nA) ), n ≥ 0, generally possesses a leading term of order n as n → ∞. In Chapter 9, we consider operators that have the same value for every trace on ℒ1,∞ , for every positive trace on ℒ1,∞ , and for every Dixmier trace on ℒ1,∞ , respectively, and what each of these conditions implies in terms of the convergence of spectral formulas, diagonal formulas, heat trace asymptotics and ζ -function residues. Chapter 9 also demonstrates the limited behavior of measurable operators under the continuous functional calculus, since the absolute value of a measurable operator need not be measurable, and Cartesian products.

6 Dixmier traces and positive traces 6.1 Introduction The two-sided ideal ℒ1,∞ of compact operators on a separable Hilbert space H corresponds, through the Calkin correspondence in Section 2.4, to the weak-l1 sequence space, ℒ1,∞ = {A ∈ ℒ(H) : μ(A) ∈ l1,∞ }.

As described in Chapter 3, ℒ1,∞ is the smallest two-sided ideal that admits infinitely many linearly independent traces such that all traces on ℒ1,∞ are singular and vanish on ℒ1 . Spectral features of classical operators from differential geometry are associated to ℒ1,∞ , making it the most important ideal for applications of singular traces, as well as the theory of singular traces. Motivation Weyl’s law describes the asymptotic spectral behavior of the Dirichlet Laplacian Δ on an open bounded domain Ω in ℝp (suppressing some absolute constants), − p2

λ(n, −Δ) ∼ Vol(Ω)

2

⋅ np ,

n → ∞,

where λ describes the discrete eigenvalues of the Laplacian ordered, with multiplicity, as an increasing sequence. The same formula holds for the eigenvalues of the Laplace–Beltrami operator on a p-dimensional compact Riemannian manifold Ω without boundary. An inverse power of the negative of the Laplacian (technically, p in the manifold case, any pseudodifferential parametrix of (1 − Δ) 2 ) has asymptotic behavior proportional to the volume and the harmonic sequence p

λ(n, (1 − Δ)− 2 ) ∼ Vol(Ω) ⋅ n−1 ,

n → ∞,

− p2

where λ(n, (1 − Δ) ), n ≥ 0, are the singular values of the positive compact operator p (1 − Δ)− 2 ∈ ℒ1,∞ on the Hilbert space L2 (Ω). Chapter 1 introduced the Dixmier trace on the Lorentz ideal ℳ1,∞ , and hence by restriction a trace on the contained weak ideal ℒ1,∞ . Defined by Dixmier originally, the Dixmier trace on ℒ1,∞ is the linear extension of the additive functional on the positive cone Trω (A) = ω(

n 1 ∑ λ(k, A)), log(2 + n) k=0

0 ≤ A ∈ ℒ1,∞ ,

where ω is a dilation-invariant extended limit on l∞ . When paired with the “density” p (1 − Δ)− 2 ∈ ℒ1,∞ , the Dixmier trace extracts the volume of sets according to Lebesgue measure (suppressing some dimensional constants), p

Trω ((1 − Δ)− 2 ) = Vol(Ω). https://doi.org/10.1515/9783110378054-008

186 | 6 Dixmier traces and positive traces From a practical perspective, the log-divergence of partial sums in Dixmier’s formula is naturally associated to volumes and to the inherent philosophy of the pseudodifferential calculus that operators of negative order on Ω represent infinitesimals. Alain Connes showed that this philosophy is borne out factually, in that it is indeed true that applying a Dixmier trace on ℒ1,∞ yields (suppressing some absolute constants) [18, 57, 167] p

Trω (A(1 − Δ)− 2 ) ∼ ∫ a(v) dv

(6.1)

S∗ Ω

for any zero order classical pseudodifferential operator A : C ∞ (Ω) → C ∞ (Ω) with principal symbol a, and dv is the Liouville measure on the sphere bundle S∗ Ω. Heuristip cally, A(1 − Δ)− 2 ∈ ℒ1,∞ is the quantization of a(v)dv and the Dixmier trace acts as summation. This is the conceptual foundation for Connes’ operator-based noncommutative integration involving the Dixmier trace on ℒ1,∞ . α The operator (1 − Δ)− 2 is of trace class for α > p and α

Tr(A(1 − Δ)− 2 )

(6.2)

provides the integral of the symbol of A, provided that the underlying manifold is a flat torus. However, if instead Δ is the Dirichlet Laplace operator on a smooth domain α Ω in ℝp then additional lower order boundary terms appear in the trace of (1−Δ)− 2 . On manifolds, additional terms relating to the derivatives of the metric on Ω also result in lower order trace class terms in the expansion of the pseudodifferential operator α (1 − Δ)− 2 and in the trace. In either case, the terms contain additional geometric information. They do not generally vanish under the trace. However, when α = p, the lower order trace class terms do vanish under the Dixmier trace since it is a singular trace, leaving the leading term relating to volume. This feature prompted renewed interest in the study of traces on ℒ1,∞ . Positive traces on ℒ1,∞ A trace φ on ℒ1,∞ is normalized if φ(diag{

1 } ) = 1. n + 1 n≥0

A normalized trace is not necessarily positive. A positive trace φ on ℒ1,∞ is a trace such that φ(A) ≤ φ(B) when 0 ≤ A ≤ B ∈ ℒ1,∞ . Hardy–Littlewood submajorization was defined in Section 2.3 as A ≺≺ B,

n

n

k=0

k=0

0 ≤ A ≤ B ∈ ℒ1,∞ ⇔ ∑ μ(k, A) ≤ ∑ μ(k, B),

n ≥ 0.

6.1 Introduction | 187

A positive trace φ on ℒ1,∞ is fully symmetric if it is monotone for submajorization, φ(A) ≤ φ(B) when 0 ≤ A ≺≺ B ∈ ℒ1,∞ . Section 6.3 proves that Pietsch’s characterization of traces in Theorem 4.1.4(b) and formula (4.2) from Chapter 4 provides a bijective correspondence between Banach limits on l∞ and normalized positive traces on ℒ1,∞ . An extended limit is a state on l∞ that vanishes on the set of sequences that converge to zero. The set of Banach limits, introduced in Section 6.2, is the set of all shift invariant extended limits on l∞ . A factorizable Banach limit is a Banach limit θ : l∞ → ℂ such that θ =γ∘C for an extended limit γ : l∞ → ℂ and the Cesaro operator C : l∞ → l∞ . The set of normalized fully symmetric traces on ℒ1,∞ turns out to be bijective with the set of factorizable Banach limits on l∞ . Theorem 6.1.1 (Traces on ℒ1,∞ ). Let φ be a trace on ℒ1,∞ . (a) φ is a normalized positive trace on ℒ1,∞ if and only if there exists a Banach limit θφ such that n+1

φ(A) = θφ (

1 2 −2 ∑ λ(k, A)), log 2 k=2n −1

A ∈ ℒ1,∞ ,

(6.3)

where λ(A) is any eigenvalue sequence of A. The map φ 󳨃→ θφ is a bijective correspondence between normalized positive traces on ℒ1,∞ and Banach limits on l∞ . (b) φ is a normalized fully symmetric trace on ℒ1,∞ if and only if θφ = γ ∘ C for some extended limit γ on l∞ . The map φ 󳨃→ θφ is a bijective correspondence between normalized fully symmetric traces on ℒ1,∞ and factorizable Banach limits on l∞ . Theorem 6.1.1 is proven in Theorems 6.3.1 and 6.3.2 in Section 6.3. ℕ The cardinality of the set of Banach limits is known, therefore there are 22 linearly independent positive traces on ℒ1,∞ . Raimi studied factorizable Banach limits to determine whether every Banach limit extended Cesaro summability. He gave an example of a Banach limit that was not factorizable. Therefore, there are positive normalized traces on ℒ1,∞ which are not fully symmetric. Dixmier traces on ℒ1,∞ Jacques Dixmier introduced the following formula for a trace on ℒ1,∞ in 1966. As described in the end notes to this chapter, it was the first formula for a singular trace on a two-sided ideal of compact operators.

188 | 6 Dixmier traces and positive traces Theorem 6.1.2 (Dixmier trace). Let ω be an extended limit on l∞ . The mapping Trω :

ℒ1,∞ → ℂ defined by the formula

Trω : A → ω(

n 1 ∑ λ(k, A)), log(n + 2) k=0

A ∈ ℒ1,∞ ,

(6.4)

where λ(A) is any eigenvalue sequence of A, is linear and is, therefore, a trace on ℒ1,∞ . Eigenvalue sequences are unitarily invariant and therefore unitary invariance of the functional Trω is evident from formula (6.4). The complication, which was discussed in Chapter 3 when lifting symmetric functionals, is linearity. Dixmier used estimates of Horn, in the method described in Section 1.3.2 of Chapter 1, to prove additivity of the functional in (6.4) on the positive cone of ℒ1,∞ . The additive formula on the positive cone was then extended by linearity to a trace called the Dixmier trace. The spectral estimates from Chapter 5 allow us to identify this linear extension with a direct definition using eigenvalue sequences for an arbitrary operator in ℒ1,∞ . Another difference between formula (6.4) and Dixmier’s original definition is that the extended limit on l∞ was assumed by Dixmier to be dilation invariant. It is evident from the formula in (6.4) that a Dixmier trace is normalized and monotone with respect to submajorization. Therefore every Dixmier trace is a normalized fully symmetric trace on ℒ1,∞ . The surprising feature of Dixmier’s construction is that the converse assertion is true. Theorem 6.1.3. Let φ be a trace on ℒ1,∞ . The following statements are equivalent: (a) φ is a Dixmier trace on ℒ1,∞ . (b) φ is a normalized fully symmetric trace. Theorems 6.1.3 and 6.1.1(b) provide two complete characterizations of Dixmier’s trace on ℒ1,∞ . The first characterization is the statement of Theorem 6.1.3, that the set of Dixmier traces and the set of normalized fully symmetric traces on ℒ1,∞ are identical. The second characterization is that the set of Dixmier traces on ℒ1,∞ is in bijective correspondence with the set of factorizable Banach limits on l∞ . Hence there are positive normalized traces on ℒ1,∞ which are not Dixmier traces. It is not known whether

the cardinality of the set of Dixmier traces is 22 , as the cardinality of the set of factorizable Banach limits is not known, but the number of linearly independent Dixmier traces is infinite. Theorems 6.1.3 and 6.1.1(b) are proved in Section 6.4 using the machinery established in Chapters 4 and 5. On page 305 of “Noncommutative Geometry” [60], Alain Connes defined the Dixmier trace as ℕ

Trω∘M (A) = (ω ∘ M)(

n 1 ∑ μ(k, A)), log(2 + n) k=0

0 ≤ A ∈ ℒ1,∞ ,

6.1 Introduction | 189

where ω is an extended limit on l∞ and M : l∞ → l∞ is the logarithmic mean (Mx)(n) =

n x(k) 1 , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

Most users of noncommutative geometry still refer to the original presentation of Dixmier’s trace by Connes. Proposition 6.4.1 in this chapter proves that the set of traces defined by Connes’ is bijective with the set of twice factorizable Banach limits of the form θ = γ ∘ C2 for an extended limit γ on l∞ . The set of Connes’ Dixmier traces is a strict subset of the set of Dixmier traces since not every twice factorizable Banach limit is a factorizable Banach limit by Example 6.4.2. Extension of traces from ℒ1,∞ to ℳ1,∞ The Dixmier trace historically was defined on ℳ1,∞ , the dual of the Macaev ideal. The two-sided ideal ℳ1,∞ is the submajorization closure of ℒ1,∞ , meaning ℳ1,∞ = {A ∈ ℒ(H) : A ≺≺ B for some B ∈ ℒ1,∞ }.

Theorem 6.1.4 shows that a normalized fully symmetric trace on ℒ1,∞ extends to a normalized fully symmetric trace on ℳ1,∞ . However, not every positive trace on ℒ1,∞ extends to a positive trace on ℳ1,∞ . Theorem 6.1.4. (a) If φ is a normalized fully symmetric trace on ℒ1,∞ , then φ extends to a normalized fully symmetric trace on ℳ1,∞ . (b) There is a positive trace φ on ℒ1,∞ which does not extend to a positive trace on ℳ1,∞ . Theorem 6.1.4 is proved in Section 6.5. While Dixmier traces extend from ℒ1,∞ to

ℳ1,∞ , care needs to be taken with the formula of the extension.

Dixmier traces on ℳ1,∞ Theorem 6.1.5 proves that every Dixmier trace on ℳ1,∞ admits the original formula due to Jacques Dixmier. Theorem 6.1.5. Let ω be a dilation invariant extended limit on l∞ . The mapping Trω :

ℳ1,∞ → ℂ defined by the formula

Trω : A → ω(

n 1 ∑ λ(k, A)), log(n + 2) k=0

is linear and is, therefore, a trace.

A ∈ ℳ1,∞ ,

(6.5)

190 | 6 Dixmier traces and positive traces Example 6.6.1 shows that the condition that the extended limit ω be dilation invariant cannot be omitted from formula (6.5) when defining a trace on ℳ1,∞ . It is evident that the Dixmier trace on ℳ1,∞ defined in (6.5) is a normalized fully symmetric trace on ℳ1,∞ . In fact, every normalized fully symmetric trace on ℳ1,∞ is described by a Dixmier trace. Theorem 6.1.6. Let φ be a trace on ℳ1,∞ . The following statements are equivalent: (a) φ is a Dixmier trace on ℳ1,∞ . (b) φ is a normalized fully symmetric trace. Theorems 6.1.5 and 6.1.6 are proved in Section 6.5. Every fully symmetric functional on ℳ1,∞ , described by formula (6.5), restricts to a fully symmetric functional on ℒ1,∞ , and every fully symmetric functional on ℒ1,∞ is described by formula (6.4). This highlights the lack of injectivity between extended limits and traces in Dixmier’s construction. Theorems 6.1.3–6.1.6 indicate that the set of Dixmier traces on ℒ1,∞ defined by extended limits is the same as the set defined by dilation-invariant extended limits. The construction still lacks injectivity on ℳ1,∞ . It has been proved that the set of dilation invariant extended limits provides the same set of Dixmier traces on ℳ1,∞ as that of translation and dilation invariant extended limits. The map from extended limits on l∞ to normalized fully symmetric functionals on ℒ1,∞ , ω → Trω , is therefore surjective but not injective, contrasting with the bijective association with factorizable Banach limits in Theorem 6.1.1.

6.2 Extended limits Chapter 1 introduced Dixmier’s construction of a singular trace on compact operators in ℳ1,∞ . Theorem 1.3.1 highlighted that the state ω : l∞ → ℂ used in the construction must be singular, meaning that ω(x) = 0 for all finitely supported sequences x in l∞ . Definition 6.2.1. An extended limit ω on l∞ is a positive linear functional such that ω(1) = 1 and ω(x) = 0 for every x ∈ c0 . The existence of extended limits follows from the Hahn–Banach theorem. Using the convention established below Example 2.2.14, when the sequence x ∈ l∞ is explicitly enumerated and the context is clear we denote the value ω(x) = ω({x(n)}n≥0 ),

x = {x(n)}n≥0 ∈ l∞

by ω(x(n)),

x ∈ l∞ , n ≥ 0.

6.2 Extended limits |

191

∗ Lemma 6.2.2. A state ω ∈ l∞ is an extended limit if and only if

ω(x) = a whenever lim x(n) = a

n→∞

for a convergent sequence x ∈ c. Proof. Suppose ω is a state on l∞ which vanishes on c0 . If x ∈ c then x − lim(x) ⋅ 1 ∈ c0 . Then ω(x) = lim(x) ⋅ ω(1) = lim(x). The converse is obvious, a state for which ω(x) = lim(x) for all x ∈ c has ω(x) = 0 for all x ∈ c0 . The formulas on ζ -function residues in Chapter 8 are more convenient to phrase and prove using the notion of an extended limit on the function space L∞ (0, 1). Definition 6.2.3. An extended limit γ on L∞ (0, 1) supported at 0 is a positive linear functional such that γ(1) = 1 and γ(x) = 0 for every x ∈ C(0, 1) such that lims→0+ x(s) = 0. Recall that l∞ can be isometrically embedded in ℒ(H), then the notation x = x ∗ indicates that x ∈ l∞ is a real-valued sequence. Lemma 6.2.4. Every extended limit ω on l∞ satisfies lim inf x(n) ≤ ω(x) ≤ lim sup x(n), n→∞

n→∞

x = x ∗ ∈ l∞ ,

and, given fixed x = x∗ ∈ l∞ , for every a ∈ (lim infn→∞ x(n), lim supn→∞ x(n)) there exists an extended limit ω on l∞ such that ω(x) = a. The same assertions are valid for extended limits on L∞ (0, 1) supported at 0 where the limit supremum and the limit infimum as n → ∞ are replaced by the limit supremum and the limit infimum as s → 0+ . The extended limits on l∞ can be embedded isometrically into the extended limits on L∞ (0, 1) supported at 0. Example 6.2.5. Define P : l∞ → L∞ (0, 1),

(Px)(t) = ∑ x(n)χ(2−(n+1) ,2−n ) (t), n≥0

x ∈ l∞ , t ∈ (0, 1)

and E : L∞ (0, 1) → l∞ ,

(Ex)(n) = 2

n+1

2−n

∫ x(s) ds,

x ∈ L∞ (0, 1), n ≥ 0.

2−(n+1)

Then E and P are continuous embeddings such that E ∘ P = 1, P ∘ E is a projection, and

192 | 6 Dixmier traces and positive traces (a) ω 󳨃→ ω∘E is an isometric injection from extended limits ω on l∞ to extended limits on L∞ (0, 1) supported at 0. (b) γ 󳨃→ γ ∘ P is a continuous surjection from extended limits γ on L∞ (0, 1) supported at 0 to extended limits on l∞ . To see that ω 󳨃→ ω ∘ E is not surjective, take ∞

x(t) = ∑ χ(2−(n+1) , 3 2−(n+1) ) (t) − χ( 3 2−(n+1) ,2−n ) (t), n=0

2

2

0 < t < 1.

Then Ex = 0 and (ω ∘ E)(x) = 0 for all extended limits ω on l∞ , but there exists an extended limit γ on L∞ (0, 1) supported at 0 such that γ(x) = a, −1 < a < 1, by Lemma 6.2.4. The map γ 󳨃→ γ ∘ P ∘ E is a proper projection of the extended limits on L∞ (0, 1) supported at 0 onto the isometric image of the extended limits on l∞ . The existence of extended limits on l∞ was first proved by S. Banach and S. Mazur. Not all the properties of the classical limit operation can be preserved by extended limits. Mazur noted that there is no extended limit ω which simultaneously obeys the shift invariance property ω(x(n + 1)) = ω(x(n)),

x ∈ l∞ , n ≥ 0,

and the multiplicativity property ω(x(n)y(n)) = ω(x(n))ω(y(n)),

x, y ∈ l∞ , n ≥ 0.

For example, taking x(n) = y(n) = (−1)n , n ≥ 0, then shift-invariance yields ω(x) = ω(y) = 0, but x(n)y(n) = 1, and hence ω(xy) = 1 ≠ ω(x)ω(y). Extended limits ω that are multiplicative in the above sense are the characters, or pure states, of the algebra l∞ that vanish on c0 . We do not consider the direction of pure states that vanish on c0 , but instead concentrate on extended limits obeying shift or additional invariances. The following result provides a useful substitute for multiplicativity. Lemma 6.2.6. Let ω be an extended limit on l∞ and let x ∈ l∞ be such that ω(x) = a and x ≥ a. We have (a) ω(xy) = ω(x)ω(y) for all y ∈ l∞ . (b) ω(1/x) = 1/ω(x), if, in addition, 1/x ∈ l∞ . The same assertions are valid for extended limits on L∞ (0, 1) supported at 0.

6.2 Extended limits |

193

Proof. By the assumptions we have that 󵄨 󵄨󵄨 󵄨󵄨ω((x − a)y)󵄨󵄨󵄨 ≤ ω(|x − a|‖y‖∞ ) = ‖y‖∞ ω(x − a) = 0. This proves the first assertion. The second assertion is a specific case of the first one by setting y = 1/x. The proof of the assertions for extended limits on L∞ (0, ∞) is identical, hence omitted. Banach and Mazur proved the existence of shift-invariant extended limits, originally termed Banach–Mazur limits but now referred to as Banach limits. Chapter 1 introduced dilation invariant extended limits on l∞ to illustrate Dixmier’s construction of a singular trace. We prove that shift invariant and dilation invariant limits exist on l∞ . On l∞ define the shift operator T : l∞ → l∞ acting by the formula T(x(0), x(1), . . . , x(n), . . .) = (0, x(0), x(1), . . . , x(n − 1), . . .),

x ∈ l∞ .

Define the dilation operator σn : l∞ → l∞ , n ∈ ℕ, acting by the formula σn (x(0), x(1), . . .) = (x(0), . . . , x(0), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x(1), . . . , x(1), . . .), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ n times

n times

x ∈ l∞ .

Definition 6.2.7. An extended limit ω on l∞ is shift invariant, or a Banach limit, if ω(x) = (ω ∘ T)(x),

x ∈ l∞ .

An extended limit ω on l∞ is dilation invariant if ω(x) = (ω ∘ σn )(x),

x ∈ l∞ for all n ≥ 0.

The existence of extended limits invariant to actions of commutative semigroups is proved using the following invariant version of the Hahn–Banach theorem. Theorem 6.2.8. Let X be a linear space. Given (a) an action g : x → g(x), g ∈ G, x ∈ X, of a commutative semigroup G on X, (b) a G-invariant subspace Y of X, (c) a convex homogeneous functional p : X → ℝ such that p ∘ g ≤ p for every g ∈ G, and (d) a G-invariant linear functional ω : Y → ℝ such that ω ≤ p, there exists a G-invariant extension ω : X → ℝ such that ω ≤ p. Corollary 6.2.9. Shift-invariant extended limits and dilation-invariant extended limits on l∞ exist. Proof. The shift semigroup generated by T n , n ∈ ℕ, and the dilation semigroup generated by σn , n ∈ ℕ, are actions of the commutative additive semigroup ℕ and the multiplicative semigroup ℕ, respectively, on the space X of real valued sequences in l∞ .

194 | 6 Dixmier traces and positive traces The subspace Y of real valued convergent sequences in c is shift and dilation invariant. The convex functional p : x → lim sup(x), x ∈ X, is also shift and dilation invariant. Define a linear functional ω on Y by setting ω(1) = 1 and ω|Y∩c0 = 0. So-defined ω is shift and dilation invariant. By Theorem 6.2.8, ω admits a shift-invariant extension ω󸀠 on X such that ω󸀠 (x) ≤ lim sup(x), x ∈ X. We have that lim inf(x) = − lim sup(−x) ≤ ω󸀠 (x) ≤ lim sup(x),

x ∈ X.

In particular, ω󸀠 is a shift-invariant state on X that vanishes on real-valued sequences c0 . An identical argument provides that ω has a dilation-invariant extension ω󸀠󸀠 that is a state on l∞ that vanishes on real-valued sequences in c0 . Linearly extending ω󸀠 and ω󸀠󸀠 to act on the real and imaginary parts of sequences in l∞ proves that there exist shift and dilation invariant states on l∞ that vanish on c0 . Note that the semigroup of endomorphisms of l∞ generated by all shifts T n , n ∈ ℕ, and all dilations σn , n ∈ ℕ, is not commutative, and the above proof does not imply the existence of extended limits that are simultaneously shift and dilation invariant. The existence of extended limits that are both shift and dilation invariant is established in Lemma 6.2.12 below. The equivalent of Lemma 6.2.4 for the set of Banach limits was studied by G. Lorentz and L. Sucheston. Definition 6.2.10. A sequence x ∈ l∞ is called almost convergent to the value c ∈ ℂ if θ(x) = c is for all Banach limits θ on l∞ . Theorem 6.2.11. (a) Let Θ denote the set of all Banach limits on l∞ . If x = x ∗ ∈ l∞ , then m+n 1 sup ∑ x(k), n→∞ n + 1 m≥0 k=m

sup θ(x) = lim θ∈Θ

m+n 1 inf ∑ x(k). n→∞ n + 1 m≥0 k=m

inf θ(x) = lim

θ∈Θ

(b) A sequence x ∈ l∞ is almost convergent to the value c ∈ ℂ if and only if 1 m+n ∑ x(k) = c n→∞ n + 1 k=m lim

uniformly in m ≥ 0. There are other operations on l∞ besides shifts and dilations that feature in formulas for Dixmier traces in noncommutative geometry, the most important being extended limits composed with the Cesaro mean operator C : l∞ → l∞ , (Cx)(n) =

n 1 ∑ x(k), n + 1 k=0

x ∈ l∞ , n ≥ 0,

6.2 Extended limits |

195

and the logarithmic mean operator M : l∞ → l∞ , (Mx)(n) =

n x(k) 1 , ∑ log(n + 2) k=0 k + 1

x ∈ l∞ , n ≥ 0.

Lemma 6.2.12. Let ω be an extended limit on l∞ . Then (a) ω ∘ C is an extended limit on l∞ which is shift invariant; (b) ω ∘ M is an extended limit on l∞ which is shift and dilation invariant. Proof. The operators C, M : l∞ → l∞ are linear and map positive sequences to positive sequences. The means C and M are also regular [127, p. 57], meaning that if x(n) → a, n → ∞, then (Cx)(n) → a, n → ∞ and (Mx)(n) → a, n → ∞. Hence ω ∘ C and ω ∘ M are extended limits on l∞ . For every x ∈ l∞ , we have (CTx − Cx)(n) =

n 1 n 1 x(n) , ∑ x(k − 1) − ∑ x(k) = − n + 1 k=1 n + 1 k=0 n+1

n ≥ 0.

Hence CTx − Cx ∈ c0 ,

x ∈ l∞ ,

and ω(CTx − Cx) = 0,

x ∈ l∞ ,

since ω vanishes on c0 . This proves that ω ∘ C is T-invariant. Similarly, for every x ∈ l∞ , (MTx − Mx)(n) =

n n 1 1 x(k − 1) x(k) − ∑ ∑ log(n + 2) k=1 k + 1 log(n + 2) k=0 k + 1

=−

1 x(n) n−1 x(k) ( +∑ ), log(n + 2) n + 1 k=0 (k + 2)(k + 1)

n ≥ 0.

Since 󵄨󵄨 󵄨󵄨 n−1 󵄨󵄨 x(n) n−1 󵄨󵄨 x(k) 1 󵄨󵄨 󵄨󵄨 ≤ ‖x‖∞ ( 1 + ∑ − ) ≤ 2‖x‖∞ , ∑ 󵄨󵄨 n + 1 󵄨 󵄨󵄨 (k + 2)(k + 1) n + 1 (k + 1)(k + 2) 󵄨󵄨 k=0 k=0 󵄨 it follows that MTx − Mx ∈ c0 ,

x ∈ l∞ .

Since ω vanishes on c0 , it follows that ω(MTx − Mx) = 0, This proves that ω ∘ M is T-invariant.

x ∈ l∞ .

196 | 6 Dixmier traces and positive traces Next, let n ≥ m ≥ 1 and denote l = ⌊ mn ⌋. We have (Mσm x)(n) = = =

=

n x(⌊ k ⌋) 1 m ∑ log(n + 2) k=0 k + 1

ml−1 x(⌊ k ⌋) n x(⌊ k ⌋) 1 1 m m + ∑ ∑ log(n + 2) k=0 k + 1 log(n + 2) k=ml k + 1 l−1 1 ∑ x(p) log(n + 2) p=0

mp+m−1

∑ k=mp

n 1 1 x(l) + ∑ k + 1 log(n + 2) k=ml k + 1

l−1 1 1 + ∑ x(p)(− log(n + 2) p=0 p+1 n

+

mp+m−1

∑ k=mp

n 1 x(l) 1 )+ ∑ k+1 log(n + 2) k=ml k + 1

n 1 x(p) 1 x(p) − . ∑ ∑ log(n + 2) p=0 p + 1 log(n + 2) p=l p + 1

Thus, 󵄨󵄨 󵄨 󵄨󵄨(Mσm x)(n) − (Mx)(n)󵄨󵄨󵄨 =

‖x‖∞ l−1 1 + ∑ (− log(n + 2) p=0 p + 1 n

+

≤

mp+m−1

∑ k=mp

1 ) k+1

n ‖x‖∞ ‖x‖∞ 1 1 + ∑ ∑ log(n + 2) k=ml k + 1 log(n + 2) p=l p + 1

‖x‖∞ l−1 m 1 − ) ∑( log(n + 2) p=0 mp + 1 p + 1

‖x‖∞ ‖x‖∞ n+1 n+1 log( )+ log( ) log(n + 2) ml log(n + 2) l ‖x‖∞ 2‖x‖∞ m−1 ≤ + log(3m). ∑ log(n + 2) p≥0 (p + 1)(mp + 1) log(n + 2) +

Hence, Mσm x − Mx ∈ c0 ,

x ∈ l∞ , m ∈ ℕ.

Since ω vanishes on c0 , it follows that ω(Mσm x − Mx) = 0,

x ∈ l∞ , m ∈ ℕ.

This proves that ω ∘ M is dilation invariant.

6.3 Positive traces on ℒ1,∞ This section provides a complete characterization of positive traces on ℒ1,∞ , and a complete characterization of those positive traces on ℒ1,∞ that are monotone for Hardy–Littlewood submajorization.

6.3 Positive traces on ℒ1,∞

| 197

Chapter 4 defined the dyadic dilation operator L : l∞ → l∞ as Lx = (x(0), x(1), x(1), ⏟⏟ x(2), . .⏟,⏟⏟⏟⏟⏟⏟⏟⏟ x(2) . . . , x(n), . . .), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟.⏟⏟⏟ ⏟⏟, . . . , x(n), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 times

2n times

4 times

x ∈ l∞ ,

and the dyadic averaging operator L󸀠 : l∞ → l∞ by 󸀠

(L x)(n) = 2

−n

2n+1 −2

∑ x(k),

k=2n −1

x ∈ l∞ , n ≥ 0.

Theorem 5.1.2(a) in Chapter 5 extends Pietsch’s formula (4.2) for traces from Chapter 4, and states that every trace φ on ℒ1,∞ has the explicit form φ(A) = ℓ(L󸀠 λ(A)),

A ∈ ℒ1,∞ ,

where L󸀠 λ(A) ∈ O is the dyadic average of an eigenvalue sequence of A, O is a shiftinvariant monotone ideal of l∞ , and the linear functional ℓ : O → ℂ satisfies ℓ ∘ T = 2ℓ where T is the right-shift operator T(x(0), x(1), . . .) = (0, x(0), x(1), . . .),

x ∈ l∞ .

There is a simple bijection R : O → l∞ , described below, that allows us to convert, for the particular case of l1,∞ , results on the shift-invariant monotone ideal O from Chapter 4 to statements on l∞ . Inserting R into the formula from Theorem 5.1.2(a), any trace φ on ℒ1,∞ has the explicit form φ(A) = (ℓ ∘ R−1 )(RL󸀠 λ(A)),

A ∈ ℒ1,∞ .

Theorem 6.3.1 below identifies the functionals on l∞ of the form ℓ ∘ R−1 . The bijection R Let z0 (n) =

1 , n+1

n ≥ 0.

Then l1,∞ is the principal ideal in l∞ generated by z0 . Let x0 (n) = 2−n ,

n ≥ 0,

and denote by O the principal ideal in l∞ generated by x0 . Note that (L󸀠 z0 )(n) ∼ 2−n (log(2n+1 − 2) − log(2n − 1)) = log(2) ⋅ x0 (n), and therefore, (Lx0 )(n) ∼

1 ⋅ z (n), log(2) 0

n ≥ 0.

198 | 6 Dixmier traces and positive traces It follows from Theorem 4.1.2 on the Pietsch correspondence between symmetric sequence spaces and shift-invariant monotone ideals that L󸀠 : l1,∞ → O is bijective. The linear bijection R : O → l∞ , (Ry)(n) = 2n y(n),

y ∈ O,

n ≥ 0,

(6.6)

composed with dyadic averaging L󸀠 : l1,∞ → O, provides the bijection RL󸀠 : l1,∞ → l∞ . The composition RL󸀠 applied to an eigenvalue sequence λ(A) ∈ L1,∞ of an operator A ∈ ℒ1,∞ has the form 󸀠

2n+1 −2

RL λ(A) = { ∑ λ(k, A)} k=2n −1

∈ l∞ .

n≥0

In particular, note that A ∈ ℒ1,∞ if and only if 2n+1 −2

RL󸀠 μ(A) = { ∑ μ(k, A)} k=2n −1

∈ l∞ .

n≥0

The operator RL󸀠 : l1,∞ → l∞ has the right inverse LR−1 : l∞ → l1,∞ given by LR−1 x = (x(0),

x(2) x(n) x(1) x(1) x(2) x(n) , , ,..., , . . . , n , . . . , n , . . .), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 2 4 4 2 2 2 times

4 times

2n

x ∈ l∞ .

times

Traces on ℒ1,∞ and shift-invariant functionals on l∞ Recall a trace φ on ℒ1,∞ is normalized if φ(diag(z0 )) = 1, and positive if φ(A) ≤ φ(B) when 0 ≤ A ≤ B ∈ ℒ1,∞ . Theorem 6.3.1. Let L󸀠 : l1,∞ → O, L : O → l1,∞ , R : O → l∞ , R−1 : l∞ → O, and T : l∞ → l∞ be the maps above. (a) If θ is a linear functional on l∞ such that θ = θ ∘ T, then n+1

2 −2 1 θ( ∑ λ(k, A)), φθ (A) = log(2) k=2n −1

defines a trace φθ on ℒ1,∞ .

A ∈ ℒ1,∞ ,

6.3 Positive traces on ℒ1,∞

| 199

(b) If φ is a trace on ℒ1,∞ , then θφ = log(2) ⋅ φ ∘ diag ∘ LR−1 is a linear functional on l∞ such that θφ = θφ ∘ T and n+1

2 −2 1 ⋅ θφ ( ∑ λ(k, A)), φ(A) = log(2) k=2n −1

A ∈ ℒ1,∞ .

(c) The mappings φ → θφ and θ → φθ are mutual inverses, and provide a bijective correspondence between the set of all traces on ℒ1,∞ and the set of all linear functionals on l∞ invariant under T. (d) φ is a positive trace on ℒ1,∞ if and only if θφ is positive on l∞ . (e) φ is a normalized trace on ℒ1,∞ if and only if θφ (1) = 1. (f) φ is a continuous trace on ℒ1,∞ if and only if θφ is continuous on l∞ . (g) φ is a positive normalized trace on ℒ1,∞ if and only if θφ is a Banach limit. Theorem 6.1.1(a) follows from (a)–(c) and (g) of Theorem 6.3.1. Proof of Theorem 6.3.1. By the definition of R, R ∘ T = 2T ∘ R,

1 R−1 ∘ T = T ∘ R−1 . 2

If θ is a linear functional on l∞ , then θ ∘ R is a linear functional on O. If θ = θ ∘ T, then (θ ∘ R) ∘ T = θ ∘ (R ∘ T) = 2θ ∘ (T ∘ R) = 2θ ∘ R. Hence θ∘R is a shift-invariant linear functional on O. Conversely, if ℓ is a shift-invariant linear functional on O, then ℓ ∘ R−1 is a linear functional on l∞ such that 1 (ℓ ∘ R−1 ) ∘ T = ℓ ∘ (R−1 ∘ T) = ℓ ∘ (T ∘ R−1 ) = ℓ ∘ R−1 . 2 Hence, the mappings ℓ 󳨃→

1 ℓ ∘ R−1 , log(2)

θ 󳨃→ log(2)θ ∘ R,

(6.7)

describe a bijection between shift-invariant linear functionals ℓ on O and linear functionals θ on l∞ such that θ = θ ∘ T. (a) Let φ be a trace on ℒ1,∞ and let ℓφ be the shift-invariant linear functional on O given by Theorem 4.1.4. It follows from (6.7) that θφ =

1 ℓ ∘ R−1 log(2) φ

200 | 6 Dixmier traces and positive traces is a linear functional on l∞ such that θφ = θφ ∘ T. The spectral formula follows from Theorem 5.1.2(a). This proves (a). (b) Let θ = θ ∘ T be a linear functional on l∞ . It follows from (6.7) that ℓθ = log(2)θ ∘ R is a shift-invariant linear functional on O. By Theorem 4.1.4, φθ = φℓθ is a trace on ℒ1,∞ . The spectral formula follows from Theorem 5.1.2(a). This proves (b) on the positive cone of ℒ1,∞ . The assertions (c) and (d) follow immediately from Theorem 4.1.4 and (6.7). (e) By Theorem 3.3.2, since z0 = μ(z0 ) ∈ D(l1,∞ ) and LR−1 1 = Lx0 = μ(Lx0 ) ∈ D(l1,∞ ), we have {

1 } − log(2) ⋅ LR−1 1 ∈ Zl1,∞ . n + 1 n≥0

Thus, φ(z0 ) = 1 if and only if θφ (1) = 1. (g) The assertion is a combination of (d) and (e). (f) If φ is continuous, then, by Theorem 4.1.10, it is a linear combination of positive traces. By (d), θφ is a linear combination of positive linear functionals. In particular, θφ is continuous. If θ is continuous, then it is a linear combination of Banach limits. By (g), φ it is a linear combination of positive normalized traces. Every positive trace on ℒ1,∞ is continuous by Lemma 2.4.12. Thus, φ is continuous. Fully symmetric traces on ℒ1,∞ and factorizable Banach limits Hardy–Littlewood submajorization was defined in Section 2.3 as A ≺≺ B,

n

n

k=0

k=0

0 ≤ A ≤ B ∈ ℒ1,∞ ⇔ ∑ μ(k, A) ≤ ∑ μ(k, B),

n ≥ 0.

A positive trace φ on ℒ1,∞ is fully symmetric if it is monotone for submajorization, φ(A) ≤ φ(B) when 0 ≤ A ≺≺ B ∈ ℒ1,∞ . Theorem 6.3.2. Let φ be a positive normalized trace on ℒ1,∞ with corresponding Banach limit θφ from Theorem 6.3.1. The following conditions are equivalent: (a) φ is fully symmetric. (b) θφ is factorizable, that is, θφ = γ ∘ C for some extended limit γ on l∞ . That the statement (b) in Theorem 6.3.2 implies the statement (a) is proved by the next lemma. Lemma 6.3.3. If a Banach limit θ is factorizable, then the trace φθ in Theorem 6.3.1(a) is monotone with respect to submajorization.

6.3 Positive traces on ℒ1,∞

|

201

Proof. Let θ = γ ∘ C for an extended limit γ on l∞ . Note that 2n+1 −2

n+1

l+1

C( ∑ μ(k, A)) = k=2n −1

1 n 2 −2 1 2 −2 ∑ ∑ μ(k, A) = ∑ μ(k, A). n + 1 l=0 l n + 1 k=0 k=2 −1

Thus, 2n+1 −2

n+1

1 2 −2 θ( ∑ μ(k, A)) = γ( ∑ μ(k, A)). n + 1 k=0 k=2n −1 Since γ is positive, it follows that the right hand side is monotone with respect to submajorization. The next lemma will prove that the Banach limit θφ in Theorem 6.3.1(b) is dominated by the sublinear functional on the real-valued sequences in l∞ , lim sup(Cx)(n), n→∞

x = x ∗ ∈ l∞ ,

when φ is fully symmetric. The Hahn–Banach theorem can then be used to show that θφ is factorizable. Lemma 6.3.4. If φ is a fully symmetric normalized trace with Banach limit θφ from Theorem 6.3.1(b), then θφ (y) ≤ θφ (x)

whenever Cy ≤ Cx, x = x ∗ , y = y∗ ∈ l∞ .

Proof. Without loss of generality, x, y ≥ 0. Recall that the operator R was defined in (6.6), and L is the dyadic dilation operator recalled above. Let a = LR−1 x and b = LR−1 y. Then a, b ∈ l1,∞ and are positive. Let 2n − 1 ≤ k < 2n+1 − 1. Fix A ⊂ ℤ+ such that |A| = 2n − 1 and such that 2n −2

∑ μ(l, b) = ∑ b(l).

l=0

l∈A

Obviously, 2n −2

∑ b(l) ≤ ∑ b(l).

l∈A l 0 be a constant such that k

k

l=0

l=0

k

k

l=0

l=0

∑ μ(l, b) ≤ ∑ μ(l, a) + N,

k ≥ 0.

Hence, ∑ μ(l, b) ≤ ∑ μ(l, a) + μ(l, Ne0 ).

Since φ is monotone with respect to the Hardy–Littlewood submajorization, it follows that φ(diag(b)) ≤ φ(diag(a)) + φ(diag(Ne0 )). As Ne0 ⊂ l1 ⊂ Zl1,∞ , it follows that φ(diag(b)) ≤ φ(diag(a)). From the statement of Theorem 6.3.1 (a), θφ = log(2) ⋅ φ ∘ diag ∘ LR−1 . By definition, a = LR−1 x and b = LR−1 y, so it follows that θφ (y) ≤ θφ (x).

6.3 Positive traces on ℒ1,∞

|

203

The next result identifies the restrictions of extended limits on l∞ to linear subsets of real-valued sequences. Lemma 6.3.5. Let X ⊂ l∞ be a linear subset of real-valued bounded sequences. Every linear functional ω : X → ℝ such that ω(z) ≤ lim sup z,

z ∈ X,

is the restriction of an extended limit on l∞ to X. Proof. By the Hahn–Banach theorem, there exists a linear extension of ω to l∞ such that ω(z) ≤ lim sup z,

z = z ∗ ∈ l∞ .

Taking −z instead of z, we obtain ω(z) ≥ lim inf z,

z = z ∗ ∈ l∞ .

Hence, z = z ∗ ∈ l∞ .

lim inf z ≤ ω(z) ≤ lim sup z,

If z ≥ 0, then ω(z) ≥ 0. If z = 1, then lim inf z = lim sup z = 1, hence ω(z) = 1. If z ∈ c0 , then lim inf z = lim sup z = 0, hence ω(z) = 0. Thus, ω is an extended limit. We can now prove that every normalized fully symmetric functional on ℒ1,∞ is associated to a factorizable Banach limit. Proof of Theorem 6.3.2. The implication (b)⇒ (a) is established in Lemma 6.3.3. Let us prove the implication (a)⇒ (b). Let φ be a fully symmetric trace on ℒ1,∞ . By Lemma 6.3.4, θφ (y) ≤ θφ (x) when Cy ≤ Cx and x = x ∗ , y = y∗ ∈ l∞ . In particular, we have Cx ≤ ‖Cx‖∞ = C(‖Cx‖∞ ). Thus, θφ (x) ≤ ‖Cx‖∞ . Since θ vanishes on finitely supported sequences, it follows that 󵄩 󵄩 θ(x) = θ(xχ(n,∞) ) ≤ 󵄩󵄩󵄩C(xχ(n,∞) )󵄩󵄩󵄩∞ . If x ≥ 0, then θ(x) ≤ sup(C(xχ(n,∞) ))(m) ≤ sup(Cx)(m). m≥0

m≥n

Letting n → ∞, we obtain θ(x) ≤ lim sup Cx,

x ≥ 0.

By adding a constant to x, the latter inequality holds for every x = x ∗ ∈ l∞ .

204 | 6 Dixmier traces and positive traces Let X ⊂ l∞ be the range of the real-valued bounded sequences in l∞ acted on by the Cesaro operator C. Define a linear functional ω on X by setting ω(Cy) = θφ (y), y = y∗ ∈ l∞ . By the preceding paragraph, ω is well defined and ω(x) ≤ lim sup x,

x ∈ X.

By Lemma 6.3.5 there exists a linear extension γ of ω to l∞ . By construction, θ = γ ∘ C for all x ∈ l∞ . The next example shows that not all Banach limits are factorizable. Hence the fully symmetric traces form a proper subset of the set of positive traces on ℒ1,∞ . Example 6.3.6. There exist nonfactorizable Banach limits on l∞ . Proof. Let x = ∑ χ[2n ,2n +n] . n≥10

Let rn = ⌈log2 (n) − 1⌉, n ≥ 2, then n

rn 2j+1

rn

k=0

j=0 k=2j

j=0

∑ x(k) ≤ ∑ ∑ x(k) = ∑ (j + 1) =

(rn + 1)(rn + 2) , 2

while 2n +n

∑ x(k) = n + 1.

k=2n

Setting m = 0 and m = 2n in Theorem 6.2.11 yields n+m n rn2 1 1 inf ∑ x(k) ≤ lim sup =0 ∑ x(k) = lim sup n→∞ n + 1 m≥0 n→∞ n + 1 n→∞ 2(n + 1) k=m k=0

lim

and n

n+m 1 1 2 +n n+1 sup ∑ x(k) ≥ lim inf = 1. ∑ x(k) = lim inf n→∞ n + 1 m≥0 n→∞ n + 1 n→∞ n + 1 n k=m k=2

lim

By Theorem 6.2.11, x is not almost convergent, and there exists a Banach limit θ such that θ(x) > 0. On the other hand, Cx ∈ c0 . Hence, ω(Cx) = 0 for every extended limit ω and every factorizable Banach limit vanishes on x.

6.4 Dixmier traces on ℒ1,∞

|

205

6.4 Dixmier traces on ℒ1,∞ This section proves that Dixmier’s trace Trω is linear, where Trω (A) = ω(

n 1 ∑ λ(k, A)), log(n + 2) k=0

A ∈ ℒ1,∞ .

Here λ(A) is any eigenvalue sequence of A and ω is any extended limit on l∞ . With the machinery of Chapter 5, the proof is a few lines. Proof of Theorem 6.1.2. Let A1 , A2 ∈ ℒ1,∞ . By Lemma 5.6.4, there exists z ∈ l1,∞ such that 󵄨 󵄨󵄨 n n 󵄨󵄨 1 󵄨󵄨󵄨󵄨 n 󵄨󵄨 ∑ λ(k, A1 ) + ∑ λ(k, A2 ) − ∑ λ(k, A1 + A2 )󵄨󵄨󵄨 ≤ μ(n, z), 󵄨󵄨 n + 1 󵄨󵄨󵄨k=0 k=0 k=0 󵄨

n ≥ 0.

