Lectures on Selected Topics in von Neumann Algebras 9783985470044

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EMS SERIES OF LECTURES IN MATHEMATICS

Fumio Hiai

Lectures on Selected Topics in von Neumann Algebras

EMS Series of Lectures in Mathematics Edited by Ari Laptev (Imperial College, London, UK) The EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series: Katrin Wehrheim: Uhlenbeck Compactness Torsten Ekedahl: One Semester of Elliptic Curves Sergey V. Matveev: Lectures on Algebraic Topology Joseph C. Várilly: An Introduction to Noncommutative Geometry Reto Müller: Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels, Sara van de Geer: Lectures on Empirical Processes Iskander A. Taimanov: Lectures on Differential Geometry Martin J. Mohlenkamp, María Cristina Pereyra: Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne, Joseph A. Thas: Finite Generalized Quadrangles Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie, Nils Henrik Risebro: Splitting Methods for Partial ­Differential Equations with Rough Solutions Koichiro Harada: “Moonshine” of Finite Groups Yurii A. Neretin: Lectures on Gaussian Integral Operators and Classical Groups Damien Calaque, Carlo A. Rossi: Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry Claudio Carmeli, Lauren Caston, Rita Fioresi: Mathematical Foundations of Supersymmetry Hans Triebel: Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration Koen Thas: A Course on Elation Quadrangles Masoud Khalkhali: Basic Noncommutative Geometry Benoît Grébert, Thomas Kappeler: The Defocusing NLS Equation and Its Normal Form Armen Sergeev: Lectures on Universal Teichmüller Space Matthias Aschenbrenner, Stefan Friedl, Henry Wilton: 3-Manifold Groups Hans Triebel: Tempered Homogeneous Function Spaces Kathrin Bringmann, Yann Bugeaud, Titus Hilberdink, Jürgen Sander: Four Faces of Number Theory Alberto Cavicchioli, Friedrich Hegenbarth, Dušan Repovš: Higher-Dimensional Generalized Manifolds: ­Surgery and Constructions Davide Barilari, Ugo Boscain, Mario Sigalotti (Eds.): Geometry, Analysis and Dynamics on sub-Riemannian Manifolds, Volume I and II Françoise Dal’Bo, François Ledrappier, Amie Wilkinson (Eds.): Dynamics Done with Your Bare Hands Hans Triebel: PDE Models for Chemotaxis and Hydrodynamics in Supercritical Function Spaces Françoise Michel, Claude Weber: Higher-Dimensional Knots According to Michel Kervaire Radha Kessar, Gunter Malle, Donna Testerman (Eds.): Local Representation Theory and Simple Groups Hans Triebel: Function Spaces with Dominating Mixed Smoothness Ciro Ciliberto: Classification of Complex Algebraic Surfaces

Fumio Hiai

Lectures on Selected T ­ opics in von Neumann Algebras

Author: Fumio Hiai Graduate School of Information Sciences Tohoku University Aoba-ku, Sendai 980-8579, Japan E-mail: [email protected]

2020 Mathematics Subject Classification: primary 46L10; secondary 46L51 Keywords: von Neumann algebra, Tomita–Takesaki theory, modular operator, standard form, Connes’ ­cocycle derivative, operator-valued weight, relative modular operator, crossed product, KMS condition, Takesaki’s duality theorem, τ-measurable operator, Haagerup’s Lp-space, conditional expectation, positive self-adjoint operator, positive quadratic form

ISBN 978-3-98547-004-4 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. Published by EMS Press, an imprint of the European Mathematical Society – EMS – Publishing House GmbH Institut für Mathematik Technische Universität Berlin Straße des 17. Juni 136 10623 Berlin, Germany https://ems.press © 2021 European Mathematical Society Typeset using the author’s LaTeX sources: Alison Durham, Manchester, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ♾ Printed on acid free paper 987654321

Preface

In the history of von Neumann algebras, originating with the work of F. J. Murray and J. von Neumann in a series of papers in the late 1930s, the biggest breakthrough is probably Tomita–Takesaki theory established around 1970. Following that, many important new developments in the subject were made throughout the 1970s by H. Araki, A. Connes, U. Haagerup, M. Takesaki and others, which are nowadays classical parts of von Neumann algebra theory. Although a few lengthy treatises on the subject have been published, it still seems difficult for beginners to actually learn the details of the theory from them. In fact, after finishing my graduate studies in 1975, I myself tried earnestly to study Connes’ papers – a quite difficult endeavor, since the speed of appearance of Connes’ new papers was much faster than my speed of studying them. This book is based on lectures that I delivered during an intensive course in April 2019 at the Department of Mathematical Analysis of the Budapest University of Technology and Economics in Hungary. The course was designed as a fast-track study of some important and classical parts of von Neumann algebra theory, from which one can gain rigorous background knowledge and can consult the original references and/or suitable, more advanced books for further details. Thus, I selected and lectured on several topics of von Neumann algebras starting from the classical developments in the 1970s. These cover the most fundamental topics in the theory, starting from Tomita–Takesaki theory and extending in other directions concerned with non-commutative integration, which are most useful in applications to mathematical/quantum physics, quantum information theory and so on. In preparing these lecture notes, I tried to make them as self-contained as possible, assuming only a basic knowledge of functional analysis and measure theory. Except for a few cases, all results are given with complete proofs, which are detailed enough to be understood by a beginner. The references, though, are restricted to what is directly needed within the text, in accordance with the aim and the character of the presentation. I thought that many people wishing to study von Neumann algebras would find my lecture notes useful, so I posted them on the arXiv in April 2020. Then I received a proposal from EMS Press to publish them in the EMS Series of Lectures in Mathematics. The present book is a corrected and enlarged version of the arXiv preprint [40]. The book consists of 11 chapters and an appendix, each chapter starting with a short introduction. Chapter 1 is an overview of von Neumann algebra theory. Chapter 2 covers Tomita–Takesaki theory. Chapters 3, 6, 7 and 8 are concerned with several fundamental classics – the standard form, Connes’ cocycle derivatives, operator-valued weights, Takesaki duality and type III structure theory – based on Tomita–Takesaki theory. Chapters 4, 5 and 9–11 treat several topics more or less concerned with non-commutative integration, such as -measurable operators, conditional expectations, relative modular operators and non-commutative Lp -spaces. While von Neumann algebras consist of bounded operators

vi

Preface

on a Hilbert space, in studying them one needs quite often to deal with unbounded closed (in particular, positive self-adjoint) operators; for instance, the (relative) modular operators are unbounded. Thus, basic facts on positive self-adjoint operators are summarized in Appendix A for the reader’s convenience. In view of the aim of this book, the classification of type III factors and the classification of AFD (= injective) factors due to Connes are not included, but only surveyed, in Sections 1.6 and 1.8 of Chapter 1. Since the 1980s, major advances have been made in von Neumann algebra theory with strong connections to other branches of mathematics; the most brilliant ones are Jones’ index theory, Voiculescu’s free probability theory and Popa’s rigidity theory. While these are beyond the scope of this book, brief surveys on them are given in the final section of Chapter 1 for the interested reader. Acknowledgments: I would like to thank Milán Mosonyi for inviting me to the aforementioned intensive course on von Neumann algebras in April 2019. In preparing for the lectures, I was able to refresh my knowledge of von Neumann algebras, which I studied over forty years ago. Without his invitation it would not have been possible for me to write these lecture notes. Also, I am grateful to Apostolos Damialis for suggesting that I publish my lecture notes in the EMS Series of Lectures in Mathematics.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Von Neumann algebras – An overview . . . . . . . 1.1 Preliminaries . . . . . . . . . . . . . . . . . 1.2 Basics of von Neumann algebras . . . . . . . 1.3 States, weights and traces . . . . . . . . . . . 1.4 Classification of von Neumann algebras . . . 1.5 Tomita–Takesaki theory . . . . . . . . . . . . 1.6 Classification of factors of type III . . . . . . 1.7 Crossed products and type III structure theory 1.8 Classification of AFD factors . . . . . . . . . 1.9 Standard form and natural positive cone . . . 1.10 Developments since the 1980s . . . . . . . .

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1 1 2 3 4 5 6 7 8 9 10

2

Tomita–Takesaki modular theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Tomita’s fundamental theorem . . . . . . . . . . . . . . . . . . . . . . . 2.2 KMS condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 20

3

Standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . 3.2 Uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 32

4

-Measurable operators . . . . . . . . 4.1 -Measurable operators . . . . . 4.2 Generalized s-numbers . . . . . 4.3 Lp -spaces with respect to a trace

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39 39 47 57

5

Conditional expectations and generalized conditional expectations . . . . . . . 5.1 Conditional expectations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Generalized conditional expectations . . . . . . . . . . . . . . . . . . . .

71 71 79

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6

Connes’ cocycle derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Basics of faithful semifinite normal weights . . . . . . . . . . . . . . . . 6.2 Connes’ cocycle derivatives . . . . . . . . . . . . . . . . . . . . . . . . .

7

Operator-valued weights . . . . . . . 7.1 Generalized positive operators . 7.2 Operator-valued weights . . . . 7.3 Pedersen–Takesaki construction

8

Takesaki duality and structure theory . . . . . . . . . . . . . . . . . . . . . . . 123 8.1 Takesaki’s duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2 Structure of von Neumann algebras of type III . . . . . . . . . . . . . . . 135

9

Haagerup’s Lp -spaces . . . . . . . . 9.1 Description of L1 .M / . . . . . 9.2 Haagerup’s Lp -spaces . . . . . 9.3 Kosaki’s interpolation Lp -spaces

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139 139 145 157

10 Relative modular operators and Connes’ cocycle derivatives (continued) . . . . 163 10.1 Relative modular operators . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.2 Connes’ cocycle derivatives (continued) . . . . . . . . . . . . . . . . . . 172 11 Spatial derivatives and spatial Lp -spaces 11.1 Spatial derivatives . . . . . . . . . 11.2 Proofs of theorems . . . . . . . . 11.3 Spatial Lp -spaces . . . . . . . . .

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181 181 192 202

A Positive self-adjoint operators and positive quadratic forms . . . . . . . . . . . 211 A.1 Positive self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . 211 A.2 Positive quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

1 Von Neumann algebras – An overview

This overview1 gives a brief survey of topics in von Neumann algebra theory, mostly developed through the 1970s, which serves as an introduction to these lecture notes. Most materials in the overview (except the final section) will be treated in detail in the main body.

1.1 Preliminaries The set B.H / of all bounded linear operators on a Hilbert space H with the inner product h
;
i is a vector space with the operator sum a C b and the scalar multiplication a (a; b 2 B.H /,  2 C) and is a Banach space with the operator norm kak WD sup¹kak W  2 H ; kk  1º: Moreover, B.H / becomes a Banach *-algebra with the operator product ab and the adjoint operation a 7! a . A subspace of B.H / is called a subalgebra if it is closed under the product, and a *-subalgebra if it is further closed under the *-operation. In general, operator algebras mean *-subalgebras of B.H /. The (operator) norm topology, the strong operator topology and the weak operator topology are defined on B.H /, which are weaker in this order. Since B.H / is the dual Banach space of the Banach space C1 .H / consisting of trace-class operators with the trace-norm, the weak topology  .B.H /; C1 .H // is also defined on B.H / and is called the -weak topology, which is particularly important in studies of von Neumann algebras. We write B.H /sa for the set of all self-adjoint a D a in B.H /. The order a  b for a; b 2 B.H /sa means that h; ai  h; bi for all  2 H , which is a partial order on B.H /sa . When a net ¹a˛ º in B.H /sa is increasing and bounded above, it has the supremum a 2 B.H /sa and a˛ ! a strongly; in this case, we write a˛ % a. We note here that an inner product h
;
i is conjugate-linear in the first variable and linear in the second according to the physics convention. 1This is an English translation extracted from my article in Encyclopedic Dictionary of Mathematics, 4th edition (in Japanese), Iwanami Publisher, Japan, 2007.

2

1 Von Neumann algebras – An overview

1.2 Basics of von Neumann algebras A *-subalgebra of B.H / is called a von Neumann algebra (also W  -algebra) if it contains the identity operator 1 and is closed in the weak topology. For S  B.H / define the commutant S 0 of S by S 0 WD ¹a 2 B.H / W ab D ba for all b 2 S º; and also S 00 WD .S 0 /0 . For a *-subalgebra M of B.H / with 1 2 M , the double commutation theorem or von Neumann density theorem says that the following three conditions are equivalent: (i) M is a von Neumann algebra (i.e., weakly closed);2 (ii) M is strongly closed; (iii) M 00 D M . The theorem shows that the polar decompositions and the spectral decompositions of operators in a von Neumann algebra M are taken inside M itself, so M contains plenty of projections. Starting from this theorem (from 1929), J. von Neumann developed basics (including the classification; see Section 1.4) of von Neumann algebra theory in a series of joint papers with F. J. Murray. C  -algebras are another major subject of operator algebras, which are faithfully represented as norm-closed *-subalgebras of B.H /. Sakai [83] gave the abstract characterization of von Neumann algebras in such a way that a C  -algebra is isomorphic to a von Neumann algebra if and only if it is the dual Banach space of some Banach space. In this case, the predual space is unique in a certain strong sense. The term W  -algebra is often used to stress this abstract (or Hilbert-space-free) situation. Although von Neumann algebras are special C  -algebras, both categories of operator algebras are quite different theoretically and methodologically. The Kaplansky density theorem is particularly useful3 in the study of von Neumann algebras, saying that if A is a *-subalgebra of B.H / containing 1, then ¹a 2 A W kak  1º is strongly dense in ¹a 2 A 00 W kak  1º. Let M , N be von Neumann algebras, and W M ! N be a *-homomorphism. If a˛ % a in M implies .a˛ / % .a/, then  is said to be normal, which is equivalent to the continuity of  with respect to the -weak topologies on M , N . A *-homomorphism W M ! B.K / (K is a Hilbert space) is called a *-representation (or simply representation). The range .M / of a normal representation  is a von Neumann algebra and its kernel is represented as M e for some central projection e (i.e., a projection in the center Z.M / WD M \ M 0 ), so  induces a *-isomorphism between M.1 e/ and .M /. Note that a faithful representation is normal automatically. 2In this monograph, when semicolons are used to separate list items, those items are understood to be equivalent conditions. 3In his book [69] Pedersen wrote, “The density theorem is Kaplansky’s great gift to mankind. It can be used every day, and twice on Sundays.”

1.3 States, weights and traces

3

For two Hilbert spaces H1 , H2 , the tensor product Hilbert space H1 ˝ H2 is defined by completing the algebraic tensor product (as a complex vector space) of H1 , H2 with respect to the inner product determined by h ˝ 2 ; 1 ˝ 2 i D h1 ; 1 ih2 ; 2 i (i ; i 2 Hi ). For any a1 2 B.H1 / and a2 2 B.H2 / the tensor product a1 ˝ a2 2 B.H1 ˝ H2 / is uniquely determined by .a1 ˝ a2 /.1 ˝ 2 / D a1 1 ˝ a2 2 (i 2 Hi ). For von Neumann algebras Mi  B.Hi /, the von Neumann algebra generated by ¹a1 ˝ a2 W a1 2 M1 ; a2 2 M2 º is denoted by M1 ˝ M2 and called the tensor product of M1 , M2 . The commutant theorem .M1 ˝ M2 /0 D M10 ˝ M20 holds for tensor products of von Neumann algebras.

1.3 States, weights and traces We write M for the set of all -weakly continuous linear functionals on a von Neumann algebra M , which is a Banach space as a closed subspace of the dual Banach space M  . Then the dual Banach space of M is isometric to M . In fact, M is a unique Banach space with this property, so it is called the predual of M . A positive linear functional ' on M is in M if and only if ' is normal (i.e., a˛ % a H) '.a˛ / % '.a/). In particular, a normal positive linear functional ' on M with '.1/ D 1 is called a normal state of M . Any ' 2 M is represented as a linear combination of normal states. A functional 'W MC D ¹a 2 M W a  0º ! Œ0; 1 is called a weight on M if it satisfies, for all a; b 2 MC and   0,  additivity: '.a C b/ D '.a/ C '.b/,  positive homogeneity: '.a/ D '.a/, where 01 WD 0. A weight ' is faithful if '.a/ > 0 for all a 2 MC n ¹0º and normal if a˛ % a H) '.a˛ / % '.a/. Let N' WD ¹a 2 M W '.a a/ < 1º and M' be the linear span of ¹a b W a; b 2 N' º. If M' is -weakly dense in M , then ' is said to be semifinite. The weight ' extends to M' as a linear functional. Similarly to the GNS (Gelfand– Naimark–Segal) construction of a C  -algebra with respect to a state, the GNS construction .H' ; ' ; ' / of M with respect to a semifinite normal weight ' is defined as follows: H' is a Hilbert space, ' W M ! B.H' / is a normal representation and ' W N' ! H' is a linear map such that ' .N' / D H' , h' .a/; ' .b/i D '.a b/ and ' .x/' .a/ D ' .xa/ for all a; b 2 N' , x 2 M . A weight  satisfying  .a a/ D  .aa / for all a 2 M is called a trace. A finite (i.e.,  .1/ < 1) trace  uniquely extends to M as a linear functional satisfying  .ab/ D  .ba/ (a; b 2 M ).

4

1 Von Neumann algebras – An overview

1.4 Classification of von Neumann algebras The notion of the Murray–von Neumann equivalence on the set Proj.M / of all projections in a von Neumann algebra M is defined as follows: e; f 2 Proj.M / is said to be equivalent (e  f ) if there is a v 2 M such that v  v D e and vv  D f . A projection e 2 Proj.M / is called an abelian projection if eM e is an abelian algebra, and a finite projection if for f 2 Proj.M /, e  f  e H) f D e. The von Neumann algebra M is said to be finite if 1 is a finite projection, and semifinite if, for every central projection e ¤ 0 (i.e., a projection in the center of M ), there is a finite projection f 2 Proj.M / such that 0 ¤ f  e. If M has no finite central projection (¤ 0), then M is said to be properly infinite. If M has no finite projection (¤ 0), then M is said to be purely infinite or of type III. A von Neumann algebra M is properly infinite if and only if M Š M ˝ B.H / for separable Hilbert spaces H , and M is semifinite if and only if M has a faithful semifinite normal trace. A von Neumann algebra M is called a factor if the center is trivial (i.e., Z.M / D C1). The factors are classified into one of the following types:  type In (n 2 N) – M is isomorphic to the matrix algebra Mn .C/ (type In for some n 2 N if M is finite and has a finite abelian projection ¤ 0),  type I1 – M is isomorphic to B.H / with dim H D 1, equivalently, M is properly infinite and has an abelian projection ¤ 0,  type II1 – M is finite and has no abelian projection ¤ 0,  type II1 – M is semifinite and properly infinite, and has no abelian projection ¤ 0,  type III – M has no finite projection ¤ 0. A finite factor has a faithful finite normal trace (unique up to positive constants). A factor of type II1 is represented as .a factor of type II1 / ˝ B.H /. Corresponding to the types of factors, the quotient set Proj.M /= (modulo the Murray–von Neumann equivalence) is identified with one of the following:  type In – ¹0; 1; : : : ; nº,  type I1 – ¹0; 1; : : : ; 1º,  type II1 – Œ0; 1,  type II1 – Œ0; 1,  type III – ¹0; 1º. Von Neumann algebras of type I (i.e., a direct sum of factors of type I) are said to be discrete or atomic. For each type II1 , II1 and III, there are continuously many non-isomorphic classes (due to McDuff, Sakai for type II1 ; see Section 1.6 for type III).

5

1.5 Tomita–Takesaki theory

Any von Neumann algebra M on a separable Hilbert space H is decomposed into factors as a direct integral (called von Neumann’s reduction theory) as follows: There exists a measurable field of factors ¹M. /; H . /º 2€ on a standard Borel space .€; / with a finite measure  such that H D

Z

˚

H . / d;

Z

˚

M D

€

Z M. / d;

€

˚

Z.M / D

C1 d .Š L1 .€; //:

€

Many issues concerning von Neumann algebras can be reduced to the case of factors by using this reduction.

1.5 Tomita–Takesaki theory The study of type III von Neumann algebras was extensively developed in the 1970s, starting with the modular theory originated by M. Tomita and further developed by M. Takesaki [91]; the theory is called Tomita–Takesaki theory. Let .H' ; ' ; ' / be the GNS construction of a von Neumann algebra M with respect to a faithful semifinite normal weight ' on M . Define a conjugate-linear operator S' on a dense subspace ' .N' \ N' / by S' ' .a/ WD ' .a /, which is closable. By taking the of the closure of S' , we define a positive self-adjoint polar decomposition S' D J' 1=2 ' operator ' called the modular operator, and a conjugate-linear unitary involution J' (J'2 D 1) called the modular conjugation. The following two statements are Tomita’s fundamental theorem: (i) J' MJ' D M 0 , it (ii) it D M (t 2 R), ' M'

where M is identified with ' .M / via the faithful representation ' . By (ii),  t' .a/ D it it ' a' (a 2 M ) defines a strongly continuous one-parameter automorphism group on M called the modular automorphism group associated with '. Takesaki’s theorem says that ' satisfies the KMS (Kubo–Martin–Schwinger) condition at ˇ D 1 as follows: For every a; b 2 N' \ N' , there exists a bounded continuous function fa;b .z/ on ˇ  Im z  0, analytic in ˇ < Im z < 0, such that fa;b .t / D '.a t' .b//;

fa;b .t C iˇ/ D '. t' .b/a/

.t 2 R/:

Furthermore, the modular automorphism group  t' is uniquely determined by this condition for '. The KMS condition was introduced by Haag–Hugenholtz–Winnink to characterize the equilibrium states in quantum systems in the C  -algebraic approach to quantum statistical mechanics. (In statistical mechanics, ˇ corresponds to the inverse temperature.) The link between the modular theory and the KMS condition was quite a remarkable occurrence.