Since μ(n, z) ≤

‖z‖1,∞ , n+1

n ≥ 0,

it follows that 󵄨󵄨 n 󵄨󵄨 n n 󵄨󵄨 󵄨 󵄨󵄨 ∑ λ(k, A1 ) + ∑ λ(k, A2 ) − ∑ λ(k, A1 + A2 )󵄨󵄨󵄨 ≤ ‖z‖1,∞ , 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 󵄨󵄨 k=0 k=0

n ≥ 0.

Thus, n n n 1 1 1 ∑ λ(k, A1 ) + ∑ λ(k, A2 ) − ∑ λ(k, A1 + A2 ) ∈ c0 . log(n + 2) k=0 log(n + 2) k=0 log(n + 2) k=0

Since ω is linear and vanishes on c0 , it follows that Trω (A1 ) + Trω (A2 ) − Trω (A1 + A2 ) = ω(

n 1 ∑ λ(k, A1 ) + λ(k, A2 ) − λ(k, A1 + A2 )) = 0. log(n + 2) k=0

Hence, Trω is linear on ℒ1,∞ . It is evident that, if n

n

k=0

k=0

∑ μ(k, A) ≤ ∑ μ(k, B),

n ≥ 0,

0 ≤ A ≤ B ∈ ℒ1,∞ ,

then Trω (A) ≤ Trω (B). Hence Dixmier traces are normalized fully symmetric traces. The following proof demonstrates the converse. With the identification of fully symmetric functionals with factorizable Banach limits in Theorem 6.1.3, the proof is also a few lines.

206 | 6 Dixmier traces and positive traces Proof of Theorem 6.1.3. Suppose φ is a normalized fully symmetric trace on ℒ1,∞ . By Theorem 6.3.2, the corresponding Banach limit θ = θφ in Theorem 6.3.1(b) is factorizable, that is, θ = ω ∘ C for some extended limit ω. As in Lemma 6.3.3, for 0 ≤ A ∈ ℒ1,∞ , we write 2n+1 −2

n+1

l+1

C( ∑ μ(k, A)) = k=2n −1

1 n 2 −2 1 2 −2 ∑ ∑ μ(k, A) = ∑ μ(k, A). n + 1 l=0 l n + 1 k=0 k=2 −1

Thus, by Theorem 6.3.1, n+1

φ(A) =

n+1

2 −2 1 1 1 2 −2 θφ ( ∑ μ(k, A)) = ω( ∑ μ(k, A)) log(2) log(2) n + 1 k=0 k=2n −1

= ω(

1

2n+1 −2

log((2n+1 − 2) + 2)

∑ μ(k, A)).

k=0

Define a new extended limit ω0 by the formula, ω0 (x) = ω(x(2n+1 − 2)),

x ∈ l∞ .

This is an extended limit, since ω0 is positive and lim x(2n+1 − 2) = 0,

n→∞

x ∈ c0 .

Using formula (6.4), we have φ(A) = Trω0 (A),

0 ≤ A ∈ ℒ1,∞ .

That φ = Trω0 on all of ℒ1,∞ follows by linearity of the trace φ and the linearity already established for Trω0 in Theorem 6.1.2. Alain Connes defined the Dixmier trace by Trω∘M (A) = (ω ∘ M)(

n 1 ∑ μ(k, A)), log(2 + n) k=0

0 ≤ A ∈ ℒ1,∞ ,

where ω is an extended limit on l∞ and M : l∞ → l∞ is the logarithmic mean operator (Mx)(n) =

n 1 x(k) , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

The logarithmic means are regular, meaning that if x(n) → a for some a ∈ ℂ, then (Mx)(n) → a. Hence, if ω is an extended limit, then ω ∘ M is also an extended limit as described in Lemma 6.2.12, and Connes’ set of Dixmier traces on ℒ1,∞ are a subset of Dixmier traces. The next proposition proves that, in the scheme of Section 6.3

6.4 Dixmier traces on ℒ1,∞

|

207

where Dixmier traces can be identified with factorizable Banach limits, Connes’ trace is bijective with twice factorizable Banach limits on l∞ of the form θ = γ ∘ C2 , where γ is an extended limit. Proposition 6.4.1. Dixmier traces on ℒ1,∞ of the form Trω∘M for an extended limit ω on l∞ are in bijective correspondence with twice factorizable Banach limits of the form θ = γ ∘ C2 for an extended limit γ on l∞ . Proof. Let 0 ≤ A ∈ ℒ1,∞ . Denote 2m+1 −2

z(m) = ∑ μ(k, A), k=2m −1

m ≥ 0.

Then z ∈ l∞ . Let 2m − 1 ≤ n < 2m+1 − 1. Since m → ∞ iff n → ∞, we have n 1 1 (Cz)(m) + o(1), ∑ μ(k, A) = log(2 + n) k=0 log(2)

n → ∞.

Thus, M(

n 1 1 (C 2 z)(m) + o(1), ∑ μ(k, A)) = log(2 + n) k=0 log(2)

n → ∞.

We therefore have m+1

Trω∘M (A) =

2 −2 1 (ω ∘ C 2 )( ∑ μ(k, A)), log(2) k=2m −1

0 ≤ A ∈ ℒ1,∞ .

That is to say, Trω∘M (A) = φω∘C2 (A) for all 0 ≤ A ∈ ℒ1,∞ where φω∘C2 is the trace from Theorem 6.3.1(a) corresponding to the twice factorizable Banach limit ω ∘ C 2 . By linearity, it follows that Trω∘M (A) = φω∘C2 (A),

A ∈ ℒ1,∞ .

The next example shows that Connes’ Dixmier traces are a proper subset of Dixmier traces on ℒ1,∞ . Example 6.4.2. Not every factorizable Banach limit is twice factorizable.

208 | 6 Dixmier traces and positive traces Proof. The proof is similar to that of Theorem 8.5.2, and so we only sketch the details. Set x = χ∪n≥0 [22n ,22n+1 ] . An elementary but lengthy calculation shows that 2 lim sup(Cx)(n) = , 3 n→∞

2

23 lim sup(C x)(n) = . 3 n→∞ 2

Let B1 (resp., B2 ) denote the set of factorizable (resp., twice factorizable) Banach limits. We have sup Bx =

B∈B1

2 , 3

2

sup =

B∈B2

23 . 3

It is now immediate that B1 ≠ B2 .

6.5 Extension of traces to ℳ1,∞ Recall that the ideal ℳ1,∞ can be described as the set of compact operators A such that there exists B ∈ ℒ1,∞ with A ≺≺ B. Let φ be a fully symmetric trace on ℒ1,∞ . Set φext (A) = inf{φ(B) : 0 ≤ B ∈ ℒ1,∞ , A ≺≺ B},

0 ≤ A ∈ ℳ1,∞ .

The functional φext is an extension of φ from the positive cone of ℒ1,∞ to the positive cone of ℳ1,∞ . The following lemmas prove that φext is additive on the positive cone of ℳ1,∞ . If A1 , A2 ∈ ℳ1,∞ , recall from Definition 2.2.11 that the direct sum A1 ⊕ A2 is defined to be A1 ⊕ A2 = B1 + B2 ∈ ℳ1,∞ , where B1 = pB1 p,

B2 = qB2 q,

and p, q ∈ ℒ(H) are projections such that pq = qp = 0 and μ(B1 ) = μ(A1 ),

μ(B2 ) = μ(A2 ).

Lemma 6.5.1. Let 0 ≤ A1 , A2 ∈ ℳ1,∞ . We have φext (A1 ⊕ A2 ) ≤ φext (A1 + A2 ) ≤ φext (diag(μ(A1 ) + μ(A2 ))).

6.5 Extension of traces to ℳ1,∞

|

209

Proof. By Theorems 2.3.7 and 2.3.5, we have A1 ⊕ A2 ≺≺ A1 + A2 ≺≺ μ(A1 ) + μ(A2 ). Thus, {0 ≤ B ∈ ℒ1,∞ : μ(A1 ) + μ(A2 ) ≺≺ B} ⊂ {0 ≤ B ∈ ℒ1,∞ : A1 + A2 ≺≺ B}

⊂ {0 ≤ B ∈ ℒ1,∞ : A1 ⊕ A2 ≺≺ B}.

Since the infimum over a smaller set is larger than the infimum over a larger set, the assertion follows. Lemma 6.5.2. Let 0 ≤ A1 , A2 ∈ ℳ1,∞ . We have φext (A1 ) + φext (A2 ) ≤ φext (A1 ⊕ A2 ). Proof. If A1 ⊕ A2 ≺≺ B, then μ(A1 ) ⊕ μ(A2 ) ≺≺ μ(B). By Lorentz–Shimogaki’s theorem [175, Theorem 1], there exist sequences 0 ≤ b1 , b2 ∈ l∞ such that μ(B) = b1 + b2 and μ(A1 ) ≺≺ b1 , μ(A2 ) ≺≺ b2 . It follows that φext (A1 ⊕ A2 ) ≥ inf{φ(b1 + b2 ) : 0 ≤ b1 , b2 ∈ l1,∞ , A1 ≺≺ b1 , A2 ≺≺ b2 } = φext (A1 ) + φext (A2 ).

Lemma 6.5.3. Let 0 ≤ A1 , A2 ∈ ℳ1,∞ . We have φext (diag(μ(A1 ) + μ(A2 ))) ≤ φext (A1 ) + φext (A2 ). Proof. We have φext (A1 ) + φext (A2 )

= inf{φ(b1 ) : b1 = μ(b1 ) ∈ l1,∞ , μ(A1 ) ≺≺ b1 }

+ inf{φ(b2 ) : b2 = μ(b2 ) ∈ l1,∞ , μ(A2 ) ≺≺ b2 }

= inf{φ(b) : b = b1 + b2 , bk = μ(bk ) ∈ l1,∞ , μ(Ak ) ≺≺ bk , k = 1, 2}. Since {b = b1 + b2 , bk = μ(bk ) ∈ l1,∞ , μ(Ak ) ≺≺ bk , k = 1, 2} ⊂ {0 ≤ b ∈ l1,∞ : μ(A1 ) + μ(A2 ) ≺≺ b}

and since the infimum over a smaller set is larger than the infimum over a larger set, the assertion follows.

210 | 6 Dixmier traces and positive traces We can now prove that φext defines a fully symmetric trace on ℳ1,∞ that extends the fully symmetric trace φ on ℒ1,∞ . Proof of Theorem 6.1.4 (a). Let φ be a fully symmetric trace on ℒ1,∞ . By Lemmas 6.5.1, 6.5.2, and 6.5.3, we have φext (A1 + A2 ) = φext (A1 ) + φext (A2 ),

0 ≤ A1 , A2 ∈ ℳ1,∞ .

Thus, φext is additive on the positive cone of ℳ1,∞ . By construction, φext (A) = φext (μ(A)),

0 ≤ A ∈ ℳ1,∞ .

Hence φext is unitarily invariant and positive homogeneous on the positive cone of ℳ1,∞ . If A ≺≺ B,

0 ≤ A, B ∈ ℳ1,∞

then, by construction, φext (A) ≤ φext (B). Thus, the linear extension of φext is a unitarily invariant linear functional on ℳ1,∞ that is monotone with respect to submajorization. If φ is a fully symmetric trace on ℒ1,∞ , then φext is an extension to a fully symmetric trace on ℳ1,∞ . The following two lemmas show that a positive trace on ℒ1,∞ generally does not have an extension to a positive trace on ℳ1,∞ . Recall from Section 2.5 that (ℳ1,∞ )0 denotes the closure with respect to the norm ‖B‖ℳ1,∞ = sup n≥0

n 1 ∑ μ(k, B), log(n + 2) k=0

B ∈ ℳ1,∞ ,

of the finite-rank operators in ℒ(H). For B ∈ (ℳ1,∞ )0 , it is necessary and sufficient that lim sup n→∞

n 1 ∑ μ(k, B) = 0. log(n + 2) k=0

Lemma 6.5.4. Let A be a positive compact operator on ℒ(H) such that μ(k, A) =

1 , { k+1 −(n+1)3

2

Then A ∈ ℒ1,∞ and A ∈ (ℳ1,∞ )0 .

3

3

k ∈ [2n , 2n ,

3

k ∈ [2

n +n

+n

),

(n+1)3

,2

),

n ≥ 0.

(6.8)

6.5 Extension of traces to ℳ1,∞

| 211

Proof. The first assertion is immediate since μ(k, A) ≤

K , k+1

k ≥ 0,

where 2

K = sup 21+2n−3n = 2. n≥0

We prove the second assertion. We have, 3

3

2(n+1) −1

n 2(l+1) −1

∑ μ(k, A) = μ(0, A) + ∑ ∑ μ(k, A)

k=0

l=0 k=2l3 n 2l

3 +l

−1

3

n 2(l+1) −1

= μ(0, A) + ∑ ∑ μ(k, A) + ∑ ∑ μ(k, A) l=0 k=2l3 l3 +l

l=0 k=2l3 +l (l+1)3

n 2 −1 3 1 + ∑ ∑ 2−(l+1) = μ(0, A) + ∑ ∑ k + 1 l=0 l=0 l3 +l l3 n 2

−1

k=2

k=2 l3 +l

n 2

≤ μ(0, A) + ∑ ∫ l=0

3

2l

3

n n(n + 1) dt + ∑ 1 = μ(0, A) + log(2) + n + 1. t l=0 2

3

If m ∈ [2n , 2(n+1) ), then 3

m

2(n+1) −1

k=0

k=0

∑ μ(k, A) ≤

2

∑ μ(k, A) = O(n2 ) = O(log 3 (m)).

Thus, m

∑ μ(k, A) = o(log(m)),

k=0

m→∞

and A ∈ (ℳ1,∞ )0 since lim sup n→∞

n 1 ∑ μ(k, A) = 0. log(n + 2) k=0

Since A ∈ (ℳ1,∞ )0 , by Lemma 2.5.3 we have that φ(A) = 0 for every positive trace on ℳ1,∞ . The next lemma shows that there is a positive trace on ℒ1,∞ that does not extend to any positive trace on ℳ1,∞ .

212 | 6 Dixmier traces and positive traces Lemma 6.5.5. There exists a positive trace φ on ℒ1,∞ such that φ(A) > 0, where the operator A is defined by (6.8). Proof. Let x = χ∪n≥0 [n3 ,n3 +n) . Let rn = ⌈n1/3 − 1⌉, n ≥ 1, then rn (j+1)3

n

rn

∑ x(k) ≤ ∑ ∑ x(k) = ∑ j = j=0 k=j3

k=0

j=0

rn (rn + 1) ≤ rn2 , 2

while n3 +n

∑ x(k) = n.

k=n3

Setting m = 0 and m = n3 in Theorem 6.2.11 yields n+m n r2 1 1 inf ∑ x(k) ≤ lim sup ∑ x(k) ≤ lim sup n = 0 n→∞ n + 1 m≥0 n→∞ n + 1 n→∞ n + 1 k=m k=0

lim

and 3

n+m 1 1 n +n n sup ∑ x(k) ≥ lim inf = 1. ∑ x(k) = lim inf n→∞ n + 1 m≥0 n→∞ n + 1 n→∞ n + 1 k=m k=n3

lim

By Theorem 6.2.11, we can choose a Banach limit θ such that θ(x) > 0. If m ∈ (n3 , n3 + n), then 2m+1 −2

2m+1 −2

1 = log(2) + o(1). k + 1 k=2m −1

∑ μ(k, A) = ∑

k=2m −1

Let φ = φθ be the positive trace on ℒ1,∞ from Theorem 6.3.1(a). Then m+1

2 −2 1 1 θ( ∑ μ(k, A)) ≥ θ(x ⋅ log(2)) = θ(x). φ(A) = log(2) k=2m −1 log(2)

Hence φ(A) ≥ θ(x) > 0. Proof of Theorem 6.1.4 (b). Consider the operator A as in (6.8). By Lemma 6.5.4, we have A ∈ ℒ1,∞ and A ∈ (ℳ1,∞ )0 . Let φ be the positive trace constructed in Lemma 6.5.5. We claim that φ does not extend to a positive trace on ℳ1,∞ . Assume the contrary, so that φ is a positive (and, therefore, continuous) trace on ℳ1,∞ . Thus, by Lemma 2.5.3, φ vanishes on (ℳ1,∞ )0 . In particular, φ(A) = 0, which contradicts Lemma 6.5.5.

6.6 Dixmier traces on ℳ1,∞

| 213

6.6 Dixmier traces on ℳ1,∞ This section proves that Dixmier’s trace Trω on ℳ1,∞ is linear, where Trω (A) = ω(

n 1 ∑ λ(k, A)), log(n + 2) k=0

A ∈ ℳ1,∞ ,

λ(A) is any eigenvalue sequence of A, and ω is any dilation-invariant extended limit on l∞ . This section also proves that all normalized fully symmetric functionals on ℳ1,∞ are described by Dixmier traces on ℳ1,∞ . The extension of a normalized fully symmetric trace discussed in Section 6.5 therefore has the form φext (A) = Trω (A),

A ∈ ℳ1,∞

for a dilation-invariant extended limit ω on l∞ . Hence, by restriction to ℒ1,∞ , for each extended limit ω0 on l∞ there exists some dilation-invariant extended limit ω on l∞ such that Trω0 (A) = Trω (A),

A ∈ ℒ1,∞ ,

and the association between extended limits and Dixmier traces on ℒ1,∞ is not injective. The condition that the extended limit be dilation invariant in the formula for a Dixmier trace on ℳ1,∞ cannot be removed. Example 6.6.1. There exists an extended limit ω on l∞ such that Trω is not additive on the positive cone of ℳ1,∞ . Proof. Suppose that Trω is additive on the positive cone of ℳ1,∞ for every extended limit ω on l∞ . Let T ∈ ℒ(H) be a positive operator such that l

μ(T) = sup 2l−2 χ[0,22l ) . l≥0

m

m+1

Let n ∈ [22 , 22

). We have l

n

m 22 −1

k=0

l=1 k=22l−1

n

∑ μ(k, T) = μ(0, T) + μ(1, T) + ∑ ∑ μ(k, T) + ∑ μ(k, T) m

l

l

l−1

= 1 + ∑ 2l−2 ⋅ (22 − 22 ) + 2m+1−2 l=1 m

m+1

l=1

l=0

m

k=22 m+1

≤ 1 + ∑ 2l + 2m+1 = ∑ 2l = 2m+2 − 1 ≤ We therefore have T ∈ ℳ1,∞ .

m

⋅ (n − 22 + 1) 4 log(n). log(2)

214 | 6 Dixmier traces and positive traces From the form of the direct sum in Definition 2.2.11, we have Trω (T ⊕ T) = 2Trω (T). Since, by Lemma 2.2.12, n

⌊ n2 ⌋

k=0

k=0

∑ μ(k, T ⊕ T) = 2 ∑ μ(k, T) + O(1),

n ≥ 0,

it follows that ⌊n⌋

2 1 1 ω( ∑ μ(k, T)) = Trω (T ⊕ T) log(n + 2) k=0 2

= Trω (T) = ω(

n 1 ∑ μ(k, T)). log(n + 2) k=0

By the linearity of ω, we have ω(

n 1 ∑ μ(k, T)) = 0. log(n + 2) k=⌊ n ⌋+1 2

Since ω is positive and n

n μ(k, T) ≥ (n − ⌊ ⌋)μ(n, T), 2 k=⌊ n ⌋+1 ∑

n ≥ 1,

2

it follows that for every extended limit ω, ω(

nμ(n, T) ) = 0. log(n + 2)

Since ω is arbitrary, Lemma 6.2.4 implies that lim

n→∞

nμ(n, T) = 0. log(n + 2)

(6.9)

l

Let n = 22 − 1, l ≥ 0. By the definition of T, we have l

μ(n, T) = 2l−2 and, therefore,

l

l

(n + 1)μ(n, T) 22 ⋅ 2l−2 1 = l . = log(n + 2) 2 ⋅ log(2) log(2)

In particular, lim sup n→∞

nμ(n, T) 1 ≥ , log(n + 2) log(2)

which contradicts (6.9). Thus, our initial assumption that Trω is linear for every extended limit ω on l∞ is incorrect. This completes the proof.

6.6 Dixmier traces on ℳ1,∞

| 215

The example showed that ω(

(n + 1)μ(n, A) ) ≠ 0 log(n + 2)

for some A ∈ ℳ1,∞ and some extended limit ω on l∞ . For dilation-invariant extended limits the situation is different. Lemma 6.6.2. Let ω be a dilation-invariant extended limit on l∞ . For every A ∈ ℳ1,∞ we have ω(

(n + 1)μ(n, A) ) = 0. log(n + 2)

Proof. Without loss of generality, A ≥ 0. We have ω(

n n 1 1 ∑ μ(k, A)) = (ω ∘ σ2 )( ∑ μ(k, A)) log(n + 2) k=0 log(n + 2) k=0

= ω(

1

⌊ n2 ⌋

∑ μ(k, A)). log(⌊ n2 ⌋ + 2) k=0

As ⌊ n2 ⌋

⌊n⌋

2 1 ∑ μ(k, A) − ∑ μ(k, A) = o(1), n log(n + 2) k=0 log(⌊ 2 ⌋ + 2) k=0

1

n → ∞,

and ω vanishes on c0 , it follows that ⌊n⌋

2 1 ω( ∑ μ(k, A)). log(n + 2) k=0

Thus, ω(

n 1 ∑ μ(k, A)) = 0. log(n + 2) k=⌊ n ⌋+1 2

Since n

∑

k=⌊ n2 ⌋+1

n μ(k, A) ≥ (n − ⌊ ⌋)μ(n, A), 2

n ≥ 0,

and n n+1 (n − ⌊ ⌋)μ(n, A) = μ(n, A) + o(1), 2 2

n → ∞,

216 | 6 Dixmier traces and positive traces we get ω(

(n + 1)μ(n, A) ) = 0. log(n + 2)

This completes the proof. To prove Theorem 6.1.5, namely that Trω is linear on ℳ1,∞ , the property that ω(

(n + 1)μ(n, A) ) = 0, log(n + 2)

A ∈ ℳ1,∞ ,

is the main requirement on the extended limit ω on l∞ . Proof of Theorem 6.1.5. Let A1 , A2 ∈ ℳ1,∞ and ω be a dilation-invariant extended limit on l∞ . By Lemma 5.6.4, there exists z ∈ m1,∞ such that 󵄨 󵄨󵄨 n n 󵄨󵄨 1 󵄨󵄨󵄨󵄨 n 󵄨󵄨 ∑ λ(k, A1 ) + ∑ λ(k, A2 ) − ∑ λ(k, A1 + A2 )󵄨󵄨󵄨 ≤ μ(n, z), 󵄨󵄨 n + 1 󵄨󵄨󵄨k=0 k=0 k=0 󵄨

n ≥ 0.

Thus, 󵄨󵄨 󵄨 󵄨󵄨Trω (A1 ) + Trω (A2 ) − Trω (A1 + A2 )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2n+1 󵄨󵄨 1 󵄨󵄨 = 󵄨󵄨󵄨ω( ∑ λ(k, A1 ) + λ(k, A2 ) − λ(k, A1 + A2 ))󵄨󵄨󵄨 󵄨󵄨 log(n + 2) k=0 󵄨󵄨󵄨 󵄨 (n + 1)μ(n, z) ). ≤ ω( log(n + 2) By Lemma 6.6.2, ω(

(n + 1)μ(n, z) ) = 0. log(n + 2)

Hence, Trω is linear on ℳ1,∞ . To show that all normalized fully symmetric traces on ℒ1,∞ are Dixmier traces, the proof of Theorem 6.3.2 used the Hahn–Banach theorem on the image under the Cesaro operator of all real valued bounded sequences in l∞ . The approach to proving that all normalized fully symmetric functionals on ℳ1,∞ are described by Dixmier traces on ℳ1,∞ is the same but it involves a more complicated subspace of real valued bounded sequences in l∞ . Recall from Chapter 3, Section 3.3, the real subspace of D(m1,∞ ) of m1,∞ defined by D(m1,∞ ) = {x ∈ m1,∞ : x = μ(a) − μ(b), a, b ∈ m1,∞ }.

6.6 Dixmier traces on ℳ1,∞

| 217

For every x ∈ D(m1,∞ ), set (Tx)(n) =

n 1 ∑ x(k), log(n + 2) k=0

n ≥ 0.

Note that T : D(m1,∞ ) → l∞ since ‖Tx‖∞ ≤ ‖a‖m1,∞ + ‖b‖m1,∞ < ∞, where x = μ(a) − μ(b), a, b ∈ m1,∞ . Lemma 6.6.3. Let φ be a normalized fully symmetric trace on ℳ1,∞ . (a) If x, y ∈ D(m1,∞ ) are such that Ty ≤ Tx, then φ(diag(y)) ≤ φ(diag(x)). (b) If x ∈ D(m1,∞ ), then φ(diag(x)) ≤ lim sup Tx. Proof. To prove (a), let x = μ(a) − μ(b) and y = μ(c) − μ(d), where a, b, c, d ∈ m1,∞ . By assumption, we have n

n

k=0

k=0

∑ μ(k, c) − μ(k, d) ≤ ∑ μ(k, a) − μ(k, b),

n ≥ 0.

Thus, μ(b) + μ(c) ≺≺ μ(a) + μ(d). Since φ is monotone with respect to submajorization, it follows that φ(μ(b)) + φ(μ(c)) ≤ φ(μ(a)) + φ(μ(d)). Consequently, φ(y) ≤ φ(x). To prove (b), set x0 = {log( k+2 )} . By construction, k+1 k≥0 (Tx0 )(n) = 1,

n ≥ 0.

Also note that n

∑ (log( k=0

k+2 1 )− ) = O(1), k+1 1+k

and hence diag(x0 ) − diag((1 + n)−1 ) ∈ [ℒ1,∞ , ℒ(H)] by Theorem 5.1.5. Since φ was normalized, φ(diag(x0 )) = φ(diag((1 + n)−1 )) = 1.

218 | 6 Dixmier traces and positive traces Assume x ∈ Dm1,∞ and let λ = lim sup Tx. Fix ϵ > 0 and choose n(ϵ) such that (Tx)(n) ≤ (λ + ϵ) = T((λ + ϵ)x0 )(n),

n ≥ n(ϵ).

Hence, there exists m(ϵ) such that Tx ≤ T((λ + ϵ)x0 + m(ϵ)e0 ), where e0 = (1, 0, 0, . . .). By part (a), we have φ(diag(x)) ≤ φ(diag((λ + ϵ)x0 + m(ϵ)e0 )). Since e0 ∈ l1 ⊂ Zm1,∞ , it follows that φ(diag(e0 )) = 0. Thus, φ(x) ≤ (λ + ϵ)φ(x0 ) = λ + ϵ. Since ϵ is arbitrarily small, it follows that φ(x) ≤ λ. This proves (b). Proof of Theorem 6.1.6. Assume that φ is a normalized fully symmetric trace on ℳ1,∞ . Let X ⊂ l∞ be the linear subset of real-valued bounded sequences defined by X = Spanℝ {Tx : x ≥ 0, x ∈ D(m1,∞ )}. Define a linear functional ω on X by setting ω(Tx) = φ(x),

x ∈ D(m1,∞ ).

The functional ω is well defined by Lemma 6.6.3(a). By Lemma 6.6.3(b), we have ω(z) ≤ lim sup z,

z ∈ X.

By Lemma 6.3.5, ω is the restriction of an extended limit to X. It will not cause any confusion to denote this extended limit again by ω. For m ≥ 1 and x ∈ Dm1,∞ , we have (σm Tx)(n) =

1

⌊ mn ⌋

log(⌊ mn ⌋ + 2)

∑ x(k)

k=0

⌊n⌋

m 1 1 = ∑ x(k) + o(1) = (T( σm x))(n) + o(1), log(n + 2) k=0 m

n → ∞.

6.7 Notes | 219

In short, (σm T − T(

1 σ ))x ∈ c0 . m m

Since ω vanishes on c0 , it follows that ω(σm Tx) = ω(T(

1 1 σ x)) = φ( σm x) = φ(x) = ω(Tx), m m m

x ∈ D(m1,∞ ).

That is, we have σm z ∈ X + c0 ,

ω(σm z) = ω(z),

z ∈ X + c0

for the extended limit ω. By Theorem 6.2.8, there exists a dilation-invariant extended limit ω0 on l∞ such that ω|X+c0 = ω0 |X+c0 . We have φ(x) = ω(Tx) = ω0 (Tx),

x ∈ D(m1,∞ ).

In particular, since μ(A) ∈ D(m1,∞ ) for every A ∈ ℳ1,∞ , φ(A) = φ(μ(A)) = ω0 (Tμ(A)) = Trω (A),

0 ≤ A ∈ ℳ1,∞ .

This completes the proof.

6.7 Notes Extended limits S. Mazur and S. Banach proved the existence of shift-invariant extended limits in 1929 and 1932, respectively [182, p. 103], [17, p. 34]. A Banach limit is often defined in terms of left shift, S : l∞ → l∞ , S(x(0), x(1), . . . , x(k), . . .) = S(x(1), x(2), . . . , x(k + 1), . . .),

x ∈ l∞ .

Since STx = x and TSx + e0 ⋅ x = x, where x ∈ l∞ , T is right shift, and e0 = (1, 0, 0, . . .), then an extended limit on l∞ is S-invariant if and only if it is T-invariant. The disjointness between multiplicative and shift-invariant extended limits was noted by Mazur in [183]. It can be proved that every character of l∞ vanishing on c0 (i. e., a multiplicative extended limit) corresponds to the limit along a nonprincipal ultrafilter on ℕ. The maps E and P between extended limits on l∞ and extended limits on L∞ (0, 1) (and, equally, L∞ (0, ∞)) in Example 6.2.5, and similar variants, feature in early works characterizing Dixmier traces [168]. The maps E and P do not neatly intertwine the shift and dilation operators between l∞ and L∞ (0, ∞) at noninteger values. A proof of Lemma 6.2.4 can be found in [39]. A proof of Theorem 6.2.8, the invariant version of the Hahn–Banach theorem, can be found in [98, Theorem 3.3.1].

220 | 6 Dixmier traces and positive traces G. Lorentz [173] introduced the notion of almost convergence when a sequence has the same value for every Banach limit. L. Sucheston [254] gave concrete criteria for almost convergence and significantly simplified the original proofs of Lorentz. For discussion of the existence of extended limits on l∞ and L∞ (0, ∞) invariant to additional operators such as Cesaro means and exponentiation, and the role they play for Dixmier traces, we refer to [7, 39, 84, 85, 239, 243, 267, 269]. The existence of extended limits invariant under such operations is provided by [39, Theorem 1.5]. Dixmier traces The main theme of this chapter has its roots in the paper of J. Dixmier [76]. To give a reader a background to this work, we include some parts from Dixmier’s letter to the conference “Singular Traces and their Applications” (Luminy, 2012, reprinted with permission and translated by V. Gayral). “…In their second big paper, Murray and von Neumann proved the existence of a trace on type II1 factors. They showed that this trace is essentially uniquely characterized by its purely algebraic properties. What happens for type I∞ or II∞ factors? In 1960, I had a clear notion of ‘normality’ for positive linear forms and traces, and it was easy to see that a normal trace is essentially unique. Let us consider the case of type I∞ factors, that is to say, ℒ(H), H being a Hilbert space that we will suppose to be separable. Is the classical trace on ℒ(H)+ characterized by its algebraic properties? Say differently, is a trace on ℒ(H)+ automatically normal? In 1950, I was convinced that the answer was yes (up to some trivial counter-examples). I didn’t solved this problem until 1965, after having studied it during 15 years — not every day, of course. I tackled this problem 4 or 5 times. Such a perseverance is quite common for researchers, but what was a bit exceptional here, is that at each attempt, I made a progress. In 1955 (I do not guarantee the absolute exactness of the following dates) I had the first idea of a counter-example: for a positive compact operator A, with eigenvalues λ(0, A) ≥ λ(1, A) ≥ ⋅ ⋅ ⋅ such that λ(n, A) = O(1/(n + 1)), I defined t(A) = lim λ(n, A)/(1/n), lim being an extended Banach limit. But I wasn’t able to prove additivity of t(A). In 1960, during a mathematical dinner, I explained my problem to my neighbor N. Aronszajn. He suggested to me to replace λ(n, A) by λ(0, A) + λ(1, A) + ⋅ ⋅ ⋅ + λ(n, A) and 1/n by 1 + 1/2 + ⋅ ⋅ ⋅ + 1/n, say differently, by log(n). The next day, I explored this possibility and I noticed that I was making progress. But I had, on one hand, to be precise which Banach limit to use (this didn’t cause a problem because I had participated a few years before in the birth of amenable groups), and on the other hand to prove certain inequalities on eigenvalues. I didn’t succeed. In 1965, I went back to the problem and succeed in proving the inequalities. A few months later, I noticed that those inequalities had been proved by J. Hersch. Then, everything could be exposed within two pages and I wrote a note in Comptes Rendus de l’Académie des Sciences. …The scene takes place at IHES, at the end of a lunch. I was talking with Alain Connes, and I told him my astonishment that singular traces don’t have applications. In my mind, singular traces are pathological monsters and they should be useful to prove monstrous behavior of operators, representations. At this time, nobody (but me) knew singular traces, Alain Connes no more than the others. But as soon as he understood (after approximately thirty seconds), he said to me: ‘this is what I need’. Effectively, a few weeks later, I saw a manuscript or a preprint (I don’t remember), which obtained from singular traces some applications not monstrous at all.”

6.7 Notes | 221

Guichardet noted [119, pp. 467–468] the similarity between Dixmier’s quest to find a trace besides Tr on operators on a Hilbert space and the original motivation of Banach for the study of extended limits: Lebesgue’s question whether the algebraic properties of the Lebesgue integral defined it uniquely in the Banach dual L∞ (0, ∞)∗ . The exposition of Dixmier’s trace given in Connes’ work [60], starting in [56] with the first reference to Dixmier’s trace by Connes in 1988 [57, 58], has influenced many researchers. From 1988 Alain Connes developed the theory of Dixmier’s trace [57, 58], with also Henri Moscovici [71]. Alain Connes coined the name “Dixmier trace” in the 1988 publication [58] and the Dixmier trace was the topic of Connes’ course at College de France the year before. An extensive body of work followed with other authors generalizing the construction and applications initiated by Connes, e. g., [5, 6, 83, 121, 122, 168, 191–193, 236]. The monograph “Elements of Noncommutative Geometry” provides a treatment of Connes’ Dixmier trace [284]. There have been numerous surveys focusing on Dixmier traces [37, 119, 205, 206, 260], including the previous edition of this book [170] and the author’s survey [172]. Theorems 6.1.3 and 6.1.6 follow the line started in [83], which viewed Dixmier traces as special cases of fully symmetric functionals on Lorentz operator ideals. This line of thought was pursued further in [84, 85] and in [154], where it was established that all fully symmetric functionals on Lorentz operator ideals are precisely Dixmier traces. Fack in 2004 noted that Dixmier’s trace, specifically, was spectral [102]. A different proof was given in 2013 in [155, Lemma 6.31], on which the present approach is based. Almost no texts in noncommutative geometry mention eigenvalues in the construction of the Dixmier trace, despite the strong interaction between spectral theory and pseudodifferential operators [246]. Spectral behavior of the Laplacian Hermann Weyl’s proved his law on the asymptotic spectral behavior of the Dirichlet Laplacian on a bounded domain in 1911 [290]. Weyl’s law originated the area of spectral geometry, the study of differential and algebraic geometry through the spectrum of differential operators such as the Dirichlet Laplacian [21, 47, 109, 227]. A treatment of Weyl’s law can be found in [123], [218, XIII.15]. Weyl’s law for closed Riemannian manifolds is derived in [24, 162], [246, p. 117]. Clark [52] and Ivrii [142] provide a history of Weyl’s law and the development of spectral asymptotic formulas. These surveys, and the previous references, discuss additional boundary terms in Weyl’s law and the lower order trace class terms in the expansion of the pseudodifferential operator (1 − Δ)−α/2 [24, 111]. Weyl’s law as the basis for viewing inverse powers of the Laplacian as densities, or infinitesimals, in an operator based calculus is the development of Connes’ spectral view of geometry [61, 65, 66]. Connes proof of (6.1), that the noncommutative integral A 󳨃→ Trω (A(1 − Δ)−p/2 ),

a ∈ ℒ(H),

222 | 6 Dixmier traces and positive traces is a quantization of integration of classical zero order symbols over the sphere bundle, can be found in [18, 57, 167]. For details on Liouville measure and the sphere bundle, see [48, VII]. That heuristically the pseudodifferential operator A(1 − Δ)−p/2 is a quantization of a(v)dv is a standard idea of pseudodifferential theory, see, e. g., [61, 117]. Constructions of traces after Dixmier The end notes to Chapter 1 and Chapter 4 indicated the contributions of Pietsch [200, 205], Kalton [148], Figiel, and Varga to the construction of traces on operator ideals. The constructions were independent of Dixmier. That the condition of shift and dilation invariance of the extended limit in Dixmier’s construction can be removed completely for ℒ1,∞ was first noted in [239] and [170, Chapter 9]. The maximal ideal such that Dixmier traces can be defined by extended limits without additional invariances has been identified [239, p. 3058] — it is not ℒ1,∞ but is a proper ideal of ℳ1,∞ . On the dual of the Macaev ideal, Dixmier’s construction still lacks injectivity between extended limits and traces [267, Theorem 40]; the set of dilation-invariant states provides the same set of Dixmier traces on ℳ1,∞ as that of translation and invariant states. Dixmier’s original article [76] deals with the class of traces Trω when ω is a translation and dilation invariant extended limit on l∞ . A. Connes in [57] used this class of ω’s (termed in [57] the “class of all means on the amenable group of upper triangular two by two matrices”). Later, it was observed by A. Connes [60], that in order to ensure that Trω is additive on ℳ1,∞ , it is sufficient to only assume that ω is a dilation invariant extended limit on l∞ . A way to generate dilation invariant ω’s was suggested by A. Connes in the monograph “Noncommutative Geometry” in 1994 [60, Section IV, 2β] by observing that for any extended limit γ on l∞ the functional ω := γ ∘ M is dilation invariant. Here, the bounded operator M : l∞ → l∞ is the logarithmic mean (Mx)(n) =

n x(k) 1 , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

The class of all Dixmier traces defined by such ω’s was termed the set of Connes– Dixmier traces in [168], and we refer the reader for additional information about this class to that paper. Dixmier traces were first considered for general Lorentz ideals in [83, 120]. Conditions for the additivity of Trω were studied in the context of fully symmetric functionals in [18, 83–85], where also various classes of extended limits γ on l∞ with additional invariance property (and corresponding subclasses of Dixmier traces) were introduced. See [170, Chapter 6] for the discussion of Dixmier traces on general Lorentz ideals. Wodzicki’s paper [294] suggested to transfer Dixmier’s construction to a much wider class of operator ideals. Questions of finiteness and linearity become very hard

6.7 Notes | 223

in this general setting. The paper [263] generalizes some of the results from [294]. The techniques used in [294] and [263] are distinct. The first edition of this monograph considered Wodzicki’s construction for the description of all fully symmetric functionals on fully symmetric ideals in [170, Chapter 6]. For concrete examples of such spaces, we refer the reader to [85]. Essentially, Dixmier traces are dense in the weak∗ -topology in the set of fully symmetric functionals [170, Theorem 6.6.1]. Which is a remarkable fact that the first construction of singular traces, Jacques Dixmier’s 1966 construction [76], turns out to provide the general construction of all fully symmetric traces on all fully symmetric ideals of compact operators. Cardinality and classification of positive traces The bijective construction describing every positive trace on ℒ1,∞ was described by Pietsch [210, Theorem 4], see also [244, Corollary 3.8, Corollary 4.2]. The spectral formula appeared in [244, Theorem 6.2]. The key element of the bijection is the result that [244, Lemma 3.6] (a variation of [207, Lemmas 5.3–5.5]) x − LL󸀠 (x) ∈ Z(l1,∞ ),

0 ≤ x ∈ l1,∞ ,

where Z(l1,∞ ) is the center of l1,∞ . Outside of l1,∞ , which again plays a unique role in the theory of traces as essentially the dyadic dilation of l∞ , characterization of shift-invariant linear functionals on the proper ideal L󸀠 J within l∞ is difficult. Even for the Sargeant sequence space m1,∞ . A complete construction of traces on the Banach space equivalent of the ideal ℳ1,∞ is detailed in [205]. That the bijective correspondence between positive normalized traces on ℒ1,∞ and Banach limits provides a very neat classification for Dixmier’s trace, and the smaller set of Connes–Dixmier traces was first noticed by Pietsch [206], see also [244]. That there was a positive trace on ℳ1,∞ not equivalent to a Dixmier trace was proved earlier using uniform submajorization [151]. Raimi studied factorizable Banach limits in 1980 and provided an example of a Banach limit that was not factorizable in [216]. That the Banach limits on ℓ∞ have cardinality of beth two is proved in [146, Theorem 3]. Therefore there are a lot of positive ℕ traces on ℒ1,∞ . That the set of positive traces on ℒ1,∞ has cardinality 22 due to the association with Banach limits was first noted by Pietsch in [209, Theorem 9.6]. The cardinality of sets of once and twice factorizable Banach limits is an open question, as is the cardinality of the set difference of Banach limits and factorizable Banach limits. Hence the cardinality of the sets of Dixmier traces discussed and the size of the difference between the sets of positive normalized traces and Dixmier traces is not precisely known, see [206]. Less precise answers about the number of linearly independent positive traces on general quasi-Banach ideals will be discussed in a later volume.

224 | 6 Dixmier traces and positive traces Most users of noncommutative geometry still refer to the original presentation of Dixmier’s trace by Connes from 1988 [57, 58] (or the monograph “Noncommutative Geometry” from 1994 [60]), and do not utilize the fact that the condition of translation and dilation invariance of the extended limit is removed for ℒ1,∞ [239], and that eigenvalues can be used instead of singular values [102, 150, 155, 240, 264]. That the set of Connes–Dixmier traces is strictly smaller than the set of Dixmier traces was first proved in [168] and identified subsequently in [206] and [268]. The identification of Connes–Dixmier traces with twice-factorizable Banach limits is [244, Theorem 5.13]. Further details on descendingly smaller sets of traces with ascendingly stronger invariance in the extended limit can be found in [243, 244, 267, 268]. One of the strongest invariances generally considered are traces Trω where ω = ω ∘ M is an extended limit invariant under M. This set of traces are characterized by what might be considered “infinitely” factorizable Banach limits where θ = θ ∘ C is a Banach limit invariant under C [244]. Various important formulae of noncommutative geometry, like Connes’ formula for a representative of the Hochschild class of the Chern character for (p, ∞)-summable spectral triples (see, e. g., [40, Theorem 7] and [18, Theorem 6]) as well as those involving heat kernel estimates and generalized ζ -functions (see, e. g., [18, 39, 40, 43, 261]) were frequently established for the smaller class of Dixmier traces defined by extended limits such that ω = ω ∘ M. Connes’ trace theorem recovering the noncommutative residue and integration of forms [57], recovery of the Lebesgue integral [18, 103, 169], and the Hochschild character formula [40, 44, 58, 70, 71, 136] have now been proven for all traces on ℒ1,∞ [45, 155, 171], negating the use in these cases of the smaller class of Dixmier traces. These results are part of the content of Volume II.

7 Diagonal formulas for traces 7.1 Introduction The trace Tr on the set of trace class operators ℒ1 is spectral by Lidskii’s formula, ∞

Tr(A) = ∑ λ(k, A), k=0

A ∈ ℒ1 ,

where λ(A) is any eigenvalue sequence of A. Chapter 5 showed that every trace on a logarithmic submajorization closed two-sided ideal of ℒ(H) admits an equivalent spectral formula. The trace Tr on ℒ1 also has a diagonal formula that extends the matrix trace as the sum of the diagonal terms of a matrix. If {ek }k≥0 is an orthonormal basis of the separable Hilbert space H, then ∞

Tr(A) = ∑ ⟨Aek , ek ⟩, k=0

A ∈ ℒ1 .

This chapter proves that Tr is the only nontrivial trace on a two-sided ideal with this diagonal property for every operator in the two-sided ideal. This chapter also proves diagonal formulas for a trace φ on ℒ1,∞ when φ is restricted to a left ideal in ℒ(H) associated to a positive operator 0 ≤ V ∈ ℒ1,∞ and an orthonormal basis {ek }k≥0 of H such that Vek = μ(k, V)ek , k ≥ 0. Motivation For the Laplacian Δ on the flat torus 𝕋p of dimension p, the formula (constants depending only on p are suppressed) α

α

Tr(Mf (1 − Δ)− 2 ) = ∫ f (s) ds ⋅ Tr((1 − Δ)− 2 ),

α > p, f ∈ L∞ (𝕋p ),

(7.1)

𝕋p

follows directly from the diagonal formula for the trace. Here, the operator α

(1 − Δ)− 2 ∈ ℒ1 (L2 (𝕋p )),

α > p,

is trace-class and the bounded operator Mf ∈ ℒ(L2 (𝕋p )), (Mf h)(s) = f (s)h(s),

f ∈ L∞ (𝕋p ), h ∈ L2 (𝕋p ),

is given by pointwise multiplication. To show (7.1), consider the expectation values for the orthonormal basis of L2 (𝕋p ) given by en (s) = e−i⟨m(n),s⟩ , https://doi.org/10.1515/9783110378054-009

n ≥ 0, s ∈ 𝕋p ,

226 | 7 Diagonal formulas for traces where m : ℕ ∪ {0} → ℤp is a bijection such that the sequence of eigenvalues 󵄨2 󵄨 λ(n, Δ) = −󵄨󵄨󵄨m(n)󵄨󵄨󵄨 ,

n ≥ 0,

of the Laplacian is nonincreasing. We have α

󵄨2 − 󵄨 ⟨Mf (1 − Δ)− 2 en , en ⟩ = ∫ f (s) ds ⋅ (1 + 󵄨󵄨󵄨m(n)󵄨󵄨󵄨 ) 2 , α

n ≥ 0,

𝕋p

and α 󵄨 󵄨2 − α ⟨(1 − Δ)− 2 en , en ⟩ = (1 + 󵄨󵄨󵄨m(n)󵄨󵄨󵄨 ) 2 ,

n ≥ 0.