6

1 Von Neumann algebras – An overview

1.6 Classification of factors of type III In the same period of time as the appearance of Tomita–Takesaki theory, Powers [76] showed the existence of continuously many non-isomorphic factors of type III, called the Powers factors. Next Araki and Woods [5] classified the so-called ITPFI factors (also called the Araki–Woods factors) defined from infinite tensor products of factors of type I. For N an infinite sequence of pairs ¹Mkn ; 'n º of matrix algebras and states, the ITPFI 1 factor N nD1 ¹Mkn ; 'n º is given by making the GNS N1construction of the infinite tensor product 1 M with respect to the tensor state k n 1 1 'n . In particular, for 0 <   1  1 0  consider the state of M2 defined by ! D Tr.D
/, where D D 1C  . Then 0 1C N1 R WD 1 ¹M2 ; ! º, 0 <  < 1, are the Powers factors, which are non-isomorphic factors of type III for different . On the other hand, when  D 1, this infinite tensor product R is the so-called hyperfinite factor of type II1 (see Section 1.8). Motivated by the idea of reconstructing the Araki–Woods factors using Tomita– Takesaki theory, Connes [14] established the classification theory of type III factors. To do so, he introduced the T -set and the S -set of M , defined by T .M / WD ¹t 2 R W  t' 2 Int.M /º .Int.M / WD the inner automorphisms of M /; \ S.M / WD ¹Sp.' / W ' is a faithful semifinite normal weight on M º; which are invariants for isomorphism classes of von Neumann algebras. For any two faithful semifinite normal weights ', on M there exists a unitary cocycle u t D .D W D'/ t (2 M ) with respect to  t' , for which  t .a/ D u t  t' .a/ut holds for all a 2 M , t 2 R. Therefore, T .M / is a subgroup of R determined independently of the choice of '. On the other hand, S.M / n ¹0º is a closed subgroup of the multiplicative group RC (D .0; 1/). A von Neumann algebra M is semifinite if and only if S.M / D ¹1º. The factors of type III are classified in terms of the S -set as  type III1 – S.M / D ¹1º,  type III (0 <  < 1) – S.M / D ¹0º [ ¹n W n 2 Zº,  type III0 – S.M / D ¹0; 1º. The T -set of type III1 factors is T .M / D ¹0º, that of III factors (0 <  < 1) is T .M / D .2= log /Z and T .M / of III0 factors is not unique. From Araki and Wood’s result, there is a unique ITPFI factor for each type III (0 <  < 1) and type III1 , and that of type III is the Powers factor R . There are continuously many ITPFI factors of type III0 .

1.7 Crossed products and type III structure theory

7

1.7 Crossed products and type III structure theory Let ˛ D ¹˛g ºg2G be a continuous action of a locally compact group G on a von Neumann algebra M  B.H /, i.e., g 2 G 7! ˛g 2 Aut.M / is a homomorphism and g 7! ˛g .a/ is strongly continuous for any a 2 M . On the Hilbert space L2 .G; H / D L2 .G/ ˝ H (with respect to the left Haar measure on G), a representation ˛ of M and a unitary representation  of G are defined by .˛ .a//.h/ WD ˛h 1 .a/.h/;

..g//.h/ WD .g

1

h/

. 2 L2 .G; H /; g; h 2 G/:

The crossed product M o˛ G of M by the action ˛ is the von Neumann algebra generated b on by ˛ .M / [ .G/. When G is abelian, a unitary representation of the dual group G L2 .G; H / is defined by ..p//.h/ WD hh; pi.h/

. 2 L2 .G; H //:

b of M o˛ G is defined by Then an action ˛ y of G ˛ yp .x/ WD .p/x.p/

b .x 2 M o˛ G; p 2 G/;

which is called the dual action. Takesaki’s duality theorem [93] is stated as b Š M ˝ B.L2 .G//: .M o˛ G/ o˛y G In particular, if M is properly infinite and G satisfies the second axiom of countability, then b Š M holds. This duality theorem was extended to the case of actions of .M o˛ G/ o˛y G non-abelian locally compact groups and more general Kac algebras (Hopf algebras with *-structure). Connes and Takesaki established the type III structure theory by use of crossed products. (a) The structure of type III factors (Connes [14]): For any factor of type III (0 <  < 1), there exist a factor N of type II1 and a  2 Aut.N / such that M Š N o Z;

 ı  D ;

where  is a faithful semifinite normal trace on N . Moreover, .N;  / is unique up to conjugacy. Furthermore, Connes [14] showed a similar (but slightly more complicated) structure theorem for type III0 factors. (b) The structure of type III von Neumann algebras (Takesaki [93]): For any von Neumann algebra of type III, there exist a von Neumann algebra N of type II1 , a faithful semifinite normal trace  on N and a continuous one-parameter action  t on N such that M Š N o R;  ı  t D e t  .t 2 R/: Moreover, .N;  / is unique up to conjugacy. Note that .N;  / is realized as the crossed product N D M o ' R by the modular automorphism group  t' and the dual action  D c' in Takesaki’s duality.

8

1 Von Neumann algebras – An overview

The above structure theorems give the crossed product decompositions of type III von Neumann algebras M by Z or R-action of type II von Neumann algebras. Thus, the study of type III structure may be reduced to that of type II and actions with trace-scaling properties such as  ı  D  and  ı  t D e t . When a properly infinite factor M is decomposed as M Š N o R as in (b) above, Connes and Takesaki [18] introduced the flow of weights of M as .X; F tM / WD .Z.N /;  t jZ.N / /; which is a non-singular ergodic flow and classifies the type of M as follows:  if X is a single point (i.e., N is a factor), then M is of type III1 ,  if .X; F tM / is a translation on Œ0; III ,

log / with the period

log , then M is of type

 if .X; F tM / is aperiodic and not the R-translation, then M is of type III0 ,  if .X; F tM / is the R-translation, then M is semifinite.

1.8 Classification of AFD factors Here, assume that a von Neumann algebra M is represented on a separable Hilbert space or, equivalently, M has the separable predual M . If M is generated by an increasing sequence of finite-dimensional *-subalgebras, then M is said to be hyperfinite or AFD (approximately finite-dimensional). The uniqueness of type II1 AFD factors is an old result by Murray and von Neumann. ITPFI factors (see Section 1.6) are obviously AFD. A von Neumann algebra M  B.H / is said to be injective if there exists a norm-one projection from B.H / onto M . In 1976, Connes [16] proved that a von Neumann algebra is AFD if and only if it is injective and, at the same time, he proved that injective factors of types II1 , II1 , III (0 <  < 1) are unique for each type, which are, respectively, R;

R0;1 WD R ˝ B.H /;

R

(see Section 1.6):

It was also shown by Connes [16] that injective factors of type III0 are only Krieger factors [61], that is, factors realized as the crossed product L1 .X / oT Z by a non-singular ergodic transformation T acting freely on a Lebesgue space .X; /. This construction is called the group measure space construction and has been known since the early stages of von Neumann algebra theory. Two Krieger factors are isomorphic if and only if two transformations are weakly equivalent, i.e., orbit equivalent (Dye, Krieger), thus completing classification of injective factors of type III0 . The class of Krieger factors is bigger than that of ITPFI factors. More general Krieger-type construction is known for the ergodic action of countable groups and for ergodic countable equivalence relations (Feldman–Moore).

1.9 Standard form and natural positive cone

9

In 1985, Haagerup [35] (also [36]) proved the uniqueness of injective factors of type III1 , that is, the III1 ITPFI factor is a unique injective factor of type III1 , thus completing the classification of AFD (= injective) factors. In other words, the conjugacy class of the flow of weights (see the end of Section 1.7) is a complete invariant for injective factors of type III. Classification of group actions on AFD factors was also well developed. When M is a factor, the outer period p0 .˛/ and the obstruction .˛/ of ˛ 2 Aut.M / are defined as follows: p0 .˛/ is the smallest integer n > 0 such that ˛ n 2 Int.M / (p0 .˛/ D 0 if such an n does not exist). Then, for a unitary u 2 M such that ˛ p0 .˛/ .a/ D uau , define .˛/ as

2 C satisfying ˛.u/ D u, where .˛/ WD 1 if p0 .˛/ D 0. Note that .˛/ is a p0 .˛/th root of 1 (a cohomological quantity). Connes proved around 1985 that .p0 .˛/; .˛// is a complete invariant of outer conjugacy (conjugacy modulo inner automorphisms) for automorphism of the AFD type II1 factor R. Moreover, for automorphisms of the AFD II1 factor R0;1 , .p0 .˛/; .˛/; mod.˛// is a complete invariant of outer conjugacy, where the module mod.˛/ is the  > 0 satisfying  ı ˛ D  for the trace  on R0;1 . Since then, there have been many studies on classification of amenable group actions on AFD factors, by Jones, Ocneanu, Takesaki and others.

1.9 Standard form and natural positive cone Let ' be a faithful semifinite normal weight on a general von Neumann algebra M . The closure P' of ¹1=4 ' ' .a/ W 0  a 2 M' º is a self-dual cone of H' , which is called the natural positive cone of M . The quadruple .M; H ; J; P/ of a von Neumann algebra M , its faithfully representing Hilbert space H D H' , the conjugate-linear unitary involution J D J' and the self-dual cone P D P' satisfy the following properties:  JMJ 0 D M 0 ,  JcJ D c  (c 2 Z.M /),  J  D  ( 2 P),  aJaJ.P/  P (a 2 M ). Such an .M; H ; J; P/ is unique up to unitary equivalence for any M , and it is called the standard form of M . The theory of standard form established independently by Araki [3], Connes [15] and Haagerup [26, 28] is important in studying von Neumann algebras. The map  2 P 7! ! WD h;
i 2 MC WD ¹' 2 M W '  0º is a homeomorphism in the norm topologies and satisfies k

k2  k!

! k  k

k k C k

.;  2 P/:

For any g 2 Aut.M / there exists a unique unitary ug 2 B.H / such that ug J D J ug , ug .P/ D P and g.a/ D ug aug (a 2 M ). The ug is a unitary representation of the

10

1 Von Neumann algebras – An overview

group Aut.M / and satisfies !ug  D ! ı g 1 (g 2 Aut.M /,  2 P). When M is a semifinite von Neumann algebra with a faithful semifinite normal trace , its standard form is realized as .M; L2 .M;  /; J; L2 .M;  /C /, where the Hilbert space L2 .M;  / is the non-commutative L2 -space with respect to  (consisting of -measurable operators a with  .a a/ < 1), M is represented on L2 .M;  / by the left multiplication and Ja D a for a 2 L2 .M;  /.

1.10 Developments since the 1980s As to developments in von Neumann algebra theory since the 1980s, there are, among others, three mainstreams having deep connections to different areas of mathematics. Although these new subjects will not be treated in the present lecture notes (indeed, they are beyond our scope and aim), for the interest of the reader we survey them here briefly. (1) Jones’ subfactor theory. In 1983, Jones [50] originated the index theory for type II1 subfactors. For a subfactor N of a type II1 factor M he introduced the index ŒM W N  in terms of the coupling constant due to Murray and von Neumann, and discovered a surprising fact that the range of possible values of ŒM W N  is ¹4 cos2 =n W n D 3; 4; : : : º [ Œ4; 1; which consists of two parts: discrete values less than 4 and continuous Œ4; 1. To prove this, Jones constructed a sequence of projections e0 ; e1 ; e2 ; : : : satisfying ei ej D ej ei (ji j j  2) and ei ei ˙1 ei D ei ( D ŒM W N  1 ), thus generating the Temperley– Lieb algebras, and a sequence of II1 factors N  M  M1 D hM; e0 i  M2 D hM1 ; e1 i 


called the basic construction. The two principal graphs are derived from the Bratteli diagrams of the increasing sequences ¹N 0 \ Mk º and ¹M 0 \ Mk º of higher relative commutant algebras (finite-dimensional if ŒM W N  < 1). In particular, when ŒM W N  < 4, Dynkin diagrams An , D2n , E6;8 appear as the principal graphs. A group-like object consisting of the principal graphs and certain data coming from .M 0 \Mk  N 0 \Mk /1 is formulated in some ways such as a paragroup (A. Ocneanu), kD1 a -lattice (S. Popa) or a planar algebra (Jones), which gives a complete invariant for “amenable” subfactors. In 1985, Jones discovered an invariant for oriented knots and links, called the Jones polynomial, via his subfactor theory. Kosaki [59] extended the Jones index to the type III setting (more precisely, to conditional expectations EW M ! N ). From the point of view of quantum field theory (the Doplicher–Haag–Roberts theory), Longo [64, 65] developed the type III subfactor theory by the sector approach (instead of the bimodule approach in the type II1 subfactor theory), where the statistical dimension of a superselection sector turns up as the square root of the index of a subfactor. Since then, the strong connection between Jones’ subfactor theory and conformal/quantum field theory has been developed; the interested reader can refer to the book [22] which includes a complete list of references on subfactor theory up to 1998.

1.10 Developments since the 1980s

11

(2) Voiculescu’s free probability theory. Around 1985, Voiculescu [106] initiated a new non-commutative probability theory, called free probability theory, related to free products of operator algebras. He introduced the notions of free independence in the setting of a non-commutative probability space .A ; '/ of a unital *-algebra with a linear functional 'W A ! C, '.1/ D 1, which is modeled after the relation of generators of free group factors .L.Fn /;  /. The semicircle (or Wigner’s) law 1 p 4 2

x2

supported on Œ 2; 2

turned up as the limit distribution in the free analogue of the classical central limit theorem, in place of the Gaussian law in the classical case. Through the late 1980s, free probability theory grew to quite a rich theory with several free analogues of classical notions, such as free additive and multiplicative convolutions and R- and S -transforms (analogues of Fourier transform); see [108]. The combinatorial approach in free probability theory was developed by Speicher [87] (see [67] for more details), where the concept of non-crossing partitions plays a key role. Along with Wigner’s old idea, Voiculescu [107] discovered that random matrices play a crucial role as asymptotic models of free independence. A continuous interpolation L.Fr / (1 < r  1), called the interpolated free group factors, was constructed based on random matrices (Dykema, Rădulescu), and it gave a deep insight into the famous non-isomorphism problem of whether L.Fn / 6Š L.Fm / if n ¤ m; although this is still unsettled. The most attractive topic in free probability theory is free entropy (also free entropy dimension) and free Fisher information, studied by Voiculescu in a series of papers (1993–1999). He introduced two kinds of free entropies .a1 ; : : : ; an / and  .a1 ; : : : ; an / for ai D ai (1  i  n) in a tracial W  -probability space .M;  /. The unicity  D  is a big open problem. Some old problems related to free group factors, for instance non-existence of Cartan subalgebras and primeness (a II1 factor is said to be prime if it cannot be isomorphic to the tensor product of two type II1 factors), were solved by using a free entropy technique. More about random matrices and free entropy are found in [45]. (3) Popa’s rigidity theory. A more recent major subject is the rigidity theory of von Neumann algebras that has been growing from Popa’s work since around 2006. Familiar constructions of type II1 factors are the group von Neumann algebra L.€/ generated by the left regular representation of a countable icc (infinite conjugacy classes) group € on `2 .€/ and the group measure space construction L1 .X / o € (the crossed product) of a free and ergodic measure-preserving action € y .X; / of a countable group € on a (standard) probability space .X; /. It is well known that when € is amenable, all II1 factors arising from the above constructions are isomorphic to the hyperfinite II1 factor R (Connes [16]) and all ergodic actions of € are orbit equivalent (Connes–Feldman–Weiss). It may be said that the arising von Neumann algebra forgets everything except being amenable. The rigidity theory of von Neumann algebra is concerned with phenomena in the opposite direction, the situation that group and action are remembered by their von

12

1 Von Neumann algebras – An overview

Neumann algebras. In the group measure space algebra setting, two free and ergodic measure-preserving actions € y .X; / and ƒ y .Y; / are said to be (a) conjugate if there exist an isomorphism ˛W .X; / ! .Y; / and a group isomorphism ıW € ! ƒ such that ˛.g
x/ D ı.g/
˛.x/ for all g 2 G and a.e. x 2 X; (b) orbit equivalent if there exists an isomorphism ˛W .X; / ! .Y; / such that ˛.€
x/ D ƒ
˛.x/ for a.e. x 2 X; (c) von Neumann equivalent if L1 .X / o € Š L1 .Y / o ƒ. The implications (a) H) (b) H) (c) generally hold. The term “rigid” refers to when some of these implications can be reversed for all € y .X; / and ƒ y .Y; / in some respective classes of actions. An action € y .X; / is “superrigid” when the same holds for an arbitrary action ƒ y .Y; /. In the group von Neumann algebra setting, the same terms are also referred to when the implication L.€/ Š L.ƒ/ H) € Š ƒ takes place in a similar fashion. A typical rigidity problem for group von Neumann algebras is the non-isomorphism problem of L.Fn / mentioned in (2) above. Another example is Connes’ famous conjecture of 1982 – whether L.€/ Š L.ƒ/ H) € Š ƒ for icc property (T) groups €; ƒ – which remains wide open. Here, note that property (T) groups often appear in various rigidity situations. The first examples of superrigid group von Neumann algebras were discovered by A. Ioana, S. Popa and S. Vaes (2013). Many examples of actions and groups satisfying the rigidity/superrigidity property have been discovered so far, for which the interested reader can consult, for example, survey articles [75, 47]. —————————————— We end the overview with a list of a few standard textbooks on von Neumann algebras.  R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, I, II, III, IV, Academic Press, 1983, 1986, 1991, 1992.  G. T. Pedersen, C  -Algebras and Their Automorphism Groups, Academic Press, 1979.  S. Sakai, C  -Algebras and W  -Algebras, Springer, 1971.  S. Strătilă and L. Zsidó, Lectures on Von Neumann Algebras, Abacus Press, 1979.  M. Takesaki, Theory of Operator Algebras, I, II, III, Springer, 1979, 2003, 2003.

2 Tomita–Takesaki modular theory

After the appearance of Tomita’s unpublished paper on the subject in the 1960s, a readable monograph was published by Takesaki [91] and a simplified proof was given by van Daele [103]. Then a proof minimizing the use of unbounded operators was also presented in [78]. In the first part of this chapter, following the expositions in [94, 104], we present a proof of Tomita’s theorem in Tomita–Takesaki theory of von Neumann algebras, in the setting of a cyclic and separating vector. In the second part we present Takesaki’s theorem on the KMS condition of the modular automorphism group, an important ingredient of Tomita–Takesaki theory in addition to Tomita’s theorem.

2.1 Tomita’s fundamental theorem Let M be a von Neumann algebra. We assume that there is a faithful ! 2 MC , which is equivalent to M being -finite, i.e., mutually orthogonal projections in M are at most countable. By making the GNS cyclic representation .H! ; ! ; / of M with respect to !, we have a faithful representation ! W M ! B.H! / with a cyclic and separating vector  2 H! for ! .M / such that !.x/ D h; ! .x/i;

x 2 M:

Thus, by identifying M with ! .M /, we may assume that M itself is a von Neumann algebra on H with a cyclic and separating vector  for M . Here,  is cyclic for M if M  D H , and  is separating for M if x 2 M , x D 0 H) x D 0, equivalently M 0  D H . When M is not -finite, the construction of the modular theory is essentially similar to below (but technically more complicated) with a faithful semifinite normal weight on M based on the left Hilbert algebra theory. We begin with the two conjugate-linear operators S0 and F0 with the dense domains D.S0 / D M  and D.F0 / D M 0  defined by S0 x WD x  ;

x 2 M;

F0 x 0  WD x 0 ;

x0 2 M 0:

For any x 2 M and x 0 2 M 0 note that hx 0 ; S0 xi D hx 0 ; x  i D hx; x 0 i D hx; F0 x 0 i;

14

2 Tomita–Takesaki modular theory

which implies that S0 and F0 are closable, F0  S0 and S0  F0 . So we set S WD S0 and F WD S  D S0 and take the polar decomposition of S as S D J1=2 ;

 WD S  S D F S:

Since the ranges of S and S  are dense, it follows that J is a conjugate-linear unitary and  is a non-singular positive self-adjoint operator. Lemma 2.1. We have (i) J D J  and J 2 D 1, (ii)  D F S and  (iii) S D J1=2 D  (iv) 

1

1

D SF , 1=2

J and F D J

D JJ and Ji t D 

it

1=2

D 1=2 J ,

J for all t 2 R,

(v) J  D  and  D . Proof. Since S 2  S02 D 1M  , it is easy to see that S D S 1 , which implies that S D J1=2 D  1=2 J  D J  J 1=2 J  . Therefore, J D J  and 1=2 D J 1=2 J  so that (i) and (iv) hold. Since F D 1=2 J , (ii) and (iii) hold. Moreover, since S  D F  D , (v) holds as well.  The next theorem is Tomita’s fundamental theorem. Theorem 2.2 (Tomita). With J and  given above, we have JMJ D M 0 ; i t M

it

D M;

(2.1) t 2 R:

(2.2)

Definition 2.3. The operator  is called the modular operator with respect to  (or !/, and J is called the modular conjugation with respect to  (or !/. By (2.2) the one-parameter automorphism group  t D  t! of M is defined by  t! .x/ WD i t x

it

;

x 2 M; t 2 R;

which is called the modular automorphism group with respect to  (or !/. Example 2.4. Consider the simple case where ! D  is a faithful normal finite trace of M , so M is a finite von Neumann algebra (see Section 1.4). Since kxk2 D  .x  x/ D  .xx  / D kx  k2 for all x 2 M , S is a conjugate-linear unitary, which means that S D J and  D 1. Hence (2.2) trivially holds. For every x; y; y1 2 M note that J xJyy1  D J xy1 y   D yy1 x   D yJ xy1  D yJ xJy1 ; so that J xJy D yJ xJ . Hence JMJ  M 0 . Moreover, for every x 2 M and x 0 2 M 0 note that hx; J x 0 i D hx 0 ; J xi D hx 0 ; x  i D hx; x 0 i; so that J x 0  D x 0 . Hence, similarly to the above, JM 0 J  M , so M 0  JMJ . Therefore, (2.1) holds. In this way, Tomita’s theorem in this case is quite easy.