Hence ∞

∞

α

α

∑ ⟨Mf (1 − Δ)− 2 en , en ⟩ = ∫ f (s) ds ⋅ ∑ ⟨(1 − Δ)− 2 en , en ⟩,

n=0

n=0

𝕋p

and formula (7.1) follows. Proving (7.1) from Lidskii’s formula requires the asymptotics of the eigenvalue sequence of the product α

λ(k, Mf (1 − Δ)− 2 ),

k ≥ 0,

which is more difficult to calculate. Let ω be an extended limit on l∞ and let Trω be a Dixmier trace as in Theorem 6.1.2 on the ideal of weak trace class operators ℒ1,∞ . Suppose that we have a diagonal formula for the Dixmier trace Trω , at least for the bounded operator Mf , f ∈ L∞ (𝕋p ), and the orthonormal basis en , n ≥ 0, of L2 (𝕋p ) with Δen = −|m(n)|2 en , n ≥ 0, p

Trω (Mf (1 − Δ)− 2 ) = ω(

n p 1 ∑ ⟨Mf (1 − Δ)− 2 ek , ek ⟩), log(2 + n) k=0

f ∈ L∞ (𝕋p ).

Weyl’s law provides the spectral asymptotics (constants depending only on p are suppressed) 1 󵄨󵄨 󵄨 󵄨󵄨m(k)󵄨󵄨󵄨 ∼ (1 + k) p ,

k → ∞,

and p

⟨Mf (1 − Δ)− 2 ek , ek ⟩ ∼ ∫ f (s) ds ⋅ 𝕋p

1 , 1+k

k → ∞.

It follows that (constants depending only on p are suppressed) p

Trω (Mf (1 − Δ)− 2 ) = ∫ f (s) ds ⋅ ω( 𝕋p

n 1 1 ) = ∫ f (s) ds. ∑ log(2 + n) k=0 1 + k p 𝕋

7.1 Introduction | 227

This result, derived from a diagonal formula, is the simplest form of Connes’ integral formula (6.1) discussed in the introduction to Chapter 6. It has also removed the requirement that f ∈ L∞ (𝕋p ) be a smooth function on the torus. Take now the Laplace–Beltrami operator Δ on a p-dimensional closed Riemannian manifold Ω. Suppose there is a diagonal formula for the Dixmier trace Trω , at least for the bounded operator Mf , f ∈ L∞ (Ω), and the orthonormal basis en , n ≥ 0, of p

p

L2 (Ω) such that (1 − Δ)− 2 en = λ(n, (1 − Δ)− 2 )en , n ≥ 0, with p

Trω (Mf (1 − Δ)− 2 ) = ω(

n p 1 ∑ ⟨Mf (1 − Δ)− 2 ek , ek ⟩), log(2 + n) k=0

f ∈ L∞ (Ω).

From Weyl’s law for the Laplace–Beltrami operator, p

Trω (Mf (1 − Δ)− 2 ) = (ω ∘ M)(⟨Mf en , en ⟩),

f ∈ L∞ (Ω),

where M : l∞ → l∞ is the logarithmic mean (Mx)(n) =

n 1 x(k) , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ ,

and ω ∘ M is the extended limit on l∞ from Lemma 6.2.12. The very simple situation of the torus 𝕋p where each eigenvector en , n ≥ 0, of the Laplacian on the torus is associated to the Radon–Nikodym derivative |en |2 = 1, n ≥ 0, in L1 (𝕋p ) and ⟨Mf en , en ⟩ = ∫ f (s) ds,

n ≥ 0,

𝕋p

is no longer true on the manifold Ω. The expectation values on the manifold, ⟨Mf en , en ⟩,

n ≥ 0,

though not a constant or convergent sequence in general, form a logarithmic mean convergent sequence, and lim M(k 󳨃→ ⟨Mf ek , ek ⟩)(n) ∼ ∫ f (s) ds,

n→∞

f ∈ L∞ (Ω), n → ∞,

Ω

where ds denotes the Riemannian volume form. From the diagonal formula for the Dixmier trace, now for any compact p-dimensional Riemannian manifold Ω, the result again is that p

Trω (Mf (1 − Δ)− 2 ) ∼ ∫ f (s) ds, Ω

f ∈ L∞ (Ω).

228 | 7 Diagonal formulas for traces The involvement of the logarithmic mean convergence of measures arises from the Dixmier trace being a singular trace. For the ordinary trace, the Dirichlet series α

∞

α

Tr(Mf (1 − Δ)− 2 ) = ∑ ⟨Mf ek , ek ⟩λ(k, (1 − Δ)− 2 ), k=0

α > p, f ∈ L∞ (Ω),

is more complicated in the manifold case. Chapter 8 examines to what extent the Dixmier trace extracts the residue at the first simple pole of a meromorphic continuation of this Dirichlet series. The state on the algebra ℒ(L2 (Ω)) given by p

A 󳨃→ Trω (A(1 − Δ)− 2 ),

A ∈ ℒ(L2 (Ω)), p

which arises from the singular trace Trω on ℒ1,∞ , and the “density” (1 − Δ)− 2 ∈ ℒ1,∞ , is an extension of the integral on the commutative algebra L∞ (Ω) embedded in ℒ(L2 (Ω)) and called the noncommutative integral in Alain Connes’ noncommutative geometry. Diagonal formulas for traces Chapter 5 introduced the spectral formula for a trace φ on a logarithmic submajorization closed two-sided ideal 𝒥 , where the eigenvalue sequence λ(A) of A belongs to the commutative core J and φ(A) = (φ ∘ diag)({λ(n, A)}n≥0 ),

A ∈ 𝒥.

The diagonal formula for the trace φ, equivalent to the formula for Tr as the sum of the diagonal matrix elements according to an orthonormal basis {en }n≥0 of the separable Hilbert space H, would be φ(A) = (φ ∘ diag)({⟨Aen , en ⟩}n≥0 ) for every A ∈ 𝒥 . In contrast to the spectral formula, the diagonal formula fails to hold for any trace except the zero trace and Tr. Theorem 7.1.1. Let 𝒥 be an ideal in ℒ(H). Let φ : 𝒥 → ℂ be a trace such that, for some orthonormal basis {en }n≥0 of H, φ(A) = (φ ∘ diag)({⟨Aen , en ⟩}n≥0 )

(7.2)

for every A ∈ 𝒥 . Then (a) φ = 0 if 𝒥 ⊄ ℒ1 ; (b) φ = α ⋅ Tr for some constant α if 𝒥 ⊂ ℒ1 . In particular, for a normalized trace φ on the ideal ℒ1,∞ there exists no orthonormal basis of a separable Hilbert space H such that φ(A) = (φ ∘ diag)({⟨Aen , en ⟩}n≥0 )

7.1 Introduction | 229

for every A ∈ ℒ1,∞ . There is no equivalent formula to (6.4) involving expectation values for Dixmier traces on the entire ideal ℒ1,∞ , in general. Theorem 7.1.1 is proved in Section 7.2. The motivation above highlighted the advantage of calculations using eigenbases of geometric operators such as the Laplacian. This led to the development of diagonal formulas that hold on subspaces of ℒ1,∞ . Modulated operators were introduced in [155] to identify sequences of energy expectation values which were modulated, or controlled, by the Laplace–Beltrami operator on a compact or noncompact manifold with no boundary. Operators modulated by the Laplace–Beltrami operator include geometrically important operators such as pseudodifferential operators of order −p on a p-dimensional manifold. The details are in Volume II. Definition 7.1.2. Let 0 ≤ V ∈ ℒ(H). If A ∈ ℒ(H) and 1 󵄩 󵄩 ‖A‖mod := sup t 2 󵄩󵄩󵄩A(1 + tV)−1 󵄩󵄩󵄩2 < ∞

t>0

where ‖ ⋅ ‖2 denotes the Hilbert–Schmidt norm on ℒ2 , then A is called a V-modulated operator. Section 7.3 shows that ‖ ⋅ ‖mod defines a norm, and that the set of V-modulated operators in ℒ(H) forms a Banach space and a left ideal of ℒ(H). Though Definition 7.1.2 appears abstract, it was motivated by the choice p

V = (1 − Δ)− 2 ∈ ℒ1,∞ (L2 (Ω)), and Volume II discusses how Definition 7.1.2 relates to a more concrete condition on the kernel of a Hilbert–Schmidt operator A ∈ ℒ(L2 (Ω)). Section 7.3 proves the following theorem on expectation values of modulated operators. Theorem 7.1.3. Let 0 ≤ V ∈ ℒ1,∞ have trivial kernel and let {en }n≥0 be an eigenbasis for V ordered so that Ven = μ(n, V)en , n ≥ 0. Let A ∈ ℒ(H) be a V-modulated operator. Then A ∈ ℒ1,∞ and (a) {⟨Aen , en ⟩}n≥0 ∈ l1,∞ ; (b) n

n

k=0

k=0

∑ λ(k, A) − ∑ ⟨Aek , ek ⟩ = O(1),

(c)

n ≥ 0,

where λ(A) is any eigenvalue sequence of A; A − diag({⟨Aen , en ⟩}n≥0 ) ∈ [ℒ1,∞ , ℒ(H)] where [ℒ1,∞ , ℒ(H)] is the commutator subspace of ℒ1,∞ .

230 | 7 Diagonal formulas for traces Section 7.3 also proves the following corollary for traces on ℒ1,∞ , indicating that formula (7.2) holds under the assumption that the operator A is a V-modulated operator. Corollary 7.1.4 (Diagonal formula for traces on ℒ1,∞ ). Let 0 ≤ V ∈ ℒ1,∞ have trivial kernel and let {en }n≥0 be an eigenbasis for V ordered so that Ven = μ(n, V)en , n ≥ 0. Let A ∈ ℒ(H) be a V-modulated operator. Then A ∈ ℒ1,∞ and (a) for every trace φ on ℒ1,∞ , φ(A) = (φ ∘ diag)({⟨Aen , en ⟩}n≥0 ); (b) for every positive trace φ on ℒ1,∞ , n+1

1 2 −2 φ(A) = θ( ∑ ⟨Aek , ek ⟩) log(2) k=2n −1 for a unique Banach limit θ on l∞ ; (c) for every Dixmier trace Trω on ℒ1,∞ , Trω (A) = ω(

n 1 ∑ ⟨Aek , ek ⟩), log(2 + n) k=0

for an extended limit ω on l∞ . Weyl asymptotics and expectations The motivation above introduced the noncommutative integral as a linear functional Eφ (A) = φ(AV),

A ∈ ℒ(H),

on the algebra ℒ(H). Here φ is a trace on ℒ1,∞ and 0 ≤ V ∈ ℒ1,∞ is a fixed “density” that is normalized φ(V) = 1. Section 7.4 proves the simple facts that Eφ is a state on ℒ(H) when φ is a positive trace, and that it is a singular state in the sense that Eφ (A) = 0 when A ∈ 𝒞0 (H) is a compact operator. It is well-known that every normal linear functional on the algebra ℒ(H) is of the form A 󳨃→ Tr(AV),

A ∈ ℒ(H),

7.1 Introduction | 231

for some trace-class density 0 ≤ V ∈ ℒ1 . The noncommutative integral is not a normal linear functional on ℒ(H). All singular states on ℒ(H) are states on the Calkin algebra and are therefore noncommutative versions of extended limits. Alain Connes exploits this property in physical applications of noncommutative geometry, linking the noncommutative integral defined by a Dixmier trace to the classical limit in quantum mechanics. Assuming that the density 0 ≤ V ∈ ℒ1,∞ satisfies a Weyl law like the Laplace– Beltrami operator, the notion of the noncommutative integral as an extended limit of expectations can be concretely realized. Section 7.4 proves the following formula where M : l∞ → l∞ is the logarithmic mean: (Mx)(n) =

n x(k) 1 , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

If (Mx)(n) → a as n → ∞, then the sequence x ∈ l∞ is said to converge in logarithmic mean, which is denoted x(n) → a (M, 1). Theorem 7.1.5 (Noncommutative integral as an expectation). Let 0 ≤ V ∈ ℒ1,∞ have trivial kernel such that λ(n, V) ∼

1 , n

n → ∞,

and let {en }n≥0 be an eigenbasis of V such that Ven = λ(n, V)en , n ≥ 0. (a) For each extended limit ω on l∞ , Eω (A) := Trω (AV),

A ∈ ℒ(H),

is a singular state on ℒ(H) such that Eω (A) = (ω ∘ M)(⟨Aen , en ⟩), for every A ∈ ℒ(H). (b) The value Eω (A) = c does not depend on the extended limit ω on l∞ if and only if ⟨Aen , en ⟩ → c

(M, 1).

The motivation described that the bounded operators Mf , f ∈ L∞ (𝕋p ), acting on the Hilbert space L2 (𝕋p ) trivially satisfy the condition ⟨Mf en , en ⟩ → ∫ f (s) ds 𝕋p

232 | 7 Diagonal formulas for traces for the eigenvectors of the Laplacian Δ on 𝕋p , and therefore the normal linear functionals on L∞ (𝕋p ) defined by f 󳨃→ ⟨Mf en , en ⟩,

f ∈ L∞ (𝕋p ),

weak-∗ converge to the Lebesgue integral. This property, in particular when a subsequence of normal linear functionals associated to the eigenvectors of the Laplace– Beltrami operator on a compact Riemannian manifold converges to the integral on the manifold, has been termed quantum unique ergodicity in geometry. The noncommutative integral of the operator Mf , Eω (Mf ) = ∫ f (s) ds, 𝕋p

is clearly independent of the extended limit ω. The notion of when the noncommutative integral provides a unique value will be explored further in Chapter 9. Example 7.4.2 in Section 7.4 proves that there always exists an operator A ∈ ℒ(L2 (H)) such that Eω (A) takes different values depending on the extended limit ω. The convergence in Theorem 7.1.5(b) is therefore a special property of certain operators in ℒ(H).

7.2 Diagonal formulation of a singular trace fails in general If {ek }k≥0 is an orthonormal basis of a Hilbert space H, then, for every A ∈ ℒ1 , we have ∞

Tr(A) = ∑ ⟨Aek , ek ⟩. k=0

This value does not depend on which orthonormal basis is chosen. Let φ be a nontrivial trace on a two-sided ideal 𝒥 of ℒ(H). In this section we prove that the formula φ(A) = (φ ∘ diag)({⟨Aek , ek ⟩}k≥0 )

(7.3)

holds for every A ∈ 𝒥 only for the trace Tr on ℒ1 . Formula (7.3) clearly holds for some operators such as diagonal operators of the form ∞

A = ∑ x(k)ek ⊗ ek , k=0

x = μ(x) ∈ J,

where J is the commutative core of 𝒥 . The requirement in this section is that (7.3) holds for every operator in the ideal 𝒥 . The central component in proving that formula (7.3) cannot hold for every operator in 𝒥 is the following theorem of Victor Kaftal and Gary Weiss [143]. Recall that for sequences 0 ≤ x, y ∈ l∞ , x majorizes y, written y ≺ x,

7.2 Diagonal formulation of a singular trace fails in general | 233

if y ≺≺ x and ∞

∞

k=0

k=0

∑ y(k) = ∑ x(k).

Theorem 7.2.1. Let 0 ≤ x, y ∈ c0 be such that y ≺≺ x and let {ek }k≥0 be an orthonormal basis of a Hilbert space H. Suppose that ker(y) = 0. (a) If y ∉ l1 , then there exists a positive compact operator A ∈ ℒ(H) such that μ(A) = μ(x) and y = {⟨Aek , ek ⟩}k≥0 . (b) If x ∈ l1 and y ≺ x, then there exists a positive compact operator A ∈ ℒ(H) such that μ(A) = μ(x) and y = {⟨Aek , ek ⟩}k≥0 . Given the result of Kaftal and Weiss, we can prove Theorem 7.1.1. Proof of Theorem 7.1.1. Suppose 𝒥 is an ideal in ℒ(H) and let φ : 𝒥 → ℂ be a trace satisfying (7.3) for every operator A ∈ ℒ(H). (a) Let 𝒥 ⊄ ℒ1 . Choose 0 ≤ x ∈ J such that x ∉ l1 and ker(x) = 0. Taking y = 21 x and using Theorem 7.2.1, there is a positive compact operator A ∈ ℒ(H) such that μ(A) = μ(x) ∈ J and 21 x = {⟨Aek , ek ⟩}k≥0 . Then A ∈ 𝒥 and, by (7.3), we have 1 (φ ∘ diag)(x) = φ(A) = (φ ∘ diag)({⟨Aek , ek ⟩}k≥0 ) = (φ ∘ diag)( x). 2 It follows that (φ ∘ diag)|J\l1 = 0. Let now z ∈ J ∩ l1 . If x ∈ J\l1 , then also x + z ∈ J\l1 . By the preceding paragraph, we obtain φ(x) = 0 and φ(x + z) = 0. By linearity, (φ ∘ diag)(z) = (φ ∘ diag)(x + z) − (φ ∘ diag)(x) = 0 − 0 = 0. It follows that (φ ∘ diag)|J∩l1 = 0. Since φ ∘ diag = 0 on J, then φ = 0 on 𝒥 . (b) Let 𝒥 ⊂ ℒ1 . Let 0 ≤ y ∈ J be such that ker(y) = 0. Let x = (Tr ∘ diag)(y)e0 where e0 = (1, 0, 0, . . .) ∈ J. Then y ≺ x. Using Theorem 7.2.1, there is a positive compact operator A ∈ ℒ(H) such that μ(A) = μ(x) and y = {⟨Aek , ek ⟩}k≥0 . By (7.2), we have (φ ∘ diag)(x) = φ(A) = (φ ∘ diag)({⟨Aek , ek ⟩}k≥0 ) = (φ ∘ diag)(y). By construction (φ ∘ diag)(x) = Tr(y) ⋅ φ(e0 ). Thus, (φ ∘ diag)(y) = Tr(y) ⋅ φ(e0 ), Setting α = φ(e0 ), we complete the proof of (b).

y ∈ J.

234 | 7 Diagonal formulas for traces

7.3 Modulated operators Section 7.2 proved that, for a trace φ on the ideal ℒ1,∞ , there exists no orthonormal basis of a separable Hilbert space H such that φ(A) = (φ ∘ diag)({⟨Aen , en ⟩}n≥0 ) for every A ∈ ℒ1,∞ . Nigel Kalton introduced modulated operators to identify, given an orthonormal basis {en }n≥0 , those operators A ∈ ℒ1,∞ for which formula (7.3) holds. In Definition 7.1.2, a bounded operator A ∈ ℒ(H) was called V-modulated if 󵄩 󵄩 ‖A‖mod := sup t 1/2 󵄩󵄩󵄩A(1 + tV)−1 󵄩󵄩󵄩2 < ∞, t>0

where V is a positive bounded operator. Note that every V-modulated operator is Hilbert–Schmidt since ‖A‖2 ≤ (1 + ‖V‖∞ )‖A‖mod . The next proposition shows that the set {A ∈ ℒ(H) : ‖A‖mod < ∞} ⊂ ℒ2 of V-modulated operators is a Banach space and a left ideal of ℒ(H). Proposition 7.3.1. Let 0 ≤ V ∈ ℒ(H). The set of all V-modulated operators is a Banach space given the norm ‖ ⋅ ‖mod . For every B ∈ ℒ(H) and for every V-modulated operator A ∈ ℒ2 , we have ‖BA‖mod ≤ ‖B‖∞ ⋅ ‖A‖mod . Proof. That ‖⋅‖mod defines a norm on the set of V-modulated operators with ‖BA‖mod ≤ ‖B‖∞ ⋅ ‖A‖mod when B ∈ ℒ(H) and A is V-modulated follows from the properties of the symmetric norm ‖ ⋅ ‖2 on ℒ2 . That the set of V-modulated operators is a normed linear space and left ideal now follows. We prove completeness of the linear space of V-modulated operators. Let An , n ∈ ℕ, be a Cauchy sequence in the space of all V-modulated operators. Note that ‖An ‖mod is a Cauchy sequence in ℝ and that An (1 + V)−1 is a Cauchy sequence in ℒ2 . Hence, An (1 + V)−1 → S ∈ ℒ2 . Setting A = S(1 + V) ∈ ℒ2 , we have 󵄩󵄩 −1 −1 󵄩 󵄩󵄩An (1 + tV) − A(1 + tV) 󵄩󵄩󵄩2 󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩An (1 + V)−1 − A(1 + V)−1 󵄩󵄩󵄩2 󵄩󵄩󵄩(1 + V)(1 + tV)−1 󵄩󵄩󵄩∞ → 0,

n → ∞.

In particular, we have 󵄩󵄩 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩An (1 + tV) 󵄩󵄩󵄩2 → 󵄩󵄩󵄩A(1 + tV) 󵄩󵄩󵄩2 ,

n → ∞.

Hence, for every t > 0, we have 1 󵄩 󵄩 󵄩 󵄩 t 2 󵄩󵄩󵄩A(1 + tV)−1 󵄩󵄩󵄩2 = lim t 1/2 󵄩󵄩󵄩An (1 + tV)−1 󵄩󵄩󵄩2 ≤ lim ‖An ‖mod .

n→∞

n→∞

The supremum over t > 0 is uniformly bounded, hence the limit A is V-modulated.

7.3 Modulated operators | 235

For the rest of this section, unless stated otherwise, V is a strictly positive bounded operator in ℒ1,∞ . Strictly positive means that V has trivial kernel. Let {en }n≥0 be an eigenbasis for V ordered so that Ven = μ(n, V)en , n ≥ 0. For a V-modulated operator A, Theorem 7.1.3 states that A ∈ ℒ1,∞ , that the sequence of expectation values of A belongs to the commutative core of ℒ1,∞ , {⟨Aen , en ⟩}n≥0 ∈ l1,∞ , and that A − diag({⟨Aen , en ⟩}n≥0 ) ∈ [ℒ1,∞ , ℒ(H)]. All traces on ℒ1,∞ vanish on the commutator subspace [ℒ1,∞ , ℒ(H)], so the V-modulated operator A satisfies the formula φ(A) = (φ ∘ diag)({⟨Aen , en ⟩}n≥0 ) for the specific basis {en }n≥0 for every trace φ on ℒ1,∞ . The following sequence of technical lemmas will be used to prove Theorem 7.1.3, and they state other properties of V-modulated operators used in Chapters 8 and 9. The first lemma provides an equivalent criterion for a Hilbert–Schmidt operator to be V-modulated in terms of the spectral projections of V. Lemma 7.3.2. Let 0 ≤ V ∈ ℒ(H). An operator A ∈ ℒ2 is V-modulated if and only if 󵄩󵄩 −1 󵄩 −1 󵄩󵄩AEV [0, t ]󵄩󵄩󵄩2 = O(t 2 ),

t > 0,

(7.4)

where EV is the spectral measure of V. Proof. By the functional calculus, we have EV [0, t −1 ] ≤ 2(1 + tV)−1 and, therefore, 󵄩󵄩 󵄩 −1 󵄩 −1 󵄩 −1 󵄩󵄩AEV [0, t ]󵄩󵄩󵄩2 ≤ 2󵄩󵄩󵄩A(1 + tV) 󵄩󵄩󵄩2 ≤ 2‖A‖mod ⋅ t 2 . We now prove the converse assertion. Suppose that A ∈ ℒ(H) satisfies (7.4) and assume for simplicity that V ≤ 1. Let t ∈ [2k , 2k+1 ) for some k ≥ 0. By the triangle inequality, k−1

󵄩󵄩 󵄩 󵄩 −1 󵄩 −k 󵄩 −j−1 −j −1 󵄩 󵄩󵄩A(1 + tV) 󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩AEV [0, 2 ]󵄩󵄩󵄩2 + ∑ 󵄩󵄩󵄩AEV (2 , 2 ](1 + tV) 󵄩󵄩󵄩2 j=0

1

k−1

−1

≤ O(t − 2 ) + ∑ (1 + t2−j−1 ) j=0

1

k−1

󵄩 󵄩 ⋅ 󵄩󵄩󵄩AEV (2−j−1 , 2−j ]󵄩󵄩󵄩2

≤ O(t − 2 ) + O(1) ⋅ ∑ (1 + 2k−j−1 ) j=0

−1

j

1

⋅ 2− 2 = O(t − 2 ).

236 | 7 Diagonal formulas for traces The set of V-modulated operators is not a right ideal. Counterexamples are indicated in the end notes to this chapter. The next lemma provides a useful estimate for the action of bounded operator on the right of a V-modulated operator. Lemma 7.3.3. Let 0 ≤ V ∈ ℒ(H) and let B ∈ ℒ(H). If A is V-modulated, then AB is |VB|-modulated. Proof. Without loss of generality, ‖V‖∞ ≤ 1 and ‖B‖∞ ≤ 1. Let pn = EV [0, 2−n ] and qn = E|VB| [0, 2−n ] for n ≥ 0. By the definition of pj , we have 󵄩󵄩 −1 󵄩 j 󵄩󵄩(1 − pj )V 󵄩󵄩󵄩∞ ≤ 2 ,

j ≥ 0.

Thus, 󵄩󵄩 󵄩 󵄩 󵄩 −1 󵄩󵄩(1 − pj )Bqk 󵄩󵄩󵄩∞ = 󵄩󵄩󵄩(1 − pj )V ⋅ VBqk 󵄩󵄩󵄩∞ 󵄩 󵄩 ≤ 󵄩󵄩󵄩(1 − pj )V −1 󵄩󵄩󵄩∞ ‖VBqk ‖∞ ≤ 2j ‖VBqk ‖∞ = 2j−k ,

j, k ≥ 0.

Since A is V-modulated, it follows from Lemma 7.3.2 that j

‖Apj ‖2 ≤ const ⋅ 2− 2 ,

j ≥ 0.

Observing that (pj−1 − pj ) = pj−1 ⋅ (1 − pj ), we obtain k

󵄩 󵄩 ‖ABqk ‖2 ≤ ‖Apk Bqk ‖2 + ∑󵄩󵄩󵄩A(pj−1 − pj )Bqk 󵄩󵄩󵄩2 j=1

k

󵄩 󵄩 ≤ ‖Apk ‖2 + ∑ ‖Apj−1 ‖2 ⋅ 󵄩󵄩󵄩(1 − pj )Bqk 󵄩󵄩󵄩∞ j=1

k

k

j

k

≤ const ⋅ (2− 2 + ∑ 2 2 −k ) = O(2− 2 ). j=1

The assertion follows from Lemma 7.3.2. The next lemma proves that V is V-modulated when V ∈ ℒ1,∞ , which is not true in general. Lemma 7.3.4. Let 0 ≤ V ∈ ℒ1,∞ . Then V is V-modulated. Proof. Since V ∈ ℒ2 , it follows, for t > 0, that ∞

󵄩󵄩 −1 󵄩2 2 −1 2 −n−1 −1 −n −1 t , 2 t ])) 󵄩󵄩VEV [0, t ]󵄩󵄩󵄩2 = Tr(V EV [0, t ]) = ∑ Tr(V EV (2 ∞

n=0

∞

≤ ∑ 2−2n t −2 Tr(EV [2−n−1 t −1 , ∞)) = ∑ 2−2n t −2 nV (2−n−1 t −1 ). n=0

n=0

7.3 Modulated operators | 237

Since V ∈ ℒ1,∞ and, therefore, nV (s) ≤ const ⋅ s−1 , s > 0. Hence, ∞

󵄩󵄩 −2n −2 n+1 −1 −1 󵄩2 󵄩󵄩VEV [0, t ]󵄩󵄩󵄩2 ≤ const ⋅ ∑ 2 t ⋅ 2 t = O(t ). n=0

Thus, V is V-modulated by Lemma 7.3.2. The next lemma proves that the operator diag({⟨Aen , en ⟩}n≥0 ) is V-modulated if A is V-modulated. Lemma 7.3.5. Let 0 < V ∈ ℒ1,∞ be strictly positive and let A ∈ ℒ(H) be V-modulated. If {en }n≥0 is an eigenbasis for V such that Ven = μ(n, V)en , n ≥ 0, then the operator diag({⟨Aen , en ⟩}∞ n=0 ) ∈ ℒ(H) is also V-modulated. Proof. Set S = diag({⟨Aen , en ⟩}∞ n=0 ). If pn is the projection on the unit vector en , then −2 󵄩 󵄩󵄩 2 −1 󵄩2 −1 󵄩2 󵄩󵄩A(1 + tV) pn 󵄩󵄩󵄩2 = 󵄩󵄩󵄩A(1 + tV) en 󵄩󵄩󵄩 = (1 + tμ(n, V)) ⋅ ‖Aen ‖ −2 󵄨 󵄩2 󵄩2 󵄩 󵄨2 󵄩 ≥ (1 + tμ(n, V)) ⋅ 󵄨󵄨󵄨⟨Aen , en ⟩󵄨󵄨󵄨 = 󵄩󵄩󵄩S(1 + tV)−1 en 󵄩󵄩󵄩 = 󵄩󵄩󵄩S(1 + tV)−1 pn 󵄩󵄩󵄩2 .

Therefore, ∞

∞

n=0

n=0

󵄩󵄩 󵄩 󵄩 󵄩 −1 󵄩2 −1 󵄩2 −1 󵄩2 −1 󵄩2 󵄩󵄩S(1 + tV) 󵄩󵄩󵄩2 = ∑ 󵄩󵄩󵄩S(1 + tV) pn 󵄩󵄩󵄩2 ≤ ∑ 󵄩󵄩󵄩A(1 + tV) pn 󵄩󵄩󵄩2 = 󵄩󵄩󵄩A(1 + tV) 󵄩󵄩󵄩2 . Hence, S is V-modulated. The next result shows that V-modulated operators satisfy a sufficient condition to prove Theorem 7.1.3. Lemma 7.3.6. Let 0 < V ∈ ℒ1,∞ be strictly positive and let A ∈ ℒ(H) be V-modulated. For every p > 2, we have AV Proof. Let S = AV

− p1

− p1

∈ℒ

p ,∞ p−1

.

. Let 2k ≤ n < 2k+1 . By Lemma 7.3.2, we have

∞ ∞ j+1 󵄩󵄩 1 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 −j−1 −j 󵄩 −j−1 −j 󵄩 󵄩󵄩SEV [0, ]󵄩󵄩󵄩 ≤ ∑󵄩󵄩󵄩SEV (2 , 2 ]󵄩󵄩󵄩2 ≤ ∑ 2 p 󵄩󵄩󵄩AEV (2 , 2 ]󵄩󵄩󵄩2 󵄩󵄩 n 󵄩󵄩2 j=k j=k ∞

≤ const ⋅ ∑ 2 j=k

j+1 p

j

⋅ 2− 2 = const ⋅ 2

j( p1 − 21 )

1

= O(n p

− 21

).

If {en }∞ n=0 is an eigenbasis for V ordered so that Ven = μ(n, V)en , n ≥ 0, set n−1

Sn x = ∑ ⟨x, ej ⟩Sej , j=0

x ∈ H.

238 | 7 Diagonal formulas for traces We have ∞

∑ μ2 (j, S) = min{‖S − B‖22 : rank(B) ≤ n} ≤ ‖S − Sn ‖22

j=n

2 ∞ 󵄩󵄩 2 1 󵄩󵄩󵄩 󵄩 −1 = ∑ ‖Sej ‖2 ≤ 󵄩󵄩󵄩SEV [0, ]󵄩󵄩󵄩 = O(n p ). 󵄩󵄩2 󵄩󵄩 n j=n

On the other hand, we have ∞

2n

j=n

j=n

∑ μ2 (j, S) ≥ ∑ μ2 (j, S) ≥ (n + 1)μ2 (2n, S).

Hence, 2

(n + 1)μ2 (2n, S) = O(n p ), −1

n ≥ 1.

In other words, 1

−1

n ≥ 1,

μ(n, S) = O(n p ), and the assertion follows.

The condition that A ∈ ℒ(H) be a V-modulated operator is sufficient to prove Theorem 7.1.3. However, weaker conditions suffice. Lemma 7.3.7 shows that the main state−1 ment of Theorem 7.1.3 holds for a bounded operator A such that AV p ∈ ℒ p ,∞ for p−1 some p ≥ 1. −1

Lemma 7.3.7. If 0 < V ∈ ℒ1,∞ is strictly positive and if A ∈ ℒ(H) is such that AV p ∈ ℒ p ,∞ for some p ≥ 1, then A ∈ ℒ1,∞ . If {en }n≥0 is an eigenbasis for V ordered so that p−1

Ven = μ(n, V)en , n ≥ 0, then n

n

k=0

k=0

∑ λ(k, ℜA) = ∑ ⟨(ℜA)ek , ek ⟩ + O(1).

Here ℜA = 2−1 (A + A∗ ) denotes the real part of A. Proof. Let fn , n ≥ 0, be a basis of H such that (ℜA)fn = λ(n, ℜA)fn , n ≥ 0. Let pn (resp., qn ) be the projection on the linear span of ek (resp., fk ), 0 ≤ k ≤ n, and let rn = pn ∨ qn . We have A ∈ ℒ1,∞ , since AV A = AV

− p1

− p1

∈ℒ 1

p ,∞ p−1

⋅Vp ∈ℒ

1

, V p ∈ ℒp,∞ , and

p ,∞ p−1

⋅ ℒp,∞ ⊂ ℒ1,∞ .

Therefore, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Tr((ℜA)(rn − qn ))󵄨󵄨󵄨 = 󵄨󵄨󵄨Tr((ℜA)(1 − qn ) ⋅ (rn − qn ))󵄨󵄨󵄨 󵄩 󵄩 ≤ ‖rn − qn ‖1 ⋅ 󵄩󵄩󵄩(ℜA)(1 − qn )󵄩󵄩󵄩∞ ≤ (n + 1)μ(n, ℜA) = O(1).

7.3 Modulated operators | 239

On the other hand, we have AV

− p1

∈ℒ

p ,∞ p−1

. Observing that

󵄨 󵄨 󵄨󵄨 ∗ 󵄨 󵄨󵄨Tr(B)󵄨󵄨󵄨 = 󵄨󵄨󵄨Tr(B )󵄨󵄨󵄨,

B ∈ ℒ1 ,

we obtain 1 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 −1 ∗ 󵄨󵄨Tr(A (rn − pn ))󵄨󵄨󵄨 = 󵄨󵄨󵄨Tr(A(rn − pn ))󵄨󵄨󵄨 = 󵄨󵄨󵄨Tr((rn − pn )(AV p ) ⋅ V p (rn − pn ))󵄨󵄨󵄨 󵄩 󵄩 󵄩 󵄩 −1 󵄩 󵄩 1 −1 󵄩 󵄩 1 ≤ 󵄩󵄩󵄩(rn − pn )(AV p )󵄩󵄩󵄩1 ⋅ 󵄩󵄩󵄩V p (rn − pn )󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩rn (AV p )󵄩󵄩󵄩1 ⋅ 󵄩󵄩󵄩V p (1 − pn )󵄩󵄩󵄩∞

2n+1

≤ ∑ μ(k, AV

− p1

k=0

󵄩 −1 󵄩 ≤ 󵄩󵄩󵄩AV p 󵄩󵄩󵄩

1

) ⋅ μ p (n, V) 1

p ,∞ p−1

2n+1

1

p ‖V‖1,∞ ⋅ ∑ (k + 1) p

−1

k=0

− p1

⋅ (n + 1)

= O(1).

Therefore, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨Tr((ℜA)(pn − qn ))󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Tr((ℜA)(rn − qn ))󵄨󵄨󵄨 + 󵄨󵄨󵄨Tr((ℜA)(rn − pn ))󵄨󵄨󵄨 󵄨 󵄨 󵄨 1󵄨 ≤ 󵄨󵄨󵄨Tr((ℜA)(rn − qn ))󵄨󵄨󵄨 + 󵄨󵄨󵄨Tr(A(rn − pn ))󵄨󵄨󵄨 + 2

1 󵄨󵄨 󵄨 ∗ 󵄨Tr(A (rn − pn ))󵄨󵄨󵄨 2󵄨

is bounded. The assertion follows now from the obvious equalities n

n

Tr((ℜA)pn ) = ∑ ⟨(ℜA)ek , ek ⟩,

Tr((ℜA)qn ) = ∑ λ(k, ℜ(A)),

k=0

k=0

n ≥ 0.

Proof of Theorem 7.1.3. Suppose A is V-modulated where 0 ≤ V ∈ ℒ1,∞ has trivial kernel and {en }n≥0 is an eigenbasis for V ordered so that Ven = μ(n, V)en , n ≥ 0. It follows from Lemma 7.3.6 that AV

− p1

∈ ℒ

p ,∞ p−1

for p > 2. Applying Lemma 7.3.7 we

have A ∈ ℒ1,∞ . From the same reasoning, since diag({⟨Aen , en ⟩}∞ n=0 ) is V-modulated by Lemma 7.3.5, then diag({⟨Aen , en ⟩}∞ ) ∈ ℒ . From the Calkin correspondence, 1,∞ n=0 Theorem 2.4.3, we obtain {⟨Aen , en ⟩}∞ ∈ l . This proves part (a). 1,∞ n=0 Applying Lemma 7.3.7 to the operators A and iA, we obtain n

n

k=0 n

k=0 n

k=0

k=0

∑ λ(k, ℜA) − ∑ ⟨(ℜA)ek , ek ⟩ = O(1),

∑ λ(k, ℑA) − ∑ ⟨(ℑA)ek , ek ⟩ = O(1).

Since A ∈ ℒ1,∞ , it follows from Theorem 5.1.5 that n

n

n

k=0

k=0

k=0

∑ λ(k, ℜA) + i ∑ λ(k, ℑA) = ∑ λ(k, A) + O(1).

240 | 7 Diagonal formulas for traces Thus, n

n

k=0

k=0

∑ λ(k, A) − ∑ ⟨Aek , ek ⟩ = O(1).

(7.5)

This proves part (b). Applying (7.5) to the V-modulated operator A − diag({⟨Aen , en ⟩}∞ n=0 ), we obtain n

∞

∑ λ(k, A − diag({⟨Aen , en ⟩}n=0 )) = O(1),

k=0

and, using Theorem 5.1.5 again, we obtain part (c). Corollary 7.1.4 is proved from Theorem 7.1.3. Proof of Corollary 7.1.4. Suppose A is V-modulated where 0 ≤ V ∈ ℒ1,∞ has trivial kernel. Let {en }n≥0 be an eigenbasis for V ordered so that Ven = μ(n, V)en , n ≥ 0. By Theorem 7.1.3, (a) and (c), we have that A ∈ ℒ1,∞ , the operator diag{⟨Aen , en ⟩}∞ n=0 ∈ ℒ1,∞ , and that A − diag{⟨Aen , en ⟩}∞ ∈ [ ℒ , ℒ (H)]. Since all traces vanish on the 1,∞ n=0 commutator subspace, the assertion (a) follows. Let us now prove (b). Denote 2n −2

z(n) = ∑ λ(k, A) − ⟨Aek , ek ⟩, k=0

n ≥ 0.

By Theorem 7.1.3(b), we have z ∈ l∞ . Note that 2n+1 −2

2n+1 −2

k=2n −1

k=2n −1

∑ λ(k, A) − ∑ ⟨Aek , ek ⟩ = z(n + 1) − z(n),

n ≥ 0.

By Theorem 6.3.1, for every positive trace φ, we have n+1

φ(A) =

2 −2 1 θ( ∑ λ(k, A)), log(2) k=2n −1

where θ is a Banach limit on l∞ . Thus, n+1

2 −2 1 1 φ(A) = θ( ∑ ⟨Aek , ek ⟩) + θ(z(n + 1) − z(n)). log(2) k=2n −1 log(2)

However, θ(z(n + 1) − z(n)) = (θ ∘ T)(z) − θ(z) = 0. This completes the proof of (b).

7.4 Weyl asymptotics and expectations | 241

By Theorem 7.1.3(b), we have n n 1 1 1 ). ∑ λ(k, A) = ∑ ⟨Aek , ek ⟩ + O( log(2 + n) k=0 log(2 + n) k=0 log(2 + n)

Let ω be an extended limit on l∞ . Since ω vanishes on c0 , it follows that ω(

n n 1 1 ∑ λ(k, A)) = ω( ∑ ⟨Aek , ek ⟩). log(2 + n) k=0 log(2 + n) k=0

By the formula for Dixmier trace in Theorem 6.1.2, the left-hand side is Trω (A). This proves (c).

7.4 Weyl asymptotics and expectations Let φ be a trace on ℒ1,∞ and 0 ≤ V ∈ ℒ1,∞ satisfy φ(V) = 1. Define the linear functional Eφ (A) := φ(AV),

A ∈ ℒ(H),

on the algebra ℒ(H). In this section, we prove Theorem 7.1.5. The next lemma proves that Eφ is a singular state on ℒ(H) when φ is a positive trace. Lemma 7.4.1. If φ is a positive trace on ℒ1,∞ , then Eφ is a state on ℒ(H) that vanishes on 𝒞0 (H). Proof. Let A ∈ ℒ(H). The operator AV is V-modulated by Lemma 7.3.4 and Proposition 7.3.1. Suppose en , n ≥ 0, is an orthonormal basis such that Ven = λ(n, V)en . Applying Corollary 7.1.4 to the operator AV, we obtain φ(AV) = (φ ∘ diag)(⟨Aen , en ⟩λ(n, V)), where φ ∘ diag is a positive symmetric functional on l1,∞ . Suppose A is positive, then ⟨Aen , en ⟩λ(n, V) ≥ 0 and φ(AV) ≥ 0. Hence Eφ is a positive linear functional on ℒ(H). Evidently, Eφ (1) = φ(V) = 1, and Eφ is a state on ℒ(H).

242 | 7 Diagonal formulas for traces Now suppose that A is a compact operator. If pk , k ≥ 0, is the projection onto the Hilbert space spanned by {e0 , . . . , ek }, 󵄨 󵄨 sup󵄨󵄨󵄨⟨(pk Apk − A)en , en ⟩󵄨󵄨󵄨 → 0,

k → ∞.

n≥0

Now {⟨pk Apk en , en ⟩λ(n, V)}n≥0 is a sequence with finitely many nonzero terms for fixed k, and {⟨(pk Apk − A)en , en ⟩λ(n, V)}n≥0 → 0,

in l1,∞ when k → ∞.

Hence {⟨Aen , en ⟩λ(n, V)}n≥0 ∈ (l1,∞ )0 and, by Lemma 2.5.3, φ(AV) = (φ ∘ diag)(⟨Aen , en ⟩λ(n, V)) = 0 since the positive symmetric functional φ ∘ diag vanishes on (l1,∞ )0 . Restricting to normalized fully symmetric traces on ℒ1,∞ , define Eω (A) := Trω (AV),

A ∈ ℒ(H),

where ω is an extended limit on l∞ and Trω is the associated Dixmier trace on ℒ1,∞ . When lim kλ(k, V) = 1,

k→∞

we prove that Eω (A) = (ω ∘ M)(⟨Aen , en ⟩),

A ∈ ℒ(H),

where {en }n≥0 is an eigenbasis of V such that Ven = λ(n, V)en , n ≥ 0, and M is the logarithmic mean on l∞ . Proof of Theorem 7.1.5. By Lemma 7.4.1, Eω is a singular state on ℒ(H). Using Lemma 7.3.4 and Proposition 7.3.1, the operator AV is V-modulated. So, Corollary 7.1.4(c) provides the formula Trω (AV) = ω(

n 1 ∑ ⟨AVek , ek ⟩). log(n + 2) k=0

Evidently, ⟨AVek , ek ⟩ = ⟨Aek , ek ⟩ ⋅ λ(k, V),

k ≥ 0.

(7.6)

7.4 Weyl asymptotics and expectations | 243

Since (k + 1)λ(k, V) → 1,

k → ∞,

it follows that 1 1 + o( ), k+1 k+1

k → ∞.

⟨Aek , ek ⟩ + o(log(n)), k+1 k=0

n → ∞.

⟨AVek , ek ⟩ = ⟨Aek , ek ⟩ ⋅ Therefore, n

n

∑ ⟨AVek , ek ⟩ = ∑

k=0

(7.7)

Combining (7.6) and (7.7), we obtain Trω (AV) = ω(

n ⟨Aek , ek ⟩ 1 ) = (ω ∘ M)(⟨Aen , en ⟩). ∑ log(n + 2) k=0 k + 1

This has proved part (a). Part (b) follows from Lemma 6.2.4, since (ω ∘ M)(⟨Aen , en ⟩) = c for every extended limit ω on l∞ if and only if (lim ∘M)(⟨Aen , en ⟩) = c. The next example provides a sequence x ∈ l∞ such that, for the diagonal operator A = diag(x) ∈ ℒ(H) with diagonal matrix elements ⟨Aen , en ⟩ = x(n),

n ≥ 0,

the value Eω (A) = (ω ∘ M)(⟨Aen , en ⟩) depends on the extended limit ω. Example 7.4.2. Define x ∈ l∞ by setting ∞

x(n) = ∑ χ[222k ,222k+1 ) (n), k=0

Then the sequence Mx does not converge.

n ≥ 0.

244 | 7 Diagonal formulas for traces Proof. Consider the subsequence 22n

(Mx)(2 ) =

=

=

2n

22

1

xk + o(1) k+1 k=0

1

n−1 22

1

n−1

2n log(22 )

∑

2m+1

1 + o(1) ∑ ∑ 2n log(22 ) m=0 22m k + 1 22n

k=2

2m

∑ log(22 ) + o(1) =

log(2 ) m=0 1 = + o(1), n ≥ 0. 3

n−1 1 ∑ 22m log(2) + o(1) 22n log(2) m=0

Similarly, 2n+1

(Mx)(22

)=

2 + o(1), 3

n ≥ 0.

Since Mx has convergent subsequences which converge to different values, Mx is not convergent. The next example shows that, when A ∈ ℒ(H) is self-adjoint and Eω (A) = c is a value independent of the extended limit ω on l∞ , it need not be true that Eω (|A|) is a value independent of ω. This aspect of the noncommutative integral for the algebra ℒ(L2 (𝕋p )) contrasts with the restriction on the subalgebra L∞ (𝕋p ), p ∈ ℕ, where, for a real-valued bounded function f on 𝕋p , f is measurable and consequently Lebesgue integrable if and only if |f | is measurable and Lebesgue integrable. Example 7.4.3. There exists 0 ≤ V ∈ ℒ1,∞ with limk→∞ kλ(k, V) = 1 and a self-adjoint operator A ∈ ℒ(H) such that (a) Eφ (A) = 0 for every trace φ on ℒ1,∞ ; (b) ETrω (|A|) depends on the extended limit ω on l∞ . Proof. Take any 0 ≤ V0 ∈ ℒ1,∞ such that limk→∞ kλ(k, V0 ) = 1, and let V = 21 V0 ⊕ V0 . Take the positive sequence x ∈ l∞ from Example 7.4.2 and set A = diag(x) ⊕ (−diag(x)). It is immediate that 2Eφ (A) = 2φ(AV) = φ(diag(x)V0 ⊕ (−diag(x)V0 )) = φ(diag(x)V0 ⊕ 0) − φ(0 ⊕ diag(x)V0 ) = 0 for every trace φ on ℒ1,∞ .

7.5 Notes | 245

On the other hand, |A| = diag(x) ⊕ diag(x) and, therefore, ETrω (|A|) = Trω (diag(x)V0 ), which depends on the extended limit ω by Example 7.4.2.

7.5 Notes Sum of the diagonal The diagonal formula for Tr follows from the fact that ∞

∑ x(k),

k=0

x ∈ l1 ,

is a symmetric functional on l1 . Therefore, once the diagonal formula is true for one ordered eigenbasis, it follows for all rearrangements. The sum n 1 ∑ x(k), log(n + 2) k=0

x ∈ l1,∞ ,

is not a symmetric functional on l1,∞ . Only with rearrangement, n 1 ∑ λ(k, diag(x)), log(n + 2) k=0

x ∈ l1,∞ ,

is the formula a symmetric functional. It should be expected, therefore, that diagonal formulas for Dixmier traces can only hold with some control on the ordering of the expectation values, which is what the modulated condition of this chapter provides. Motivation and modulated operators That the Laplace–Beltrami operator Δ : C ∞ (Ω) → C ∞ (Ω) on a closed Riemannian manifold extends to a positive self-adjoint unbounded operator Δ : H2 (Ω) → L2 (Ω) with compact resolvent can be found in any book on spectral geometry, e. g., [162, p. 198], [246, Section 8], [227, p. 32], and [218, p. 255] for the Dirichlet Laplacian on a bounded open set of ℝp . Here H2 (Ω) denotes the Sobolev Hilbert space. It follows that there exists an orthonormal basis of smooth functions {en }n≥0 where Δen = λ(n, Δ)en , n ≥ 0 for a decreasing sequence of numbers λ(n, Δ) called the eigenvalues of Δ, e. g., [162, p. 198], [246, Theorem 8.3], and [218, Theorem XIII.64]. The asymptotic properties of the smoothness and Lp -bounds of the eigenfunctions en as n → ∞ are well studied, see, e. g., [185, 273, 297] and the lecture notes [303]. In the case of the torus, the sequence {⟨Mf en , en ⟩}n≥0 trivially converges for every f ∈ L1 (𝕋p ) (and, hence, for f ∈ C ∞ (𝕋p )). For many compact manifolds Ω, it is known that {⟨Mf en , en ⟩}n≥0 ,

f ∈ C ∞ (Ω),

246 | 7 Diagonal formulas for traces has different convergent subsequences, which converge to different measures other than the measure induced by the volume form on Ω [87]. Theorem 7.1.5(b), for smooth functions on manifolds, repeats the known fact that the expectation values always converge in Cesaro mean to the measure induced by the volume form [87, Lemma 2.2]. Hence, see the notes on divergent series below, it is well known for smooth functions that the expectation values converge in logarithmic mean to the measure induced by the volume form. The standard approach to proving this fact utilizes the heat trace. These results are consequences of what is known as microlocal spectral asymptotics in spectral geometry [54, 142, 272]. If f ∈ L∞ (Ω), then Mf ∈ ℒ(L2 (Ω)). This chapter proved the estimate for the partial sums n

n

p

p

∑ λ(k, Mf (1 − Δ)− 2 ) − ∑ ⟨Mf (1 − Δ)− 2 )ek , ek ⟩ = O(1),

k=0

k=0

n ≥ 0.

Here {en }n≥0 is the sequence of eigenvectors of the Laplace–Beltrami operator Δ such that Δen = λ(n, Δ)en , n ≥ 0, for the decreasing sequence of eigenvalues λ(n, Δ). Even though the eigenvectors of the Laplace–Beltrami operator are difficult to specify for Ω in general, Volume II proves a fundamental formula between sums of expectation values and an integral of an L2 -symbol of a Hilbert–Schmidt operator over cylinders in phase space. The concept of a modulated operator was introduced by Kalton, with the first and second authors, in [155]. The estimates for singular values and investigation that we know of which is closest to the modulation condition is in analysis using Gabor frames, see [132]. The expectation valued approach to traces has also been used in Pietsch’s work on traces on Banach ideals. Operators that are modulated by the Laplace–Beltrami operator are studied further in Volume II. There, see also Proposition 11.7.3 in combination with Example 11.3.8 in [170], an example is given of a bounded function f on the circle such that (1 − Δ)−1/2 Mf is not (1 − Δ)−1/2 -modulated. Given an orthonormal basis {en }n≥0 of a separable Hilbert space, the diagonal formulas in Theorem 7.1.3 and Corollary 7.1.4 hold for all operators A ∈ ℒ(H) such that ∞

∑ ‖Aek ‖2 = O(n−1 ).

k=n+1

(7.8)

Let 0 ≤ V ∈ ℒ1,∞ be such that Ven = μ(n, V)en , n ≥ 0, for the chosen basis and μ(n, V) ∼ n−1 (Weyl asymptotics). Then A ∈ ℒ(H) is V-modulated if and only if (7.8) holds [170, Lemma 12.2.3], [155, Proposition 5.13]. Integrals as expectations involving singular traces In [60, §VI], [61, 63, 71], Connes introduced his quantized calculus with the compact 1 operator ⟨D⟩−1 := (1 + D2 )− 2 on a separable Hilbert space H as the analogue of an infinitesimal length element ds. The self-adjoint operator D is a generalization of a Dirac

7.5 Notes | 247

type operator [162]. The pair (D, H) have been termed unbounded Fredholm modules over Frechet ∗ -algebras 𝒜 ⊂ ℒ(H) such that [D, A] ∈ ℒ(H), A ∈ 𝒜 [16, 59], or K-cycles over the C∗ -closure of 𝒜 [60, p. 546]. The Connes–Moscovici approach to pseudodifferential calculus where D2 replaces the role of the Laplacian operator was introduced in [71] and [61]. We refer the reader to [42] and [41] for semifinite versions of this calculus. Connes introduced the formula [60, p. 545] Trω (A⟨D⟩−p ),

A ∈ 𝒜,

as a noncommutative integral with density ⟨D⟩−p ∈ ℒ1,∞ , also termed an infinitesimal of order one. That the noncommutative integral extends integration for smooth functions on closed manifolds was shown in [57, pp. 157–158] by Connes’ extension of Wodzicki’s noncommutative residue. The term “noncommutative integral” was established by Connes and expositions such as [18, 69, 116, 284], where [18, p. 34] states “This led Connes to introduce the Dixmier trace as the correct operator theoretical substitute for integration of infinitesimals of order one in non-commutative geometry.”

Observations between the noncommutative integral and the logarithmic mean of expectation values were first noted in [167]. The expectation formula for the noncommutative integral appeared in [170, Theorem 12.1.2]. It extended [167] and [155]. For the notion of quantum ergodicity, see [53, 86, 87, 231, 245, 302, 303]. Divergent series Chapters 7 and 9 refer to arithmetic and logarithmic mean convergence of bounded sequences. Hardy in [127, §3.8] illustrates the differences between the Cesaro and logarithmic means. In particular, the logarithmic means are less trivial than the Cesaro means, meaning that more sequences converge logarithmically than arithmetically. If p = {p(n)}∞ n=0 > 0 and ∑ p(n) diverges, Hardy defines the means (N, p) : l∞ → l∞ [127, p. 57], ((N, p)(x))(n) =

∑nj=0 x(j)p(j) ∑nj=0 p(j)

,

∞

x = {x(j)}j=0 ∈ l∞ , n ≥ 0.

As ∑ p(n) diverges more quickly, the mean (N, p) becomes more trivial, until reaching the point that only convergent sequences end up being (N, p)-convergent [127, Theorem 15]. The Cesaro means are the (N, 1) means and the logarithmic means are the (N, (1 + n)−1 ) means. The term “logarithmic mean” originates from Hardy [127, p. 59]. If x(n) → a in (N, 1) then x(n) → a in (N, (1 + n)−1 ) [127, Theorem 14]. The logarithmic means are

248 | 7 Diagonal formulas for traces equivalent to Riesz’s typical means (R, log(n + 1), 1) [127, Theorem 37], and the Cesaro means are equivalent to Riesz’s typical means (R, n, 1) [127, Theorem 58]. Theorem 7.1.5(b) is a form of quantum ergodicity in the sense of the uniqueness of the noncommutative integral when the sequence {⟨Aen , en ⟩}n≥0 is (M, 1), or, equivalently, (N, (1 + n)−1 )-convergent. There are stronger forms of convergence of {⟨Aen , en ⟩}n≥0 for some operators A ∈ ℒ(H), which are explored further in Chapter 9. An exact statement for when a logarithmically convergent sequence is a convergent sequence is known [189, Theorem 5]: a logarithmically convergent sequence x = {x(n)}∞ n=0 ∈ l∞ is a convergent sequence if and only if 󵄨󵄨 󵄨 [nr ] 󵄨󵄨 x(k) − x(n) 󵄨󵄨󵄨󵄨 1 lim+ lim sup󵄨󵄨󵄨 r ∑ 󵄨󵄨 = 0. 󵄨󵄨 k r→1 n→∞ 󵄨󵄨 ([n ] − n) log(1 + n) k=n+1 󵄨 󵄨 Here [nr ] denotes the ceiling of nr , n ≥ 1, r > 1. The more testable Tauberian criterion of slowly oscillating real sequences was shown by B. Kwee in [161]. Equivalent criterion for slow oscillation is given in [189].

8 Heat trace and ζ-function formulas 8.1 Introduction The calculation of a Dixmier trace is seldom done using the spectral formula of Chapter 6, or the diagonal formula of Chapter 7, directly. Generally, neither the eigenvalue sequence nor the sequence of eigenfunctions of physical operators such as the Laplace–Beltrami operator are known. The kernels of these physical operators involve the eigenvalues and eigenfunctions, and information on the spectral asymptotic behavior can be extracted from the kernel by integration. These observations go back to Mercer and Hilbert and were exploited in the ζ -function and heat kernel approach of Minakshisundaram and Pleijel to spectral geometry. Many applications in Alain Connes’ noncommutative geometry follow the same approach as Minakshisundaram and Pleijel, at least up to first order, to calculate the Dixmier trace of operators on the ideal ℒ1,∞ . This chapter shows that the Dixmier trace extends the residue at the first simple pole of a spectral ζ -function, and it extends the leading term in an asymptotic expansion of a heat trace. Motivation Let Δ be the Laplace–Beltrami operator on a closed p-dimensional Riemannian manifold Ω, with nonincreasing eigenvalue sequence λ(n, Δ), n ≥ 0, and a smooth basis of eigenfunctions en ∈ C ∞ (Ω), n ≥ 0, such that Δen = λ(n, Δ)en . If f : [0, ∞) → [0, ∞) is a positive continuous decreasing function such that f (−Δ) ∈ ℒ1 , the square integrable kernel K ∈ L2 (Ω, Ω) of the Hilbert–Schmidt operator f (−Δ) is ∞

K(x, y) = ∑ f (λ(n, −Δ))en (x)en (y), n=0

x, y ∈ Ω,

by the spectral theorem. Since en are smooth, the kernel is sufficiently well behaved on the diagonal of Ω × Ω so that Tr(f (−Δ)) = ∫ K(x, x) dx. Ω

Integration over the kernel replaces direct calculations involving eigenvalues and eigenfunctions, and can extract information about the spectral behavior of the operator f (−Δ). The integration over Ω is concrete in the sense that it reduces locally to integration of symbols on ℝp involving the metric. With a concrete method to calculate Tr(fs (−Δ)) for each s > 0 for a parametrized family {fs }s>0 of functions with fs (−Δ) ∈ ℒ1 , asymptotic spectral behavior can be studied by the behavior s → 0+ of the value Tr(fs (−Δ)) for particular choices of fs . https://doi.org/10.1515/9783110378054-010

250 | 8 Heat trace and ζ -function formulas Most readers will recognize formula (6.2) from the introduction to Chapter 6 as the Minakshisundaram and Pleijel ζ -function p

ζ (s) = Tr((1 − Δ)− 2 (1+s) ),

s > 0,

p

where (1 − Δ)− 2 belongs to ℒ1,∞ by Weyl’s law. The ζ -function is of the form described p above, where the family of functions is fs (t) = (1 + t)− 2 (1+s) , t > 0. Minakshisundaram and Pleijel proved in 1949 that the ζ -function had a meromorphic continuation to the entire complex plane. Residues at the poles of the meromorphic extension relate to the noncommutative residue in mathematics, and zetafunction regularization and Seeley–DeWitt numbers in mathematical physics. The residue at the first pole of ζ describes the asymptotic behavior p

Ress=0 ζ (s) = lim+ sTr((1 − Δ)− 2 (1+s) ). s→0

It was well-known, again from Weyl’s law, that the residue at the first pole is proportional to the volume Vol(Ω) which, as described in Chapter 6, is equal to the value p p Trω ((1 − Δ)− 2 ) provided by a Dixmier trace applied to the operator (1 − Δ)− 2 ∈ ℒ1,∞ . Seeley extended Minakshisundaram and Pleijel’s ζ -function in 1967 by proving that the zeta function of any order −p strictly positive elliptic pseudodifferential operator on Ω has a meromorphic extension from Re(s) > 0 with a simple pole at s = 0. This was extended to any classical pseudodifferential operator of order −p on Ω by the noncommutative residue Res developed by Wodzicki in 1983 and Guillemin in 1985. Let P be an order −p strictly positive elliptic classical pseudodifferential operator on Ω (satisfying certain technical conditions which are not important here). Then the ζ -function ζQ,P (s) = Tr(QP 1+s ),

Q ∈ Ψ0 (Ω) ⊂ ℒ(L2 (Ω)),

is meromorphic with simple poles at s = − pk , k ≥ 0. Here Ψ0 (Ω) denotes the algebra of order 0 classical pseudodifferential operators on Ω. If Q ∈ Ψ0 (Ω), then Q : C ∞ (Ω) → C ∞ (Ω) has a unique extension to a bounded operator, denoted by the same symbol, Q : L2 (Ω) → L2 (Ω). Hearing of the existence of singular traces from Dixmier in 1985 – an exchange described by Dixmier in the end notes to Chapter 6 – Connes effectively showed in 1988 that Trω (QP) = lim+ sζQ,P (s), s→0

Q ∈ Ψ0 (Ω) ⊂ ℒ(L2 (Ω)).

(8.1)

The formula in (8.1) is called the ζ -function residue formula for the Dixmier trace. Minakshisundaram published a shorter proof in 1953 of the meromorphicity of the zeta function of the Laplace–Beltrami operator Δ on a closed manifold Ω. He used the Mellin transform to relate the asymptotics of the diagonal of the heat kernel and

8.1 Introduction | 251

the poles of the meromorphic continuation of the zeta function. For each s > 0, the operator esΔ : L2 (Ω) → C ∞ (Ω) is infinitely smoothing, is of trace class, and has smooth real kernel K(s, ⋅, ⋅) ∈ C ∞ (Ω × Ω) of the form ∞

K(s, x, y) = ∑ e−sλ(n,−Δ) en (x)en (y), n=0

s > 0, x, y ∈ Ω,

using the family of functions fs (t) = e−st , t > 0, in the argument above. Minakshisundaram identified an asymptotic expansion in s of the heat kernel on the diagonal of Ω × Ω, p

∞

K(s, x, x) ∼ s− 2 ∑ ak (x)sk , k=0

s → 0+ , x ∈ Ω,

where ak (x) ∈ C ∞ (Ω), k ≥ 0, can be expressed combinatorially in terms of the derivatives of the coefficients of the symbol of Δ. The heat kernel coefficients ak (x), k ≥ 0, are invariant under isometry and hence local invariants of the manifold Δ. The asymptotic expansion of the kernel becomes an asymptotic expansion of the heat trace p

∞

Tr(esΔ ) = ∫ K(s, x, x) dx ∼ s− 2 ∑ ak sk , k=0

Ω

s → 0+ ,

where the sequence of Seeley–DeWitt numbers ak = ∫ ak (x) dx,

k ≥ 0,

Ω

are the integrals of the heat kernel coefficients. Minakshisundaram could calculate the first two coefficients for the Laplace–Beltrami operator p

p

Tr(esΔ ) ∼ (4π)− 2 s− 2 (Vol(Ω) +

1 ∫ Rg (x) dx ⋅ s + O(s2 ) + ⋅ ⋅ ⋅), 6

s → 0+ ,

(8.2)

Ω

where Rg (x) denotes the scalar curvature of the metric g at the point x ∈ Ω. So the invariants of dimension, volume, and total scalar curvature of a closed Riemannian manifold are contained in the expansion. Further coefficients depend on combinatorial terms and derivatives of the metric. The leading term in the expansion is proportional to the volume Vol(Ω) and, hence, as discussed in Chapter 6, is equal to the value p p Trω ((1 − Δ)− 2 ) provided by a Dixmier trace applied to the operator (1 − Δ)− 2 ∈ ℒ1,∞ . Seeley showed that Minakshisundaram’s heat kernel expansion holds for a positive elliptic pseudodifferential operator of order 2 on the closed manifold Ω. Let P be

252 | 8 Heat trace and ζ -function formulas an order −p strictly positive elliptic classical pseudodifferential operator on Ω, so that

the power P

− p2

is of order 2. From the calculations of Gilkey, Tr(Qe−sP

− p2

Q ∈ Ψ0 (Ω) ⊂ ℒ(L2 (Ω)),

),

p

also has an asymptotic expansion as s → 0+ with coefficients ak (Q, P) of s− 2 +k , k ≥ 0, in the expansion. It follows by the Mellin transform that Trω (QP) =

2 p 1 −sP − p 2 ), p lim+ s Tr(Qe Γ(1 + 2 ) s→0

Q ∈ Ψ0 (Ω) ⊂ ℒ(L2 (Ω)),

(8.3)

and the leading term in the heat trace expansion is proportional to the Dixmier trace. The coefficient involving Γ(1 + p2 ) in (8.3) can be absorbed into the heat trace formula by the changes of variables, e−t

−2/p

p

󳨃→ e−t , t ≥ 0, and s 󳨃→ s− 2 , s > 0, −1

1

Trω (QP) = lim+ sTr(Qe−( s P) ), −1

s→0

Q ∈ Ψ0 (Ω) ⊂ ℒ(L2 (Ω)).

(8.4)

The formula in (8.4) is called the heat trace formula for the Dixmier trace. Heat trace formula The motivation showed in formula (8.4) on a p-dimensional closed Riemannian manifold Ω that the Dixmier trace of QP, where Q ∈ Ψ0 (Ω) is a classical zero-order pseudodifferential operator with extension to a bounded operator in ℒ(L2 (Ω)) and P ∈ ℒ1,∞ is the extension of a strictly positive elliptic pseudodifferential operator of order −p, coincides with the leading term in the heat trace expansion. This chapter proves that the Dixmier trace of a product AV ∈ ℒ1,∞ is associated to a similar formula, where A ∈ ℒ(H) is any bounded operator, and 0 ≤ V ∈ ℒ1,∞ is any positive weak trace class operator. When 0 ≤ V ∈ ℒ1,∞ , we can define the operator −1

e−(nV) ∈ ℒ1 ,

n ≥ 0,

by the functional calculus, understanding that e−(nV) η = 0 for every η ∈ ker(nV). By −1 this convention, e−(nV) = 0 when n = 0, and, for convenience of notation, we set −1

−1 1 Tr(e−(nV) A) = 0, n

n = 0, A ∈ ℒ(H).

Definition 8.1.1. For an extended limit ω on l∞ , define a positive functional ξω on the positive cone of ℒ1,∞ by −1 1 ξω (V) := ω( Tr(e−(nV) )), n

0 ≤ V ∈ ℒ1,∞ .

For A ∈ ℒ(H), define a functional on the positive cone of ℒ1,∞ by −1 1 ξω,A (V) := ω( Tr(e−(nV) A)), n

We call ξω and ξω,A heat trace functionals.

0 ≤ V ∈ ℒ1,∞ .

8.1 Introduction | 253

Example 6.2.5 describes the embedding of extended limits on l∞ within extended limits on L∞ (0, 1) supported at 0. The right-hand side of (8.4) is therefore generalized by a heat trace functional 1

ξω,Q (P) = lim+ sTr(Qe−( s P) ), −1

s→0

Q ∈ Ψ0 (Ω),

for any extended limit ω, where P ∈ ℒ1,∞ is the extension of a strictly positive elliptic pseudodifferential operator of order −p. Evidently, ξω = ξω,1 . For arbitrary A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ , that the sequence −1 1 { Tr(e−(nV) A)} ∈ l∞ n n≥0

is bounded and ξω,A (V) is well defined is demonstrated in Lemma 8.3.1. Let M : l∞ → l∞ denote the logarithmic mean (Mx)(n) =

n 1 x(k) , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

The composition ω ∘ M is an extended limit on l∞ by Lemma 6.2.12. Theorem 8.1.2 (Heat trace formula). Suppose A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ . (a) For each extended limit ω on l∞ , −1 1 Trω (AV) = ξω∘M,A (V) = (ω ∘ M)( Tr(Ae−(nV) )), n

where Trω is a Dixmier trace from Theorem 6.1.2. (b) Setting A = 1, Trω (V) = ξω∘M (V),

0 ≤ V ∈ ℒ1,∞ ,

and each heat trace functional ξω∘M , ω being an extended limit on l∞ , defines a trace on ℒ1,∞ by linear extension from the positive cone of ℒ1,∞ to the whole ideal. The set of heat trace functionals ξω∘M , ω being an extended limit on l∞ , coincides with the set of Dixmier traces on ℒ1,∞ . Theorem 8.1.2 is proved for A = 1 in Section 8.2, and for arbitrary A ∈ ℒ(H) in Section 8.3. Theorem 8.1.2 provides the third characterization of Dixmier traces discussed in Chapter 6, they correspond to the set of heat trace functionals defined by extended limits of the form ω ∘ M for an extended limit ω on l∞ . Definition 8.1.1 defined a heat trace functional ξω on the positive cone of ℒ1,∞ for any extended limit ω on l∞ . The functional ξω need not be additive on the positive cone of ℒ1,∞ . The set of heat trace functionals of the form ξω∘M , with ω an extended limit on l∞ , describe all the heat trace functionals that are additive on the positive cone of

254 | 8 Heat trace and ζ -function formulas ℒ1,∞ , as explained in the end notes of this chapter. Thus, the Dixmier trace, generally,

is the “leading term of the asymptotic expansion” of the heat trace −1

Tr(e−sV ),

s → 0+ , 0 ≤ V ∈ ℒ1,∞ ,

when there are no well-defined asymptotics. We changed variables to convert the Minakshisundaram form of the heat trace formula for closed manifolds in (8.3) into a normalized version (8.4). Theorem 8.3.2 in Section 8.3 shows that a similar change of variable holds for heat trace functionals. It demonstrates that −1

f (t) = e−t ,

t > 0,

is a convenient choice, but the formula is the same, up to normalization, using any bounded function f ∈ C 2 [0, ∞) such that f (0) = f 󸀠 (0) = 0. Using −q

f (t) = e−t ,

t > 0, q > 0,

in Theorem 8.3.2 we can revert back to the form in (8.3). Corollary 8.1.3. If A ∈ ℒ(H), 0 ≤ V ∈ ℒ1,∞ , and ω is an extended limit on l∞ , then Trω (AV) =

1

Γ(1 +

1 ) q

−q 1 (ω ∘ M)( Tr(Ae−(nV) )), n

q > 0.

Here Γ is the Gamma function. ζ-function residue formula The motivation showed in the ζ -function residue formula (8.1) on a p-dimensional closed Riemannian manifold Ω that the Dixmier trace of QP, where Q ∈ Ψ0 (Ω) is a classical zero-order pseudodifferential operator with extension to a bounded operator in ℒ(L2 (Ω)) and P ∈ ℒ1,∞ is the extension of a strictly positive elliptic pseudodifferential operator of order −p, coincides with the residue at s = 0 of the first simple pole of the meromorphic continuation of the ζ -function ζQ,P . This chapter proves that similar behavior of the function ζA,V (s) = Tr(AV 1+s ),

s > 0,

where A ∈ ℒ(H) is any bounded operator, and 0 ≤ V ∈ ℒ1,∞ is any positive weak trace class operator, defines a Dixmier trace for the product AV ∈ ℒ1,∞ . The operator V 1+s ∈ ℒ1 , is defined by the functional calculus.

s > 0,

8.1 Introduction | 255

Definition 8.1.4. (a) For A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ , the function ζA,V (s) := Tr(AV 1+s ),

s > 0,

is called the ζ -function associated to A and V. (b) For an extended limit γ on L∞ (0, 1) concentrated at 0, define a positive functional on the positive cone of ℒ1,∞ by ζγ (V) := γ(sζ1,V (s)),

0 ≤ V ∈ ℒ1,∞ .

For A ∈ ℒ(H), define a functional on the positive cone of ℒ1,∞ by ζγ,A (V) := γ(sζA,V (s)),

0 ≤ V ∈ ℒ1,∞ .

We call ζγ and ζγ,A a ζ -function residue functional. Extended limits on L∞ (0, 1) concentrated at 0 were introduced in Definition 6.2.3. The term ζ -function arises in Definition 8.1.4(a) from the fact that, for A = 1 and V the positive diagonal operator diag({

∞

1 } ) ∈ ℒ1,∞ , n + 1 n=0

we have ζ (1 + s) = Tr(diag({

∞

1+s

1 } ) n + 1 n=0

),

s > 0,

where ζ is the Riemann ζ -function. The right-hand side of (8.1) is generalized by a ζ -function residue functional ζγ,Q (P) = lim+ sTr(QP 1+s ), s→0

Q ∈ Ψ0 (Ω),

for any extended limit γ on L∞ (0, 1) supported at 0, where P ∈ ℒ1,∞ is the extension of a strictly positive elliptic pseudodifferential operator of order −p. Evidently, ζγ = ζγ,1 . For arbitrary A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ , that the function sTr(AV 1+s ) ∈ L∞ (0, 1),

0 < s < 1,

is bounded and ζγ,A (V) is well defined is demonstrated in Lemma 8.4.1. Chapter 6, in Theorems 6.1.1 and 6.1.3, explained the bijective correspondence between Dixmier traces on ℒ1,∞ and factorizable Banach limits on l∞ . In particular, the formula n+1

φθ (A) =

2 −2 1 ⋅ θ( ∑ λ(k, A)), log(2) k=2n −1

A ∈ ℒ1,∞ ,

256 | 8 Heat trace and ζ -function formulas describes every Dixmier trace on ℒ1,∞ where θ = ω ∘ C, for some extended limit ω on l∞ , is a factorizable Banach limit. Theorem 8.1.2 shows that every Dixmier trace is associated to a heat trace functional. The situation for ζ -function residues is more complicated, and not every Dixmier trace is described by a ζ -function residue. Theorem 8.1.5 (ζ -function formula). Suppose A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ . (a) For each extended limit γ on L∞ (0, 1) concentrated at 0, there is a factorizable Banach limit θγ (x) := log(2) ⋅ γ(s ∑ 2−ks x(k)), k≥0

x ∈ l∞ ,

such that φθγ (AV) = ζγ,A (V) = γ(sTr(AV 1+s )) for the Dixmier trace φθγ on ℒ1,∞ . (b) Setting A = 1, φθγ (V) = ζγ (V),

0 ≤ V ∈ ℒ1,∞ ,

and each ζ -function residue describes a trace on ℒ1,∞ by linear extension from the positive cone of ℒ1,∞ to the whole ideal. The set of ζ -function residues ζγ , γ an extended limit on L∞ (0, 1) concentrated at 0, forms a subset of Dixmier traces φθγ , γ an extended limit on L∞ (0, 1) concentrated at 0. (c) There exists a factorizable Banach limit θ on l∞ such that θ ≠ θγ for any extended limit γ on L∞ (0, 1) supported at 0. Hence, the set of Dixmier traces is strictly larger than the set of ζ -function residues. Theorem 8.1.5(b) is proved in Section 8.4. Theorem 8.1.5, (a) and (c), are proved in Section 8.5. Theorem 8.1.5 provides the identification of every Dixmier trace that can be written as a ζ -function residue formula using Pietsch’s formula for a Dixmier trace. The end notes to this chapter discuss Dixmier’s formula for a Dixmier trace, that is, when, for certain extended limits ω on l∞ , there is an extended limit γω on L∞ (0, 1) supported at 0 such that Trω (V) = ζγω (V), 0 ≤ V ∈ ℒ1,∞ .

8.2 Heat trace functionals and Dixmier traces This section proves that a heat trace functional −1 1 ξω∘M (V) := (ω ∘ M)( Tr(e−(nV) )), n

0 ≤ V ∈ ℒ1,∞ ,

8.2 Heat trace functionals and Dixmier traces | 257

from Definition 8.1.1, where ω is an extended limit on l∞ and M : l∞ → l∞ is the logarithmic mean (Mx)(n) =

n 1 x(k) , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0,

is equal to the value of the Dixmier trace Trω (V) = ω(

n 1 ∑ λ(k, V)) log(n + 2) k=0

from Theorem 6.1.2. There are several steps to the proof that the heat trace functional and the Dixmier trace coincide on the positive cone of ℒ1,∞ , and it involves two intermediate expressions for the Dixmier trace which can be observed in Lemmas 8.2.2 and 8.2.4 below. When the positive compact operator 0 ≤ V ∈ ℒ1,∞ does not have trivial kernel, the following conventions are assumed. When writing e−(nV) , we assume that this operator acts trivially on ker(nV) when n ≥ 0. Additionally, we assume −1

−1 1 Tr(e−(nV) ) = 0, n

n = 0,

to save the notational burden of assigning a different formula to the n = 0 term whose contribution will always vanish under an extended limit on l∞ . The following lemma shows that, given our conventions, the sequence −1 1 ∈ l∞ { Tr(e−(nV) )} n n≥0

and the heat functional is well defined by an extended limit on l∞ applied to this bounded sequence. Lemma 8.2.1. If 0 ≤ V ∈ ℒ1,∞ , then −1

Tr(e−(nV) ) = O(n),

n ≥ 0.

Proof. Assume, without loss of generality, that ‖V‖1,∞ ≤ 1. That is, μ(k, V) ≤ The function

1 , k+1 −1

t → e−t ,

k ≥ 0.

t > 0,

is increasing. Thus, Tr(e−(nV) ) = Tr(e−(n diag(μ(V))) ) = ∑ e−(nμ(k,V)) −1

−1

−1

k≥0

≤ ∑e k≥0

n −1 −( k+1 )

= ∑e k≥0

− k+1 n

∞

t

≤ ∫ e− n dt = n. 0

258 | 8 Heat trace and ζ -function formulas Lemma 8.2.2 below provides a more useful formula for the heat trace functional. Lemma 8.2.2. If ω is an extended limit on l∞ , then ξω∘M (V) = ω(

−1 1 Tr(Ve−(nV) )), log(n + 2)

0 ≤ V ∈ ℒ1,∞ .

Proof. By the definition of logarithmic mean M, we have M(k 󳨃→

n −1 −1 1 1 1 Tr(e−(kV) ))(n) = Tr(e−(kV) ). ∑ k log(n + 2) k=0 k(k + 1)

(8.5)

The 0th summand on the right-hand side is considered to be 0, using the convention that −1 1 Tr(e−(kV) ) = 0, k

k = 0.

Step 1. We claim that n

−1 −1 1 Tr(e−(kV) ) ≥ Tr(Ve−(nV) ) + O(1), k(k + 1) k=0

n → ∞.

∑

Clearly, −1

−1

Tr(e−(kV) ) ≥ Tr(e−(tV) ),

t ∈ (k − 1, k).

Integrating over t ∈ (k − 1, k), we obtain k

k

k−1

k−1

−1 dt −1 −1 1 1 Tr(e−(kV) ) ≥ ∫ Tr(e−(tV) ) dt ≥ ∫ Tr(e−(tV) ) 2 . (k − 1)2 (k − 1)2 t

Summing over 2 ≤ k ≤ n and substituting t = s−1 , we arrive at n

n

−1 −1 dt 1 Tr(e−(kV) ) ≥ ∫ Tr(e−(tV) ) 2 ∑ 2 (k − 1) t k=2

1

1

1

−1

−1

= ∫ Tr(e−sV ) ds = Tr(∫ e−sV ds) 1 n

1 n

∞

= Tr( ∫ e

∞ −sV −1

−1

ds) − Tr( ∫ e−sV ds). 1

1 n

Taking into account that ∞ −1

−1

∫ e−sV ds = Ve−uV , u

(8.6)

8.2 Heat trace functionals and Dixmier traces | 259

where the integral is understood in the Bochner sense in ℒ1 , we arrive at n

−1 −1 −1 1 Tr(e−(kV) ) ≥ Tr(Ve−(nV) ) − Tr(Ve−V ). 2 (k − 1) k=2

∑

By Lemma 8.2.1, we have n −1 −1 1 1 Tr(e−(kV) ) + O(1), Tr(e−(kV) ) = ∑ 2 k(k + 1) (k − 1) k=0 k=2 n

n ∈ ℕ.

∑

This completes the proof of (8.6). Step 2. We claim that n

−1 −1 1 Tr(e−(kV) ) ≤ Tr(Ve−(nV) ) + O(1), k(k + 1) k=0

∑

n ∈ ℕ.

(8.7)

Clearly, −1

−1

Tr(e−(kV) ) ≤ Tr(e−(tV) ),

t ∈ (k, k + 1).

Integrating over t ∈ (k, k + 1), we obtain k+1

k+1

k

k

−1 dt −1 −1 1 1 Tr(e−(kV) ) ≤ ∫ Tr(e−(tV) ) dt ≤ ∫ Tr(e−(tV) ) 2 . 2 2 (k + 1) (k + 1) t

Summing over 0 ≤ k ≤ n, we arrive at n+1

n

−1 dt −1 1 Tr(e−(kV) ) ≤ ∫ Tr(e−(tV) ) 2 ∑ 2 (k + 1) t k=0

0

1 n

∞

= ∫ Tr(e

−sV −1

) ds = ∫ Tr(e

1 n+1

∞ −sV −1

−1

) ds + ∫ Tr(e−sV ) ds

1 n+1

1 n

−1

= Tr(Ve−((n+1)V) ) + O(1). By Lemma 8.2.1, we have n

n −1 1 1 −(kV)−1 Tr(e ) = Tr(e−(kV) ) + O(1), ∑ 2 k(k + 1) (k + 1) k=0 k=0

∑

n ∈ ℕ.

This completes the proof of (8.7). Step 3. Combining Steps 1 and 2, we arrive at n

−1 −1 1 Tr(e−(kV) ) = Tr(Ve−(nV) ) + O(1), k(k + 1) k=0

∑

n ∈ ℕ.

260 | 8 Heat trace and ζ -function formulas In particular, n −1 −1 1 1 1 Tr(e−(kV) ) = Tr(Ve−(nV) ) + o(1), ∑ log(n + 2) k=0 k(k + 1) log(n + 2)

n → ∞.

Since ω vanishes on c0 , the assertion follows. The advantage of the formula in Lemma 8.2.2 involving the sequence of trace −1 class operators Ve−(nV) , n ≥ 0, is seen in Lemma 8.2.4. Advantages to the formula in Lemma 8.2.2 will also be observed in Chapter 9 and Volume II. Lemma 8.2.4 proves that −1

Tr(Ve−(nV) ) − Tr((V −

1 ) ) = O(1), n +

n ≥ 0.

Here (V − n1 )+ is the finite-rank operator (V −

1 1 1 ) = (V − )EV [ , ∞), n + n n

n ≥ 0,

where EV is the spectral projection of the positive operator V from Chapter 2. Since V is compact then EV [ n1 , ∞) is a finite-rank projection for every n ≥ 0. Lemma 8.2.3 provides technical estimates for the proof of Lemma 8.2.4. In the statement of Lemma 8.2.3, the operator Φ((nV)−1 ),

n ≥ 1,

for a bounded function Φ : (0, ∞) → (0, 1) is defined using the bounded functional calculus. For η ∈ H but η ∉ ker(V), define ∞ −1

Φ((nV) )η = ∫ Φ((nλ)−1 ) dEV (λ) ⋅ η, 0

where EV (λ) is the projection-valued spectral measure of V from Theorem 2.1.6 in Chapter 2. For η ∈ ker(V), set Φ((nV)−1 )η = 0. Similarly, 1 min{VΦ((nV)−1 ), }, n

n ≥ 1,

is defined using the bounded functional calculus. For η ∈ H but η ∉ ker V, define ∞

1 1 min{VΦ((nV) ), }η = ∫ min{λΦ((nλ)−1 ), } dEV (λ) ⋅ η. n n −1

0

8.2 Heat trace functionals and Dixmier traces | 261

For η ∈ ker(V), set 1 min{VΦ((nV)−1 ), }η = 0. n Lemma 8.2.3. Let Φ : (0, ∞) → (0, 1) be a function such that (i) Φ is convex, decreasing, and positive; (ii) Φ(0) = 1 and ∞

dt < ∞, ∫ Φ(t) t 1

1

dt < ∞. t

∫(1 − Φ(t)) 0

For every 0 ≤ V ∈ ℒ1,∞ , we have (a) 󵄩󵄩 󵄩 󵄩󵄩 󵄩 −1 1 󵄩 󵄩󵄩min{VΦ((nV) ), }󵄩󵄩󵄩 = O(1), n 󵄩󵄩1 󵄩󵄩

n → ∞,

(b) 󵄩 󵄩󵄩 1 󵄩 󵄩󵄩 −1 󵄩 󵄩󵄩(V − ) (1 − Φ((nV) ))󵄩󵄩󵄩 = O(1), 󵄩󵄩1 󵄩󵄩 n +

n → ∞.

Proof. For simplicity of computations, let ‖V‖1,∞ = 1. Let W ∈ ℒ1,∞ be an operator 1 commuting with V such that 0 ≤ V ≤ W and such that μ(k, W) = k+1 , k ≥ 0. (a) Since Φ is decreasing, it follows that the mapping x 󳨃→ x −1 Φ(x) decreases. Thus, the mapping x 󳨃→ xΦ(x−1 ) increases on (0, ∞). In particular, taking into account that V and W commute, 1 1 0 ≤ min{VΦ((nV)−1 ), } ≤ min{WΦ((nW)−1 ), }. n n Thus, 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 −1 1 󵄩 −1 1 󵄩 󵄩󵄩min{VΦ((nV) ), }󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩min{WΦ((nW) ), }󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 n 󵄩1 󵄩 n 󵄩󵄩1 ∞ k+1 1 ), } = ∑ min{(k + 1)−1 Φ( n n k=0 ∞

≤ ∑ (k + 1)−1 Φ( k=n ∞

n−1 k+1 1 )+ ∑ n n k=0 ∞

t dt dt ≤ ∫ Φ( ) + 1 = ∫ Φ(t) + 1. n t t n

1

262 | 8 Heat trace and ζ -function formulas By the assumptions on Φ, we have 󵄩󵄩 󵄩 󵄩󵄩 󵄩 −1 1 󵄩 󵄩󵄩min{VΦ((nV) ), }󵄩󵄩󵄩 = O(1), 󵄩󵄩 n 󵄩󵄩1

n → ∞.

(b) Let ψ = 1 − Φ. By assumption, ψ is concave, increasing, and positive on (0, ∞). In particular, the mapping x 󳨃→ x−1 ψ(x) is decreasing on (0, 1). Hence, the mapping x 󳨃→ (x−1 − 1)ψ(x) = x−1 ψ(x) − ψ(x) is decreasing on (0, 1) and the mapping x 󳨃→ (x −1 − 1)+ ψ(x) is decreasing on (0, ∞). Composing this function with the decreasing function x 󳨃→ x−1 , x > 1, we infer that the function x 󳨃→ (x − 1)+ (1 − Φ(x−1 )) is increasing on (1, ∞) and, therefore, on (0, ∞). In particular, 0 ≤ (V −

1 1 ) (1 − Φ((nV)−1 )) ≤ (W − ) (1 − Φ((nW)−1 )). n + n +

Thus, 󵄩 󵄩󵄩 󵄩 󵄩󵄩 1 1 󵄩 󵄩 󵄩 󵄩󵄩 −1 󵄩 −1 󵄩 󵄩󵄩(V − ) (1 − Φ((nV) ))󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(W − ) (1 − Φ((nW) ))󵄩󵄩󵄩 󵄩󵄩1 󵄩󵄩 󵄩󵄩1 󵄩󵄩 n + n + n−1

= ∑( k=0 n

= ∑( k=1

1 k+1 1 − )(1 − Φ( )) k+1 n n 1 1 k − )(1 − Φ( )) k n n

n

1

0

0

1 1 t 1 ≤ ∫( − )(1 − Φ( )) dt = ∫( − 1)(1 − Φ(t)) dt. t n n t By the assumptions on Φ, we have 󵄩󵄩 󵄩 1 󵄩󵄩 󵄩 −1 󵄩 󵄩󵄩(V − ) (1 − Φ((nV) ))󵄩󵄩󵄩 = O(1), 󵄩󵄩 󵄩󵄩1 n +

n → ∞.

Lemma 8.2.4. Let Φ be as in Lemma 8.2.3. If 0 ≤ V ∈ ℒ1,∞ , then 󵄩󵄩 1 󵄩󵄩󵄩 󵄩󵄩 −1 󵄩󵄩VΦ((nV) ) − (V − ) 󵄩󵄩󵄩 = O(1), 󵄩󵄩 n + 󵄩󵄩1

n → ∞.

8.2 Heat trace functionals and Dixmier traces | 263

Proof. For simplicity of computations, let ‖V‖1,∞ = 1. Note that (V − (V −

1 1 ) ) = min{V, }. n + n

Then VΦ((nV)−1 ) − (V − = (V −

1 ) n +

1 1 ) (Φ((nV)−1 ) − 1) + min{V, }Φ((nV)−1 ). n + n

By the triangle inequality, we have 󵄩󵄩 1 󵄩󵄩󵄩 󵄩󵄩 −1 󵄩󵄩VΦ((nV) ) − (V − ) 󵄩󵄩󵄩 󵄩󵄩 n + 󵄩󵄩1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 1 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(V − ) (Φ((nV)−1 ) − 1)󵄩󵄩󵄩 + 󵄩󵄩󵄩min{V, }Φ((nV)−1 )󵄩󵄩󵄩 . 󵄩󵄩1 󵄩󵄩 󵄩󵄩1 󵄩󵄩 n + n Since Φ((nV)−1 ) ≤ 1, it follows that 1 1 min{V, }Φ((nV)−1 ) ≤ min{VΦ((nV)−1 ), }. n n Consequently, 󵄩󵄩 1 󵄩󵄩󵄩 󵄩󵄩 −1 󵄩󵄩VΦ((nV) ) − (V − ) 󵄩󵄩󵄩 󵄩󵄩 n + 󵄩󵄩1 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(V − ) (Φ((nV)−1 ) − 1)󵄩󵄩󵄩 + 󵄩󵄩󵄩min{VΦ((nV)−1 ), }󵄩󵄩󵄩 . 󵄩󵄩1 󵄩󵄩 󵄩󵄩 n + n 󵄩󵄩1 The assertion follows now from Lemma 8.2.3. The final step in the proof that a heat trace functional of the form ξω∘M , ω being an extended limit on l∞ , equates to a Dixmier trace on the positive cone of ℒ1,∞ , is approximation of the partial sums of an eigenvector sequence Tr((V −

n 1 ) ) − ∑ λ(k, V) = O(1), n + k=0

n ≥ 1,

for a positive operator 0 ≤ V ∈ ℒ1,∞ with trivial kernel. The next lemma uses the properties of modulated operators from Theorem 7.1.3 in Chapter 7 to prove the approximation. Section 9.4, in Chapter 9, will use Lemma 8.2.5 for an arbitrary A ∈ ℒ(H). This section only requires the result for A = 1. Lemma 8.2.5. Let 0 ≤ V ∈ ℒ1,∞ satisfy ‖V‖1,∞ = 1, ker(V) = {0}, and let A ∈ ℒ(H). We have n

∑ λ(k, AV) = Tr(A(V −

k=0

1 ) ) + O(1), n +

n ≥ 1.