15

2.1 Tomita’s fundamental theorem

The proof of Theorem 2.2 taken from [104] needs several lemmas as below. 1. For every x 0 2 M 0 there exists a

Lemma 2.5. Let  2 C with jj D 1 and  ¤ (unique) x 2 M such that x 0  D . C /x; Hence . C /

1

i.e.;

. C /

1 0

x  D x:

(2.3)

M 0   M . 1

Proof. We may assume that 0  x 0  1. Let ˛ WD .1 C / ; x 2 M for x 2 Msa (the set of self-adjoint x 2 M ), by .y/ WD hx 0 ; y i;

x .y/

; then ˛ C ˛ D 1. Define

WD !.˛xy C ˛yx/;

y 2 M:

Clearly, 0   ! and x 2 Msa (the set of self-adjoint ' 2 M ) since x .y/ D !.˛y  x C ˛xy  / D x .y  /. Let us first prove that there exists an x 2 Msa such that D x . Since x 2 Msa 7! x 2 Msa is  .M; M /– .M ; M /-continuous, it follows that V WD ¹ x W x 2 Msa ; kxk  1º is  .M ; M /-compact and convex in Msa . Now assume, on the contrary, that Then by the Hahn–Banach separation theorem, there is a y0 2 Msa such that .y0 / > sup¹

x .y0 /

62 V .

W x 2 Msa ; kxk  1º:

Take the Jordan decomposition y0 WD y0C y0 and let x0 WD s.y0C / s.y0 /, where s.y0C / is the support projection of y0C . Since kx0 k  1 and x0 y0 D y0 x0 D y0C C y0 , we have .y0 / 

.y0C C y0 /  !..˛ C ˛/x0 y0 /

D !.˛x0 y0 C ˛y0 x0 / D a contradiction. Hence an x 2 Msa with

D

x

x0 .y0 /


0 and 1=2  s  1=2, 2e j jR 2e jjR s s k E k kxk k E k  n2 kxk ! 0 n n e R e R e R e R as R ! 1, and the residue of f at z D 0 is kf .s ˙ iR/k 

1 En xEn :  Use the residue theorem and let R ! 1 to obtain Z 1  Z 1  1 1 1 En xEn D f it C i dt f it  2 i 2 1 1 lim zf .z/ D

z!0

 1 i dt : 2

Since 1  1 2i e i.i t˙ 2 / 1 1 f it ˙ D  .i t˙ 2 / En xEn i t˙ 2 1 1 2 e i .i t˙ 2 / e i .i t˙ 2 / ˙2i e  t e ˙i=2 1=2 D  En  i t xi t En ˙1=2 ; e t C e  t

18

2 Tomita–Takesaki modular theory

one has iEn xEn D 1=2 En xzEn  where

1=2

1=2

C 

En xzEn 1=2 ;

(2.6)

1

e  t e i=2 i t  x i t dt: t C e t 1 e On the other hand, since (2.5) holds for all 1 ; 2 2 En H , one has Z

xz WD

En xEn D 1=2 En yEn 

1=2

1=2

C 

En yEn 1=2 :

(2.7)

It follows from (2.6) and (2.7) that 0 D 1=2 En .y

i xz/En 

1=2

C 

1=2

i xz/En 1=2 :

En .y

(2.8)

Now consider the operators a, b acting on B.H / defined by ax WD .En /x and bx WD x. 1 En / for x 2 B.H /. Let A be the unital commutative Banach subalgebra of B.B.H // generated by a, b. Then (2.8) means that abv D v, where v WD En .y i xz/En 2 B.H /. Assume that v ¤ 0; then  2  .ab/, the spectrum of ab in A. In view of the Gelfand transform (see [82, Thm. 11.9]), we can easily see that  .ab/   .a/ .b/. Since  .a/;  .b/  Œ0; 1/ obviously, we have  2  .ab/  Œ0; 1/, contradicting jj D 1 with  ¤ 1. Therefore, v D 0, i.e., En .y i xz/En D 0. Letting n ! 1 gives y D i xz.  Lemma 2.8. Let  D e i with  <  < . Then Z 1 e  t e i=2 1=2 1  . C / D i  t C e t 1 e

it

dt;

(2.9)

where the integral converges in the strong operator topology. Proof. Apply the proof of Lemma 2.7 to f .z/ WD Since limz!0 zf .z/ D

e i z  sin z

z

En :

1 E ,  n

one obtains Z 1  1 1 1 En D f it C i dt  2 i 2 1

Z

1

 1 i dt ; 2

 f it

1

which is specified as Z

1

iEn D 1

Since 1=2 C 

1=2

D

e  t e i=2 1=2 . C  e t C e  t

1=2

/

En dt:

1=2

. C /, one has Z 1 e  t e i=2 1=2 . C / 1 En D i  t C e t 1 e

Letting n ! 1 gives (2.9).

it

it

 dt En : 

19

2.1 Tomita’s fundamental theorem

Proof of Theorem 2.2. Let x 0 2 M 0 and  D e i with  <  < . By Lemma 2.5 there exists an x 2 M satisfying (2.3), so by Lemma 2.6, (2.4) holds for all 1 ; 2 2 D.1=2 / \ D. 1=2 /. Then by Lemma 2.7 applied to x 0 , J x  J (in place of x, y), it follows that Z 1 e  t e i=2  i t x 0 i t dt: (2.10) J xJ D i t C e t 1 e Therefore, for every y 0 2 M 0 we have by (2.3) and (2.10), y 0 J1=2 . C /

1 0

x  D y 0 J1=2 x D y 0 x   D x  y 0  Z 1 e t J i t x 0 i t Jy 0  dt: D i e i=2 t C e t e 1

On the other hand, by Lemma 2.8 we have 0

y J

1=2

. C /

1 0

xD

ie

i=2

Z

1 1

Combining the two identities above gives Z 1 e t .J i t x 0 i t Jy 0  t C e t 1 e Since

y 0 J

e t e t C e

it

t

y 0 J

x 0 / dt D 0;

it

x 0  dt:

 <  < :

1

e zt .J i t x 0 i t Jy 0  y 0 J i t x 0 / dt t C e t 1 e is analytic in  < Re z <  as easily verified, it follows from analytic continuation that Z 1 e i st .J i t x 0 i t Jy 0  y 0 J i t x 0 / dt D 0; s 2 R: t C e t 1 e Z

z7 !

The injectivity of the Fourier transform yields J

it

x 0 i t Jy 0  D y 0 J

it

x 0 ;

t 2 R:

(2.11)

Now, for every y10 ; y20 2 M 0 , using (2.11) twice we have y10 J

it

x 0 i t Jy20  D y10 y20 J

it

x 0  D J

it

x 0 i t Jy10 y20 ;

which implies that y10 J i t x 0 i t J D J i t x 0 i t Jy10 so that J i t x 0 i t J 2 M 00 D M . Letting t D 0 gives J x 0 J 2 M . Therefore, JM 0 J  M and so M 0  JMJ . Furthermore, for every x; y 2 M and x 0 2 M 0 we find that hx 0 ; yJ xi D hy  ; x 0 J xi D hy  ; J.J x 0 J /xi D hy  ; JS x  J x 0 J i

(since J x 0 J 2 M )

D hy  ; FJ x  J x 0 i

(by Lemma 2.1(iii))

D hJ x  J x 0 ; Sy  i D hx 0 ; J xJy i:

20

2 Tomita–Takesaki modular theory

Therefore, J xJy  D yJ x: For every y1 ; y2 2 M we hence have y1 J xJy2  D y1 y2 J x D J xJy1 y2 ; which implies that J xJ 2 M 0 , so JMJ  M 0 . Thus, (2.1) follows. Moreover, since J it M 0 it J  M as proved above, we have  i t M 0 i t  JMJ D M 0 and hence M 0  it M 0  i t as well. Thus, i t M 0  i t D M 0 and (2.2) follows.  Remark 2.9. From (2.1) note that JM  D JMJ  D M 0  and for every x 0 2 M 0 , JS0 J x 0  D JS0 .J x 0 J / D JJ x 0 J  D x 0  D F0 x 0 ; so that F0 D JS0 J and hence F0 D JSJ D F D S0 . In this way, we have complete symmetry between S0 and F0 . If we start from .M 0 ; ; F0 / in place of .M; ; S0 /, then the modular operator is  1 and the modular conjugation is the same J . A direct proof of F0 D S0 without the use of (2.1) can be found in, e.g., [8, Prop. 2.5.9]. Proposition 2.10. If x 2 M \ M 0 (the center of M ), then J xJ D x  ;

 t .x/ D i t x

it

D x;

t 2 R:

Proof. Assume that x 2 M \ M 0 ; then S x D x   and F x   D x. Hence by Lemma 2.1(ii), x D F S x D x. Hence, since  t .x/ D i t x D x, one has  t .x/ D x. Moreover, since J xJ  D J1=2 x D S x D x  , one has J xJ D x  . 

2.2 KMS condition In this section we present Takesaki’s theorem on the KMS condition of the modular automorphism group. Let ˛ t (t 2 R) be a one-parameter weakly continuous automorphism group of M . It is worth noting that ˛ t is automatically strongly* continuous. Indeed, for any x 2 M and  2 H , k.˛ t .x/

x/k2 D h; .˛ t .x/

x/ .˛ t .x/



D h; ˛ t .x x/i !0

x/i

h˛ t .x/; xi

hx; ˛ t .x/i C h; x  xi

.t ! 0/;

and also k.˛ t .x/ x  /k ! 0 (t ! 0). For ˇ 2 R with ˇ ¤ 0, if ˇ < 0, set Dˇ WD ¹z 2 C W 0 < Im z < and if ˇ > 0, Dˇ WD ¹z 2 C W

ˇº;

Dˇ WD ¹z 2 C W 0  Im z 

ˇ < Im z < 0º and Dˇ is similar.