264 | 8 Heat trace and ζ -function formulas Proof. Let {ek }k≥0 be an eigenbasis for V ordered such that Vek = λ(k, V)ek , k ≥ 0. By Proposition 7.3.1 and Lemma 7.3.4, the operator AV is V-modulated. Theorem 7.1.3 states that as n → ∞ we have n

n

n

k=0

k=0

k=0

∑ λ(k, AV) = ∑ ⟨AVek , ek ⟩ + O(1) = ∑ ⟨Aek , ek ⟩λ(k, V) + O(1).

For n ≥ 1, let 1 }. n

m(n) = max{k ∈ ℕ : λ(k, V) > Using this notation, we write m(n)

∑ ⟨Aek , ek ⟩λ(k, V) = ∑ ⟨Aek , ek ⟩λ(k, V). k=0

λ(k,V)> n1

We have λ(k, V) ≤

1 k+1

for every k ≥ 0 and, therefore,

m(n) ≤ max{k ∈ ℤ+ : On the other hand, we have λ(k, V) ≤

1 n

1 1 > } = n − 2 < n. k+1 n

for all k > m(n). Thus,

󵄨󵄨 n 󵄨󵄨 n 󵄨󵄨 󵄨 󵄨󵄨 ∑ ⟨Aek , ek ⟩λ(k, V)󵄨󵄨󵄨 ≤ ‖A‖∞ ∑ λ(k, V) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨k=m(n)+1 k=m(n)+1 ≤

‖A‖∞ n

n

∑ k=m(n)+1

1≤

‖A‖∞ n ∑ 1 = O(1). n k=1

Consequently, n

n

k=0

k=0

∑ λ(k, AV) = ∑ ⟨Aek , ek ⟩λ(k, V) + O(1) m(n)

= ∑ ⟨Aek , ek ⟩λ(k, V) + O(1) = k=0

∑ ⟨Aek , ek ⟩λ(k, V) + O(1).

λ(k,V)> n1

Also, we have 1 ∑ ⟨Aek , ek ⟩λ(k, V) = Tr(AVEV ( , ∞)) n 1

λ(k,V)> n

= Tr(A(V −

1 1 1 ) ) + Tr(AEV ( , ∞)). n + n n

8.2 Heat trace functionals and Dixmier traces | 265

Since ‖V‖1,∞ = 1, then μ(n − 1, V) ≤ n−1 for n ≥ 1. By Remark 2.2.3, 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 −1 󵄨󵄨Tr(AEV ( , ∞))󵄨󵄨󵄨 ≤ ‖A‖∞ nV (n ) ≤ ‖A‖∞ nV (μ(n − 1, V)) = ‖A‖∞ n. 󵄨󵄨 󵄨󵄨 n Hence 1 1 Tr(AEV ( , ∞)) = O(1), n n

n → ∞,

and ∑ ⟨Aek , ek ⟩λ(k, V) = Tr(A(V −

λ(k,V)> n1

1 ) ) + O(1), n +

n → ∞.

Thus n

∑ λ(k, AV) = Tr(A(V −

k=0

1 ) ) + O(1), n +

n → ∞,

and the assertion is proved. We can now prove Theorem 8.1.2 for the case A = 1. Proof of Theorem 8.1.2(b). Let Φ(t) = e−t , t > 0. The function Φ(t) is positive, convex and decreasing, with ∞

∫ t −1 e−t dt < ∞ 1

and 1

∫ t −1 (1 − e−t ) dt < ∞. 0

Hence Φ satisfies the assumptions in Lemma 8.2.3. Let 0 ≤ V ∈ ℒ1,∞ . Without loss of generality, we may assume that ker(V) = {0}. By Lemma 8.2.4, we have −1

Tr(Ve−(nV) ) = Tr((V −

1 ) ) + O(1), n +

n ≥ 0.

By Lemma 8.2.5, we have −1

n

Tr(Ve−(nV) ) = ∑ λ(k, V) + O(1), k=0

n ≥ 0.

266 | 8 Heat trace and ζ -function formulas Hence n −1 1 1 1 Tr(Ve−(nV) ) = ), ∑ λ(k, V) + O( log(n + 2) log(n + 2) k=0 log(n + 2)

n ≥ 0.

1 ) term vanishes under ω, hence Let ω be any extended limit on l∞ . The O( log(n+2)

ω(

n −1 1 1 Tr(Ve−(nV) )) = ω( ∑ λ(k, V)). log(n + 2) log(n + 2) k=0

By Lemma 8.2.2 and Theorem 6.1.2, ξω∘M (V) = Trω (V),

0 ≤ V ∈ ℒ1,∞ .

Equality on the positive cone of ℒ1,∞ between ξω∘M and the Dixmier trace Trω , for every extended limit ω, proves that ξω∘M is additive on the positive cone of ℒ1,∞ . Hence, the heat trace functional ξω∘M can be linearly extended to ℒ1,∞ . By the description of Dixmier traces in Theorem 6.1.2, for any given Dixmier trace there exists a heat kernel functional whose linear extension coincides with that Dixmier trace.

8.3 General heat trace functionals Theorem 8.1.2(b), proved in the previous section, shows that Trω (V) = ξω∘M (V),

0 ≤ V ∈ ℒ1,∞ ,

for every extended limit ω on l∞ . Here Trω is a Dixmier trace on ℒ1,∞ given by Trω (V) = ω(

n 1 ∑ λ(k, V)) log(n + 2) k=0

from Theorem 6.1.2, and −1 1 ξω∘M (V) := (ω ∘ M)( Tr(e−(nV) )), n

0 ≤ V ∈ ℒ1,∞ ,

is the heat trace functional of Definition 8.1.1 for the extended limit ω ∘ M on l∞ from Lemma 6.2.12 for the logarithmic mean M : l∞ → l∞ . The functional ξω∘M on the positive cone of ℒ1,∞ is therefore unitarily invariant, positive homogenous, and additive for every extended limit ω on l∞ . The linear extension of ξω∘M to the whole ideal ℒ1,∞ , which we also denote when required by ξω∘M , is the Dixmier trace Trω . Let A ∈ ℒ(H) be a bounded operator written as a linear combination of four positive operators A = iA1 − A2 − iA3 + A4 ,

0 ≤ Aj ∈ ℒ(H), j = 1, . . . , 4.

8.3 General heat trace functionals | 267

Then Trω (AV) = ξω∘M (AV),

0 ≤ V ∈ ℒ1,∞ ,

where the right-hand side of this equation is 4

4

j=1

j=1

1/2 ξω∘M (AV) = ∑ ij ξω∘M (Aj V) = ∑ ij ξω∘M (A1/2 j VAj ) 4

1/2 1/2 −1 1 = ∑ ij (ω ∘ M)( Tr(e−(nAj VAj ) )) n j=1

for any 0 ≤ V ∈ ℒ1,∞ . We want to prove that this expression for the trace ξω∘M can be simplified. If we prove, for 0 ≤ Aj ∈ ℒ(H), j = 1, . . . , 4, that 1/2 1/2 −1 −1 1 1 (ω ∘ M)( Tr(e−(nAj VAj ) )) = (ω ∘ M)( Tr(Aj e−(nV) )), n n

(8.8)

then we obtain, by linearity of ω ∘ M and Tr, −1 1 ξω∘M (AV) = (ω ∘ M)( Tr(Ae−(nV) )). n

This is a simpler expression for the trace ξω∘M on ℒ1,∞ , which provides the heat trace formula of the Dixmier trace stated in Theorem 8.1.2(a). This section also considers a change of variables. Theorem 8.3.2 demonstrates that the function −1

f (t) = e−t ,

t > 0,

is a convenient choice, but the heat trace formula for a Dixmier trace on ℒ1,∞ is the same, up to normalization, using any bounded function f ∈ C 2 [0, ∞) such that f (0) = f 󸀠 (0) = 0. Reduction of general heat trace functionals Let f ∈ C 2 [0, ∞) be a bounded function such that f (0) = f 󸀠 (0) = 0. Consider the general heat trace functional on the positive cone of ℒ1,∞ defined by the formula 1 ξω,A,f (V) := ω( Tr(f (nV)A)), n

0 ≤ V ∈ ℒ1,∞ , A ∈ ℒ(H).

(8.9)

The operator f (nV), n ≥ 0, is defined by the functional calculus for compact operators, with the caveat that 1 Tr(f (nV)) = 0, n

n = 0, 0 ≤ V ∈ ℒ1,∞ .

The next lemma shows that general heat trace functionals are well defined.

268 | 8 Heat trace and ζ -function formulas Lemma 8.3.1. Let f ∈ C 2 [0, ∞) be a bounded function such that f (0) = f 󸀠 (0) = 0. (a) The value ∞

∫ f (s) 0

ds s2

is finite. (b) For A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1.∞ , Tr(f (nV)A) = O(n),

n → ∞.

Proof. (a) Without loss of generality, suppose f is real-valued. We have ∞

∞

1

1

ds 󵄨 󵄨 ds ∫ 󵄨󵄨󵄨f (s)󵄨󵄨󵄨 2 ≤ ‖f ‖∞ ∫ 2 = ‖f ‖∞ . s s Since f ∈ C 2 [0, ∞) and f (0) = f 󸀠 (0) = 0, using l’Hôpital’s rule lim+

s→0

f (s) f 󸀠 (s) 1 󸀠󸀠 = lim+ = f (0). 2 2s 2 s→0 s

Hence f (s) ∈ L∞ (0, 1) s2 and 1

|f (s)| 󵄨 󵄨 ds < ∞. ∫󵄨󵄨󵄨f (s)󵄨󵄨󵄨 2 ≤ sup 2 s s∈(0,1) s 0

(b) Without loss of generality, suppose f is real-valued, ‖A‖∞ ≤ 1 and ‖V‖1,∞ ≤ 1. Let f 󸀠 ∈ C 1 [0, ∞) have the decomposition into positive functions f 󸀠 = (f 󸀠 )+ − (f 󸀠 )− . Define min{t,1}

f1,± (t) :=

∫ (f 󸀠 )± (s) ds,

t > 0.

0

Set g = f − f1,+ + f1,− . Clearly, f1,± are increasing functions. Let n ≥ 1. By the triangle inequality, we have 󵄨󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄨󵄨Tr(f (nV)A)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩f (nV)󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩f1,+ (nV)󵄩󵄩󵄩1 + 󵄩󵄩󵄩f1,− (nV)󵄩󵄩󵄩1 + 󵄩󵄩󵄩g(nV)󵄩󵄩󵄩1 .

8.3 General heat trace functionals | 269

Since g is bounded and supported on (1, ∞), it follows that 1 󵄩 󵄩󵄩 󵄩󵄩g(nV)󵄩󵄩󵄩1 ≤ ‖g‖∞ nV ( ) ≤ ‖g‖∞ ‖V‖1,∞ n, n

n ≥ 1.

Since f1,± is an increasing function and (k + 1)μ(k, V) ≤ ‖V‖1,∞ ≤ 1, k ≥ 0, it follows that n 󵄩󵄩 󵄩 ) 󵄩󵄩f1,± (nV)󵄩󵄩󵄩1 = ∑ f1,± (nμ(k, V)) ≤ ∑ f1,± ( k+1 k≥0 ∞

∞

0

0

k≥0

ds n ≤ ∫ f1,± ( ) dt = n ∫ f1,± (s) 2 . t s This completes the proof, since ∞

∞

∫ f1,± (s) 0

ds 󵄨 󵄨 ds ≤ ∫ 󵄨󵄨󵄨f (s)󵄨󵄨󵄨 2 < ∞ s2 s 0

by part (a). A general heat trace functional can be reduced, up to a constant, to the heat trace functional from Section 8.2. In the statement of Theorem 8.3.2 below, M : l∞ → l∞ is the logarithmic mean and the extended limit ω ∘ M on l∞ , where ω is an extended limit on l∞ , is described in Lemma 6.2.12. Theorem 8.3.2. Let f ∈ C 2 [0, ∞) be a bounded function such that f (0) = f 󸀠 (0) = 0. Let ω be an extended limit on l∞ and A ∈ ℒ(H). Then ∞

ξω∘M,A,f (V) = ( ∫ f (s) 0

ds )ξ (AV), s2 ω∘M

0 ≤ V ∈ ℒ1,∞ ,

(8.10)

where ξω∘M,A,f is the general heat trace functional from (8.9), and ξω∘M is the linear extension to ℒ1,∞ of the heat trace functional on the positive cone of ℒ1,∞ from Definition 8.1.1 and Section 8.2. Let A ∈ ℒ(H) and let 0 ≤ V ∈ ℒ1,∞ . Corollary 8.1.3 in Section 8.1 indicated that Theorem 8.3.2 contains the result that, for any q > 0, Trω (AV) =

1

Γ(1 +

1 ) q

−q 1 (ω ∘ M)( Tr(Ae−(nV) )) n

for the Dixmier trace Trω associated to the extended limit ω on l∞ .

270 | 8 Heat trace and ζ -function formulas Proof of Corollary 8.1.3. Theorem 8.3.2 can be applied to the function f (t) = e−t , t > 0, observing that −q

∞

∫ f (s) 0

1 ds = Γ(1 + ). q s2

By Theorem 8.1.2(b), proved in Section 8.2, the Dixmier trace Trω is the extension to

ℒ1,∞ of the heat kernel functional ξω∘M on the positive cone of ℒ1,∞ .

The rest of this section proves Theorem 8.3.2. Proof of the heat trace formula for general heat trace functionals Several operator inequalities proved in Appendix A.2 are used below. The following lemmas break up the proof of Theorem 8.3.2. Lemma 8.3.3. If 0 ≤ V ∈ ℒ1,∞ and A ∈ ℒ(H), then ω(

1 1 Tr((V − ) A)) = ξω∘M (AV) log(n + 2) n +

(8.11)

for every extended limit ω on l∞ . Proof. We assume, without loss of generality, that 0 ≤ A ≤ 1. 1 1 Assume first that ϵ ≤ A ≤ 1 for some ϵ > 0. Let W = A 2 VA 2 . Since ξω∘M is a Dixmier trace, it follows from Lemmas 8.2.2 and 8.2.4 (taken with Φ(t) = e−t ) that ξω∘M (AV) = ξω∘M (W) = ω( Clearly, the function u 󳨃→ (u − Lemma A.2.3 that

1 ) n +

Tr((W −

1 1 Tr((W − ) )). log(n + 2) n +

is convex. Since 0 ≤ A ≤ 1, it follows from

1 1 ) ) ≤ Tr(A(V − ) ). n + n +

Thus, ξω∘M (AV) ≤ ω(

1 1 Tr(A(V − ) )). log(n + 2) n +

(8.12)

Since ξω∘M is a Dixmier trace, it also follows from Lemmas 8.2.2 and 8.2.4 (taken with Φ(t) = e−t ) that ξω∘M (AV) = ϵξω∘M (ϵ−1 W) = ϵ ⋅ ω(

1 1 Tr((ϵ−1 W − ) )). log(n + 2) n +

8.3 General heat trace functionals | 271

Since ϵ−1 A ≥ 1, it follows from Lemma A.2.3 that Tr((ϵ−1 W −

1 1 ) ) ≥ Tr(ϵ−1 A(V − ) ). n + n +

Thus, ξω∘M (AV) ≥ ω(

1 1 Tr(A(V − ) )). log(n + 2) n +

(8.13)

The assertion for ϵ ≤ A ≤ 1 follows by combining (8.12) and (8.13). Now we show that one can dispense with the condition ϵ ≤ A ≤ 1. Let Aϵ = max{A, ϵ}. By the preceding paragraph, we have ω(

1 1 Tr((V − ) Aϵ )) = ξω∘M (Aϵ V). log(n + 2) n +

Clearly, 󵄨󵄨 󵄨 󵄨󵄨ξω∘M (Aϵ V) − ξω∘M (AV)󵄨󵄨󵄨 ≤ ‖A − Aϵ ‖∞ ξω∘M (V) ≤ ϵ‖V‖1,∞ and 󵄨󵄨 󵄨󵄨 1 1 1 1 󵄨󵄨 󵄨 Tr((V − ) Aϵ )) − ω( Tr((V − ) A))󵄨󵄨󵄨 󵄨󵄨ω( 󵄨󵄨 log(n + 2) 󵄨󵄨 n + log(n + 2) n + 1 1 Tr((V − ) )) ≤ ϵ‖V‖1,∞ . ≤ ‖A − Aϵ ‖∞ ω( log(n + 2) n + Thus, 󵄨󵄨 󵄨󵄨 1 1 󵄨󵄨 󵄨 Tr((V − ) A)) − ξω∘M (AV)󵄨󵄨󵄨 ≤ 2ϵ‖V‖1,∞ . 󵄨󵄨ω( 󵄨󵄨 log(n + 2) 󵄨󵄨 n + Letting ϵ → 0, we complete the proof. The formula in Lemma 8.3.3 for the heat trace functional in terms of the finiterank operator (V − n1 )+ can be reduced further to a formula involving the finite-rank spectral projections of V. Reducing the formula to spectral projections in Lemma 8.3.4 will allow us to build up the operator f (nV) using step-function approximations to f in Lemma 8.3.5. Lemma 8.3.4. If 0 ≤ V ∈ ℒ1,∞ and A ∈ ℒ(H), then 1 s (ω ∘ M)( Tr(EV ( , ∞)A)) = s−1 ξω∘M (AV) n n for every extended limit ω on l∞ and for every s > 0.

272 | 8 Heat trace and ζ -function formulas Proof. Let n ≥ 0 and s > 0. Step 1. We expand the logarithmic mean. We have M(k 󳨃→

n 1 s 1 1 s Tr(EV ( , ∞)A))(n) = Tr(EV ( , ∞)A) ∑ k k log(n + 2) k=1 k(k + 1) k

=

n 1 1 s Tr(A ⋅ ∑ EV ( , ∞)). log(n + 2) k(k + 1) k k=1

Step 2. We want to approximate the expression in Step 1. We prove that 󵄩 󵄩󵄩 n 󵄩󵄩 s s 󵄩󵄩󵄩󵄩 1 −1 󵄩󵄩 ∑ E ( , ∞) − s VE ( , s] 󵄩 = O(1). V 󵄩󵄩 k(k + 1) V k n 󵄩󵄩󵄩󵄩1 󵄩󵄩k=1 Using summation by parts, we obtain n

s 1 EV ( , ∞) k(k + 1) k k=1 ∑

n

n−1 n 1 1 s s EV (s, ∞) + ∑ ( ∑ )EV ( , ] k(k + 1) l(l + 1) k+1 k k=1 k=1 l=k+1

=∑ n

n−1 1 1 s 1 s s EV (s, ∞) − EV ( , s] + ∑ EV ( , ]. k(k + 1) n+1 n k+1 k+1 k k=1 k=1

=∑

It follows from the triangle inequality that 󵄩 󵄩󵄩 n 󵄩󵄩 s s 󵄩󵄩󵄩󵄩 1 −1 󵄩󵄩 ∑ E ( , ∞) − s VE ( , s] 󵄩 V 󵄩󵄩 k(k + 1) V k n 󵄩󵄩󵄩󵄩1 󵄩󵄩k=1

n−1󵄩 󵄩󵄩 1 s s s 󵄩󵄩󵄩󵄩 1 s nV (s) + nV ( ) + ∑ 󵄩󵄩󵄩(V − )EV ( , ]󵄩 . 󵄩 k(k + 1) n+1 n k+1 k + 1 k 󵄩󵄩󵄩1 k=1󵄩 k=1 n

≤∑

By the Hölder inequality, 󵄩󵄩 s s 󵄩󵄩󵄩󵄩 s 󵄩󵄩 )EV ( , ]󵄩 󵄩󵄩(V − 󵄩󵄩 k+1 k + 1 k 󵄩󵄩󵄩1 󵄩󵄩 s s 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 s s s 󵄩󵄩󵄩󵄩 󵄩 )EV ( , ]󵄩󵄩 󵄩󵄩EV ( , ]󵄩 ≤ 󵄩󵄩󵄩(V − 󵄩 󵄩󵄩 󵄩 k+1 k + 1 k 󵄩∞ 󵄩 k + 1 k 󵄩󵄩󵄩1 󵄩󵄩 󵄩󵄩 s 󵄩󵄩󵄩EV ( s , s ]󵄩󵄩󵄩 . ≤ k(k + 1) 󵄩󵄩󵄩 k + 1 k 󵄩󵄩󵄩1 Thus, 󵄩󵄩 n 󵄩 󵄩󵄩 1 s s 󵄩󵄩󵄩󵄩 −1 󵄩󵄩 ∑ E ( , ∞) − s VE ( , s] 󵄩 V V 󵄩󵄩 k(k + 1) k n 󵄩󵄩󵄩󵄩1 󵄩󵄩k=1 n

n−1 󵄩󵄩 1 1 s s s s 󵄩󵄩󵄩󵄩 󵄩󵄩 nV (s) + nV ( ) + ∑ , ]󵄩 . 󵄩󵄩EV ( k(k + 1) n+1 n k(k + 1) 󵄩󵄩 k + 1 k 󵄩󵄩󵄩1 k=1 k=1

≤∑

(8.14)

8.3 General heat trace functionals | 273

Again using summation by parts, we obtain 󵄩󵄩 s s 󵄩󵄩󵄩󵄩 n−1 s s s s 󵄩󵄩 , ]󵄩󵄩 = ∑ (nV ( ) − nV ( )) 󵄩󵄩EV ( 󵄩 󵄩 k(k + 1) 󵄩 k + 1 k 󵄩1 k=1 k(k + 1) k+1 k k=1

n−1

∑

=

n−1 s s 2s s s nV ( ) − nV (s) + ∑ nV ( ). (n − 1)n n 2 (k − 1)k(k + 1) k k=2

Since s nV ( ) = O(k), k

k ∈ ℕ,

it follows that n−1

󵄩󵄩 s s s 󵄩󵄩󵄩󵄩 󵄩󵄩 , ]󵄩 = O(1). 󵄩󵄩EV ( k(k + 1) 󵄩󵄩 k + 1 k 󵄩󵄩󵄩1 k=1 ∑

Hence we obtain (8.14). Step 3. Substituting (8.14) into Step 1, we obtain M(k 󳨃→

s 1 s 1 Tr(EV ( , ∞)A))(n) = (Tr(A ⋅ s−1 VEV ( , s]) + O(1)) k k log(n + 2) n 1 s = (Tr(A ⋅ s−1 VEV ( , ∞)) + O(1)) log(n + 2) n 1 1 = (Tr(A ⋅ s−1 VEV ( , ∞)) + O(1)) log(n + 2) n 1 1 = (Tr(A ⋅ s−1 (V − ) ) + O(1)). log(n + 2) n +

Hence, we have, 1 s 1 1 (ω ∘ M)( Tr(EV ( , ∞)A)) = s−1 ω( Tr((V − ) A)). n n log(n + 2) n + The assertion follows from Lemma 8.3.3. We can now build up the operator f (nV) using step-function approximations to f on a finite interval. Lemma 8.3.5. Let 0 < a < b. Let f be a bounded function on [0, ∞) such that f |[a,b] ∈ C 2 ([a, b]) and let f = 0 on ℝ+ \[a, b]. Let A ∈ ℒ(H) and let ω be an extended limit on l∞ . Then b

ξω∘M,A,f (V) = (∫ f (s) a

ds )ξ (AV), s2 ω∘M

0 ≤ V ∈ ℒ1,∞ ,

where ξω∘M,A,f is the general heat trace functional defined in (8.9).

274 | 8 Heat trace and ζ -function formulas Proof. On the interval [a, b], decompose f 󸀠 ∈ C 1 [a, b] into positive and negative parts f 󸀠 = f+󸀠 − f−󸀠 . Then we can write f = f1 − f2 , where f1 = f2 = 0 on ℝ+ \[a, b], f1 , f2 ∈ C 2 [a, b] on the interval [a, b], and f1󸀠 (x) = f+󸀠 (x) ≥ 0 and f2󸀠 (x) = f−󸀠 (x) ≥ 0 for x ∈ [a, b]. Hence, without loss of generality, we may assume that f is increasing on [a, b] and that A ≥ 0. Let a = a0 ≤ a1 ≤ a2 ≤ ⋅ ⋅ ⋅ ≤ am = b define a partition of [a, b]. Since f is increasing on [a, b], we have, for every n ∈ ℕ, m−1

∑ f (ak )EV (

k=0

m−1 a a ak ak+1 , ] ≤ f (nV) ≤ ∑ f (ak+1 )EV ( k , k+1 ]. n n m m k=0

Therefore, m−1 a a 1 ξω∘M,A,f (V) ≤ ∑ f (ak+1 )(ω ∘ M)( Tr(EV ( k , k+1 ]A)) n n n k=0

and m−1 a a 1 ξω∘M,A,f (V) ≥ ∑ f (ak )(ω ∘ M)( Tr(EV ( k , k+1 ]A)). n n n k=0

We have EV (

a a ak ak+1 , ] = EV ( k , ∞) − EV ( k+1 , ∞). n n n n

It follows from Lemma 8.3.4 that a a 1 1 1 (ω ∘ M)( Tr(EV [ k , k+1 )A)) = ( − )ξ (AV). n n n ak ak+1 ω∘M Hence, m−1

( ∑ f (ak )( k=0

1 1 − ))ξω∘M (AV) ≤ ξω∘M,A,f (V) ak ak+1 m−1

≤ ( ∑ f (ak+1 )( k=0

1 1 − ))ξω∘M (AV). ak ak+1 b

Both left- and right-hand sides in the latter formula tend to ∫a f (s)s−2 ds when m → ∞. The proof of Theorem 8.3.2 can now be given. Proof of Theorem 8.3.2. Without loss of generality, f ≥ 0. By assumption, see the proof of Lemma 8.3.1(a), |f (t)| ≤ const ⋅ min{1, t 2 }, t > 0. For every m ∈ ℕ, we have 0 ≤ f − fχ[ 1 ,m] ≤ const ⋅ gm , m

8.3 General heat trace functionals | 275

where the constant is independent of m and where gm (t) = min{t 2 , m−2 } + min{1, t 2 m−2 },

t > 0.

We have 0 ≤ ξω∘M,A,f (V) − ξω∘M,A,fχ

1 ,m] [m

(V) ≤ const ⋅ ‖A‖∞ ξω∘M,1,gm (V).

By definition, ξω∘M,1,gm (V)

1 1 = (ω ∘ M)( Tr(min{(nV)2 , m−2 })) + (ω ∘ M)( Tr(min{1, m−2 (nV)2 })) n n 1 1 ≤ lim sup Tr(min{(nV)2 , m−2 }) + lim sup Tr(min{1, m−2 (nV)2 }). n→∞ n n→∞ n

Since lim sup is dilation invariant, it follows that ξω∘M,1,gm (V) ≤

2 1 ⋅ lim sup Tr(min{(nV)2 , 1}). m n→∞ n

The upper limit in the right-hand side is finite since 2

Tr(min{(nV)2 , 1}) = ∑ min{(nμ(k, V)) , 1} k≥0

≤ ∑ min{( k≥0

2

n‖V‖1,∞ ) , 1} = O(n), k+1

n ∈ ℕ.

Thus, ξω∘M,A,f (V) − ξω∘M,A,fχ

1 ,m] [m

(V) = O(m−1 ),

m ∈ ℕ.

By Lemma 8.3.5, we have m

ξω∘M,A,fχ

1 ,m] [m

(V) = (∫ f (s) 1 m

ds )ξ (AV). s2 ω∘M

Consequently, m

ξω∘M,A,f (V) − (∫ f (s) 1 m

ds )ξ (AV) = O(m−1 ), s2 ω∘M

Taking the limit m → ∞, we complete the proof.

m ∈ ℕ.

276 | 8 Heat trace and ζ -function formulas Theorem 8.3.2 allows us to complete the proof of Theorem 8.1.2 on the heat trace formula for Dixmier traces on ℒ1,∞ . Proof of Theorem 8.1.2(a). Let ω be an extended limit on l∞ and 0 ≤ V ∈ ℒ1,∞ . Without loss, take 0 ≤ A ∈ ℒ(H). By Theorem 8.1.2(b), Trω (AV) = ξω∘M (A1/2 VA1/2 ) where ξω∘M denotes the heat trace functional on the positive cone of ℒ1,∞ . Setting f (t) = e−t , t > 0, in Theorem 8.3.2 implies that −1

−1 1 ξω∘M (A1/2 VA1/2 ) = ξω∘M,A (V) = (ω ∘ M)( Tr(Ae−(nV) )). n

This completes the proof.

8.4 ζ-function residues and Dixmier traces This section proves that a ζ -function residue functional from Definition 8.1.4, ζγ (V) := γ(sTr(V 1+s )),

0 ≤ V ∈ ℒ1,∞ ,

where γ is an extended limit on L∞ (0, 1) concentrated at 0, is an additive positive functional on the positive cone of ℒ1,∞ which is monotone for Hardy–Littlewood submajorization and ζγ (diag{

∞

1 } ) = 1. n + 1 n=0

That is, the functional ζγ on the positive cone of ℒ1,∞ has an extension to the whole ideal ℒ1,∞ , and the extension is a normalized fully symmetric trace on ℒ1,∞ in the terminology of Chapter 6. This section also proves that the normalized fully symmetric extension of ζγ to ℒ1,∞ , which we also denote by ζγ , satisfies the formula ζγ (AV) = ζγ,A (V) := γ(sTr(AV 1+s )),

A ∈ ℒ(H), 0 ≤ V ∈ ℒ1,∞ ,

where ζγ,A was also defined in Definition 8.1.4 for γ an extended limit on L∞ (0, 1) concentrated at 0. Both the functionals ζγ and ζγ,A from Definition 8.1.4 are well defined on the positive cone of ℒ1,∞ by the next lemma.

8.4 ζ -function residues and Dixmier traces | 277

Lemma 8.4.1. Let A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ . (a) The functions s 󳨃→ sTr(V 1+s ),

s 󳨃→ sTr(V 1+s A),

s ∈ (0, 1),

are bounded. (b) For any extended limit γ on L∞ (0, 1) concentrated at 0, ζγ (diag{

∞

1 } ) = lim+ sζ (1 + s) = 1 n + 1 n=0 s→0

where ζ is the Riemann ζ -function. Proof. (a) Let s ∈ (0, 1). By the Hölder inequality, we have 󵄨󵄨 1+s 󵄨 1+s 󵄨󵄨sTr(V A)󵄨󵄨󵄨 ≤ ‖A‖∞ ⋅ sTr(V ). It is clear that μ(k, V) ≤

‖V‖1,∞ , k+1

k ≥ 0.

Thus, Tr(V 1+s ) = ∑ μ(k, V)1+s ≤ ‖V‖1+s 1,∞ ⋅ ζ (1 + s). k≥0

We now have 󵄨󵄨 1+s 󵄨 1+s 󵄨󵄨sTr(V A)󵄨󵄨󵄨 ≤ ‖A‖∞ ⋅ ‖V‖1,∞ ⋅ sζ (1 + s). Since the function s → sζ (1 + s), s ∈ (0, 1), is bounded, the assertion follows. (b) Let z = {(1 + n)−1 }∞ n=0 ∈ l1,∞ . Then ∞

Tr(diag(z)1+s ) = ∑ (1 + n)−(1+s) = ζ (1 + s), n=0

s ∈ (0, 1).

A standard estimate of the sum for s > 0 and Lemma 6.2.4 yield ζγ (diag(z)) = lim+ sζ (1 + s) = 1 s→0

for every γ.

278 | 8 Heat trace and ζ -function formulas Extension of a ζ-function residue is a normalized fully symmetric trace Let γ be an extended limit on L∞ (0, 1) concentrated at 0. The next two results are the core of the proof that ζγ (V) = γ(sTr(V 1+s )),

0 ≤ V ∈ ℒ1,∞ ,

is an additive positive functional on the positive cone of ℒ1,∞ . Lemma 8.4.2. Let 0 ≤ V1 , V2 ∈ ℒ1,∞ . (a) For every s > 0, Tr(V11+s ) + Tr(V21+s ) ≤ Tr((V1 + V2 )1+s ). (b) For every s > 0, Tr((V1 + V2 )1+s ) ≤ 2s Tr(V11+s ) + 2s Tr(V21+s ). Proof. By Theorem 2.3.7, we have V1 ⊕ V2 ≺≺ V1 + V2 . Applying Theorem 2.3.9 to the function F : t 󳨃→ t 1+s , t > 0, we obtain Tr((V1 ⊕ V2 )1+s ) ≤ Tr((V1 + V2 )1+s ). Noting that Tr((V1 ⊕ V2 )1+s ) = Tr(V11+s ) + Tr(V21+s ), due to our definition of the direct sum in Definition 2.2.11, we complete the proof of (a). By Theorem 2.3.5, we have V1 + V2 ≺≺ μ(V1 ) + μ(V2 ). Applying Theorem 2.3.9 to the function F : t 󳨃→ t 1+s , t > 0, we obtain Tr((V1 + V2 )1+s ) ≤ Tr((μ(V1 ) + μ(V2 ))

1+s

).

Using the numerical inequality (a + b)1+s ≤ 2s (a1+s + b1+s ),

a, b, s > 0,

we have that (μ(V1 ) + μ(V2 ))

1+s

≤ 2s μ(V1 )1+s + 2s μ(V2 )1+s .

Taking the trace, we obtain Tr((μ(V1 ) + μ(V2 ))

1+s

) ≤ 2s Tr(V11+s ) + 2s Tr(V21+s ).

Combining these inequalities, we complete the proof of (b).

8.4 ζ -function residues and Dixmier traces | 279

Corollary 8.4.3. Let 0 ≤ V1 , V2 ∈ ℒ1,∞ . We have Tr((V1 + V2 )1+s ) = Tr(V11+s ) + Tr(V21+s ) + O(1),

s ∈ (0, 1).

Proof. By Lemma 8.4.2, we have 0 ≤ Tr((V1 + V2 )1+s ) − Tr(V11+s ) − Tr(V21+s ) ≤ (2s − 1)Tr(V11+s ) + (2s − 1)Tr(V21+s ). By Lemma 8.4.1, the functions s 󳨃→

(2s − 1) sTr(V11+s ), s

s 󳨃→

(2s − 1) sTr(V21+s ), s

s ∈ (0, 1),

are bounded. We can now prove that each ζ -function residue extends to a normalized fully symmetric trace on ℒ1,∞ , and prove Theorem 8.1.5(b). The estimate in Corollary 8.4.3 allows us to prove additivity of a ζ -function residue below. Theorem 2.3.9 in Chapter 2, used twice in the proofs above, will also show that the ζ -function residue is monotone for Hardy–Littlewood submajorization. Proof of Theorem 8.1.5(b). Let γ be an extended limit on L∞ (0, 1) concentrated at 0. Set ζγ (V) = γ(sTr(V 1+s )),

0 ≤ V ∈ ℒ1,∞ .

We prove that ζγ extends to a normalized fully symmetric trace on ℒ1,∞ . Since ∞

Tr(V 1+s ) = ∑ μ(k, V)1+s , k=0

s > 0,

and the singular values of V are unitarily invariant by Corollary 2.2.10(b), we obtain ∞

∞

k=0

k=0

1+s

Tr(V 1+s ) = ∑ μ(k, V)1+s = ∑ μ(k, U ∗ VU)

1+s

= Tr((U ∗ VU)

),

s > 0,

for any unitary U ∈ ℒ(H). Hence ζγ (V) = ζγ (U ∗ VU), and ζγ is unitarily invariant on the positive cone of ℒ1,∞ . Let 0 ≤ V1 , V2 ∈ ℒ+1,∞ . We prove that ζγ is additive and monotone for submajorization on the positive cone of ℒ1,∞ . It follows from Corollary 8.4.3 that ζγ (V1 + V2 ) − ζγ (V1 ) − ζγ (V2 )

= γ(s(Tr((V1 + V2 )1+s ) − Tr(V11+s ) − Tr(V21+s ))

= lim+ sO(1) = 0. s→0

Hence ζγ is additive.

280 | 8 Heat trace and ζ -function formulas Applying Theorem 2.3.9 to the function F : t 󳨃→ t 1+s , t > 0, we obtain Tr(V21+s ) ≤ Tr(V11+s ),

s > 0,

when V2 ≺≺ V1 . Since γ is positive, we have ζγ (V2 ) = γ(sTr(V21+s )) ≤ γ(sTr(V11+s )) = ζγ (V1 ) when V2 ≺≺ V1 . Hence ζγ is monotone for submajorization. The functional ζγ is positive, additive, positively homogeneous, and unitarily invariant on the positive cone of ℒ1,∞ . The linear extension to ℒ1,∞ , also denoted by ζγ , is therefore a positive trace on ℒ1,∞ . The positive trace ζγ is normalized by Lemma 8.4.1(b) and monotone for submajorization. Hence ζγ is a normalized fully symmetric trace on ℒ1,∞ . By Theorem 6.1.3, ζγ is a Dixmier trace on ℒ1,∞ . Formula for the extension of a ζ-function residue Let γ be an extended limit on L∞ (0, 1) concentrated at 0. By the previous section, the extension to ℒ1,∞ of the functional ζγ (V) = γ(sTr(V 1+s )),

0 ≤ V ∈ ℒ1,∞ ,

on the positive cone of ℒ1,∞ is a normalized fully symmetric trace. The extension is also denoted ζγ . Let A ∈ ℒ(H) be a bounded operator written as a linear combination of four positive operators A = iA1 − A2 − iA3 + A4 ,

0 ≤ Aj ∈ ℒ(H), j = 1, . . . , 4.

Then, by the trace properties of ζγ on ℒ1,∞ , 4

4

j=1

j=1

1

1

ζγ (AV) = ∑ ij ζγ (Aj V) = ∑ ij ζγ (Aj2 VAj2 ) 4

1

1

= ∑ ij γ(sTr((Aj2 VAj2 )

1+s

j=1

))

for any 0 ≤ V ∈ ℒ1,∞ . We want to prove that this expression for the trace ζγ can be simplified. If we prove that 1

1

γ(sTr((Aj2 VAj2 )

1+s

)) = γ(sTr(Aj V 1+s )),

0 ≤ Aj ∈ ℒ(H), j = 1, . . . , 4,

then we obtain, by linearity of γ and Tr, ζγ (AV) = γ(sTr(AV 1+s )).

(8.15)

8.4 ζ -function residues and Dixmier traces | 281

This is a simpler expression for the trace ζγ on ℒ1,∞ , which will allow us to prove Theorem 8.1.5(a) in Section 8.5. To prove (8.15), we require some standard estimates on positive operators. Recall that a continuous function f : [0, ∞) → [0, ∞) is operator convex if f (tA + (1 − t)B) ≤ tf (A) + (1 − t)f (B),

0 ≤ A, B ∈ ℒ(H), t ∈ (0, 1),

where the bounded positive operators f (tA + (1 − t)B), f (A), and f (B) are defined by the bounded functional calculus on ℒ(H), and the inequality is in the sense of positive operators on ℒ(H). Lemma 8.4.4. Let f : [0, ∞) → [0, ∞) be an operator convex function and let 0 ≤ A, V ∈ ℒ(H).

(a) If A ≤ 1, then 1

1

1

1

1

1

1

1

f (A 2 VA 2 ) ≤ A 2 f (V)A 2 . (b) If A ≥ 1, then f (A 2 VA 2 ) ≥ A 2 f (V)A 2 . Proof. The first inequality is Theorem V.2.3 in [27]. The second inequality follows by 1 1 applying the first one to V0 = A 2 VA 2 and A0 = A−1 . Corollary 8.4.5. Let 0 ≤ V ∈ ℒ1,∞ and let 0 ≤ A ∈ ℒ(H). Then 1

1+s

1

Tr(AV 1+s ) = Tr((A 2 VA 2 )

) + O(1),

s ∈ (0, 1).

Proof. Step 1. Suppose first that A ≥ 1. Let s ∈ (0, 1). The function t 󳨃→ t 1+s , t > 0, is operator convex as a consequence of the Löwner–Heinz inequality (see Theorems V.2.9 and V.2.10 in [27]). If M = ‖A‖∞ , then Lemma 8.4.4 implies that 1

1

1+s

M −s (A 2 VA 2 )

1

1

1

1

1+s

≤ A 2 V 1+s A 2 ≤ (A 2 VA 2 )

,

s ∈ (0, 1).

Therefore, 1

1

0 ≤ Tr((A 2 VA 2 )

1+s

1

1

1

1

) − Tr(A 2 V 1+s A 2 ) ≤ (1 − M −s )Tr((A 2 VA 2 )

1+s

By Lemma 8.4.1, we have 1

1

1+s

(1 − M −s )Tr((A 2 VA 2 )

) = O(1),

s ∈ (0, 1).

Therefore, 1

1

Tr((A 2 VA 2 ) 1

1+s

) − Tr(AV 1+s ) 1

1+s

= Tr((A 2 VA 2 )

1

1

) − Tr(A 2 V 1+s A 2 ) = O(1),

s ∈ (0, 1).

).

282 | 8 Heat trace and ζ -function formulas Step 2. Let A ≥ 0. We want to prove that 1+s

1

1

Tr((A 2 VA 2 )

1

1

) = Tr((A + 1) 2 V(A + 1) 2 )

1+s

) − Tr(V 1+s ) + O(1),

so that we can use Step 1 on the operator A + 1 ≥ 1. 1 1 1 1 Using Lemma 8.4.2, since V 2 (A + 1)V 2 = V 2 AV 2 + V, 1

1+s

1

Tr((V 2 AV 2 )

1

1+s

1

) + Tr(V 1+s ) ≤ Tr((V 2 (A + 1)V 2 )

)

and 1

1

Tr((V 2 (A + 1)V 2 )

1+s

1

1

) ≤ 2s Tr((V 2 AV 2 )

1+s

) + 2s Tr(V 1+s ).

Hence 1 1+s 1 1 1+s 1 󵄨󵄨 1+s 󵄨 󵄨󵄨Tr((V 2 (A + 1)V 2 ) ) − Tr((V 2 AV 2 ) ) − Tr(V )󵄨󵄨󵄨 1

1

1+s

≤ (1 − 2s )Tr((V 2 (A + 1)V 2 )

) + (1 − 2s )Tr(V 1+s ).

By Lemma 8.4.4(b), 1

1

(1 − 2s )Tr((V 2 (A + 1)V 2 )

1+s

) + (1 − 2s )Tr(V 1+s ) = O(1).

Hence 1

1

Tr((V 2 AV 2 )

1+s

1

1

1+s

1

1

) = Tr(V 2 (A + 1)V 2 )

) − Tr(V 1+s ) + O(1).

The estimate (8.16) follows by noting that 1

1

1

1

1

1+s

2

2

1

1

μ(A 2 VA 2 ) = μ(A 2 V 2 ) = μ(V 2 A 2 ) = μ(V 2 AV 2 ). Thus, 1

Tr((A 2 VA 2 )

1

1

1+s

) = Tr((V 2 AV 2 )

).

Similarly, 1

1

1+s

Tr(((A + 1) 2 V(A + 1) 2 )

1

1

1+s

) = Tr((V 2 (A + 1)V 2 )

).

Step 3. Let A ≥ 0. By Step 2, 1

1+s

1

Tr((A 2 VA 2 )

1

1

) = Tr((A + 1) 2 V(A + 1) 2 )

1+s

) − Tr(V 1+s ) + O(1).

By Step 1, we have 1

1

Tr(((A + 1) 2 V(A + 1) 2 )

1+s

) = Tr((A + 1)V 1+s ) + O(1),

s ∈ (0, 1).

Thus, 1

1

Tr((A 2 VA 2 )

1+s

) = Tr((A + 1)V 1+s ) − Tr(V 1+s ) + O(1) = Tr(AV 1+s ) + O(1),

s ∈ (0, 1).

(8.16)

8.5 Not every Dixmier trace is a ζ -function residue | 283

With Corollary 8.4.5, we can now prove the simplified formula for the trace extension of a ζ -function residue. Theorem 8.4.6. Let γ be an extended limit on L∞ (0, 1) concentrated at 0 and A ∈ ℒ(H). Then ζγ (AV) = ζγ,A (V),

0 ≤ V ∈ ℒ1,∞ ,

where ζγ,A is the ζ -function residue functional associated to γ and A from Definition 8.1.4, ζγ,A (V) = γ(sTr(AV 1+s )),

0 ≤ V ∈ ℒ1,∞ ,

and ζγ is the normalized fully symmetric trace that extends the ζ -function residue functional ζγ,1 on the positive cone of ℒ1,∞ . Proof. By linearity of ζγ,A (V) and ζγ (AV) in the bounded operator A, we may assume, without loss of generality, that A ≥ 0. It follows from Corollary 8.4.5 that 1

1

ζγ,A (V) − ζγ (AV) = ζγ,A (V) − ζγ (A 2 VA 2 ) 1

1

1+s

= γ(s(Tr(AV 1+s ) − Tr((A 2 VA 2 ) = lim(s(Tr(AV s→0

1+s

1 2

1 2

)))

1+s

) − Tr((A VA )

))) = 0.

8.5 Not every Dixmier trace is a ζ-function residue The previous section proved that every ζ -function residue functional on the positive cone of ℒ1,∞ is an additive functional that is normalized and monotone for Hardy– Littlewood submajorization. That is, for each extended limit γ on L∞ (0, 1) concentrated at 0, the ζ -function residue functional ζγ coincides with a normalized fully symmetric trace on the positive cone of ℒ1,∞ . Theorem 6.1.3 identified all the normalized fully symmetric traces on ℒ1,∞ with Dixmier traces on ℒ1,∞ . Hence, every ζ -function residue ζγ extends, by the linear extension discussed in the previous section, to a Dixmier trace. Chapter 6, in Theorems 6.1.1 and 6.1.3, explained the bijective correspondence between Dixmier traces on ℒ1,∞ and factorizable Banach limits on l∞ . In particular, the formula n+1

φθ (A) =

2 −2 1 ⋅ θ( ∑ λ(k, A)), log(2) k=2n −1

A ∈ ℒ1,∞ ,

describes every Dixmier trace on ℒ1,∞ where θ = ω ∘ C, for some extended limit ω on l∞ , is a factorizable Banach limit.