ˇº;

21

2.2 KMS condition

Definition 2.11. A functional ' 2 MC is said to satisfy the KMS (Kubo–Martin– Schwinger) condition with respect to ˛ t at ˇ, or the .˛ t ; ˇ/-KMS condition, if for every x; y 2 M there is a bounded continuous function fx;y .z/ on Dˇ , analytic in Dˇ , such that fx;y .t / D '.˛ t .x/y/; fx;y .t iˇ/ D '.y˛ t .x//; t 2 R: This condition (proposed by Haag, Hugenholtz and Winnink) serves as a mathematical formulation of equilibrium states in quantum statistical mechanics, which is also defined and more useful in C  -algebraic dynamical systems (see [9]). To illustrate this, given a Hamiltonian H 2 B.H /sa in a finite-dimensional H , consider the Gibbs state '.x/ D Tr e ˇH x= Tr e ˇH and the corresponding dynamics ˛ t .x/ D e i tH xe i tH (t 2 R). For any x; y 2 B.H / the entire function fx;y .z/ WD Tr.e ˇH e izH xe izH y/= Tr e ˇH satisfies fx;y .t / D Tr.e fx;y .t

ˇH i tH

e

iˇ/ D Tr.e i tH xe

xe

i tH

e

i tH

y/= Tr e

ˇH

D '.˛ t .x/y/;

ˇH

y/= Tr e

ˇH

D '.y˛ t .x//;

t 2 R;

so that ' satisfies the .˛ t ; ˇ/-KMS condition. Moreover, the Gibbs state ' is a unique state satisfying the .˛ t ; ˇ/-KMS condition (an exercise). The following lemma is another justification for the KMS condition to describe equilibrium states. Lemma 2.12. If ' 2 MC satisfies the .˛ t ; ˇ/-KMS condition, then ' ı ˛ t D ' for all t 2 R. Proof. Let x 2 M . From the KMS condition applied to x and y D 1, the function f .t / D '.˛ t .x// can extend to a bounded continuous function f .z/ on Dˇ , analytic in Dˇ , such that f .t / D f .t iˇ/, t 2 R. From the Schwarz reflection principle, f can further extend to an entire function with period iˇ, which is bounded. Hence the Liouville theorem says that f is a constant function, so f .t /  f .0/, i.e., '.˛ t .x// D '.x/, t 2 R.  An element x 2 M is said to be ˛ t -analytic (entire) if there is an M -valued entire function x.z/ in the strong* topology such that x.t / D ˛ t .x/ for all t 2 R. In this case, we write ˛z .x/ for x.z/, z 2 C. Lemma 2.13. The set of ˛ t -analytic elements is an ˛ t -invariant *-subalgebra of M and it is strongly* dense in M . Moreover, if x 2 M is ˛ t -analytic, then ˛ .x/ is also ˛ t -analytic for every  2 C. Proof. Write M.˛/ for the set of ˛ t -analytic elements x 2 M . If x; y 2 M.˛/ with analytic extensions ˛z .x/ and ˛z .y/, then we have xy; x  ; ˛ t .x/ 2 M.˛/ with ˛z .xy/ D ˛z .x/˛z .y/, ˛z .x  / D ˛z .x/ and ˛z .˛ t .x// D ˛zCt .x/. Hence M.˛/ is an ˛ t invariant *-subalgebra of M . Since ˛ t .˛ .x// D ˛ tC .x/, we also have ˛ .x/ 2 M.˛/

22

2 Tomita–Takesaki modular theory

with ˛z .˛ .x// D ˛zC .x/. To prove the strong* denseness of M.˛/, for every x 2 M and n 2 N define r Z 1 n 2 e nt ˛ t .x/ dt: xn WD  1 It is easy to verify that xn ! x strongly* as n ! 1. Moreover, define r Z 1 n 2 xn .z/ WD e n.t z/ ˛ t .x/ dt; z 2 C:  1 It then follows that xn .z/ is entirely analytic in the strong* topology and xn .s/ D ˛s .xn / for all s 2 R.  Takesaki’s theorem in [91] is the following. Theorem 2.14 (Takesaki). In the same situation as in Theorem 2.2, ! satisfies the . t ; 1/KMS condition. Furthermore,  t D  t! is uniquely determined as a weakly continuous one-parameter automorphism group of M for which ! satisfies the KMS condition at ˇ D 1. Proof. For any x; y 2 M , since x; y  2 D.1=2 /, it follows from Theorem A.7 of Appendix A.1 that  iz=2 x and  iz=2 y  are bounded continuous on 0  Im z  1 and analytic in 0 < Im z < 1 in the norm on H . Therefore, f .z/ WD hi z=2 x; 

iz=2

y i

is bounded continuous on 0  Im z  1 and analytic in 0 < Im z < 1. For every t 2 R compute f .t / D hi t =2 x; 

i t =2

y i D hi t x

it

; y i (by Lemma 2.1(v))



D h t .x/; y i D !. t .x /y/; f .t C i / D h1=2 i t =2 x; 

i t =2

1=2 y i D h1=2 i t x

it

; 1=2 y i

D hJ1=2 y ; J1=2  t .x/i D hy  ;  t .x  /i D !.y t .x  //: Hence ! satisfies the . t ; 1/-KMS condition. To prove the uniqueness assertion, let ˛ t be a weakly (hence strongly) continuous oneparameter automorphism group of M such that ! satisfies the .˛ t ; 1/-KMS condition. By Lemma 2.12, ! is ˛ t -invariant, so one can define U t x WD ˛ t .x/ for x 2 M to obtain a strongly continuous one-parameter unitary group U t on H such that ˛ t .x/ D U t xU t for all x 2 M and t 2 R. Let x 2 M be an ˛ t -analytic element with the analytic extension ˛z .x/, and y 2 M be a  t -analytic element with the analytic extension z .y/. Define f .z/ WD !.˛z .x/z .y//, which is an entire function. For any s 2 R let yz WD s .y/. Since the entire function fx;yz .z/ WD !.˛z .x/y/ z satisfies fx;yz .t / WD !.˛ t .x/y/; z

fx;yz .t C i / WD !.y˛ z t .x//;

t 2 R;

23

2.2 KMS condition

from the .˛; 1/-KMS condition of !, we have !.˛z .x/y/ z D !.y˛ z z i .x// for all z 2 C. Therefore, !.˛ t .x/s .y// D !.s .y/˛ t i .x//; t; s 2 R: (2.12) Similarly, for any s 2 R let xz WD ˛s i .x/. Since the entire function fy;zx .z/ WD !.z .y/z x/ satisfies fy;zx .t / D !. t .y/z x /; fy;zx .t C i / D !.z x  t .y//; t 2 R; from the .; 1/-KMS condition of !, we have !.z .y/z x / D !.z x z i .y// for all z 2 C. Therefore, !. t .y/˛s i .x// D !.˛s i .x/ t i .y//; t; s 2 R: (2.13) Combining (2.12) and (2.13) gives f .t / D !.˛ t .x/ t .y// D !. t .y/˛ t i .x// D !.˛ t i .x/ t i .y// D f .t

i /;

t 2 R;

that is, f has period i. From this and the three-lines theorem it follows that f is bounded on 0  Im z  1 and hence bounded on the whole of C. By the Liouville theorem, f  f .0/ so that !.˛ t .x/ t .y// D !.xy/ for all t 2 R. By Lemma 2.13 (applied to ˛ t and  t ), this equality can extend to all x; y 2 M , so we have hx  ; U t i t y i D h˛ t .x  /;  t .y/i D !.˛ t .x/ t .y// D !.xy/ D hx  ; y i for all x; y 2 M . Therefore, U t i t D 1, i.e., U t D i t , which implies that ˛ t D  t .  Definition 2.15. The centralizer of ! is defined as M! WD ¹x 2 M W !.xy/ D !.yx/; y 2 M º: It is obvious that M! is a von Neumann subalgebra of M including the center M \ M 0 . Proposition 2.16. The centralizer M! coincides with the fixed-point algebra of  t! , i.e., M! D ¹x 2 M W  t! .x/ D x; t 2 Rº: Proof. Let x 2 M . For every  t -analytic element y 2 M , the entire function fy;x .z/ WD !.z .y/x/ satisfies fy;x .t / D !. t .y/x/ D !.y t .x//;

fy;x .t C i / D !.x t .y//;

t 2 R;

from ! ı  t D ! (by Lemma 2.12) and the .; 1/-KMS condition of !. If  t .x/ D x for all t 2 R, then fy;x .z/  !.yx/ so that !.xy/ D !.yx/. By Lemma 2.13, x 2 M! follows. Conversely, if x 2 M! , then fy;x .t / D fy;x .t C i / for all t 2 R so that fy;x  fy;x .0/ as in the proof of Theorem 2.14. Therefore, hy  ;  t .x/i D !.y t .x// D !.yx/ D hy  ; xi; which implies by Lemma 2.13 that  t .x/ D x for all t 2 R.



24

2 Tomita–Takesaki modular theory

By Proposition 2.16 we see that  t .x/ D x for all x 2 M \ M 0 and t 2 R, which was shown in Proposition 2.10. Also, it follows that ! is a trace if and only if  t D id for all t 2 R.

3 Standard form

The theory of the standard form of von Neumann algebras was developed, independently, by Araki [3] and Connes [15] in the case of -finite von Neumann algebras, and by Haagerup [26, 28] in the general case. In this chapter we present a concise exposition of the theory, mainly following [26, 28] with certain simplifications from [8, §2.5.4].

3.1 Definition and basic properties Let M be a -finite von Neumann algebra, thus represented on a Hilbert space H with a cyclic and separating vector , for which we have the modular operator  and the modular conjugation J , as explained in Chapter 2. Let j W M ! M 0 be the conjugate-linear *-isomorphism defined by j.x/ WD J xJ , x 2 M . Definition 3.1. The natural positive cone P D P \ in H associated with .M; / is defined by P WD ¹xj.x/ W x 2 M º D ¹xJ x W x 2 M º: (3.1) In addition, define P ] WD MC ;

P [ WD MC0 :

Theorem 3.2. We have (i) P D 1=4 MC  D 1=4 P ] D  closed cone,

1=4 M 0  C

D

1=4 P [ ;

in particular, P is a

(ii) J  D  for all  2 P, (iii) it P D P for all t 2 R, (iv) xj.x/P  P for all x 2 M , (v) if f is a positive definite function on R, then f .log /P  P, (vi) P is self-dual, i.e., P D ¹ 2 H W h; i  0;  2 Pº:

26

3 Standard form

Proof. (i) Let  t be the modular automorphism group associated with .M; /. Write M. / for the set of  -analytic elements x 2 M . For every x 2 M. / let y WD i=4 .x/; then y 2 M. / and Tx D  i=4 .y/. Note that, by Lemma 2.13 and Theorem A.7, for any  2 C,  .y/ 2 z2C D.z / and z  .y/ D  izC .y/. We hence have xj.x/ D 

i=4 .y/J  i=4 .y/ 1=2

D

i=4 .y/J

D

i=4 .y/ i=4 .y

D

i=4 .y/J  i=2Ci=4 .y/

i=4 .y/ D  

/ D 

 i=4 .y/i=4 .y/ 

i=4 .yy



/ D 1=4 yy  :

(3.2)

By Lemma 2.13 and the Kaplansky density theorem (see Section 1.2) this implies that P  1=4 MC   1=4 P ] : Conversely, for every  2 P ] , from Lemma 2.13 and the Kaplansky density theorem again, there is a sequence ¹yn º in M. / such that yn yn  ! . Then Syn yn  D yn yn  ! , so  2 D.S / D D.1=2 /  D.1=4 / and S  D . Hence 1=2 yn yn  D Jyn yn  ! J  D 1=2 ; which implies that k1=4 .yn yn 

/k2 D hyn yn 

; 1=2 .yn yn 

/i ! 0:

Letting xn WD  i=4 .yn / we have 1=4 yn yn  D xn j.xn / similarly to (3.2), so 1=4  2 P. Therefore 1=4 P ]  P, which implies P D 1=4 MC  D 1=4 P ] . When we replace .M; / with .M 0 ; /, the modular conjugation is the same J and the modular operator is  1 (see Remark 2.9). Moreover, the natural positive cone is the same P due to (2.1). Therefore, P D  1=4 MC0  D  1=4 P [ as well. (ii) follows from J xj.x/ D Jj.x/x D xj.x/;

x 2 M:

(iii) follows from (i) and i t 1=4 MC  D 1=4 i t MC 

it

 D 1=4 MC :

(iv) For every x; y 2 M , xj.x/yj.y/ D xyj.x/j.y/ D xyj.xy/ 2 P: R1 (v) Bochner’s theorem says that f has the form f .t / D 1 Re i st d.s/ with a finite 1 positive Borel measure  on R. We hence write f .log / D 1 i s d.s/, which implies that f .log /P  P by (iii). Before proving (vi) we give the following lemma.

3.1 Definition and basic properties

27

Lemma 3.3. The cones P ] and P [ are mutually dual in H , i.e., ® ¯ ® ¯ P [ D  2 H W h; i  0;  2 P ] ; P ] D  2 H W h; i  0;  2 P [ : Proof. Write P ]_ and P [_ for the dual cones of P ] and P [ , respectively. For every x 2 MC and x 0 2 MC0 , hx; x 0 i D h; x 1=2 x 0 x 1=2 i  0; which implies that P [  P ]_ and P ]  P [_ . To prove the converse, let  2 P ]_ and define an operator T0 W M  ! H by T0 x WD x for x 2 M . Since hx; T0 xi D hx; xi D hx  x; i  0; it follows that T0 is a densely defined positive symmetric operator. So one has the Friedrichs extension (the largest positive self-adjoint extension) T of T0 (see [79, §124]). For every unitary u 2 M , T0 ux D ux D uT0 x for x 2 M , which means that u T0 u D T0 . From the construction of the Friedrichs extension, it follows that u T u D T for all unitaries u 2 M , so T is affiliated with M 0 (see the first paragraph of Section 4.1). Taking the spectral projection en0 of T corresponding to Œ0; n, one has xn0 WD T en0 2 MC0 and xn0  ! T  D  so that  2 P [ . Hence P [ D P ]_ , and P ] D P [_ is similar.  Proof of Theorem 3.2(vi). Since h1=4 x; 

1=4 0

x i D hx; x 0 i  0;

x 2 MC ; x 0 2 MC0 ;

it follows from (i) that h; i  0 for all ;  2 P. Conversely, assume that  2 H 2 2 and h; i  0 for all  2 P. For each n 2 N let fn .t / WD e t =2n (t 2 R). Since 0 < fn .log t / % 1 (n % 1) for every t > 0, we have n WD fn .log / ! . For any ˛ 2 R, note that Z 1 Z 1  .log t /2  t 2˛ dkE.t /n k2 D dkE.t /k2 t 2˛ exp 2 n 0 0 Z 1  .log t /2  D exp 2˛ log t dkE.t /k2 < 1; 2 n 0 T where E.
/ is the spectral measure of . Therefore, by Theorem A.7, n 2 z2C D.z /. Furthermore, it is well known that fn is a positive definite function on R, so by (v) and by assumption, h; n i D hfn .log /; i  0 for all  2 P. Therefore, for every x 2 MC , since 1=4 x 2 P by (i), we have hx; 1=4 n i D h1=4 x; n i  0. By Lemma 3.3 this implies that 1=4 n 2 P [ . So n 2  1=4 P [  P by (i). Letting n ! 1 gives  2 P.  Remark 3.4. The cones P ] and P [ were first introduced by Takesaki [91], where Lemma 3.3 was proved. In [3] Araki introduced a one-parameter family of cones V˛ WD ˛ MC  for ˛ 2 Œ0; 1=2. Note that V0 D P ] , V1=4 D P \ and V1=2 D P [ . In [3] it was shown, 1

among other properties, that the dual of V˛ is V2

˛

.

28

3 Standard form

Summing up the discussions so far in this section and in Section 2.1, we conclude that any (-finite) von Neumann algebra is faithfully represented on a Hilbert space H with a conjugate-linear involution J and a self-dual cone P such that (a) JMJ D M 0 (Theorem 2.2), (b) J xJ D x  for all x 2 M \ M 0 (Proposition 2.10), (c) J  D  for all  2 P, (d) xj.x/P  P for all x 2 M , where j.x/ WD J xJ . Definition 3.5. A quadruple .M; H ; J; P/ such as above, satisfying conditions (a)–(d), is called a standard form of a von Neumann algebra M . This is the abstract (or axiomatic) definition, and we have shown the existence of a standard form for any  -finite von Neumann algebra. Examples 3.6. (1) Let .X; X ; / be a -finite (or more generally, localizable) measure space. The commutative von Neumann algebra M D L1 .X; / is faithfully represented on the Hilbert space L2 .X; / as multiplication operators .f / WD f  for f 2 L1 .X; / and  2 L2 .X; /. The standard form of M is .L1 .X; /; L2 .X; /; J  D ; L2 .X; /C /; where L2 .X; /C is the cone of non-negative functions  2 L2 .X; /. (2) Let M D B.H /, a factor of type I. Let C2 .H / be the space of Hilbert–Schmidt operators (i.e., a 2 B.H / with Tr a a < 1), which is a Hilbert space with the Hilbert– Schmidt inner product ha; bi WD Tr a b for a; b 2 C2 .H /. Then M D B.H / is faithfully represented on C2 .H / as left multiplication operators .x/a WD xa for x 2 B.H / and a 2 C2 .H /. The standard form of M is
B.H /; C2 .H /; J D  ; C2 .H /C ; where J D  is the adjoint operation and C2 .H /C is the cone of positive a 2 C2 .H /. In this case, note that for any x 2 B.H /, J xJ is the right multiplication of x  on C2 .H / and xj.x/C2 .H /C D xC2 .H /C x   C2 .H /C . The following gives geometric properties of the cone P. Proposition 3.7. Let .M; H ; J; P/ be a standard form. Then (1) P is a pointed cone, i.e., P \ . P/ D ¹0º, (2) if  2 H and J  D , then  has a unique decomposition  D 1 1 ; 2 2 P and 1 ? 2 , (3) H is linearly spanned by P.

2 with

29

3.1 Definition and basic properties

Proof. (1) If  2 P \ . P/, then the self-duality of P implies that h; i  0, hence  D 0. (2) Assume that  2 H and J  D . Since P is a closed convex set in H , there is a unique 1 2 P such that k1

k D inf¹k

k W  2 Pº:

Set 2 WD 1 . For any  2 P and  > 0, since 1 C  2 P, one has k1 k1 C  k2 , i.e., k2 k2  k2 C k2 so that 2 Reh2 ; i C 2 kk2  0;

k2 

 > 0:

Therefore, Reh2 ; i  0. Since J 2 D J 1 J  D 2 , one has h2 ; i D hJ 2 ; J i D h; 2 i, so that h2 ; i 2 R and h2 ; i  0 for all  2 P. Hence 2 2 P. Next show that 1 ? 2 . For any  2 .0; 1/, since .1 /1 2 P, one has k1 k2  k.1 /1 k2 , i.e., k2 k2  k2 1 k2 so that 2h2 ; 1 i C 2 k1 k2  0;

 2 .0; 1/:

Therefore, h2 ; 1 i  0 and so 1 ? 2 since h2 ; 1 i  0 for 1 ; 2 2 P. To show the uniqueness of the decomposition, besides  D 1 2 , let  D 1 2 with 1 ; 2 2 P and 1 ? 2 . Then 1 1 D 2 2 and so k1

1 k2 D h1

1 ; 2

2 i D

h1 ; 2 i

h1 ; 2 i  0;

implying 1 D 1 and 2 D 2 . (3) For any  2 H , let  WD . C J /=2 and  0 WD . J /=2i ; then J  D ,  J  D  0 and  D  C i  0 . By (2),  D .1 2 / C i.3 4 / with i 2 P. 0

The next proposition gives the description of the standard form of a reduced von Neumann algebra eM e, where e is a projection in M . Proposition 3.8. Let .M; H ; J; P/ be a standard form. Let e 2 M be a projection and set q WD ej.e/. Then (1) exe 7! qxq is a *-isomorphism of eM e onto qM q and, in particular, e ¤ 0 ” q ¤ 0, (2) .qM q; qH ; qJ q; qP/ is a standard form of qM q Š eM e. Proof. (1) Note that the commutant of eM e on eH is M 0 e. As is well known, the central support cM 0 e .q/ of q 2 M 0 e is the projection onto M 0 eqH D eM 0 j.e/H .

30

3 Standard form

Hence cM 0 e .q/ D ecM 0 .j.e//, where cM 0 .j.e// is the central support of j.e/ 2 M 0 . Since J commutes with projections in M \ M 0 (see Proposition 2.10), we have cM 0 .j.e// D JcM 0 .j.e//J  Jj.e/J D e so that cM 0 e .q/ D e. This means that x 2 eM e 7! xq 2 .eM e/q is a *-isomorphism. (2) Since J q D JeJeJ D qJ , J leaves qH invariant. Hence qJ D J q is a conjugate-linear involution on qH . Since qP  P by (d), it is obvious that h; i  0 for all ;  2 qP. Assume that  2 qH and h; i  0 for all  2 qP. Then for every  2 P, 0  hq; i D h; qi D h; i so that  2 P and  D q 2 qP. Therefore, qP is a self-dual cone in qH . Now it remains to show conditions (a)–(d) for .qM q; qH ; qJ q; qH /. (a) follows since .qJ /.qM q/.qJ / D qJMJ q D qM 0 q D q.eM e/0 q D .eM eq/0 D .qM q/0 : (b) The center of qM q is Z.qM q/ D Z.eM e/ej.e/ D Z.M /ej.e/ D Z.M /q; for every z 2 Z.qM q/, writing z D xq for some x 2 Z.M / one has .qJ /z.qJ / D q.J xJ /q D qx  q D z  : (c) For every  2 P one has qJ.q/ D qJ  D q. (d) For every x 2 M one has .qxq/.qJ /.qxq/.qJ /qP D qxqJ xJ qP D qxej.e/j.x/j.e/P D q.exe/j.exe/P  qP:



Here, let us introduce some simple notation for later use. For each  2 H we denote by ! (2 MC ) the vector functional x 7! h; xi on M . We write e./ for the projection onto M 0 . Note that e./ 2 M and e./ D s.! /, the support projection of ! (an exercise). Lemma 3.9. Let .M; H ; J; P/ be a standard form. (1) If  2 P, then  is cyclic for M if and only if  is separating for M . (2) If M is  -finite, then there exists a  2 P that is cyclic and separating for M . Proof. (1) Let  2 P. If  is cyclic for M , then  D J  is cyclic for JMJ D M 0 , so  is separating for M . The converse is similar.