284 | 8 Heat trace and ζ -function formulas This section identifies the factorizable Banach limits that correspond to Dixmier traces arising as ζ -function residues, and proves that the factorizable Banach limits that correspond to ζ -function residues do not cover the set of all factorizable Banach limits. Factorizable Banach limits that correspond to ζ-function residues Lemma 8.5.1 identified the factorizable Banach limits that correspond to Dixmier traces arising as ζ -function residues. Lemma 8.5.1. If γ is an extended limit on L∞ (0, 1) concentrated at 0, then ζγ corresponds to a Banach limit θγ given by the formula θγ (x) = log(2) ⋅ γ(s ∑ 2−ks x(k)),

x ∈ l∞ .

k≥0

Proof. Section 6.3 defines the map LR−1 : l∞ → l1,∞ , given by LR−1 x = (x(0),

x(n) x(2) x(n) x(1) x(1) x(2) , , ,..., , . . . , n , . . . , n , . . .), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 2 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 4 4 2 2 2 times

2n

4 times

x ∈ l∞ .

times

Let ζγ be the Dixmier trace on ℒ1,∞ associated to γ. Theorem 6.3.1(b) gives the formula for the Banach limit associated to ζγ , θζγ = log(2) ⋅ ζγ ∘ diag ∘ LR−1 . Take x ∈ l∞ such that 0 ≤ x ≤ 1. We have 1+s

θζγ (x) = log(2) ⋅ γ(sTr(diag(LR−1 x)

)).

We write Tr(diag(LR−1 x)

1+s

) = ∑ 2k ⋅ (2−k x(k)) k≥0

1+s

= ∑ 2−ks x(k)1+s . k≥0

We have, θγ (x) − θζγ (x) = γ(s ∑ 2−ks (x(k) − x(k)1+s )). k≥0

Since, 1

0 ≤ x(k) − x(k)1+s ≤ sup (t − t 1+s ) = s ⋅ (1 + s)− s −1 ≤ s, t∈[0,1]

it follows that 0 ≤ θγ (x) − θζγ (x) ≤ γ(s2 ∑ 2−ks ) = 0. k≥0

Consequently, θγ (x) = θζγ (x) for every x ∈ l∞ such that 0 ≤ x ≤ 1. This completes the proof.

8.5 Not every Dixmier trace is a ζ -function residue

| 285

Not all factorizable Banach limits correspond to ζ-function residues Lemma 8.5.1 identified the Banach limits that correspond to Dixmier traces arising as ζ -function residues. Theorem 8.5.2 proves that the factorizable Banach limits that correspond to ζ -function residues do not cover the set of all factorizable Banach limits. Hence, unlike the set of heat trace functionals in Theorem 8.1.2, the set of all ζ -function residues is strictly smaller than set of all Dixmier traces. Theorem 8.5.2. There exists a Banach limit θ on l∞ such that θ(x) ≠ log(2) ⋅ γ(s ∑ 2−ks x(k)), k≥0

x ∈ l∞ ,

for any extended limit γ on L∞ (0, 1) concentrated at 0. We prove Theorem 8.5.2 by constructing an operator C1 : l∞ → l∞ such that the Banach limits θγ , γ being an extended limit on L∞ (0, 1) concentrated at 0, described in Lemma 8.5.1 are of the form θγ = ωγ ∘ C1 for some extended limit ωγ on l∞ . We then prove that extended limits of the form ω∘C1 cannot provide every extended limit on l∞ of the form ω ∘ C where C is the Cesaro operator. Lemma 8.5.3. Let γ be an extended limit on L∞ (0, 1) concentrated at 0. Let θγ be the Banach limit associated to γ in Lemma 8.5.1. There exists an extended limit ωγ on l∞ such that θγ = ωγ ∘ C1 where the operator C1 : l∞ → l∞ is defined by (C1 x)(n) =

n 2 ∑ (n − l + 1)x(l), (n + 1)(n + 2) l=0

x ∈ l∞ , n ≥ 0.

Proof. Set ω(x) = ωγ (x) :=

1 log3 (2) ⋅ γ(s3 ∑ (k + 1)(k + 2)2−ks x(k)), 2 k≥0

x ∈ l∞ .

A short argument confirms that ω is a positive functional on l∞ , ω(1) = 1 and ω vanishes on c0 . Hence, ω is an extended limit on l∞ . By the definition of C1 , we have k

ω(C1 x) = log3 (2) ⋅ γ(s3 ∑ 2−ks ∑ (k − l + 1)x(l)). k≥0

l=0

286 | 8 Heat trace and ζ -function formulas By interchanging the order of summation, k

∑ 2−ks ∑ (k − l + 1)x(l) = ∑ x(l) ∑ 2−ks (k − l + 1)

k≥0

l=0

l≥0

k≥l

= ∑ 2−ls x(l) ⋅ ∑ 2−ms (m + 1) = m≥0

l≥0

1 ∑ 2−ls x(l). (1 − 2−s )2 l≥0

Hence ω(C1 x) = log3 (2) ⋅ γ(

s3 ∑ 2−ls x(l)) (1 − 2−s )2 l≥0

for every x ∈ l∞ . Since 1 − 2−s > 0, s > 0, and lim+

s→0

1 1 − 2−s = , s log(2)

then, by Lemma 6.2.6, γ(

2

log3 (2) ⋅ s3 log(2) ⋅ s ) ⋅ log(2)γ(s ∑ 2−ls x(l)) ∑ 2−ls x(l)) = lim ( s→0+ 1 − 2−s (1 − 2−s )2 l≥0 l≥0 = log(2)γ(s ∑ 2−ls x(l)) = θγ (x). l≥0

Lemma 6.2.4 showed that, for z ∈ l∞ and every a ∈ (lim infn→∞ z(n), lim supn→∞ z(n)), there exists an extended limit ω on l∞ such that ω(z) = a. The next lemma gives an example of a positive sequence 0 ≤ x ∈ l∞ such that lim sup(C1 x)(n) < lim sup(Cx)(n), n→∞

n→∞

where the inequality is strict. This will be sufficient, in combination with Lemma 6.2.4, to prove that extended limits of the form ω ∘ C1 , with ω an extended limit on l∞ , do not provide all factorizable Banach limits of the form ω ∘ C, ω being an extended limit on l∞ . Lemma 8.5.4. Let A = ⋃n≥0 [22n , 22n+1 ). We have lim sup(CχA )(k) = k→∞

2n

2n+1

Proof. If k ∈ [2 , 2

2 , 3

lim sup(C1 χA )(k) = k→∞

5 . 9

), then 2m+1

(C1 χA )(k) = =

n−1 2 −1 k 2 ( ∑ ∑ (k − l + 1) + ∑ (k − l + 1)) (k + 1)(k + 2) m=0 l=22m l=22n n−1 2 (k − 22n + 2)(k − 22n + 1) ( ∑ (2k − 3 ⋅ 22m + 3) ⋅ 22m−1 + ) (k + 1)(k + 2) m=0 2

8.5 Not every Dixmier trace is a ζ -function residue | 287

4n − 1 16n − 1 1 (2k ⋅ − + 4n − 1 + (k − 22n + 2)(k − 22n + 1)) (k + 1)(k + 2) 3 5 1 4n 16n 2 = (2k ⋅ − + (k − 22n ) + O(k)) (k + 1)(k + 2) 3 5 =

2

=

2

2 4n 1 4n 4n ⋅ − ⋅ ( ) + (1 − ) + O(k −1 ). 3 k 5 k k

If k ∈ [22n−1 , 22n ), then 2m+1

n−1 2 −1 2 ( ∑ ∑ (k − l + 1)) (C1 χA )(k) = (k + 1)(k + 2) m=0 l=22m

=

n−1 2 ( ∑ (2k − 3 ⋅ 22m + 3) ⋅ 22m−1 ) (k + 1)(k + 2) m=0

4n − 1 16n − 1 1 (2k ⋅ − + 4n − 1) (k + 1)(k + 2) 3 5 1 4n 16n = (2k ⋅ − + O(k)) (k + 1)(k + 2) 3 5

=

2

=

2 4n 1 4n ⋅ − ⋅ ( ) + O(k −1 ). 3 k 5 k

Therefore, lim sup(C1 χA )(k) = max{ sup k→∞

t∈[ 21 ,1]

2 1 2 1 5 t − t 2 + (1 − t)2 , sup t − t 2 } = . 3 5 5 9 t∈[1,2] 3

This proves the second equality. The proof of the first equality is similar. Proof of Theorem 8.5.2. Let A be the set in Lemma 8.5.4. Let B be the class of Banach limits which correspond to Dixmier traces. By Lemma 6.3.3, this class contains all Banach limits θ = ω ∘ C, where ω is an extended limit on l∞ . By Lemma 8.5.4, sup θ(χA ) ≥ sup ω(CχA ) = lim sup(CχA )(n) = θ∈B

ω

n→∞

2 . 3

Let B1 be the class of Banach limits which correspond to Dixmier traces arising from ζ -function residues. By Lemmas 8.5.1 and 8.5.3, this set of Banach limits is contained in the class of all Banach limits of the form θ = ω ∘ C1 . By Lemma 8.5.4, sup θ(χA ) ≤ sup ω(C1 χA ) = lim sup(C1 χA )(n) =

θ∈B1

ω

n→∞

By comparing these inequalities, sup θ(χA ) < sup θ(χA ),

θ∈B1

and we must have B1 ⊊ B.

θ∈B

5 . 9

288 | 8 Heat trace and ζ -function formulas We can now prove the rest of Theorem 8.1.5. Proof of Theorem 8.1.5, (a) and (c). (a) Let γ be an extended limit on L∞ (0, 1) concentrated at 0. Theorem 8.1.5(b) proves that the ζ -function residue ζγ on the positive cone of ℒ1,∞ has a linear extension to a normalized fully symmetric trace that we also denote by ζγ on ℒ1,∞ . By Theorem 6.1.3, the linear extension is, therefore, a Dixmier trace on ℒ1,∞ . Theorem 8.4.6 proves that the Dixmier trace ζγ has the formula ζγ (AV) = ζγ,A (V),

A ∈ ℒ(H), 0 ≤ V ∈ ℒ1,∞ .

According to Theorem 6.1.1, the Dixmier trace ζγ , as a positive trace on ℒ1,∞ , corresponds to a unique factorizable Banach limit θγ and, therefore, ζγ (AV) = φθγ (AV), where θγ is the Banach limit identified in Lemma 8.5.1. (c) Theorem 8.5.2 proves that there is a factorizable Banach limit θ on l∞ such that θ ≠ θγ , where θγ is the Banach limit in Lemma 8.5.1, for any extended limit γ on L∞ (0, 1) concentrated at 0. By the bijective correspondence between factorizable Banach limits and Dixmier traces in Theorem 6.1.1, there exists a Dixmier trace on ℒ1,∞ that is not the linear extension of a ζ -function residue.

8.6 Notes Noncommutative residue of Wodzicki and Guillemin For notes on the spectral ζ -function of the Laplace–Beltrami operator, see [186, 217]; and for elliptic operators generally, see [137]. Minakshisundaram and Pleijel published their proof on the meromorphic extension of the spectral ζ -function in 1949 [186]. From Seeley in 1967 [241], it was known that the zeta function of a −p/m power of an order m elliptic differential operator on a closed Riemannian manifold Ω of dimension p has a meromorphic extension from Re(s) > 0 with a simple pole at s = 0 [246, p. 112], [137]. That the poles of the meromorphic extension of the ζ -function relate to the noncommutative residue can be seen in [292, 293]. For the connection with zeta-function regularization, see Hawking’s original paper [130]; for Seeley–DeWitt coefficients, see [276]; and for the connection with the Einstein–Hilbert action, see [2, 157]. Wodzicki in 1983 and Guillemin 1985 [292], [293, p. 384], [123] independently extended the residue at the first pole to any classical pseudodifferential operator of order −p on Ω by the noncommutative residue Res. Wodzicki was motivated by Adler and Manin’s work of spectral asymmetry in one dimension [3, 180]. The noncommutative residue appeared as an end note in Wodzicki’s study of spectral asymmetry, a fundamental concept to the Atiyah–Singer–Patodi index theory [13, 14]. Wodzicki mainly

8.6 Notes | 289

concentrated on local densities, which when integrated provide the noncommutative residue. The reference [116] features an exposition on densities. Connes’ trace theorem linking the noncommutative residue and the Dixmier trace appeared as [57, Theorem 1]. Connes’ theorem and its extensions will be discussed in Volume II. That the noncommutative residue of a classical pseudodifferential operator Q of order −p, normally written as an integral of the principal symbol of Q over the sphere bundle, is equivalent to the residue p

d ⋅ Ress=0 Tr(Q0 (1 − Δ)− 2 (1+s) ),

p

Q = Q0 (1 − Δ)− 2

is shown on [292, p. 387]. p For a classical pseudodifferential operator, at least, the value Tr(Q0 (1 − Δ)− 2 (1+s) ), p s > 0, can be calculated from local integrals of the principal symbol of Q0 (1 − Δ)− 2 (1+s) . Using the pseudodifferential calculus, this principal symbol is the product of the principal symbol of Q0 and the − p2 (1 + s)-power of the symbol of the Laplace–Beltrami operator, the latter of which only requires knowledge of the metric on Ω. For classical pseudodifferential operators, this approach would seldom be used. Wodzicki’s noncommutative residue Res(Q) can be calculated directly by integrating the principal symbol of Q over the cosphere bundle of Ω. Volume II discusses a direct formula for a Dixmier trace by integration of symbols over disc bundles in the cotangent bundle. Pleijel and Minakshisundaram heat kernel For Minakshisundaram’s use of the Mellin transform to obtain meromorphicity of the zeta-function of the Laplace–Beltrami operator Δ on a closed Riemannian manifold Ω of dimension p from the asymptotics of heat kernel, see [185]. For a proof, for any s > 0, that the operator esΔ : L2 (Ω) → C ∞ (Ω) is infinitely smoothing, that is, the kernel K(s, ⋅, ⋅) of esΔ belongs to C ∞ (Ω, Ω), see [241]. That the operator esΔ is of trace class is discussed on [246, p. 203]. Proofs that the heat kernel K(s, x, y), s > 0, x, y ∈ Ω decomposes into a sum of the eigenvalues of the Laplace–Beltrami operator and the diagonal matrix elements of eigenfunctions are found in [246, Theorem 8.3], [110, Lemma 1.6.3]. Shubin and Gilkey also prove Minakshisundaram’s asymptotic expansion in s of the kernel on the diagonal of Ω × Ω, ∞

K(s, x, x) ∼ s−p/2 ∑ ak (x)sk , k=0

s → 0+ , x ∈ Ω,

in [110, Lemma 1.7.4], [246, p. 119]. The Seeley–DeWitt coefficient functions ak (x) have a long history in spectral geometry [109], [110, Section 4.9], [22, IV]. For calculation of the Seeley–DeWitt terms, see [110, Theorem 4.8.18].

290 | 8 Heat trace and ζ -function formulas Under the Mellin transform, the scalar coefficients ak (P) in the asymptotic expansion of the kernel of e−sP , s > 0, correspond to the residues of the zeta-function of a general positive elliptic differential operator P of order m [110, Lemma 1.10.1], ak (P) = Ress= p−2k Γ(s)Tr(P −s ), m

k ≥ 0.

For 0 ≤ k < p/2, the leading poles are in the positive half-plane where the Gamma function Γ is regular. In terms of the noncommutative residue Res [293, p. 384], ak (P) = p−1 Γ(

p−2k p − 2k )Res(P − m ), m

0 ≤ k < p/2.

That Tr(Qe−sP ),

Q ∈ Ψ0 (Ω),

has an asymptotic expansion as s → 0+ with coefficients such that ak (Q, P) = Ress= p−2k Γ(s)Tr(QP −s ), m

k ≥ 0,

is proved in [110, Lemma 1.7.7]. Gilkey has a comprehensive discussion of heat kernel asymptotics (total heat content) on domains and boundary conditions [112]. Heat traces formulas The results of this chapter improve on Chapter 8 in [170]. In Definition 8.1.1 the heat trace functional ξω is defined for any extended limit ω. Combining Theorems 8.2.3 and 8.2.4 of [170] and Theorem 8.1.2, if ω is an extended limit on l∞ such that ξω is additive on the positive cone of ℒ1,∞ , then there exists an extended limit ω0 on l∞ such that ξω = ξω0 ∘M on the positive cone of ℒ1,∞ . So the heat trace functionals of the form ξω∘M in Theorem 8.1.2 describe all heat trace functionals that are equal to a trace on the positive cone of ℒ1,∞ . Removing the operator M from Theorem 8.1.2 is not possible. There exist dilationinvariant extended limits such that ξω ≠ Trω , see Theorem 8.2.9 in [170]. The behavior of the function −1

s 󳨃→ sTr(e−sV ),

s → 0+ , 0 ≤ V ∈ ℒ1,∞ ,

is bounded if and only if 0 ≤ V ∈ ℒ1,∞ [44, 238]. That it is not convergent in general is shown in Chapter 9. This function is not bounded for arbitrary 0 ≤ A ∈ ℳ1,∞ , and the logarithmic mean is applied to obtain a bounded function in this case [43, 238]. ζ-function formulas The ζ -function ζγ (V) of Definition 8.1.4 does not have a meromorphic continuation in general. Counterexamples are provided by [38, Lemma 17] or [155, Corollary 6.34], and in Chapter 9. The function sTr(AV 1+s ) is bounded for s ≥ 0 if and only if 0 ≤ V ∈ ℳ1,∞ , [43, Theorem 4.5].

8.6 Notes | 291

Development of the formulas The formulas between the heat trace, ζ -function, and Dixmier trace were established in [18, 39, 43, 60] using a dilation invariant extended limit ω on L∞ (0, ∞) assumed to be M-invariant, where M is the continuous version of the logarithmic mean. The original development followed Connes’ observation in [60, p. 306]. The class of M-invariant extended limits was first introduced in [39], see also [85], and further studied and used in [15, 18, 43]. Theorems 8.1.2 and 8.1.5 follow a line of results from [170, Chapter 8] and [238, Theorem 5], [269, Theorem 16], [267, Theorem 33], [244, p. 599], [270, Theorem 3.10], and their precedents in [18, 38, 39, 42, 43, 116, 238, 262]. In most of these references the formulas were developed simultaneously for the semifinite von Neumann algebra theory and the separable Hilbert space theory. Theorem 8.3.2 extends the results of [39, 43] and gives an affirmative answer to the question stated in [18]. The formulas in Theorem 8.1.5 strengthen and extend the results of [43, Theorem 4.11] and [39, Theorem 3.8]. Proofs in previous articles [39, 43, 240] were based on the weak-∗ Karamata theorem. A statement of the weak-∗ Karamata theorem can be found in Theorem 8.6.7 in [170]. Errata to results in [39, 43, 240, 262] are discussed in the end notes to Chapter 8 of [170]. In [127] Hardy proved the Hardy–Littlewood Tauberian theorem [127, Theorem 95, p. 155] following Karamata [156], [127, Theorem 98, p. 156]. Karamata’s Tauberian theorem provides the equivalence between logarithmically diverging series and zeta functions. The Mellin transform relates zeta-functions to heat trace asymptotics. This method underlies the proofs of the formulas from [39] to [238], where a weak-∗ Karamata theorem was proved for extended limits and applied to zeta functions [39, Theorem 2.2] using substitution or used directly for heat traces [238, Theorem 2]. The origin of the approach to this chapter is [262], as developed in Chapter 8 of [170]. In the 2010 paper [262], by which time the results in Chapter 6 that Dixmier traces provide all Hardy–Littlewood submajorization monotone positive traces were known, the method transferred to identification of heat and zeta functionals with submajorization monotone traces. Different computations yield the association of heat trace functionals with Dixmier traces, and the Mellin transform is not used to relate the heat trace and the ζ -function residue. The formula identifying that the set of extended limits of the form ω ∘ M is sufficient for the ideal ℒ1,∞ in this chapter is an improvement on the results for ℳ1,∞ in Chapter 8 of [170]. The proof in this chapter uses the diagonal formulas from Chapter 7 and the representation of the heat kernel from [45]. The proofs for the ζ -function formula in Section 8.4 do not use the Mellin transform and are different to Chapter 8 in [170]. That a zeta functional is a fully symmetric trace is broadly the same proof as Chapter 8 in [170]. The identification of zeta residue functionals inside the set of Dixmier traces uses Pietsch’s bijective association with factorizable Banach limits described in Chapter 6. The bijective association with factorizable Banach limits was first used to identify the set of zeta residue functions in [270]. Theorem 8.1.5 can be seen in [270, Lemma 3.5] and [270, Theorem 3.10].

292 | 8 Heat trace and ζ -function formulas There are statements for the ζ -function residue formula in Theorem 8.1.5(a) in terms of Dixmier’s format (equation (6.4)) for the Dixmier trace. The statements involving extended limits and Trω , instead of the Pietsch identification with factorizable Banach limits, do not classify all Dixmier traces that are ζ -function residues and require assumptions on the extended limit ω to enable a weak-∗ Karamata theorem. The neatest of these conditions is exponentiation invariance. Let ω be an extended limit on L∞ (0, ∞). There is an embedding of extended limits on l∞ as extended limits on L∞ (0, ∞), for example, using the projection P ∘ E from Example 6.2.5 using the partition {[n, n+1)}n≥0 of [0, ∞) in the place of the partition {(2−(n+1) , 2−n ]}n≥0 of (0, 1]. Define the continuous Dixmier trace formula for an extended limit ω on L∞ (0, ∞) by t

τω (V) = ω(

1 ∫ μ(s, V) ds), log(1 + t)

0 ≤ V ∈ ℳ1,∞ , t > 0.

0

That Trω (V) = τω∘E (V),

0 ≤ V ∈ ℳ1,∞ ,

for an extended limit ω on l∞ can be observed in [170, Theorem 6.4.4]. Define (ω ∘ log)(f (t)) = ω(f (log+ (t))),

f ∈ L∞ (0, ∞), t > 0,

for an extended limit ω on L∞ (0, ∞). An extended limit ω on L∞ (0, ∞) is exponentiation invariant, [269], if ω(f (t)) = ω(f (t a )),

f ∈ L∞ (0, ∞), t > 0,

for all a > 0. Then it is known that 1 1 τω (V) = ζω∘log (V) = (ω ∘ log)( Tr(V 1+ t )), t

0 ≤ V ∈ ℳ1,∞ ,

for an exponentiation-invariant extended limit ω on L∞ (0, ∞) [269, Theorem 16]. Higher order terms in the heat trace expansion For the Laplace–Beltrami operator Δ on a closed Riemannian manifold Ω the second term a1 in the heat trace expansion (8.2) is 1 ∫ Rg (x) dx, 6 Ω

which is proportional to the Einstein–Hilbert action whose variation describes the vacuum field equations of general relativity. Replacing −Δ acting on C ∞ (Ω) by a generalized Laplacian D2 where D is a general Dirac operator on the smooth sections C ∞ (Ω, E)

8.6 Notes | 293

of a Clifford bundle E over the closed manifold Ω [162, Section 5] only changes the constant of proportionality and introduces a cosmological constant to the action; this is noted by [147, Section 5] and seen in [110, Theorem 4.8.18]. Therefore the second term 2 in the expansion of Tr(e−tD ) for any standard Laplacian in differential geometry is equivalent to the vacuum gravity action. Following Connes [63] and Kastler’s computation in [157], Kalau and Walze [147, Section 5] further noted that the heat trace for the square of the Dirac operator of the Connes–Lott model of a four-dimensional manifold tensored by a matrix algebra representing the standard model [60, VI] has second term proportional to the vacuum Einstein–Hilbert action. In 1996 Chammseddine– Connes introduced the spectral action, where the first, second, and third terms of an asymptotic expansion of the heat trace for the Connes–Lott model were associated to a (bosonic) action incorporating the standard model, gravity, and additional terms [46]. Given the heat trace Tr(e−sH ), s > 0, of an arbitrary positive operator H with compact resolvent and H −p/2 ∈ ℒ1,∞ for some p ≥ 4, Dixmier’s trace extracts the “leading term of the asymptotic expansion”. Presently there are no analogue formulas involving singular traces that extract the second or higher terms. It was noted in [293, p. 389], [147], and [46], and reiterated in [2] after [157], that in the case of a Dirac operator D on a closed Riemannian manifolds, as above, the noncommutative residue Res((D2 )

− p2 +1

)

calculates the second term and hence Einstien–Hilbert action. However, for general positive operators such that H −p/2 ∈ ℒ1,∞ and p ≥ 4, there is no analogue formula for the noncommutative residue of H −p/2+1 in the same manner as the Dixmier trace is the noncommutative residue of H −p/2 . Using singular traces to extract the second and higher terms of the asymptotics of the heat trace for more general geometries than those associated to elliptic differential operators is an open problem [250]. Further calculation formulas Chapter 9 observes that the ζ -function and heat trace approach to the Dixmier trace relate to a Tauberian type result on Abel and logarithmic mean convergence n ∞ 1 ∑ λ(k, P) = lim+ (t − 1) ∑ λ(k, P)t . n→∞ log(2 + n) t→1 k=0 k=0

lim

Here P is as a strictly positive elliptic classical pseudodifferential operator of order −p on a p-dimensional closed Riemannian manifold Ω. The left-hand side of the formula is the Dixmier trace involving partial sums, and the right-hand side involving the infinite sum is the residue of the ζ -function. The Mellin transform converts the right-hand side to an equivalent form of Abel convergence, explained by Hardy [127], and so the

294 | 8 Heat trace and ζ -function formulas heat trace approach to the Dixmier trace is traditionally based on results of Tauberian type. The results of Chapter 9, and even Chapter 8 using the concept of extended limits, can be viewed as generalizations of these Tauberian-type results. Volume II describe further formulas for traces and operators on closed manifolds, where the partial sums ∑nk=0 λ(k, P), n ≥ 0, are directly approximated by integration of symbols.

9 Criteria for measurability 9.1 Introduction Chapter 6 characterized all positive normalized traces and Dixmier traces on the ideal ℒ1,∞ of weak trace class compact operators on a separable Hilbert space H. This chapter characterizes the operators in ℒ1,∞ that take the same value for every trace, every positive normalized trace, or every Dixmier trace, and considers expansion of the heat trace and meromorphic continuation of the ζ -function for such operators. Motivation Chapter 6 described Weyl’s law and that the spectral behavior of the inverse power of the Laplace–Beltrami operator is proportional to the harmonic sequence and the measure on a p-dimensional closed Riemannian manifold Ω (suppressing some constants depending on the dimension p) p

λ(n, (1 − Δ)− 2 ) ∼ Vol(Ω) ⋅ n−1 ,

n ≥ 0.

In Chapter 7, a trivial corollary of diagonal formulas for traces on ℒ1,∞ shows that (suppressing some dimensional constants) p

φ(Mf (1 − Δ)− 2 ) = ∫ f (s) ds,

f ∈ L∞ (𝕋p )

𝕋p

for every normalized trace φ on ℒ1,∞ . Here 𝕋p is the flat p-torus and the essentially bounded function f acts as by pointwise multiplication as the operator Mf on L2 (𝕋p ). Volume II develops the above results and proves Connes’ formula for a closed manifold Ω introduced in (6.1) (suppressing some dimensional constants) p

φ(Q(1 − Δ)− 2 ) = ∫ q(v) dv,

Q ∈ Ψ0 (Ω),

(9.1)

S∗ Ω

where φ is any normalized trace on ℒ1,∞ . Here Ψ0 (Ω) denotes the algebra of zero-order classical pseudodifferential operators Q : C ∞ (Ω) → C ∞ (Ω), q is the principal symbol of Q, and dv is the Liouville measure on the sphere bundle S∗ Ω. The operators p

Mf (1 − Δ)− 2 ∈ ℒ1,∞ ,

f ∈ L∞ (𝕋p ),

and p

Q(1 − Δ)− 2 ∈ ℒ1,∞ , https://doi.org/10.1515/9783110378054-011

Q ∈ Ψ0 (Ω),

296 | 9 Criteria for measurability have the property that the value p

φ(A(1 − Δ)− 2 ),

A = Mf , Q

is equivalent to a Radon measure on 𝕋p and S∗ Ω, respectively, and is independent of which normalized trace φ on ℒ1,∞ is applied. Having a large and central class of operators in measure theory and differential geometry where the value of that operator under a trace does not depend on which trace is chosen warrants the notion of a measurable operator. Connes, originally working with Dixmier traces on ℒ1,∞ , introduced the terminology that a measurable operator A ∈ ℒ1,∞ satisfies φ(A) = c for a constant c ∈ ℂ for all φ ∈ X, where X is a set of normalized traces on ℒ1,∞ . Chapter 8 observed that the equality p

Trω (Q(1 − Δ)− 2 ) = c,

Q ∈ Ψ0 (Ω),

for every Dixmier trace Trω on ℒ1,∞ , where c = ∫ q(v) dv S∗ Ω

is the value from (9.1) above, could be observed from the behavior of the associated ζ -function and heat trace expansion. Minakshisundaram and Pleijel’s zeta-function for the Laplacian, generalized by Seeley and the noncommutative residue of Wodzicki and Guillemin, p

ζQ,Δ (s) = Tr(Q(1 − Δ)− 2 (1+s) ),

s > 0,

has a meromorphic extension from Re(s) > 0 and a simple pole at s = 0 such that Ress=0 ζQ,Δ (s) = c,

(9.2)

where c is the same value as the constant in (9.1) above. Minakshisundaram’s asymptotic expansion for the heat trace p

∞

Tr(QesΔ ) ∼ s− 2 ∑ ak sk , n=0

s → 0+ ,

which is equivalent to Tr(Qe−(s(1−Δ)

p 2 −2 −p )

∞

) ∼ sa0 + ∑ ak s k=1

1− 2k p

,

s → ∞,

9.1 Introduction | 297

leads to the formula p 2 −2 −p 1 Tr(Qe−(s(1−Δ) ) ) = a0 , s→∞ s

lim

(9.3)

identifying the leading term of the heat trace expansion. From the Mellin transform, we get a0 =

c , Γ(1 + p2 )

which again identifies the same value c as in (9.1) and (9.2) above. Hence the existence of a leading term in the asymptotic expansion of the heat trace Tr(Qe−(s(1−Δ)

p 2 −2 −p

)

)

and the analyticity of ζQ,Δ (s) −

c s

in the half-plane Re(s) > −ϵ, ϵ > 0, both infer that the trace of the operator p

Q(1 − Δ)− 2 ∈ ℒ1,∞ is independent of the trace on ℒ1,∞ used, at least for Dixmier traces. The examples of pseudodifferential operators lead naturally to the question whether, in general, for A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ , the existence of the residue of the ζ -function from Chapter 8, ζA,V (s),

s > 0,

at s = 0 implies measurability of the operator AV ∈ ℒ1,∞ , and what behavior of the ζ -function measurability implies. Similarly, whether the existence of a leading term of order t in an expansion of the heat trace of Chapter 8, −α

Tr(Ae−(tV) ),

t > 0, α > 0,

implies measurability of the operator AV ∈ ℒ1,∞ , and what behavior of the heat trace measurability implies. Spectral criteria for measurability of operators in ℒ1,∞ Chapter 1, in Definition 1.2.10, defined a trace φ on ℒ1,∞ as a linear functional φ such that φ(U ∗ AU) = φ(A) for all unitary operators U and A ∈ ℒ1,∞ . A trace φ is normalized if φ(diag{

1 } ) = 1. n + 1 n≥0

298 | 9 Criteria for measurability A positive trace φ on ℒ1,∞ is a trace such that φ(A) ≤ φ(B) when 0 ≤ A ≤ B ∈ ℒ1,∞ . Chapter 6 identified the formula for every normalized positive trace on ℒ1,∞ , n+1

φ(A) = θφ (

1 2 −2 ∑ λ(k, A)), log 2 k=2n −1

A ∈ ℒ1,∞ ,

where θφ is a Banach limit on l∞ and λ(A) is an eigenvalue sequence of the compact operator A. The correspondence φ → θφ between positive normalized traces and Banach limits is bijective. Chapter 6 also proved that every normalized fully symmetric trace φ on ℒ1,∞ , that is, a normalized trace φ such that φ(A) ≤ φ(B) when 0 ≤ A ≺≺ B ∈ ℒ1,∞ , is a Dixmier trace Trω (A) = ω(

n 1 ∑ λ(k, A)), log(2 + n) k=0

A ∈ ℒ1,∞ ,

where ω is an extended limit on l∞ . The correspondence Trω → ω between Dixmier traces and extended limits on l∞ is surjective but not injective. Definition 9.1.1. An operator A ∈ ℒ1,∞ is called (a) universally measurable if all normalized traces take the same value on A; (b) measurable if all positive normalized traces take the same value on A; (c) Dixmier measurable if all Dixmier traces take the same value on A. A continuous trace φ in the quasinorm of ℒ1,∞ , that is, a trace φ on ℒ1,∞ satisfying 󵄨 󵄨 sup 󵄨󵄨󵄨φ(A)󵄨󵄨󵄨 < ∞,

‖A‖1,∞ ≤1

where ‖A‖1,∞ = sup(n + 1)μ(n, A), n≥0

decomposes into four positive traces by the results of Chapter 4. Definition 9.1.1(b) is equivalent to asking that all continuous normalized traces take the same value on A. Chapter 6 discussed extended limits, in particular Lemma 6.2.4 shows that a sequence x ∈ l∞ is convergent if and only if ω(x) is the same value for every extended limit ω on l∞ . Following Banach and Mazur, G. Lorentz introduced the notion of almost convergence of a sequence x ∈ l∞ , see Definition 6.2.10 and Theorem 6.2.11. A

9.1 Introduction | 299

sequence x ∈ l∞ is almost convergent if θ(x) is the same value a ∈ ℂ for every Banach limit θ on l∞ , and almost convergence of x ∈ l∞ is denoted x(n) → a a. c. The Cesaro operator C : l∞ → l∞ is defined by (Cx)(n) =

n 1 ∑ x(k), n + 1 k=0

x ∈ l∞ , n ≥ 0.

The sequence x ∈ l∞ converges (C, 1) if Cx is convergent to a ∈ ℂ, denoted x(n) → a (C, 1). The spectral characterization of positive traces and Dixmier traces on ℒ1,∞ in Chapter 6 provides direct spectral criteria for measurability of operators. Theorem 9.1.2. Let A ∈ ℒ1,∞ and let λ(A) be an eigenvalue sequence of A. Then (a) A is universally measurable if and only if there exists a constant c ∈ ℂ such that n

∑ λ(k, A) = c ⋅ log(n + 2) + O(1),

k=0

n ≥ 0.

(b) A is measurable if and only if there exists a constant c ∈ ℂ such that 2n+1 −2

∑ λ(k, A) → c ⋅ log(2)

k=2n −1

a. c.

(c) A is Dixmier measurable if and only if there exists a constant c ∈ ℂ such that n 1 ∑ λ(k, A) → c, log(n + 2) k=0

n → ∞,

or, equivalently, 2n+1 −2

∑ λ(k, A) → c ⋅ log(2) (C, 1).

k=2n −1

In each case the constant c ∈ ℂ determines the value of the normalized trace of A, the normalized positive trace of A, or the Dixmier trace of A, respectively. Writing the spectral criterion for Dixmier measurability as n

∑ λ(k, A) = c ⋅ log(n + 2) + o(log(n + 2)),

k=0

n ≥ 0,

300 | 9 Criteria for measurability makes overt that the difference between Dixmier and universal measurability of an operator A ∈ ℒ1,∞ lies in the rate of divergence of the remainder term. The difference between Dixmier measurability and measurability is in the type of convergence of the bounded sequence of dyadic averages 2n+1 −2

∑ λ(k, A),

k=2n −1

n ≥ 0.

Theorem 9.1.2 is proved in Section 9.2. Corollary 9.2.4 in Section 9.2 frames the equivalent criteria in Theorem 9.1.2 in terms of expectation values, based on the diagonal formulas for traces on ℒ1,∞ from Chapter 7. Section 9.3 provides a list of examples of measurable operators. The examples demonstrate that the set of Dixmier measurable elements is strictly larger than the set of measurable elements and that the set of measurable elements is strictly larger than the set of universally measurable elements. Chapter 6 proved that the sets of Dixmier traces, positive normalized traces, and normalized traces are distinct. That the respective sets of measurable elements are strictly contained in each other does not follow automatically from the fact that the respective sets of traces are strictly contained in each other. Strict subsets of Dixmier traces on ℒ1,∞ may admit the same set of Dixmier measurable operators in ℒ1,∞ . On page 305 of “Noncommutative Geometry” [60], Alain Connes defined the Dixmier trace as Trω∘M (A) = (ω ∘ M)(

n 1 ∑ μ(k, A)), log(2 + n) k=0

0 ≤ A ∈ ℒ1,∞ ,

where ω is an extended limit on l∞ and M : l∞ → l∞ is the logarithmic form of the Cesaro operator (Mx)(n) =

n x(k) 1 , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0,

which implies, see Proposition 6(a) on page 307 in [60], that a positive operator A ∈ ℒ1,∞ takes the same value for every trace defined by Connes if M(k 󳨃→

k 1 ∑ μ(j, A))(n) log(2 + k) j=0

converges as n → ∞. Most users of noncommutative geometry still refer to the original presentation of measurability by Connes. Chapter 6 proved that Dixmier traces are bijective with factorizable Banach limits of the form θ =γ∘C

9.1 Introduction | 301

for an extended limit γ on l∞ , and that Connes’ trace is bijective with twice factorizable Banach limits of the form θ = γ ∘ C2 for an extended limit γ on l∞ . Hence Connes’ Dixmier traces form a strict subset of Dixmier traces. Despite this fact, Proposition 9.2.3 proves that an operator A ∈ ℒ1,∞ is Dixmier measurable if and only if the operator A takes the same value for every Connes’ Dixmier trace. This result follows from a Tauberian theorem of Hardy. Measurability criteria and the heat trace Theorem 9.1.2 provides criteria for measurability in terms of eigenvalue sequences and, in practice, the eigenvalues λ(A) of an operator A ∈ ℒ1,∞ may be inaccessible. Heat trace and ζ -function residue formulas from Chapter 8 offer additional measurability criteria for an operator A ∈ ℒ1,∞ . In applications, criteria involving the trace Tr on ℒ1 have proven to be more accessible. The following theorem provides exact criteria for universal and Dixmier measurability in terms of the heat semigroup. Theorem 9.1.3. Let 0 ≤ V ∈ ℒ1,∞ and let A ∈ ℒ(H). (a) For every α > 0, the following conditions are equivalent: i. φ(AV) = c for every normalized trace φ on ℒ1,∞ . ii. We have −α

Tr(AVe−(nV) ) = c ⋅ log(n + 2) + O(1),

n ≥ 0.

(b) For every α > 0, the following conditions are equivalent: i. Trω (AV) = c for every Dixmier trace Trω on ℒ1,∞ . ii. There exists a limit lim

n→∞

−α 1 Tr(AVe−(nV) ) = c. log(n + 2)

iii. There exists a limit lim M(k 󳨃→

n→∞

−α 1 Tr(Ae−(kV) ))(n) = c, k

where M : l∞ → l∞ is the logarithmic mean. The equivalence of the statements in Theorem 9.1.3(b)(i) and Theorem 9.1.3(b)(iii) is straightforward from Theorem 8.1.2, which indicated that −α 1 Trω (AV) = (ω ∘ M)( Tr(Ae−(nV) )), n

A ∈ ℒ1,∞ ,

302 | 9 Criteria for measurability for every extended limit ω on l∞ . Theorem 9.1.4(b) below shows that the logarithmic mean M cannot be removed from the equivalent conditions in Theorem 9.1.3(b). The introduction to Chapter 8 and the above motivation described the example p of the operator V = P − 2 ∈ ℒ1,∞ , where P is an order 2 strictly positive elliptic classical pseudodifferential operator on a closed Riemannian manifold of dimension p. For the example of the order 2 strictly positive elliptic operator P, the heat trace has an asymptotic expansion of the form − p2

Tr(e−(tV) ) = c ⋅ t + O(t 1−2/p ),

t→∞

for a constant c > 0. It is natural to ask whether a Dixmier measurable operator, or a universally measurable operator, admits a similar asymptotic expansion. Generally, this is not the case. The notion of a leading term in the asymptotic expansion of the heat trace is a stronger condition than either Dixmier or universal measurability. Theorem 9.1.4. Let 0 ≤ V ∈ ℒ1,∞ and let A ∈ ℒ(H). (a) The condition Tr(Ae−(tV) ) = c ⋅ t + O(t 1−ϵ ), −α

t→∞

(9.4)

for some α > 0 and ϵ > 0 implies that AV is universally measurable and φ(AV) =

c Γ(1 + α−1 )

for every normalized trace φ on ℒ1,∞ . However, there exists a universally measurable operator AV such that (9.4) is not satisfied for any α > 0. (b) The condition −α

Tr(Ae−(tV) ) = c ⋅ t + o(t),

t→∞

(9.5)

for some α > 0 implies that AV is Dixmier measurable and Trω (AV) =

c Γ(1 + α−1 )

for every extended limit ω. However, there exists a Dixmier measurable operator AV such that (9.5) is not satisfied for any α > 0. Theorems 9.1.3 and 9.1.4 are proved in Section 9.4. Measurability criteria and the ζ-function Theorem 9.1.5 below relates measurability of operators of the form AV, A ∈ ℒ(H), 0 ≤ V ∈ ℒ1,∞ , to the nature of the singularity at s = 0 of the ζ -function of Chapter 8, ζA,V (s) = Tr(AV 1+s ),

s > 0.

9.1 Introduction | 303

Recall that if f is a meromorphic function in a neighborhood of z = 0 in the complex plane with a simple pole at z = 0, then Resz=0 f (z) denotes the coefficient of z −1 in the Laurent expansion of f , or, equivalently, the value of the analytic function zf (z) at z = 0. Theorem 9.1.5. Let 0 ≤ V ∈ ℒ1,∞ and let A ∈ ℒ(H). (a) If the function ζA,V (s) − cs ,

s > 0,

has an analytic continuation to the set {z ∈ ℂ : ℜ(z) > −ε} for some ε > 0,

(9.6)

then AV is universally measurable and φ(AV) = Resz=0 ζA,V (z) = c for every normalized trace φ on ℒ1,∞ . However, there exists a universally measurable operator AV such that (9.6) is not satisfied.

(b) The following conditions are equivalent: i. Trω (AV) = c for every Dixmier trace Trω on ℒ1,∞ . ii. There exists the limit lim sζA,V (s) = c.

s→0+

Theorem 9.1.5 is proved in Section 9.5. Condition (9.4) of the first order term in the heat trace expansion is the strongest condition imposed on the operator AV, A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ , of the conditions in Theorems 9.1.4 and 9.1.5. The end notes to this chapter describe that, from the Mellin transform, we get Tr(Ae−(tV) ) = Γ(1 + α−1 )c ⋅ t + O(t 1−ϵ ), −α

t → ∞,

which implies that c ζA,V (s) − , s

s > 0,

has an analytic continuation for ℜ(z) > −ϵ. The existence of meromorphic continuation of the ζ -function in (9.6) implies, from the inverse Mellin transform, the condition (9.5), namely −α

Tr(Ae−(tV) ) = Γ(1 + α−1 )c ⋅ t + o(t),

t → ∞.

304 | 9 Criteria for measurability By Theorem 9.1.4, the leading term in the heat trace expansion in condition (9.5) implies Dixmier measurability in the form ζA,V (s) −

1 c = o( ), s s

s > 0,

seen in Theorem 9.1.5(b). Measurability criteria in ℳ1,∞ Chapter 6 proved that Dixmier traces on ℒ1,∞ extend to the Dixmier–Macaev ideal ℳ1,∞ , Trω (A) = ω(

n 1 ∑ λ(k, A)), log(n + 2) k=0

A ∈ ℳ1,∞ .

However, the formula for the extension requires that the extended limit ω on l∞ be dilation invariant. The notion of a measurable operator in ℳ1,∞ can be extended from ℒ1,∞ . An operator A ∈ ℳ1,∞ is called Dixmier measurable if Trω (A) takes the same value for all dilation-invariant extended limits ω on l∞ . Theorems 6.1.4 and 6.1.6 together prove that this definition coincides on ℒ1,∞ with Definition 9.1.1. There are differences between the form of Dixmier measurable operators in ℳ1,∞ and the form of Dixmier measurable operators in ℒ1,∞ . Theorem 9.6.1 proves that a positive operator 0 ≤ A ∈ ℳ1,∞ is Dixmier measurable if and only if there exists the limit n 1 ∑ μ(k, A). n→∞ log(2 + n) k=0

lim

(9.7)

However, unlike in ℒ1,∞ , for an arbitrary compact operator A ∈ ℳ1,∞ it is not true that A being Dixmier measurable implies that the limit n 1 ∑ λ(k, A) n→∞ log(2 + n) k=0

lim

exists, where λ(A) is an eigenvalue sequence of A. Proposition 9.6.5 provides a counterexample. Behavior of measurable operators and criteria for products The reader is cautioned against a literal translation of measurable operators. Sections 9.3 and 9.7 show that the properties of measurable functions and the Fubini theorem fail to translate to measurable operators.

9.1 Introduction | 305

Chapter 7 introduced singular states on ℒ(H) of the form Eφ (A) = φ(AV),

A ∈ ℒ(H),

where 0 ≤ V ∈ ℒ1,∞ is a density operator normalized such that φ(V) = 1 and φ is a positive trace on ℒ1,∞ . Chapter 7 described the states Eφ as the extension in Alain Connes’ noncommutative geometry of integration of functions. That 0 ≤ V ∈ ℒ1,∞ is measurable does not imply that AV is measurable for a given A ∈ ℒ(H). Chapter 7 provided counterexamples. This is problematic, since in noncommutative geometry it is desirable that the value Eφ (A) be independent of the normalized trace. The analogous property of measurability of a function is measurability of the product AV,

A ∈ ℒ(H),

so that the value Eφ (A) is independent of the trace φ on ℒ1,∞ , the positive trace φ on ℒ1,∞ , or the Dixmier trace φ on ℒ1,∞ , respectively, depending on whether the focus is on universal measurability, measurability, or Dixmier measurability. If f is a realvalued measurable function on a measure space then |f | is a measurable function, and the positive and negative parts f+ = max{f , 0} and f− = max{0, −f } are measurable functions where f = f+ − f− . Example 7.4.3 in Chapter 7 provided an example of a bounded self-adjoint operator A ∈ ℒ(H) such that AV ∈ ℒ1,∞ is a Dixmier measurable operator but |A|V is not a Dixmier measurable operator for H = L2 (𝕋) and V = (1 − Δ𝕋 )−1/2 for the Laplacian Δ𝕋 on the circle. The set 𝒜V := {A ∈ ℒ(H) : AV is Dixmier measurable}

is a closed linear subspace of ℒ(H) that is not an algebra and is not closed under taking the absolute value or decomposition into positive parts. The spectral characterization of measurability is of limited value to determine whether a given A ∈ ℒ(H) belongs to the subspace 𝒜V . The spectrum of products is generally inaccessible. The next theorem collates the results in Theorem 7.1.4 and Theorems 9.1.3 and 9.1.5 for the reader, phrasing some conditions in terms of logarithmic mean convergence. The logarithmic mean operator M : l∞ → l∞ is defined by (Mx)(n) =

n x(k) 1 , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

306 | 9 Criteria for measurability A sequence x ∈ l∞ converges in logarithmic mean to a ∈ ℂ, denoted x(n) → a (M, 1), if Mx is convergent to a. The following conditions from Theorems 9.1.3, 9.1.5, and Corollary 9.2.4 can determine measurability of general products in Alain Connes’ noncommutative geometry. Conceptually, they are equivalent forms of generalized convergence of the bounded sequence of expectation values ⟨Aen , en ⟩,

n ≥ 0,

where en , n ≥ 0, is the ordered eigenbasis such that Ven = λ(n, V)en . Theorem 9.1.6. Suppose A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ . Let en , n ≥ 0, denote an orthonormal basis of H such that Ven = λ(n, V)en . The following statements are equivalent: (a) AV is Dixmier measurable and there is a constant c ∈ ℂ such that Trω (AV) = c (b)

for any extended limit ω on l∞ . ⟨Aen , en ⟩ n λ(n, V) → c

(M, 1).

(c) −1 1 Tr(AVe−(nV) ) → c, log(2 + n)

n → ∞.

(d) sTr(AV 1+s ) → c,

s → 0+ .

Exact criteria for (M, 1)-convergence of a sequence x ∈ l∞ are known and described in the end notes to Chapter 7. Fubini theorem Chapter 7 showed that multiplication operators on the torus are universally measurable operators. Let Δ𝕋p : H2 (𝕋p ) → L2 (𝕋p ) be the flat Laplacian on the torus 𝕋p , p ∈ ℕ. Let Mf , f ∈ L∞ (𝕋), be the bounded operator on L2 (𝕋p ) given by (Mf h)(s) = f (s)h(s), h ∈ L2 (𝕋p ). It was evident from the diagonal formula for an arbitrary trace φ on ℒ1,∞ (L2 (𝕋p )) that φ(Mf (1 − Δ𝕋p )

− p2

p

π2 )= ∫ f (s) ds. Γ(1 + p2 ) p 𝕋

9.1 Introduction | 307

The Laplacian on Δ𝕋2p acting on the Hilbert space L2 (𝕋p × 𝕋p ) can be written as the operator Δ𝕋p ⊗ 1 + 1 ⊗ Δ𝕋p acting on the Hilbert space L2 (𝕋p ) ⊗ L2 (𝕋p ). Fubini’s theorem on the Lebesgue integral of the Cartesian product indicates that φ((Mf ⊗ Mg )(1 − Δ𝕋p ⊗ 1 − 1 ⊗ Δ𝕋p )−p ) =

Γ(1 + p2 )Γ(1 + p2 ) Γ(1 + p)

p

p

φ(Mf (1 − Δ𝕋p )− 2 ) ⋅ φ(Mg (1 − Δ𝕋p )− 2 ),

f , g ∈ L∞ (𝕋p ),

for every trace φ on ℒ1,∞ . To discuss the Fubini theorem in Alain Connes’ noncommutative geometry, we introduce the equivalent of the Laplacian operator. We require the notion of an unbounded operator with compact resolvent. Let a(n), n ≥ 0, be an unbounded sequence of real numbers such that |a(n)|, n ≥ 0, is increasing. Given some orthonormal basis en , n ≥ 0, of a separable Hilbert space H, the operation Den = a(n)en ,

n ≥ 0,

defines a linear operator D : dom(D) → H, ∞

Dξ = ∑ a(n)⟨ξ , en ⟩en , n=0

x ∈ dom(D),

where dom(D) = {ξ ∈ H : a(n)⟨ξ , en ⟩ ∈ l2 }. The operator D : dom(D) → H is self-adjoint, meaning that it is: (a) a closed operator (when {ξm }m≥0 ⊂ dom(D) such that ξm → ξ and Dξm → η for ξ , η ∈ H, then ξ ∈ dom(D) and Dξ = η); (b) if ξ ∈ H satisfies |⟨Dη, ξ ⟩| ≤ c‖η‖ for some constant c and all η ∈ dom(D), then ξ ∈ dom(D) and ⟨Dη, ξ ⟩ = ⟨η, Dξ ⟩. Conversely, for any linear operator D : dom(D) → H satisfying conditions (a) and (b) such that (D + i)−1 is a compact operator on H, there exists a sequence a(n), n ≥ 0, of real numbers with |a(n)| increasing and an orthonormal basis en , n ≥ 0, of H such that Den = a(n)en ,

n ≥ 0.

308 | 9 Criteria for measurability The numbers a(n) are called the eigenvalues of D and denoted λ(n, D), n ≥ 0. For any decreasing continuous function f : ℝ → ℝ+ such that lims→∞ f (s) = 0, the operator f (D) : H → H defined by f (D)en = f (λ(n, D))en ,

n ≥ 0,

is a compact operator such that λ(n, f (D)) = f (λ(n, D)),

n ≥ 0.

Given a self-adjoint operator D : dom(D) → H with compact resolvent, the self-adjoint operator D2 : dom(D2 ) → H is defined by 󵄨 󵄨2 D2 en = 󵄨󵄨󵄨a(n)󵄨󵄨󵄨 en ,

n ≥ 0.

Connes’ noncommutative geometry is based on a self-adjoint operator D : dom(D) → H with compact resolvent, where D2 : dom(D2 ) → H is analogous to p the Laplacian operator, and the existence of p > 0 such that (1 + D2 )− 2 ∈ ℒ1,∞ is analogous to Weyl’s law and indicates the dimension of the geometry associated to D. If D1 : dom(D1 ) → H1 is a self-adjoint operator on a separable Hilbert space H1 with compact resolvent such that (D1 + i)−1 ∈ ℒp1 ,∞ (H1 ) for p1 > 0, and D2 : dom(D2 ) → H2 is a self-adjoint operator on a separable Hilbert space H2 with compact resolvent such that (D2 + i)−1 ∈ ℒp2 ,∞ (H2 ) for p2 > 0, then the positive operator D21 ⊗ 1 + 1 ⊗ D22 : H2 (D1 , D2 ) → H1 ⊗ H2 with domain H2 (D1 , D2 ) = {ξ ∈ H1 ⊗ H2 : (λ(n, D1 )2 + λ(m, D2 )2 )⟨ξ , e1,n ⊗ e2,m ⟩ ∈ l2 } corresponds to the Laplacian of the Cartesian product of geometries in noncommutative geometry. The dimension of the product is p1 + p2 from the property that (1 + D21 ⊗ 1 + 1 ⊗ D22 )

− 21

∈ ℒp1 +p2 ,∞ (H1 ⊗ H2 ).

Condition (9.4) of a leading term in the asymptotic expansion of the heat trace was proved to be a stronger condition than universal measurability in Theorem 9.1.4. Condition (9.4) is sufficient for the Fubini theorem to hold in noncommutative geometry.

9.1 Introduction | 309

Theorem 9.1.7 (Fubini theorem). Let H1 and H2 be separable Hilbert spaces. Suppose A1 ∈ ℒ(H1 ) and D1 : dom(D1 ) → H1 is a self-adjoint operator with compact resolvent such that (D1 + i)−1 ∈ ℒp1 ,∞ and Tr(A1 e−t

2

D21

) = c1 ⋅ t −p1 + O(t ϵ−p1 ),

t → 0+ ,

for some p1 > 0 and ϵ > 0. Suppose A2 ∈ ℒ(H2 ), and D2 : dom(D2 ) → H2 is a self-adjoint operator with compact resolvent such that (D2 + i)−1 ∈ ℒp2 ,∞ and Tr(A2 e−t

2

D22

) = c2 ⋅ t −p2 + O(t ϵ−p2 ),

t → 0+ ,

for some p2 > 0 and ϵ > 0. Then p1 (a) V1 := (1 + D21 )− 2 ∈ ℒ1,∞ (H1 ) with φ(A1 V1 ) =

c1 Γ(1 + p21 ) p2

for every normalized trace φ on ℒ1,∞ , and V2 := (1 + D22 )− 2 ∈ ℒ1,∞ (H2 ) with φ(A2 V2 ) =

c2 Γ(1 + p22 )

for every normalized trace φ on ℒ1,∞ . (b) A1 ⊗ A2 ∈ ℒ(H1 ⊗ H2 ) satisfies Tr((A1 ⊗ A2 )e−t

2

(D21 ⊗1+1⊗D22 )

) = c1 c2 ⋅ t −p1 −p2 + O(t ϵ−p1 −p2 ),

t → 0+ .

(c) For every normalized trace φ on ℒ1,∞ , φ((A1 ⊗ A2 )V1,2 ) = φ(A1 V1 ) ⋅ φ(A2 V2 ), where V1,2 :=

Γ(1 + Γ(1 +

p1 +p2 ) 2

p1 )Γ(1 2

+

p2 ) 2

⋅ (1 + D21 ⊗ 1 + 1 ⊗ D22 )

−

p1 +p2 2

∈ ℒ1,∞ (H1 ⊗ H2 ).

Proposition 9.7.1 in Section 9.7 shows that there is a Hilbert space H = H1 = H2 , a self-adjoint operator D = D1 = D2 , and a bounded operator A = A1 = A2 ∈ ℒ(H) such that A1 V1 ∈ ℒ1,∞ (H1 ) is universally measurable, A2 V2 ∈ ℒ1,∞ (H2 ) is universally measurable, and (A1 ⊗ A2 )V1,2 ∈ ℒ1,∞ (H1 ⊗ H2 ) is universally measurable, but φ((A1 ⊗ A2 )V1,2 ) ≠ φ(A1 V1 ) ⋅ φ(A2 V2 ) for any trace φ on ℒ1,∞ . The operators A1 and A2 are, essentially, Fourier multipliers on the torus. It follows that universal measurability is not a sufficient notion to guarantee that the Fubini theorem holds for the integral in noncommutative geometry.

310 | 9 Criteria for measurability Conditions for the Fubini theorem when normalized traces are replaced by Dixmier traces, that is, when Trω ((A1 ⊗ A2 )V1,2 ) = Trω (A1 V1 ) ⋅ Trω (A2 V2 ), for an extended limit ω on l∞ , are discussed in the end notes to this chapter.

9.2 Spectral description of measurability Spectral and diagonal formulas for traces on ℒ1,∞ in Chapters 6 and 7 provide direct criteria for measurability of operators. Spectral criteria This section proves Theorem 9.1.2. The proof of Theorem 9.1.2(a) uses the spectral characterization of the commutator subspace [ℒ1,∞ , ℒ(H)] in Chapter 5 to show that A ∈ ℒ1,∞ and φ(A) = c for every normalized trace φ on ℒ1,∞ if and only if n

n

c = O(1), k +1 k=0

∑ λ(k, A) − ∑

k=0

n ≥ 0.

Proof of Theorem 9.1.2(a). Suppose that φ(A) = c for every normalized trace φ on ℒ1,∞ . Let A0 = diag{

1 } . n + 1 n≥0

We have that φ(A − cA0 ) = 0 for every normalized trace φ. Let now φ be an arbitrary trace (not necessarily normalized). If φ(A0 ) ≠ 0, then φ is a scalar multiple of a normalized trace. By the preceding paragraph, we have that φ(A − cA0 ) = 0. If φ(A0 ) = 0, then, φ + φ0 is normalized for every normalized trace φ0 . Thus, φ(A − cA0 ) = (φ + φ0 )(A − cA0 ) − φ0 (A − cA0 ) = 0 − 0 = 0. Thus, φ(A − cA0 ) = 0 for an arbitrary trace φ. Since the commutator subspace is the common kernel of all traces, we have A − cA0 ∈ [ℒ1,∞ , ℒ(H)].

9.2 Spectral description of measurability | 311

By Theorem 5.1.5, there exists x ∈ l1,∞ such that 󵄨 󵄨󵄨 󵄨󵄨Cλ(A) − c ⋅ Cλ(A0 )󵄨󵄨󵄨 ≤ μ(x). Thus, 󵄨󵄨 󵄨 n 󵄨󵄨 1 c n 1 󵄨󵄨󵄨󵄨 ‖x‖1,∞ 󵄨󵄨 λ(k, A) − , ∑ ∑ 󵄨≤ 󵄨󵄨 n + 1 n + 1 k=0 k + 1 󵄨󵄨󵄨󵄨 n+1 󵄨󵄨 k=0

n ≥ 0.

Multiplying by n + 1, we obtain n

n

1 = O(1), k+1 k=0

∑ λ(k, A) − c ⋅ ∑

k=0

n ≥ 0.

Conversely, let n

n

1 = O(1), k + 1 k=0

∑ λ(k, A) − c ⋅ ∑

k=0

n ≥ 0.

Dividing by n + 1, we obtain 󵄨󵄨 󵄨 n n 󵄨󵄨 1 1 c 󵄨󵄨󵄨󵄨 const 󵄨󵄨 , λ(k, A) − ∑ ∑ 󵄨≤ 󵄨󵄨 n + 1 n + 1 k=0 k + 1 󵄨󵄨󵄨󵄨 n + 1 󵄨󵄨 k=0

n ≥ 0.

Thus, Cλ(A) − Cλ(cA0 ) ∈ l1,∞ . By Theorem 5.1.5, we have A − cA0 ∈ [ℒ1,∞ , ℒ(H)]. The proof of Theorem 9.1.2(b) is straightforward from the identification of normalized positive traces on ℒ1,∞ with Banach limits in Chapter 6 and the definition of an almost convergence sequence. Proof of Theorem 9.1.2(b). By Theorem 6.1.1(a), we have φ(A) = c for every positive normalized trace φ if and only if 2n+1 −2

θ( ∑ λ(k, A)) = c ⋅ log(2) k=2n −1

for every Banach limit θ on l∞ . This is exactly the condition of almost convergence in Theorem 9.1.2(b). The proof of Theorem 9.1.2(c) is straightforward from the fact a Dixmier trace on

ℒ1,∞ is defined in Chapter 6 using an eigenvalue sequence and an extended limit

on l∞ . The equivalent condition of Cesaro convergence of the dyadic averages in Theorem 9.1.2(c) follows from the identification of Dixmier traces with factorizable Banach limits.

312 | 9 Criteria for measurability Proof of Theorem 9.1.2(c). Let ω be an extended limit on l∞ . By Theorem 6.1.2, Trω (A) = ω(

n 1 ∑ λ(k, A)). log(n + 2) k=0

Thus, A is Dixmier measurable if and only if ω(

n 1 ∑ λ(k, A)) = c log(n + 2) k=0

for every extended limit ω on l∞ . This occurs if and only if n 1 ∑ λ(k, A) = c, n→∞ log(n + 2) k=0

lim

by Lemma 6.2.4. By Theorem 6.3.2, we have Trω (A) = c for every Dixmier trace if and only if 2n+1 −2

γ ∘ C(n 󳨃→ ∑ λ(k, A)) = c ⋅ log(2) k=2n −1

for every extended limit γ on l∞ . Again, by Lemma 6.2.4, this occurs if and only if 2n+1 −2

z(n) = ∑ λ(k, A), k=2n −1

n ≥ 0,

is Cesaro convergent to c ⋅ log(2). We now show that the criteria that an operator A ∈ ℒ1,∞ takes the same value for the smaller set of Connes’ Dixmier traces of the form Trω∘M (A) = (ω ∘ M)(

n 1 ∑ λ(k, A)), log(2 + n) k=0

A ∈ ℒ1,∞ ,

results in the same set of Dixmier measurable operators. Here M : l∞ → l∞ is the logarithmic mean (Mx)(n) =

n x(k) 1 , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

To prove this result we require a Tauberian theorem of Hardy. Lemma 9.2.1 ([128, Theorem 63]). Let x ∈ l∞ and let a ∈ ℂ. If n(x(n + 1) − x(n)) = O(1),

n ≥ 0,

and if (Cx)(n) → a as n → ∞, then x(n) → a as n → ∞.

9.2 Spectral description of measurability | 313

Lemma 9.2.2. Let z ∈ l∞ and let a ∈ ℂ. If (C 2 z)(n) → a as n → ∞, then (Cz)(n) → a as n → ∞. Proof. Let x(n) = (Cz)(n), n ≥ 0. Then (Cx)(n) → a as n → ∞. We have y(n) = n(x(n + 1) − x(n)) = n( =

n 1 n+1 1 ∑ z(k) − ∑ z(k)) n + 2 k=0 n + 1 k=0

n+1 n z(n + 1) + n( − 1)(Cz)(n), n+2 n+2

n ≥ 0.

Hence ‖y‖∞ ≤ ‖z‖∞ + ‖Cz‖∞ ≤ 2‖z‖∞ . By Lemma 9.2.1, x(n) → a as n → ∞, that is, (Cz)(n) → a as n → ∞. Proposition 9.2.3. Let A ∈ ℒ1,∞ and M : l∞ → l∞ denote the logarithmic mean. The following statements are equivalent: (a) Trω∘M (A) = c for every extended limit ω on l∞ . (b) Trω (A) = c for every extended limit ω on l∞ . Proof. That (b) implies (a) is evident since ω∘M is an extended limit on l∞ . We prove (a) implies (b). Let 2n+1 −2

z(n) = ∑ λ(k, A), k=2n −1

n ≥ 0.

We have λ(A) ∈ l1,∞ by Theorem 5.6.1. Then 󵄨 󵄨 󵄩 󵄩 ‖z‖∞ ≤ sup(2n − 1)󵄨󵄨󵄨λ(2n − 1, A)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩λ(A)󵄩󵄩󵄩1,∞ . n≥0

From Proposition 6.4.1, Connes’ Dixmier traces are bijective with extended limits of the form {γ ∘ C 2 : γ is an extended limit on l∞ }. Then Trω∘M (A) =

1 (γ ∘ C 2 )(z) = c log(2)

for every extended limit ω implies that (C 2 z)(n) → c ⋅ log(2),

n → ∞,

by Lemma 6.2.4. Hence (Cz)(n) → c ⋅ log(2) as n → ∞ by Lemma 9.2.2. By Theorem 9.1.2(c), Trω (A) = c for every Dixmier trace Trω .

314 | 9 Criteria for measurability Diagonal criteria Theorem 7.1.3 in Chapter 7 relates partial sums of an eigenvalue sequence to partial sums of a sequence of expectation values for a particular orthonormal basis of the separable Hilbert space H. The spectral criteria in Theorem 9.1.2 can be phrased in terms of expectation values. From Definition 7.1.2, a bounded operator A ∈ ℒ(H) is V-modulated for an operator 0 ≤ V ∈ ℒ1,∞ with trivial kernel if 󵄩 󵄩 sup t 1/2 󵄩󵄩󵄩A(1 + tV)−1 󵄩󵄩󵄩2 < ∞. t>0

The following result is a corollary to Theorems 9.1.2 and 7.1.3. Corollary 9.2.4. Let 0 ≤ V ∈ ℒ1,∞ have trivial kernel and let A ∈ ℒ(H) be a V-modulated operator. If {en }∞ n=0 is an eigenbasis for V ordered so that Ven = μ(n, V)en , n ≥ 0, then A ∈ ℒ1,∞ and (a) A is universally measurable if and only if there exists a constant c ∈ ℂ such that n

∑ ⟨Aek , ek ⟩ = c ⋅ log(n + 2) + O(1),

k=0

n ≥ 0.

(b) A is measurable if and only if there exists a constant c ∈ ℂ such that 2n+1 −2

∑ ⟨Aek , ek ⟩ → c ⋅ log(2) a. c.

k=2n −1

(c) A is Dixmier measurable if and only if there exists a constant c ∈ ℂ such that n 1 ∑ ⟨Aek , ek ⟩ → c, log(n + 2) k=0

n → ∞,

or, equivalently, 2n+1 −2

∑ ⟨Aek , ek ⟩ → c ⋅ log(2) (C, 1).

k=2n −1

Proof. From Theorem 7.1.3(b), n

n

k=0

k=0

∑ λ(k, A) − ∑ ⟨Aek , ek ⟩ = O(1),

n ≥ 0.

Hence the criteria in part (a) and the first display in part (c) of the corollary hold if and only if the criteria in part (a) and the first display in part (c) in Theorem 9.1.2 hold. From Theorems 7.1.3(c) and 6.1.1(a), 2n+1 −2

2n+1 −2

k=2n −1

k=2n −1

θ( ∑ λ(k, A) − ∑ ⟨Aek , ek ⟩) = 0 for every Banach limit θ on l∞ and every factorizable Banach limit. Hence the criteria in part (b) and the second display in part (c) of the corollary hold if and only if the criteria in part (b) and the second display in part (c) in Theorem 9.1.2 hold.

9.3 Examples of measurability | 315

9.3 Examples of measurability Operators that are measurable and not measurable according to normalized traces, positive normalized traces, and Dixmier traces, respectively, can be illustrated using the spectral criteria for measurability in Theorem 9.1.2. Volume II will discuss classical and nonclassical pseudodifferential operators on a closed Riemannian manifold that exhibit the following spectral behavior. The difference in Volume II is that measurability and nonmeasurability will be determined by examination of symbols. Theorem 3.1.4 in Chapter 3 indicated the singular nature of all traces on ℒ1,∞ . Naturally, since every trace on ℒ1,∞ vanishes on ℒ1 , a trace class operator is universally measurable. Example 9.3.1. Every operator from ℒ1 is universally measurable. Proof. Since φ(A) = 0 for every A ∈ ℒ1 and for every trace φ on ℒ1,∞ by Theorem 3.1.4(b), it follows that A ∈ ℒ1 is universally measurable. Embedding a sequence x from l∞ as a diagonal operator diag(x) in ℒ1,∞ , as described in Example 2.2.14 in Chapter 2 and the Calkin correspondence in Section 3.1, provides most of the examples. We have λ(n, diag(x)) = x(n),

x ∈ l∞ , n ≥ 0,

if x ∈ l∞ is ordered such that |x| is decreasing. Universal measurability Evidently, if x(n) =

1 , n

n ≥ 0,

then diag(x) is universally measurable and takes the same value on all normalized traces. The dyadic dilation operator L : l∞ → l∞ from Chapter 4, . . . , x(n), . . .), L(x) = (x(0), x(1), x(1), x(2), . .⏟,⏟⏟⏟⏟⏟⏟⏟⏟ x(2) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟.⏟⏟⏟ ⏟⏟, . . . , x(n), ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2 times

2n times

4 times

x ∈ l∞ ,

will be used to provide examples of operators that are measurable but not Dixmier measurable. Set x0 (n) = 2−n ,

n ≥ 0.

Example 9.3.2. The operators diag(Lx0 ), diag(y) ∈ ℒ1,∞ are universally measurable, where (Lx0 )(n) = ∑ 2−k χ[2k −1,2k+1 −1) (n), k≥0

y(n) = ∑ 2−k χ[2k ,2k+1 ) (n), k≥0

n ≥ 0.

316 | 9 Criteria for measurability Proof. Let x = Lx0 . Let m ≥ 0 be such that 2m − 1 ≤ n ≤ 2m+1 − 2. Then m 2j+1 −2

n

m+1 2j+1 −2

j=0 k=2j −1

k=0

j=0 k=2j −1

m 2j+1 −2

m

j=0 k=2j −1

j=0

∑ ∑ x(k) ≤ ∑ x(k) ≤ ∑ ∑ x(k).

Now ∑ ∑ x(k) = ∑ 1 + O(1) = m + O(1).

Hence n

∑ x(k) = m + O(1) = log2 (n + 1) + O(1).

k=0

Clearly, x = μ(x) and n

∑ x(k) =

k=0

log(n + 1) + O(1), log(2)

n ≥ 0.

By Theorem 9.1.2, diag(x) is universally measurable. Since x − y ∈ l1 , we get that diag(x) − diag(y) ∈ ℒ1 is universally measurable. Hence diag(y) is universally measurable. The next example demonstrates that there exists a universally measurable operator whose heat trace does not have the leading term expansion (9.4), and whose ζ -function does not have a meromorphic extension. Example 9.3.3. There exists a universally measurable operator 0 ≤ V ∈ ℒ1,∞ such that the mapping z → Tr(V 1+z ),

ℜ(z) > 0,

does not have a meromorphic extension to any neighborhood of z = 0. Proof. Let 0 ≤ V ∈ ℒ1 be such that V log(V −1 ) ∉ ℒ1 . For simplicity of notations, assume that 0 ≤ V ≤ 1. Since V ∈ ℒ1 , it follows that φ(V) = 0 for every trace φ on ℒ1,∞ . In particular, V is universally measurable. We claim that the mapping z → Tr(V 1+z ),

ℜ(z) > 0,

(9.8)

does not have a meromorphic extension to any neighborhood of z = 0. Assume the contrary. The point z = 0 is either regular or a pole. Since lim Tr(V 1+s ) = Tr(V) < ∞, s↓0

9.3 Examples of measurability | 317

it follows that z = 0 cannot be a pole. Hence, it is regular. In particular, there exists a (finite) limit lim s−1 ⋅ (Tr(V) − Tr(V 1+s )). s↓0

For every n ≥ 0, we have ∞

n

k=0

k=0

Tr(V) − Tr(V 1+s ) = ∑ μ(k, V) − μ(k, V)1+s ≥ ∑ μ(k, V) − μ(k, V)1+s . Thus, n

lim s−1 ⋅ (Tr(V) − Tr(V 1+s )) ≥ lim s−1 ∑ μ(k, V) − μ(k, V)1+s s↓0

s↓0

k=0

n

= ∑ lim s−1 (μ(k, V) − μ(k, V)1+s ) k=0 n

s↓0

= ∑ μ(k, V) log( k=0

1 ). μ(k, A)

Letting n → ∞, we arrive at lim s−1 ⋅ (Tr(V) − Tr(V 1+s )) ≥ ∑ μ(k, V) log( s↓0

k≥0

1 󵄩 󵄩 ) = 󵄩󵄩󵄩V log(V −1 )󵄩󵄩󵄩1 . μ(k, V)

However, the left-hand side is a finite number and the right-hand side is +∞. This contradiction shows that our assumption was incorrect. The function (9.8) cannot have a meromorphic extension. Example 9.3.4. There exists a universally measurable operator 0 ≤ V ∈ ℒ1,∞ such that (9.4) fails for A = 1. Proof. To lighten the notation, we assume that α = 1. Suppose that −1

Tr(e−tV ) =

c + O(t ϵ−1 ), t

t ∈ (0, 1),

for some ϵ > 0. From the Mellin transform, V

1+z

∞

−1 1 = ∫ t z e−tV dt. Γ(1 + z)

0

Thus, Tr(V

1+z

∞

−1 1 )= ∫ t z Tr(e−tV ) dt Γ(1 + z)

0

=

∞

1

1

0

−1 −1 1 1 c c (∫ t z (Tr(e−tV ) − ) dt + ). ∫ t z Tr(e−tV ) dt + Γ(1 + z) Γ(1 + z) t z

318 | 9 Criteria for measurability The function ∞

z → ∫ t z Tr(e−tV ) dt, −1

ℜ(z) > 0,

1

extends to an entire function. The function 1

−1 c ∫ t z (Tr(e−tV ) − ) dt, t

ℜ(z) > 0,

0

admits an analytic extension to the half-plane {ℜ(z) > −ϵ}. Hence, the function z → Tr(V 1+z ),

ℜ(z) > 0

admits an analytic extension to a punctured neighborhood of 0 with 0 at worst a simple pole. Now, take V as in Example 9.3.3. Assuming that V satisfies (9.4) contradicts Example 9.3.3. Measurability The next example provides measurable operators that are not universally measurable, in the sense of Definition 9.1.1. Example 9.3.5. Let 0 ≤ x ∈ (l1,∞ )0 \ l1 . The operator diag(x) ∈ ℒ1,∞ is measurable but not universally measurable. Proof. By definition, every operator in (ℒ1,∞ )0 is a limit of finite rank operators; in particular, every continuous trace vanishes on (ℒ1,∞ )0 . Hence every positive trace vanishes on diag(x) and the operator diag(x) is measurable. Since diag(x) vanishes on positive traces, diag(x) is universally measurable if and only if diag(x) vanishes for every trace on ℒ1,∞ . Suppose diag(x) vanishes for every trace on ℒ1,∞ . It follows from Theorem 9.1.2(a) that μ(x) = μ(diag(x)) ∈ l1 , since n

∑ μ(k, diag(x)) = O(1).

k=0

Hence x ∈ l1 , which is a contradiction, and so diag(x) cannot be universally measurable. The next example provides an operator that is not a measurable operator, but is a Dixmier measurable operator. Example 9.3.6. The sequence x(n) = ∑ χ[2k ,2k +k) (n), k≥0

n≥0

9.3 Examples of measurability | 319

is Cesaro convergent but is not almost convergent. Hence, the operator diag(LR−1 x) ∈ ℒ1,∞ is Dixmier measurable, but is not measurable. The operation LR−1 : l∞ → l1,∞ was defined in Section 6.3. Proof. If n ∈ [2k , 2k+1 ), k ≥ 1, then n

2k+1 −1

m=0

m=0

k 2l +l−1

k

∑ x(m) ≤ ∑ x(m) = ∑ ∑ 1 = ∑ l = l=0 m=2l

l=0

k(k + 1) ≤ k2 . 2

Thus, n

∑ x(m) = O(log2 (n + 2)),

m=0

n ≥ 0.

In particular, (Cx)(n) → 0 as n → ∞. However, x is not almost convergent since it fails the criteria in Theorem 6.2.11 as described in Example 6.3.6. By Theorem 6.3.1, φ(diag(LR−1 x)) = θφ (x) for a normalized positive trace and Banach limit θφ . Since x is not almost convergent, the value θφ (x) must depend on the Banach limit θφ . Hence φ(diag(LR−1 x)) depends on the trace φ. By Theorem 6.3.2, for a Dixmier trace φ, the corresponding Banach limit θφ is factorizable. Since Cx ∈ c0 , it follows that θφ (x) = 0. Thus, φ(diag(LR−1 x)) = θφ (x) = 0 for every Dixmier trace φ. Dixmier measurability The following example shows that there is a Dixmier measurable operator in ℒ1,∞ , satisfying the criterion in Theorem 9.1.2(c), that does not satisfy condition (9.5) in Theorem 9.1.4. Example 9.3.7. There exists a positive operator V ∈ ℒ1,∞ such that n 1 ∑ μ(k, V) = 0 n→∞ log(2 + n) k=0

lim

and −1 1 lim sup Tr(e−(nV) ) > 0. n→∞ n

320 | 9 Criteria for measurability Proof. Let n

n

k ∈ [22 , n ⋅ 22 ), n ≥ 0,

k , { { { −2n+1 μ(k, V) = {2 , { { −1 {2 , −1

n

n+1

k ∈ [n ⋅ 22 , 22 ), n ≥ 0, k = 0, 1.

It is clear that μ(k, V) ≤ k −1 , k ≥ 1. Hence V ∈ ℒ1,∞ . n Let m = 22 . We have 2n

−1 1 n⋅2 −1 −(mμ(k,V)−1 ) 1 ∞ −(mμ(k,V)−1 ) 1 ≥ . Tr(e−(mV) ) = ∑e ∑ e m m k=0 m 2n

k=2

By the definition of V, we have n

n⋅22

n

n⋅22 −1

1 1 Tr(e−(mV) ) ≥ ∑ e m m 2n −1

− mk

k=2

≥

n

s 1 ∫ e− m ds = ∫ e−u du. m

1

n

22

Letting n → ∞, we conclude that lim sup m→∞

−1 1 Tr(e−(mV) ) ≥ e−1 > 0. m

On the other hand, we have n+1

22

2n+1

n

n⋅22 −1

1 2 −1 −2n+1 + ∑ 2 ≤ ∑ μ(k, V) = ∑ k 2n 2n 2n −1

k=2

k=n⋅2

k=2

n

n⋅22 −1

∫ n

22 −1

ds + 1 ≤ 2 + log(n). s

Thus, n+1

22

l+1

n 22

−1

n

−1

∑ μ(k, V) = 1 + ∑ ∑ μ(k, V) ≤ 1 + ∑ (2 + log(l)) = O(n log(n)).

k=0

l=0

l=0 k=22l

n

n+1

Thus, if m ∈ [22 , 22 ], then m

n+1

22

−1

∑ μ(k, V) ≤ ∑ μ(k, V) = O(n log(n)) = o(log(m)).

k=0

k=0

The next example uses the criterion of (C, 1) convergence in Theorem 9.1.2(c) to find an operator in ℒ1,∞ that is not Dixmier measurable. Example 9.3.8. The sequence x(n) = ∑ χ[22k ,22k+1 ] (n) k≥0

is not Cesaro convergent. Hence, the operator diag(LR−1 x) ∈ ℒ1,∞ is not Dixmier measurable. The operation LR−1 : l∞ → l1,∞ was defined in Section 6.3.

9.3 Examples of measurability | 321

Proof. We have (Cx)(22k ) =

1 k−1 2l ∑ (2 + O(1)) = 22k + 1 l=0

4 3

⋅ 22k−2 + O(k) 22k

=

1 + o(1) 3

=

2 + o(1). 3

and (Cx)(22k+1 ) =

1

22k+1

k

∑ (22l + O(1)) =

+ 1 l=0

4 3

⋅ 22k + O(k) 22k+1

Thus, lim sup Cx ≥

2 , 3

lim inf Cx ≤

1 . 3

In particular, x is not Cesaro convergent. Since x ∈ l∞ , by Section 6.3, we have that LR−1 x ∈ l1,∞ and diag(LR−1 x) ∈ ℒ1,∞ . That diag(LR−1 x) is not Dixmier measurable follows from Theorem 9.1.2(c), since the sequence 2n+1 −1

∑ λ(k, diag(LR−1 x)) = (RL󸀠 LR−1 x)(n) = x(n),

k=2n −1

n ≥ 0,

is not Cesaro convergent. The decomposition of a self-adjoint Dixmier measurable operator A ∈ ℒ1,∞ into Dixmier measurable parts A = A+ − A− fails, as shown by the following two examples. The first example shows that if |A| is a Dixmier measurable operator in ℒ1,∞ then A may not be a Dixmier measurable operator. It uses the convergence criteria in Theorem 9.1.2. Example 9.3.9. The operator diag(x) ∈ ℒ1,∞ where x(n) =

ei log(log(n+e)) , n+e

n ≥ 0,

is not Dixmier measurable. However, |diag(x)| is universally measurable. Proof. We may take λ(n, diag(x)) = x(n), n ≥ 0. We have n

n

k=0

e

∑ x(k) = ∫

ei log(log(t)) dt + O(1) t 1+i

= (log(n + e))

+ O(1).

Hence ω(

n 1 ∑ λ(n, diag(x))) = ω(log(n + e)i ). log(2 + n) k=0

322 | 9 Criteria for measurability Since log(n + e)i is not convergent as n → ∞, we get that diag(x) is not Dixmier measurable by Theorem 9.1.2(c). Clearly, 1 󵄨 󵄨󵄨 ) 󵄨󵄨diag(x)󵄨󵄨󵄨 = diag(|x|) = diag( n+e and φ(|diag(x)|) = 1 for every normalized trace φ on ℒ1,∞ . The second example shows that if A is a Dixmier measurable operator in ℒ1,∞ then |A| may not be a Dixmier measurable operator. Example 9.3.10. Let diag(LR−1 x) ∈ ℒ1,∞ be the positive operator from Example 9.3.8 that is not Dixmier measurable and let A = diag(LR−1 x) ⊕ (−diag(LR−1 x)) ∈ ℒ1,∞ . The operator A is universally measurable, but the operator |A| is not Dixmier measurable. Proof. For brevity, set B = diag(LR−1 x) ∈ ℒ1,∞ . By linearity, we have φ(A) = φ(B ⊕ 0) − φ(0 ⊕ B) = φ(B) − φ(B) = 0 for every trace on ℒ1,∞ . In particular, A is universally measurable. On the other hand, Trω (|A|) = Trω (B ⊕ B) = 2Trω (B) for every extended limit ω on l∞ . However, B is not Dixmier measurable and, hence, neither is |A|.

9.4 Measurability criteria and heat traces Theorem 9.1.3 is proved in this section using heat trace formulas from Chapter 8 and diagonal formulas from Chapter 7. Universal measurability and heat traces Let A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ . We want to prove that φ(AV) = c for every normalized trace φ on ℒ1,∞ , and that −α

Tr(AVe−(nV) ) = c ⋅ log(n + 2) + O(1), for some α > 0, are equivalent conditions. In the terminology of Chapter 7, the operator AV ∈ ℒ1,∞

n → ∞,

9.4 Measurability criteria and heat traces | 323

is V-modulated by Proposition 7.3.1 and Lemma 7.3.4. The next lemma connects the partial sums of the eigenvalue sequence of the modulated operator AV to the key result for the heat trace formula in Lemma 8.2.4. Following Lemma 8.2.3 in Chapter 8, let Φ : (0, ∞) → (0, 1) be a function such that: (i) Φ is convex, decreasing, and positive; (ii) Φ(0) = 1 and ∞

∫ Φ(t) 1

dt < ∞, t

1

∫(1 − Φ(t)) 0

dt < ∞. t

The following lemma proves Theorem 9.1.3(a) for a function Φ satisfying the conditions above. Lemma 9.4.1. Let 0 ≤ V ∈ ℒ1,∞ and let A ∈ ℒ(H). Let Φ be as in Lemma 8.2.3. The following conditions are equivalent: (a) φ(AV) = c for every normalized trace φ on ℒ1,∞ . (b) We have Tr(AVΦ((nV)−1 )) = c ⋅ log(n) + O(1),

n → ∞.

Proof. Let P = supp(V). Evidently, φ(PAP ⋅ V) = φ(A ⋅ PVP) = φ(AV) and Tr(AVΦ((nV)−1 )) = Tr(PAP ⋅ VΦ((nV)−1 )),

n → ∞.

Thus, we may consider PAP instead of A. By restricting to the subspace PH, we may assume, without loss of generality, that ker(V) = 0. For simplicity of notations, we may assume ‖V‖1,∞ = 1. For n ≥ 1, it is clear that 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 −1 󵄨󵄨Tr(AVΦ((nV) )) − Tr(A(V − ) )󵄨󵄨󵄨 󵄨󵄨 n + 󵄨󵄨 󵄩󵄩 1 󵄩󵄩󵄩 󵄩 ≤ ‖A‖∞ 󵄩󵄩󵄩VΦ((nV)−1 ) − (V − ) 󵄩󵄩󵄩 . 󵄩󵄩 n + 󵄩󵄩1 From Lemma 8.2.4, we obtain Tr(AVΦ((nV)−1 )) − Tr(A(V −

1 ) ) = O(1), n +

n → ∞.

324 | 9 Criteria for measurability By Lemma 8.2.5, we have n

Tr(AVΦ((nV)−1 )) − ∑ λ(k, AV) = O(1),

n → ∞.

k=0

The assertion of equivalence of the condition Tr(AVΦ((nV)−1 )) = c ⋅ log(n) + O(1),

n → ∞,

with universal measurability now follows from the equivalence in Theorem 9.1.2(a). Theorem 9.1.3(a) can be proved by confirming that the function α

Φ(t) = e−t , t > 0, satisfies the conditions above. α

Proof of Theorem 9.1.3(a). The function Φ(t) = e−t , α > 0 is positive, convex, and decreasing. Using the substitution v = t α , we have ∞

−1 −t α

∫t e 1

∞

1 1 dt = ∫ v−1 e−v dv ≤ α α 1

and 1

−1

∫ t (1 − e

−t α

0

1

1 1 ) dt = ∫ v−1 (1 − e−v ) dv ≤ . α α 0

Hence Φ satisfies the conditions in Lemma 8.2.3. The equivalent conditions in Theorem 9.1.3(a) follow from Lemma 9.4.1. Leading term in a heat trace asymptotic expansion Let 0 ≤ V ∈ ℒ1,∞ and let A ∈ ℒ(H). We want to prove that condition (9.4), namely Tr(Ae−(tV) ) = c ⋅ t + O(t 1−ϵ ), −α

t → ∞,

implies that AV is universally measurable, and that the reverse implication is false. Proof of Theorem 9.1.4(a). Set Φ(s) =

1

Γ(α−1

∞

+ 1)

α

∫ e−t dt, s

s > 0.

9.4 Measurability criteria and heat traces | 325

We have 1

1

Γ(α−1 + 1)

∫ e−t

α

V −α

dt = V(Φ(sV −1 ) − Φ(V −1 )),

s

where the integral is understood in the Bochner sense in ℒ1 . In particular, we have 1

Γ(α−1

1

+ 1)

∫ Tr(Ae−(t

−1

V)−α

) dt = Tr(AVΦ(sV −1 )) − Tr(AVΦ(V −1 )).

s

Integrating both sides in (9.4) over t ∈ [s, 1] and dividing by Γ(1 + α−1 ), we infer that Tr(AVΦ(sV −1 )) =

1 c log( ) + O(1), s Γ(α−1 + 1)

s → 0.

Observe that Φ satisfies the conditions of Lemma 8.2.3. The assertion that AV is universally measurable follows now from Lemma 9.4.1 setting s = n−1 , n → ∞. Setting A = 1 and V = diag(x), where x is the sequence in Example 9.3.4, provides an example where AV is universally measurable but condition (9.4) is not satisfied. We now want to prove that condition (9.5), namely −α

Tr(Ae−(tV) ) = c ⋅ t + o(t),

t → ∞,

for some α > 0, implies that AV is Dixmier measurable, and that the reverse implication is false. The first implication follows from the regularity of logarithmic convergence. Proof of Theorem 9.1.4(b). Condition (9.5) implies that lim

n→∞

−α 1 Tr(Ae−(nV) ) = c. n

By regularity of the logarithmic mean, e. g., [127, Theorem 12, p. 58], M(k 󳨃→

−α 1 Tr(Ae−(kV) ))(n) = c. k

By Corollary 8.1.3, Trω (AV) =

−α 1 1 c ω ∘ M( Tr(Ae−(nV) )) = −1 n Γ(1 + α ) Γ(1 + α−1 )

for every extended limit ω. Hence AV is Dixmier measurable. Setting A = 1 and taking V as in Example 9.3.7 provides an example where AV is Dixmier measurable but condition (9.5) is not satisfied.

326 | 9 Criteria for measurability Dixmier measurability and heat traces Let A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ . Theorem 9.1.3(b) requires us to prove that Trω (AV) = c for every extended limit ω on l∞ , and that −α

Tr(AVe−(nV) ) = c ⋅ log(n + 2) + o(log(n + 2)),

n → ∞,

for some α > 0, are equivalent conditions. The following lemma allows us to replace α with the value 1 in the condition above, so that we can directly use Lemma 8.2.2. Lemma 9.4.2. If 0 ≤ V ∈ ℒ1,∞ , then 󵄩󵄩 1 󵄩󵄩󵄩 󵄩󵄩 −(nV)−α − (V − ) 󵄩󵄩󵄩 = O(1), 󵄩󵄩Ve 󵄩󵄩 n + 󵄩󵄩1

n ≥ 0, α > 0.

Proof. We have sup t −1 e−t

t∈[0,1]

−α

< ∞,

󵄨 󵄨 −α sup t α 󵄨󵄨󵄨e−t − 1󵄨󵄨󵄨 < ∞.

t∈(1,∞)

Thus, 󵄩󵄩 −1 1 󵄩󵄩󵄩 1 󵄩󵄩 −(nV)−α 󵄩 󵄩 − Ve−(nV) 󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩EV ( , ∞)󵄩󵄩󵄩 󵄩󵄩Ve 󵄩󵄩1 n 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 −α −α 1 󵄩󵄩󵄩 󵄩󵄩󵄩 1 󵄩 󵄩 + 󵄩󵄩󵄩Ve−(nV) EV [0, ]󵄩󵄩󵄩 + 󵄩󵄩󵄩V(e−(nV) − 1)EV ( , ∞)󵄩󵄩󵄩 󵄩󵄩1 󵄩󵄩 n 󵄩󵄩1 󵄩󵄩 n 󵄩 󵄩󵄩 −α 󵄩 1 󵄩 󵄩 ≤ ‖V‖1,∞ + 󵄩󵄩󵄩V ⋅ (nV)EV [0, ]󵄩󵄩󵄩 ⋅ sup t −1 e−t 󵄩󵄩 n 󵄩󵄩1 t∈[0,1] 󵄩󵄩 󵄩󵄩 1 󵄨 󵄩 󵄩 󵄨 −α + 󵄩󵄩󵄩V ⋅ (nV)−α EV ( , ∞)󵄩󵄩󵄩 ⋅ sup t α 󵄨󵄨󵄨e−t − 1󵄨󵄨󵄨. 󵄩󵄩1 t∈(1,∞) 󵄩󵄩 n We have 2 2 󵄩󵄩 󵄩󵄩 1 󵄩󵄩󵄩 1 1 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩V ⋅ (nV)EV [0, ]󵄩󵄩󵄩 ≤ n󵄩󵄩󵄩min{V, } 󵄩󵄩󵄩 = n ∑ min{μ(k, V), } 󵄩󵄩 󵄩󵄩 n 󵄩󵄩1 n 󵄩󵄩1 n k≥0

≤ n ∑ min{ k≥0

2

‖V‖1,∞ 1 , } = O(1). k+1 n

Also, 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 1 1 󵄩󵄩 󵄩 󵄩 󵄩 1−α −α −α 󵄩 󵄩󵄩V ⋅ (nV) EV ( , ∞)󵄩󵄩󵄩 = n 󵄩󵄩󵄩V EV ( , ∞)󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩1 󵄩󵄩1 󵄩󵄩 n n If 0 < α < 1, then 1−α

1 1 1 V 1−α EV ( , ∞) ≤ (V − ) + nα−1 EV ( , ∞). n n + n

9.5 Measurability criteria and ζ -functions | 327

Thus, for 0 < α < 1, we have 1−α 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 1 1 1 󵄩 󵄩󵄩 󵄩 −α −α 󵄩 󵄩󵄩V ⋅ (nV) EV ( , ∞)󵄩󵄩󵄩 ≤ n 󵄩󵄩󵄩(V − ) 󵄩󵄩󵄩 + ⋅ nV ( ) 󵄩󵄩 󵄩󵄩 󵄩󵄩1 n n + 󵄩󵄩1 n n 1−α 1 = n−α ∑ (μ(k, V) − ) + O(1) n + k≥0

≤ n−α ∑ ( k≥0

1−α

‖V‖1,∞ 1 − ) + O(1) = O(1). k+1 n +

If α ≥ 1, then 1 1 V 1−α EV ( , ∞) ≤ nα−1 EV ( , ∞). n n Thus, for α ≥ 1, we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 1 1 󵄩󵄩 󵄩 󵄩 󵄩 −α −1 󵄩 󵄩󵄩V ⋅ (nV) EV ( , ∞)󵄩󵄩󵄩 ≤ n 󵄩󵄩󵄩EV ( , ∞)󵄩󵄩󵄩 = O(1). 󵄩󵄩 󵄩󵄩1 󵄩󵄩1 󵄩󵄩 n n Combining these estimates, we complete the proof. Proof of Theorem 9.1.3(b). By Lemma 9.4.2, we have 󵄨󵄨 󵄨󵄨 1 󵄨 󵄨󵄨 −(nV)−α ) − Tr(A(V − ) )󵄨󵄨󵄨 󵄨󵄨Tr(AVe 󵄨󵄨 n + 󵄨󵄨 󵄩󵄩 −α 1 󵄩󵄩󵄩 󵄩 ≤ ‖A‖∞ ⋅ 󵄩󵄩󵄩Ve−(nV) − (V − ) 󵄩󵄩󵄩 = O(1). 󵄩󵄩 n + 󵄩󵄩1 Hence, the condition in Theorem 9.1.3(b) (ii) is equivalent to lim

n→∞

1 1 Tr(A(V − ) ) = c. log(n + 2) n +

In other words, ω(

1 1 Tr(A(V − ) )) = c log(n + 2) n +

for every extended limit ω on l∞ . By Lemma 8.3.3, this is equivalent to ξω∘M (AV) = c for every extended limit ω on l∞ . This proves the equivalence between Theorem 9.1.3(b) (ii) and Theorem 9.1.3(b) (iii). The equivalence between (b) (i) and (b) (iii) is stated in Theorem 8.1.2.

9.5 Measurability criteria and ζ-functions Let A ∈ ℒ(H) and 0 ≤ V ∈ ℒ1,∞ . Theorem 9.1.5 is proved in this section using Tauberian theorems relating Abel convergence, and a Tauberian theorem of Subhankulov.

328 | 9 Criteria for measurability Universal measurability When A = A∗ ∈ ℒ(H), Theorem 9.1.5(a) is proved by defining a measure of bounded variation on (0, ∞) associated to n

∑ λ(k, AV) − c ⋅ log(n),

k=0

n ≥ 1, c ∈ ℝ,

and whose Laplace transform is

c f (s) = ζA,V (s) − , s

where

ζA,V (s) = Tr(AV 1+s ),

s > 0,

s > 0,

is the ζ -function of Chapter 8. A theorem of Subhankulov, Theorem A.1.2 in Appendix A, will then relate analytic continuation of f to the universal measurability condition n

∑ λ(k, AV) − c ⋅ log(n) = O(1),

n → ∞,

k=0

of Theorem 9.1.2. Lemma 9.5.1. Let A = A∗ ∈ ℒ(H) and let 0 ≤ V ∈ ℒ1,∞ be such that ker(V) = {0}. Given an orthonormal basis {ek }k≥0 of H such that Vek = μ(k, V)ek , k ≥ 0, and c ∈ ℝ, there is a signed Borel measure b on [0, ∞) such that b([0, t]) := −ct +

∑ μ(k,V)>e−t

⟨Aek , ek ⟩μ(k, V),

t ≥ 0,

and sup b+ ([s, s + 1]) + b− ([s, s + 1]) < ∞.

(9.9)

s≥0

1 Proof. Let αk := log( μ(k,V) ) and bk := ⟨Aek , ek ⟩μ(k, V) for k ≥ 1. The measure b is a sum of Lebesgue measure and atoms concentrated at the points αk of magnitude bk for k ≥ 1. We have

sup b+ ([s, s + 1]) + b− ([s, s + 1]) ≤ |c| + s≥0

∑ e−x−1 ≤μ(k,V)≤e−x

≤ |c| + ‖A‖∞ ≤ |c| + ‖A‖∞ ≤ |c| + ‖A‖∞ and so condition (9.9) is shown.

󵄨󵄨 󵄨 󵄨󵄨⟨Aek , ek ⟩󵄨󵄨󵄨μ(k, V)

∑ e−x−1 ≤μ(k,V)≤e−x

∑ e−x−1 ≤μ(k,V)≤e−x −x

∑

e−x−1 ≤μ(k,V)

e

μ(k, V) e−x = O(1),

9.5 Measurability criteria and ζ -functions | 329

The signed Borel measure b in Lemma 9.5.1 satisfies the variation condition (A.1) in Appendix A. Proof of Theorem 9.1.5(a) . Assume that the function z → Tr(AV 1+z ) −

c z

admits an analytic continuation to {z : ℜ(z) > −ε}. If f admits an analytic extension to {ℜ(z) > −ϵ, z ≠ 0}, then so does the function ̄ Hence, so do the functions z → f (z). 1 ̄ f1 : z → (f (z) + f (z)), 2

f2 : z →

1 ̄ (f (z) − f (z)). 2i

If f has a simple pole at 0, then so do f1 and f2 . Moreover, we have Resz=0 f1 (z) = ℜ(Resz=0 f (z)),

Resz=0 f2 (z) = ℑ(Resz=0 f (z)).

Now, set f (z) = Tr(AV 1+z ). Using that Tr(X) = Tr(X ∗ ), we obtain f (z)̄ = Tr((AV 1+z ) ) = Tr(V 1+z A∗ ) = Tr(A∗ V 1+z ). ̄ ∗

Thus, f1 (z) = Tr((ℜA)V 1+z ),

f2 (z) = Tr((ℑA)V 1+z )

are analytic on {ℜ(z) > −ϵ, z ≠ 0}. Hence, we only need to prove the assertion for ℜ(A) and ℑ(A) instead of A. We may assume, without loss of generality, that A = A∗ and that ‖A‖∞ = 1. If the function z → Tr(AV 1+z ) admits an analytic continuation to {z ≠ 0 : ℜ(z) > −ε}, ε > 0, then, for every positive constant C, so does the function z → Tr(A(CV)1+z ). Thus, we may assume, without loss of generality, that ‖V‖1,∞ = 1. We may also assume, without loss of generality, that ker(V) = 0. Indeed, let P = EV (0, ∞). We have φ(AV) = φ(A ⋅ PVP) = φ(PAP ⋅ V) for every trace φ on ℒ1,∞ . We also have Tr(AV 1+z ) = Tr(A ⋅ PV 1+z P) = Tr(PAP ⋅ V 1+z ). Therefore, we can work, without loss of generality, on the Hilbert space PH.

330 | 9 Criteria for measurability 1 ) and bk := Let b be the Borel measure defined in Lemma 9.5.1. Let αk := log( μ(k,V)

⟨Aek , ek ⟩μ(k, V) for k ≥ 1. Let ℜ(z) > 0. Using the identity e−αk z = μ(k, V)z , we have ∞

∞

∫ e−zt db(t) = ∑ e−αk z ⋅ bk − c ∫ e−zt dt k≥0

0

0

∞

= ∑ μ(k, V)z ⋅ ⟨Aek , ek ⟩μ(k, V) − c ∫ e−zt dt k≥0

0

∞

= ∑ ⟨AV 1+z ek , ek ⟩ − c ∫ e−zt dt k≥0

= Tr(AV

1+z

c )− . z

0

By assumption, the above right-hand side has analytic continuation to the set {z : ℜ(z) > −ε}. Thus, since b satisfies (A.1) by Lemma 9.5.1, the assumptions of Theorem A.1.2 are satisfied, and we conclude that b(t) = O(1), for t → ∞. By the definition of b, ∑ μ(k,V)>e−t

⟨Aek , ek ⟩μ(k, V) = ct + O(1),

t → ∞.

Setting e−t = n1 , we obtain ∑ μ(k,V)> n1

⟨Aek , ek ⟩μ(k, V) = c log(n) + O(1),

n → ∞.

In other words, we have 1 Tr(AVEV ( , ∞)) = c log(n) + O(1), n

n → ∞.

It follows that Tr(A(V −

1 ) ) = c log(n) + O(1), n +

n → ∞.

By Lemma 8.2.5, we have n

∑ λ(k, AV) = c log(n) + O(1),

k=0

The assertion follows now from Theorem 9.1.2.

n → ∞.

9.5 Measurability criteria and ζ -functions | 331

Dixmier measurability We now prove Theorem 9.1.5(b): that AV is Dixmier measurable with Trω (AV) = c for every extended limit ω on l∞ , and that lim sζA,V (s) = c,

s→0+

are equivalent conditions. The proof relies on a Tauberian theorem stating when Abel convergence implies Cesaro convergence. Lemma 9.5.2 ([128, Theorem 95]). Let z ∈ l∞ and let a ∈ ℂ. If lim (1 − t) ∑ z(k)t k = a

t→1−

k≥0

then (Cz)(n) → a as n → ∞. In the next lemma, Abel convergence is related to an operator having the same value for every ζ -function residue ζγ in the sense of Chapter 8, and Cesaro convergence relates to Dixmier measurability. Here γ is an extended limit on L∞ (0, 1) concentrated at 0, as defined in Definition 6.2.3. Lemma 9.5.3. Let A ∈ ℒ1,∞ . The following conditions are equivalent: (a) Trω (A) = c for every extended limit ω on l∞ . (b) ζγ (A) = c for every extended limit γ on L∞ (0, 1) concentrated at 0. Proof. By Theorem 8.1.5, every ζγ is a Dixmier trace. Thus, (a) implies (b). Suppose now that A ∈ ℒ1,∞ satisfies (b). Define x ∈ l∞ by setting 2n+1 −2

x(n) = ∑ λ(k, A), k=2n −1

n ≥ 0.

Let Bγ be the factorizable Banach limit from Theorem 6.3.2 that corresponds to the normalized fully symmetric trace ζγ . Then 1 B (x) = c log(2) γ for every extended limit γ on L∞ (0, 1) concentrated at 0. By Lemma 8.5.1, we have 1 B (x) = γ(s ∑ 2−ks x(k)). log(2) γ k≥0 Thus, γ(s ∑ 2−ks x(k)) = c k≥0

332 | 9 Criteria for measurability for every extended limit γ on L∞ (0, 1) concentrated at 0. In other words, lim s ∑ 2−ks x(k) = c.

s→0+

k≥0

Setting t = 2−s , we rewrite the latter equality as lim (1 − t) ∑ x(k)t k = c ⋅ log(2).

t→1+

k≥0

By Lemma 9.5.2, x is Cesaro convergent to c ⋅ log(2). By the equivalent condition in Theorem 9.1.2(c), A is Dixmier measurable. Thus, (b) implies (a). Proof of Theorem 9.1.5(b). We have lim sTr(AV 1+s ) = c

s→0

if and only if γ(sTr(AV 1+s )) = c for every extended limit γ on L∞ (0, 1) concentrated at 0. In the notations of Definition 8.1.4, this means ζγ,A (V) = c for every extended limit γ on L∞ (0, 1) concentrated at 0. By Theorem 8.4.6, this is equivalent to ζγ (AV) = c for every extended limit γ on L∞ (0, 1) concentrated at 0. By Lemma 9.5.3, the latter condition is equivalent to Trω (AV) = c for every extended limit ω on l∞ .

9.6 Measurability criteria in ℳ1,∞ For positive operators in ℳ1,∞ , Dixmier measurability looks similar to Dixmier measurability on ℒ1,∞ . However, a counterexample for an operator that is not positive demonstrates the difference between Dixmier measurability in ℒ1,∞ and ℳ1,∞ . Theorem 9.6.1. Let 0 ≤ A ∈ ℳ1,∞ The following conditions are equivalent:

9.6 Measurability criteria in ℳ1,∞

| 333

(a) There exists a constant c ∈ ℂ such that Trω (A) = c for every Dixmier trace Trω on ℳ1,∞ . (b) There exists the limit n 1 ∑ μ(k, A) = c n→∞ log(2 + n) k=0

lim

for the singular value sequence μ(A) of A. To prove the theorem, we start with the one-sided integral variant of Lemma 9.2.1. For this section the upright symbol C denotes the arithmetic mean C : L∞ (0, ∞) → L∞ (0, ∞) given by t

(Cx)(t) =

1 ∫ x(s) ds, t

x ∈ L∞ (0, ∞), t > 0.

0

Lemma 9.6.2 ([128, Theorem 64]). Let x ∈ L∞ (0, ∞) be a positive differentiable function and let a ∈ ℝ. If tx 󸀠 (t) is bounded from below and if (Cx)(t) → a as t → ∞, then x(t) → a as t → ∞. For this section the upright symbol M denotes the logarithmic mean M : L∞ (0, ∞) → L∞ (0, ∞) given by t

(Mx)(t) =

1 ds ∫ x(s) , log(t) s

x ∈ L∞ (0, ∞), t > 0.

1

The next lemma plays the main role in the proof of Theorem 9.6.1. Lemma 9.6.3. Let y be a positive locally integrable function on (0, ∞). If My ∈ L∞ (0, ∞) and if (M2 y)(t) → a for some a ∈ ℝ as t → ∞, then (My)(t) → a as t → ∞. Proof. Set x = (My) ∘ exp ∈ L∞ (0, ∞). We claim that x satisfies the assumption of Lemma 9.2.1. We have et

t

1

0

du 1 1 = ∫ x(s) ds, (M y)(e ) = ∫(My)(u) t u t 2

t

where we used the substitution u = es in the second equality. In other words, M2 y ∘ exp = Cx. It follows now from the assumption that (Cx)(t) → a as t → ∞. It remains to shows that tx 󸀠 (t) is bounded from below.

334 | 9 Criteria for measurability Since y is locally integrable, it follows that My is (locally) absolutely continuous (and, hence, so is x). We have et

et

0

0

1 ds 1 t 1 ds x (t) = − 2 ∫ y(s) + y(e ) ≥ − 2 ∫ y(s) . s t s t t 󸀠

Multiplying by t, we obtain et

1 ds tx (t) ≥ − ∫ y(s) = −x(t). t s 󸀠

0

Since x is bounded, it follows that tx 󸀠 (t) is bounded from below. By Lemma 9.2.1, we have x(t) → a as t → ∞ and, therefore, (My)(t) → a as t → ∞. We convert the integral mean results into results on sequences. The logarithmic mean operator M : l∞ → l∞ is defined by (Mx)(n) =

n 1 x(k) , ∑ log(2 + n) k=0 1 + k

x ∈ l∞ , n ≥ 0.

The next lemma plays a role similar to that of Lemma 9.2.2. The difference from Lemma 9.2.2 is that the sequence z in the next lemma is not assumed to be bounded. Lemma 9.6.4. Let z be a positive sequence such that z(n) = O(log(n)),

n → ∞.

If Mz ∈ l∞ is bounded and if (M 2 z)(n) → a for some a ∈ ℝ as n → ∞, then (Mz)(n) → a as n → ∞. Proof. Define a positive locally integrable function y on (0, ∞) by setting y = ∑ z(n)χ[n,n+1) . n≥0

Let us show that y satisfies the assumptions in Lemma 9.6.3. Firstly, n

k

n 1 y(s)ds 1 y(s)ds (My)(n) = = ⋅∑ ∫ ∫ log(n) s log(n) k=2 s 1

n

= =

k−1

n−1 1 k 1 1 ⋅ ∑ z(k − 1) ⋅ log( )= ⋅ ∑ z(k) ⋅ log(1 + ) log(n) k=2 k−1 log(n) k=1 k n−1 n−1 1 z(k) 1 1 1 ⋅∑ + ⋅ ∑ z(k) ⋅ (log(1 + ) − ). log(n) k=1 k + 1 log(n) k=1 k k+1

9.6 Measurability criteria in ℳ1,∞

| 335

Since z(n) = O(log(n)) for n ≥ 2, it follows that n−1

1 1 )− ) = O(1), k k+1

∑ z(k) ⋅ (log(1 +

k=1

n ≥ 1.

Thus, (My)(n) = =

n−1 1 z(k) ⋅∑ + o(1) log(n) k=1 k + 1

n−1 log(n + 1) 1 z(k) ⋅ ⋅∑ + o(1) log(n) log(n + 1) k=1 k + 1

= (1 + o(1)) ⋅ (Mz)(n − 1) + o(1) = (Mz)(n − 1) + o(1),

n → ∞.

A similar, though longer, argument shows that (M2 y)(n) = (M 2 z)(n − 2) + o(1),

n → ∞.

If t ∈ (n, n + 1), then 󵄨󵄨 󵄨 󵄨󵄨(My)(t) − (My)(n)󵄨󵄨󵄨 ≤(

t

n

1 ds 1 ds 1 − ) ∫ y(s) + ∫ y(s) log(n) log(t) s log(t) s n

1

≤(

=(

n

n+1

1

n

1 1 ds 1 ds − ) ∫ y(s) + ∫ y(s) log(n) log(n + 1) s log(n) s 1 z(n) 1 1 − ) ⋅ log(n) ⋅ (My)(n) + ⋅ log(1 + ). log(n) log(n + 1) log(n) n

Thus, My is bounded. If t ∈ (n, n + 1), then 󵄨󵄨 2 󵄨 2 󵄨󵄨(M y)(t) − (M y)(n)󵄨󵄨󵄨 ≤(

n

t

1 1 ds 1 ds − ) ∫(My)(s) + ∫(My)(s) log(n) log(t) s log(t) s 1

n

n

n+1

1

n

1 1 ds 1 ds ≤( − ) ∫(My)(s) + ∫ (My)(s) log(n) log(n + 1) s log(n) s 1 1 1 1 ≤( − ) ⋅ ‖My‖∞ log(n) + ⋅ ‖My‖∞ log(1 + ). log(n) log(n + 1) log(n) n Thus, lim (M2 y)(t) = lim (M2 y)(n) = lim (M 2 z)(n) = a.

t→∞

n→∞

n→∞

336 | 9 Criteria for measurability By Lemma 9.6.3, we have lim (My)(t) = a.

t→∞

Thus, lim (Mz)(n − 1) = lim (My)(n) = a.

n→∞

n→∞

This completes the proof. Proof of Theorem 9.6.1. The implication (b) ⇒ (a) follows from the fact that ω is an extended limit on l∞ . Set z(n) = (n + 1)μ(n, A),

n ≥ 0.

Then (Mz)(n) =

n 1 ∑ μ(k, A), log(2 + n) k=0

n ≥ 0.

We have ‖Mz‖∞ = ‖A‖ℳ1,∞ and Mz ∈ l∞ . Suppose that Trω (A) = a for every dilation invariant extended limit ω on l∞ . It follows that Trγ∘M (A) = a for every extended limit γ on l∞ . Hence γ(M 2 z) = (γ ∘ M 2 )(z) = a for every extended limit γ on l∞ . It follows from Lemma 6.2.4 that (M 2 z)(n) → a,

n → ∞.

By Lemma 9.6.4, we infer that (Mz)(n) → a as n → ∞. Hence, (a) ⇒ (b). The next proposition shows that, for an arbitrary compact operator A ∈ ℳ1,∞ it is not true that A is Dixmier measurable implies that the limit n 1 ∑ λ(k, A) n→∞ log(2 + n) k=0

lim

exists, where λ(A) is an eigenvalue sequence of A. This contrasts with the result of Theorem 9.1.2(c) for operators in ℒ1,∞ .

9.6 Measurability criteria in ℳ1,∞

| 337

Proposition 9.6.5. There exists an operator A = A∗ ∈ ℳ1,∞ such that φ(A) = 0 for every trace φ on ℳ1,∞ and such that the sequence {

n 1 ∑ λ(k, A)} log(2 + n) k=0

n≥0

does not converge. Proof. Let X ∈ ℳ1,∞ be a positive operator such that m

μ(X) = sup 2m−2 χ[0,22m ) . m≥0

Set A = X ⊕ (−X). Then, φ(A) = φ(X ⊕ 0) − φ(0 ⊕ X) = φ(X) − φ(X) = 0 for every trace φ on ℳ1,∞ . To prove the divergence assertion on the eigenvalue sequence of A note that we can choose a particular rearrangement of the eigenvalue sequence of X (so that absolute values still decrease). Namely, we ask that every positive eigenvalue precedes every negative eigenvalue with the same absolute value. We have m

m−1

22 −1

22

{λ(k, A)}k=0 = {−μ(k, X)}k=0

−1

m

m−1

22 −22

∪ {μ(k, X)}k=0

−1

.

Thus, m

m−1

22 −1

22

m

−1

∑ λ(k, A) = − ∑ μ(k, X) +

k=0

k=0

m

m−1

22 −22

=

−1

∑ m−1

m−1

22 −22

−1

∑ k=0

μ(k, X) m

m−1

22 −22

m−2m

μ(k, X) = 2

⋅

∑ m−1

k=22

−1

1 = 2m + o(1).

k=22

Letting m → ∞, we obtain lim sup n→∞

n 1 1 . ∑ λ(k, A) ≥ log(2 + n) k=0 log(2)

On the other hand, we have m

2⋅22 −1

{λ(k, A)}k=0

m

m

22 −1

22 −1

= {−μ(k, X)}k=0 ∪ {μ(k, X)}k=0 .

Thus, m

m

m

2⋅22 −1

22 −1

22 −1

k=0

k=0

k=0

∑ λ(k, A) = − ∑ μ(k, X) + ∑ μ(k, X) = 0.

338 | 9 Criteria for measurability Letting m → ∞, we obtain lim inf n→∞

n 1 ∑ λ(k, A) ≤ 0. log(2 + n) k=0

9.7 A Fubini Theorem This section proves that a leading term in the asymptotic expansion of the heat trace, in the form of Condition (9.4), is sufficient for the Fubini Theorem to hold in noncommutative geometry. We also prove that the condition of universal measurability is not sufficient for the same Fubini Theorem to hold, and that the notion of the volume in noncommutative geometry is not well behaved under Cartesian products. Heat trace expansion is sufficient for a Fubini theorem Let H1 and H2 be separable Hilbert spaces. Suppose A1 ∈ ℒ(H1 ), and D1 : dom(D1 ) → H1 is a self adjoint operator with compact resolvent such that Tr(A1 e−t

2

D21

) = c1 ⋅ t −p1 + O(t ϵ−p1 ),

t → 0+ ,

(9.10)

with V1 = (1 + D21 )

−

p1 2

∈ ℒ1,∞ (H1 ),

for some p1 > 0 and ϵ > 0. Suppose A2 ∈ ℒ(H2 ), and D2 : dom(D2 ) → H2 is a self adjoint operator with compact resolvent such that Tr(A2 e−t

2

D22

) = c2 ⋅ t −p2 + O(t ϵ−p2 ),

t → 0+ ,

with V2 = (1 + D22 )

−

p2 2

∈ ℒ1,∞ (H2 ),

for some p2 > 0 and ϵ > 0. We want to prove that φ((A1 ⊗ A2 )V1,2 ) = φ(A1 V1 ) ⋅ φ(A2 V2 ) for every normalized trace φ on ℒ1,∞ , where V1,2 :=

Γ(1 + Γ(1 +

p1 +p2 ) 2

p1 )Γ(1 2

+

p2 ) 2

⋅ (1 + D21 ⊗ 1 + 1 ⊗ D22 )

−

p1 +p2 2

∈ ℒ1,∞ (H1 ⊗ H2 ).

(9.11)

9.7 A Fubini Theorem

| 339

Proof of Theorem 9.1.7(a). Part (a) follows from Theorem 9.1.4. Substituting t with t −1/p1 in (9.10), we have that − p2 1

Tr(A1 e−(tV1 )

Hence, setting α =

) = Tr(A1 e−t 2 p1

− p2 1

D21

) + O(1) = c1 ⋅ t + O(t

1− pϵ

1

t → ∞.

),

> 0 in Theorem 9.1.4, φ(A1 V1 ) =

c1 . Γ(1 + p21 )

φ(A2 V2 ) =

c2 . Γ(1 + p22 )

Similarly, from (9.11),

Proof of Theorem 9.1.7, (b) and (c). For bounded operators A1 ∈ ℒ(H1 ) and A2 ∈ ℒ(H2 ), (A1 ⊗ A2 )e−t

2

(D21 ⊗1+1⊗D22 )

= A1 e−t

2

D21

⊗ A2 e−t

2

D22

.

Hence (A1 ⊗ A2 )e−t

2

(D21 ⊗1+1⊗D22 )

∈ ℒ1 (H1 ⊗ H2 ),

t > 0,

and Tr((A1 ⊗ A2 )e−t

2

(D21 ⊗1+1⊗D22 )

) = Tr(A1 e−t

2

D21

)Tr(A2 e−t

2

D22

t > 0.

),

Hence, by conditions (9.10) and (9.11), Tr((A1 ⊗ A2 )e−t

2

(D21 ⊗1+1⊗D22 )

)=

c1 c2 + O(t ϵ−(p1 +p2 ) ), t p1 +p2

t → 0+ .

Equivalently, 2 − p +p 1 2

Tr((A1 ⊗ A2 )e−(tV1,2 )

)=

p1 +p2 ) 2 c1 c2 p1 )Γ(1 + p21 ) 2

Γ(1 + Γ(1 +

⋅ t + O(t

1− p

ϵ

1 +p2

t → ∞.

),

Let us now prove that V1,2 ∈ ℒ1,∞ . Indeed, nV1,2 (t) =

1

∑ (1+λ(m,D1

=

)2 +λ(n,D

2

)2 )−

p1 +p2 2

>t

1≤

∑ λ(m,D1

)2 +λ(n,D

=(

2

− 2 )2 −ϵ}, it follows that 󵄨󵄨 󵄨󵄨 1 󵄨 󵄨 sup sup 󵄨󵄨󵄨f0 ( + it)󵄨󵄨󵄨 < ∞. 󵄨󵄨 x x≥1 t∈[−1,1] 󵄨󵄨

(A.10)

The assertion of Lemma A.1.4 is now written as follows: 1

1

−1

−1

1 1 1 (1 − t 2 )2 1 1 2 f (0)e( x +it)x dt ∫ (1 − t 2 ) f0 ( + it)e( x +it)x dt + ∫ 1 2π x 2π + it x

= b(x) + O(1),

x → ∞.

(A.11)

The first summand on the left-hand side is bounded for x ≥ 1 due to (A.10). By Proposition A.1.3, we have 1

(1 − t 2 )2 ( x1 +it)x 1 e dt = 1 + O(x−2 ), ∫ 1 2π + it x

x → ∞.

(A.12)

−1

Combining (A.11) and (A.12), we get b(x) + O(1) = f (0) + O(1),

x → ∞.

So b(x) = O(1), as x → ∞.

A.2 Operator inequalities The following lemmas on operator inequalities were used to prove Theorem 8.3.2 on heat trace functionals and Dixmier traces. Similar results were presented in [31, 34, 39]. For a self-adjoint operator A ∈ ℒ(H), the positive part A+ is defined using the functional calculus or the spectral projection EA [0, ∞) as A+ = max{A, 0} = AEA [0, ∞).

362 | A Miscellaneous results Lemma A.2.1. Let A0 , A1 ∈ ℒ(H) be self-adjoint operators such that A1 ≤ A0 . It follows that μ((A1 )+ ) ≤ μ((A0 )+ ). Proof. There exists a projection p ∈ ℒ(H) such that (A1 )+ = pA1 p. We have (A1 )+ = pA1 p ≤ pA0 p ≤ p(A0 )+ p. Hence, from Lemma 2.2.5, μ((A1 )+ ) ≤ μ(p(A0 )+ p) ≤ μ((A0 )+ ). Lemma A.2.2. Let 0 ≤ A, V ∈ ℒ(H). Suppose that V is compact and that V ≥ supp(V). If A ≥ 1, then 1

1

1

1

(A 2 VA 2 − 1)+ ≥ A 2 (V − 1)+ A 2 . Proof. If t ≥ nV (0), then μ(t, V) = 0. It follows that 1

1

μ(t, A 2 VA 2 ) ≤ ‖A‖∞ μ(t, V) = 0. If t < nV (0), then μ(t, V) ≥ 1 and, therefore 1

1

1

1

1 ≤ μ(t, V) ≤ ‖A−1 ‖∞ ⋅ μ(t, A 2 VA 2 ) ≤ μ(t, A 2 VA 2 ). It follows that 1

1

spec(A 2 VA 2 ) ⊂ {0} ∪ [1, ∞). In other words, 1

1

1

1

1

1

A 2 VA 2 ≥ supp(A 2 VA 2 ).

(A.13)

Similarly, 1

1

A 2 supp(V)A 2 ≥ supp(A 2 supp(V)A 2 ).

(A.14)

We have 1

1

1

1

1

ker(A 2 VA 2 ) = {ξ ∈ H : A 2 VA 2 ξ = 0} = A− 2 ker(V). Similarly, 1

1

1

ker(A 2 supp(V)A 2 ) = A− 2 ker(V). Therefore, 1

1

1

1

supp(A 2 VA 2 ) = supp(A 2 supp(V)A 2 ).

(A.15)

A.2 Operator inequalities | 363

It follows from (A.14) and (A.15) that 1

1

1

1

supp(A 2 VA 2 ) ≤ A 2 supp(V)A 2 .

(A.16)

It follows that from (A.13) that 1

1

1

1

1

1

1

1

(A 2 VA 2 − 1)+ = A 2 VA 2 − supp(A 2 VA 2 ).

(A.17)

On the other hand, we have 1

1

1

1

A 2 (V − 1)+ A 2 = A 2 VA 2 − A 2 supp(V)A 2 .

(A.18)

The assertion follows from (A.16), (A.17), and (A.18). Lemma A.2.3. Let 0 ≤ A, V ∈ ℒ(H). (a) If 0 ≤ A ≤ 1, then 1

1

1

1

1

1

1

1

μ((A 2 VA 2 − 1)+ ) ≤ μ(A 2 (V − 1)+ A 2 ). (b) If A ≥ 1, then μ((A 2 VA 2 − 1)+ ) ≥ μ(A 2 (V − 1)+ A 2 ). Proof. Let us prove the first assertion. Set V0 = 1 + (V − 1)+ . Since V ≤ V0 and A ≤ 1, it follows that 1

1

1

1

1

1

1

1

A 2 VA 2 − 1 ≤ A 2 V0 A 2 − 1 ≤ A 2 (V0 − 1)A 2 = A 2 (V − 1)+ A 2 . The first assertion follows now from Lemma A.2.1. Let us now prove the second assertion. Suppose that V ≥ supp(V) (we do not require that V be compact). Let {pk }k≥0 be the sequence of eigenprojections of V so that Vpk = μ(k, V)pk . Define n

Vn := ∑ μ(k, V)pk . k=0

We have V ≥ Vn . Clearly, 1

1

1

1

A 2 VA 2 − 1 ≥ A 2 Vn A 2 − 1. It follows now from Lemmas A.2.1 and A.2.2 that 1

1

1

1

1

1

μ((A 2 VA 2 − 1)+ ) ≥ μ((A 2 Vn A 2 − 1)+ ) ≥ μ(A 2 (Vn − 1)+ A 2 ).

364 | A Miscellaneous results Clearly, (Vn − 1)+ → (V − 1)+

1

1

1

1

A 2 (Vn − 1)+ A 2 → A 2 (V − 1)+ A 2

and, therefore,

in ℒ(H) (the convergence is understood in the uniform norm). Therefore, 1

1

1

1

μ(A 2 (Vn − 1)+ A 2 ) → μ(A 2 (V − 1)+ A 2 ). We conclude that 1

1

1

1

μ((A 2 VA 2 − 1)+ ) ≥ μ(A 2 (V − 1)+ A 2 ). This proves the first assertion for the case when V ≥ supp(V). Let X = Vχ[1,∞) (V). Clearly, X ≤ V and, therefore, 1

1

1

1

A 2 VA 2 − 1 ≥ A 2 XA 2 − 1. It follows from Lemma A.2.1 that 1

1

1

1

μ((A 2 VA 2 − 1)+ ) ≥ μ((A 2 XA 2 − 1)+ ). Since X ≥ supp(X), it follows from the preceding paragraph that 1

1

1

1

1

1

μ((A 2 VA 2 − 1)+ ) ≥ μ(A 2 (X − 1)+ A 2 ) = μ(A 2 (V − 1)+ A 2 ). This completes the proof of the second assertion.

A.3 Matrix inequalities This section contains auxiliary results on matrices used in Section 5.4. In Section 5.3, if u : ℂ → ℝ is continuous, we defined a function û : Mn (ℂ) → ℝ by setting ̂ u(A) = ∑ u(λ). λ∈σ(A)

Set log+ (x) := max{log(|x|), 0},

x ∈ ℝ.

̂ (A) ≤ log ̂ (|A|). Lemma A.3.1. For every matrix A ∈ Mn (ℂ), we have log + + Proof. Let 󵄨 󵄨 N1 = 󵄨󵄨󵄨{λ ∈ σ(A) : |λ| > 1}󵄨󵄨󵄨,

󵄨 󵄨 N2 = 󵄨󵄨󵄨{λ ∈ σ(|A|) : |λ| > 1}󵄨󵄨󵄨.

A.3 Matrix inequalities | 365

It follows from Lemma 1.1.21 that ∏ λ∈σ(A),|λ|>1

N1

N1

k=0

k=0

|λ| = ∏ |λ(k, A)| ≤ ∏ μ(k, A).

If N2 ≥ N1 , then μ(k, A) > 1 for every k ∈ (N1 , N2 ]. Therefore, N1

N2

k=0

k=0

∏ μ(k, A) ≤ ∏ μ(k, A) =

∏

|λ|.

(A.19)

|λ|.

(A.20)

λ∈σ(|A|),|λ|>1

If N2 < N1 , then μ(k, A) ≤ 1 for every k ∈ (N2 , N1 ]. Therefore, N1

N2

k=0

k=0

∏ μ(k, A) ≤ ∏ μ(k, A) =

∏ λ∈σ(|A|),|λ|>1

Combining (A.19) and (A.20), we conclude the proof. In Section 2.2 we defined the distribution function of an operator A ∈ ℒ(H) (or a matrix) as nA (s) := Tr(E|A| (s, ∞)),

s ≥ 0.

Lemma A.3.2. For every normal operator A ∈ Mn (ℂ) and every α ∈ ℂ such that |α| ≤ 1, we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 λ−α λ󵄨󵄨󵄨 ≤ nA (1). ∑ ∑ 󵄨󵄨 󵄨󵄨 󵄨 λ∈σ(αA),|λ|>1 λ∈σ(A),|λ|>1 󵄨 Proof. It is clear that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 αλ󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 1󵄨󵄨󵄨 = nA (1). λ−α λ󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ ∑ ∑ ∑ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 −1 λ∈σ(A),|λ|>1 󵄨 λ∈σ(αA),|λ|>1 λ∈σ(A),|λ|>1 λ∈σ(A),11 󵄨 Proof. Let pi = E|Ai | (1, ∞), i = 1, 2, 3. Since every Ai is normal, it follows that pi = EAi ({z ∈ ℂ, |z| > 1}). Hence ∑ λ∈σ(Ai ),|λ|>1

λ = Tr(pi Ai pi ),

∀i = 1, 2, 3.

366 | A Miscellaneous results Set p = ⋁ pi . i=1,2,3

We have Tr(pi Ai pi ) = Tr(pAi p) − Tr((p − pi )Ai (p − pi )). Since A1 + A2 + A3 = 0, it follows that 3

3

i=1

i=1

∑ Tr(pi Ai pi ) = − ∑ Tr((p − pi )Ai (p − pi )). We have ‖(p − pi )Ai (p − pi )‖ ≤ ‖(1 − pi )Ai (1 − pi )‖ ≤ 1 and rank((p − pi )Ai (p − pi )) ≤ Tr(p − pi ) = Tr(p) − Tr(pi ). It follows from the obvious inequality |Tr(A)| ≤ ‖A‖ ⋅ rank(A) that Tr((p − pi )Ai (p − pi )) ≤ Tr(p) − Tr(pi ). Therefore, 󵄨󵄨 3 󵄨󵄨 3 󵄨󵄨 󵄨 λ󵄨󵄨󵄨 ≤ ∑ Tr(p) − Tr(pi ). ∑ 󵄨󵄨 ∑ 󵄨󵄨 󵄨 i=1 λ∈σ(Ai ),|λ|>1 󵄨 i=1 It is clear that Tr(pi ) = nAi (1) for i = 1, 2, 3 and Tr(p) ≤ nA1 (1) + nA2 (1) + nA3 (1). The assertion follows immediately.

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Index ζ -function 254 – Formula 250, 256 – Meromorphic continuation 250, 303 – Minakshisundaram and Pleijel 250, 288 – Regularization 288 – Residue 250, 255, 303 – Minakshisundaram and Pleijel XV – Residue XXIII

Eigenvalue 5, 308 Eigenvalue sequence XVII, 6, 82, 141, 225 – of the Laplacian 227, 245, 246, 249 Expectation values 228, 229, 235, 314 Extended limit XVIII, 20, 190, 219, 298 – Dilation invariant 19, 193, 304 – on L∞ (0, 1) 191 – Shift invariant see Banach limit

Almost convergent 194, 299 Aoki–Rolewicz theorem 59, 64, 68

Figiel–Kalton Theorem 79, 86, 92, 100, 144 Fubini Theorem 304, 307, 308, 338, 353

Banach limit XIX, 187, 193, 199, 219, 283, 298 – Factorizable XX, 187, 200, 255, 301

Hahn–Banach Theorem – invariant version of 193 Heat kernel 251, 289, 353 Heat trace – Asymptotic expansion XV, XXII, 251, 302 – Formula 183, 252, 253 – Functional 252, 256 – General functional 267 Hermitian operator see Operator, Self adjoint Hilbert space 3 – Inner product 3 – Norm 3 – Separable 4

C ∗ -algebra 30 Calkin algebra XXII, 24, 231 Calkin correspondence XXIV, 11, 56, 73, 79, 83 – for traces XXV, 85, 99, 142 Cesaro mean XIX, 82, 86, 92, 100, 143, 194, 247, 299 – Convergence 299 Commutative core XXIV, 12, 56, 79, 115 – Quasi-normed 84 Commutator XVI, 15, 27, 175 Commutator subspace 28, 82, 143, 175, 235 Compact operator XVI, 5 – Absolute value 7 – Decomposition into normal and quasi-nilpotent see Ringrose decomposition – Finite rank 4 – Schmidt decomposition 7, 24, 138 – Spectral decomposition 7 Decreasing rearrangement – of a function 35 – of a sequence see Singular value, Sequence Diagonal formula see Trace, Diagonal formula Dirac operator 292, 352 Dyadic – Averaging XXIV, 80, 114, 197 – Dilation XXIV, 80, 114, 197, 315 Eigenbasis 225, 307 – for the Laplacian 227, 245, 246, 249 Eigenspace 6

Ideal – Banach 61 – Dixmier–Macaev 15, 25, 189, 208, 213, 304 – Finite rank 13 – Fully symmetric 62 – Logarithmic submajorization closed 63, 141, 177 – Lorentz 14, 71, 75, 221 – Marcinkiewicz see Ideal, Lorentz – Monotone 114 – of bounded sequences 114 – of compact operators XVI, 11, 55, 65 – Positive cone 13 – Quasi-Banach 61, 68, 84 – Quasi-normed 59 – Schatten (Lp −) 13, 66, 75 – Separable part 69 – Shift-invariant XXIV, 114 – Trace class XVII, 17, 26, 66, 86, 104, 106, 182, 186, 225, 228, 252, 254 – Weak Schatten (Lp,∞ −) 69

382 | Index

– Weak trace class XVII, 13, 69, 104, 116, 181, 185, 229, 297 Integral on a manifold 227 Invariance – Permutation 113, 115 – Shift XXIV, 114, 115, 138, 193, 199 – Unitary XVIII, 15 Kernel of a Hilbert–Schmidt integral operator 249 Laplace–Beltrami operator 185, 227, 245, 249 Laplacian XV, 221, 225, 295, 308 – Dirichlet 185, 221, 245 Lebesgue – Measurable 244 – Measure 185 Lidskii formula XXVII, 22, 23, 28, 141, 142, 225 Logarithmic mean 183, 195, 247, 253, 257, 300, 305 – Convergence 227, 231, 248, 306 Measurable operator 184, 244, 298, 304, 315, 349 – Dixmier 298 – Dixmier on ℳ1,∞ 304 – Positive 298 – Universal 298, 309 Mellin Transform XXIII, 250, 289, 303 Modulated operator 229, 234, 246, 314 Noncommutative geometry XV, 246, 307, 308, 352 Noncommutative integral XXI, 186, 228, 230, 231, 241, 247, 305 Noncommutative residue 250, 288, 293 Norm – Fatou 63 – Modulated operator 234 – Operator 3 – Quasi- 58 – Shift-invariant monotone quasi- 116 – Symmetric quasi- 59, 83, 117 Operator – Absolute value 33 – Adjoint 5 – Bounded 3 – Compact see Compact operator

– Diagonal 6 – Finite rank see Compact operator, Finite rank – Hilbert–Schmidt 234, 249 – Identity operator 5 – Multiplier by a function 31, 225 – Nilpotent 8 – Normal 7 – Partial isometry 34 – Polar decomposition 34 – Positive 5, 30 – Projection see Projection – Quasi-nilpotent 8 – Self adjoint 5, 307 – Self adjoint as linear combination of positive 33 – Strictly positive 235 – Strong topology 4, 29 – Trace class 18 – Unbounded 307 – Uniform topology 4, 29 – Unitary 12 – Weak topology 30 Orthonormal basis 4, 12, 225 Permutation group 11, 113 Pietsch correspondence XV, XXIV, 27, 80, 109, 113 – for traces XXVI, 115, 198 Projection 30 – Range 30 – Rank 30 – Spectral 33, 235 – Support 33 – Valued measure 32 Pseudodifferential operator 186 – Elliptic 250, 251, 302 – Principal symbol 289 – Zero order 250 Quantum Unique Ergodicity 232, 247 Resolvent 5 Riemannian manifold 185, 227, 249 – Einstein–Hilbert action 288, 292 – Scalar curvature 251, 292 – Sphere bundle 186 Ringrose decomposition 10, 24, 170 Seeley–DeWitt coefficients 251, 289

Index | 383

Sequences – embedded in the diagonal of a separable Hilbert space 42 Shift operator XIX, XXIV, 81, 113, 193, 197 Shift-invariant functional XXVI, 115, 197 – Continuous 118 – Hermitian 117 – Regular 117 Singular state 18, 231 Singular value – Function 34 – Sequence 7, 35 Spectral – Action 293 – Asymmetry 288 – Distribution function 35 – Geometry XV, 249 – Measure 33 Spectral Theorem 7, 23, 33, 249 Spectrum 5 Submajorization – Hardy–Littlewood 21, 44, 200 – Logarithmic XXVII, 23, 50, 73, 141, 177 Symmetric functional XXVI, 16, 57, 82, 84, 99, 108, 142 – Continuous 61 – Fully 63 – Positive 57 Symmetric sequence space XXIV, 11, 56 – Banach 61 – Center 16, 101, 144 – lp 66 – lp,∞ 69 – mp,∞ 70 – Positive cone 13 – Quasi-Banach 61

Tauberian 293, 301 – Subhankulov Theorem 355 – Theorem on Abel Convergence 331 – Theorem on Cesaro Convergence 312 – Theorem on Logarithmic Convergence 248, 334 Trace XVI, 15, 18, 84, 141 – Connes–Dixmier 188, 206, 222, 300 – Continuous 61, 81, 111, 118, 199 – Diagonal formula XX, 225, 230, 314 – Dixmier XV, XVIII, 18, 27, 181, 185, 187, 213, 221, 231, 249, 253, 257, 269, 298, 304 – Fully symmetric 187, 196, 200, 298 – Hermitian 117 – Jordan decomposition 81, 118, 139 – Lattice of regular 118 – Matrix 10 – Normalized XIX, 186, 198, 297 – on trace class operators 18, 26, 86 – Positive 57, 117, 186, 187, 196, 198, 223 – Regular 81, 117, 139 – Singular XV, 21, 86, 104, 111, 293 – Spectral 141 – Spectral formula XXVII, 22, 81, 142, 299 – Spectral invariant 28, 84, 141 Unitarily equivalent 12, 56, 141 Von Neumann algebra 30, 72 – Commutative 31 – Factor 30 – of bounded sequences 31 – of essentially bounded functions 31 Weyl’s law 185, 221, 226, 227, 250, 295

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