3.1 Definition and basic properties

31

(2) Take a maximal family .i /i2I of non-zeroPvectors in P such that e.i / (i 2 I ) are mutually orthogonal. Assume that e WD 1 i 2I e.i / ¤ 0. By Proposition 3.8, q WD ej.e/ ¤ 0 and qP is a self-dual cone in qH , so qP ¤ ¹0º. Since qP  P, one 0 can choose a  2 P such that  D q ¤ 0, so  D e. Since eM 0  D MP e D M 0 , one has e./  e, which contradicts the maximality of .i /i2I . Therefore, i2I e.i / D 1. Since M is -finite, the index set P I is at most countable. So we may assume that P 2 0 0 k (i ¤ k) and i 2I ki k < 1. Now set  WD i 2I i 2 P. Since M i ? M  P 0 M D JMJ , we have M i ? M k (i ¤ k). Hence it follows that ! D i2I !i and _ _ X e./ D s.! / D s.!i / D e.i / D e.i / D 1; i2I

i2I

i2I

which means that  is cyclic for M 0 or, equivalently,  is separating for M . By (1),  is cyclic and separating for M .  The next proposition shows the universality of .J; P/ for the choice of a cyclic and separating vector  in P. Proposition 3.10. Let .M; H ; J; P/ be a standard form. If  2 P is cyclic and separating for M , then J D J; P D P; where J is the modular conjugation and P is the natural positive cone associated with .M; /. Proof. Let S be the closure of x 7! x   for x 2 M , and F be the closure of x 0  7! x 0  for x 0 2 M 0 ; then F D S (see Remark 2.9). For every x 2 M one has JF J x D JF .J xJ / D J.J xJ /  D x   D S x; so S  JF J , and F  JS J by a symmetric argument. Hence JS D F J D .JS / , so JS is self-adjoint. Moreover, for every x 2 M one has hx; JS xi D hx; J x  i D h; x  j.x  /i  0; since ; x  j.x  / 2 P by (d). Since M  is a core of JS , it follows that JS is a positive self-adjoint operator. Take the polar decomposition S D J 1=2 . Since  J.JS / D J 1=2 , it follows from the uniqueness of the polar decomposition that J D J .  From the definition of P as in (3.1) and J D J , we have P D ¹xJ x W x 2 M º  P thanks to (d). From the self-duality of P (see Theorem 3.2(vi)) and P, we hence have P  P as well, so P D P.  In later sections we will sometimes consider the tensor product M ˝M2 .C/ D M2 .M / of M with the 2  2 matrix algebra M2 D M2 .C/. The next example gives a description of the standard form of M2 .M /.

32

3 Standard form

Example 3.11. Let .M; H ; J; P/ be a standard form. We write M .2/ for the tensor product M ˝ M2 of M and M2 . Choose a cyclic and separating vector  2 .2/ P and set !.x/ 2 .M .2/ /C  de
xi, x 2 M . Consider a faithful !  WD h; .2/ x11 x12 WD !.x11 / C !.x22 /. Then the GNS cyclic representation fined by ! x21 x22 .H .2/ ;  .2/ ; .2/ / of M .2/ with respect to ! .2/ is given as   °h i ±  0 12 .2/ H .2/ D H ˚ H ˚ H ˚ H D 11 W  2 H ;  D ; (3.3) ij 21 22 0   
  11 12  x12 x12 and  .2/ xx11 acts like a 2  2 matrix product as xx11 , whose 4  4 21 x22 21 x22 21 22 representation is 2 32 3 2 3 x11 0 x12 0 11 11 6 0 x11 7 612 7 612 7 0 x 12 6 7 6 7 for 6 7 2 H .2/ : (3.4) 4x21 421 5 0 x22 0 5 421 5 0 x21 0 x22 22 22 Write S .2/ and .2/ for .M .2/ ; .2/ / as well as S and  for .M; /; see Section 2.1. Since        x11 x12 x11  x21  .2/ .2/ S  D  ;  x21 x22 x12  x22  one can write 3 2 3 2  0 0 0 S 0 0 0 60  0 07 60 0 S 07 .2/ 7 6 7 (3.5) S .2/ D 6 4 0 S 0 0 5 and  D 4 0 0  0 5 : 0 0 0  0 0 0 S From this with S D J1=2 one has the polar decomposition S .2/ D J .2/ ..2/ /1=2 where 2 3 J 0 0 0 60 0 J 07 7 J .2/ D 6 (3.6) 40 J 0 05 : 0 0 0 J Therefore, the standard form of M .2/ is given as .M .2/ ; H .2/ ; J .2/ ; P .2/ / with identifications (3.3), (3.4) and (3.6). Moreover, by (3.1) one has ® ¯ P .2/ D xJ .2/ x.2/ W x 2 M .2/ :   ®  ¯ In particular, restricting to x D x01 x02 one has P .2/  0 0 W ;  2 P .

3.2 Uniqueness theorem The following is the most important property of standard forms, establishing the relation between P and MC .

33

3.2 Uniqueness theorem

Theorem 3.12. Let .M; H ; J; P/ be a standard form. For every ' 2 MC there exists a  2 P such that ' D ! , i.e., '.x/ D h; xi for all x 2 M . Furthermore, the following estimates hold: k

k2  k!

! k  k

k k C k;

;  2 P:

Consequently, the map  7! ! is a homeomorphism from P onto MC when P and MC are equipped with the norm topology. The  2 P satisfying '.x/ D h; xi (x 2 M ) is called the vector representative of ' 2 MC . A fundamental property of standard forms is universality (uniqueness) in the sense of unitary equivalence given as follows. Theorem 3.13. Let .M; H ; J; P/ and .M1 ; H1 ; J1 ; P1 / be standard forms of von Neumann algebras of M and M1 , respectively. If ˆW M ! M1 is a *-isomorphism, then there exists a unique unitary U W H ! H1 such that (1) ˆ.x/ D UxU  for all x 2 M , (2) J1 D UJ U  , (3) P1 D U P. The above theorems were proved in [26, 28] for general von Neumann algebras, but below we assume that von Neumann algebras are -finite. We first prove Theorem 3.13 while assuming that Theorem 3.12 holds true. The proof of the latter will be presented later on. Proof of Theorem 3.13. First we prove the uniqueness of U . Assume that U; V W H ! H1 are unitaries satisfying (1)–(3). For every  2 P, by (3) and (1) we have U ; V  2 P1 and !U  .ˆ.x// D h; U  ˆ.x/U i D h; xi D h; V  ˆ.x/V i D !V  .ˆ.x//;

x 2 M:

Since  2 P1 7! ! 2 .M1 /C  is injective by Theorem 3.12, it follows that U  D V  for all  2 P. Hence U D V by Proposition 3.7(3). (Note that condition (2) is unnecessary for the uniqueness of U .) Next we prove the existence of U . By Lemma 3.9(2) there exists a cyclic and separating vector 0 2 P for M . By Theorem 3.12 there exists an 0 2 P1 such that !0 D !0 ı ˆ 1 , i.e., !0 .ˆ.x// D !0 .x/ for all x 2 M . Then 0 is separating for M1 , so it is also cyclic by Lemma 3.9(1). Note that kˆ.x/0 k2 D !0 .ˆ.x  x// D !0 .x  x/ D kx0 k2 ;

x 2 M:

Hence an isometry U W M 0 ! M1 0 is defined by Ux0 WD ˆ.x/0 for x 2 M , which can extend by continuity to a unitary U W H ! H1 . We now show that U satisfies (1)–(3).

34

3 Standard form

(1) Let  2 M1 0 , so  D ˆ.y/0 D Uy0 for some y 2 M . For every x 2 M , ˆ.x/ D ˆ.xy/0 D Uxy0 D UxU  : Since M1 0 D H1 , one has ˆ.x/ D UxU  . (2) Let S0 (resp., S0 ) be the closure of S00 x0 D x  0 for x 2 M (resp., S00 y0 D  y 0 for y 2 M1 ). Since US00 U  ˆ.x/0 D US00 x0 D Ux  0 D ˆ.x/ 0 D S00 ˆ.x/0 ; one has S00 D US00 U  and hence S0 D US0 U  . Taking the polar decompositions  S0 D J0 1=2 and S0 D J0 1=2 0 , one has J0 D UJ0 U . Since J D J0 and 0 J1 D J0 by Proposition 3.10, J1 D UJ U  follows. (3) By Proposition 3.10 one has

P1 D P0 D ¹yJ1 y0 W y 2 M1 º D ¹.UxU  /.UJ U  /.UxU  /0 W x 2 M º D U ¹xJ x0 W x 2 M º D U P0 D U P:



In the rest of the section we give the proof of Theorem 3.12, which we divide into several lemmas. Lemma 3.14. The linear map ˆW Msa ! Hsa , where Hsa WD P

P, defined by

ˆ.x/ WD 1=4 x is an order isomorphism from Msa onto the set L WD ¹ 2 Hsa W

˛    ˛ for some ˛ > 0º;

where the orders on Msa and Hsa are induced by the cones MC and P, respectively. Proof. By Theorem 3.2(i), if x 2 MC , then 1=4 x 2 P. Conversely, if x 2 Msa and 1=4 x 2 P, then for any x 0 2 M 0 , hx 0 ; xx 0 i D hjx 0 j2 ; xi D h

1=4

jx 0 j2 ; 1=4 xi  0;

hence x  0. Since ˆ.1/ D  and ˆ is clearly injective, it follows that ˆW Msa ! ˆ.Msa /  L is an order isomorphism. Hence it remains to show that ˆ.Msa / D L . First we show that ˆ is continuous with respect to the  .M; M /-topology and the weak topology on H . For any x 2 M note that .1 C 1=2 /x D x C J x   so that 1=4 x D 1=4 .1 C 1=2 / Since .1=4 C 

1=4

/

1

1

.x C J x  / D .1=4 C 

1=4

/

1

.x C J x  /:

is bounded and

h; 1=4 xi D h.1=4 C  the above-stated continuity of ˆ follows.

1=4

/

1

; x C J x  i;

2H;

35

3.2 Uniqueness theorem

Now let  2 L . We may assume that 0    . Put n WD fn .log /, where 2 fn .t / WD e t =2n . Since fn .log /P  P by Theorem 3.2(v), one has 0  n D fn .log /  fn .log / D ; where the last equalityT is immediate from  D  and fn .log 1/ D 1. Furthermore, one has n !  and n 2 z2C D.z / as in the proof of Theorem 3.2(vi). For any  2 P [ , since  1=4  2 P by Theorem 3.2(i), it follows that h;  1=4 n i D h 1=4 ; n i  0, so  1=4 n 2 P ] by Lemma 3.3. Similarly,   1=4 n D  1=4 . n / 2 P ] . Let n WD  1=4 n and define an operator Tn W M 0  ! H by Tn x 0  WD x 0 n for x 0 2 M 0 , which is affiliated with M as in the proof of Lemma 3.3. Since hx 0 ; Tn x 0 i D hx 0 x 0 ; n i  0; hx 0 ; x 0 i

hx 0 ; Tn x 0 i D hx 0 x 0 ; 

n i  0;

one has 0  hx 0 ; Tn x 0 i  hx 0 ; x 0 i for all x 0 2 M 0 . Hence Tn is extended to an xn 2 M with 0  xn  1, so n D 1=4 n D 1=4 xn  D ˆ.xn /. Therefore, ¹n º  K WD ¹ˆ.x/ W x 2 Msa ; 0  x  1º: But K is weakly compact due to the continuity of ˆ shown above. Thus  D limn n 2 K  ˆ.Msa /.  Lemma 3.15. For every ;  2 P we have k

k2  k!

! k  k

k k C k:

Proof. The second inequality holds for all ;  2 H since .!

! /.x/ D 21 Œh

; x. C /i C h C ; x.

/i:

To prove the first inequality, assume first that  C  is cyclic and separating for M , so PC D P by Proposition 3.10. Since . C /  

   C ;

there exists, by Lemma 3.14 applied to  C  (in place of ), an x 2 Msa with 1  x  1 such that   D 1=4 x. C /; C where C is the modular operator with respect to  C . Therefore, k!

! k  .!

! /.x/ D h; xi h; xi ˝ 1=4 D Reh ; x. C /i D  ; C .

˛ / :

36

3 Standard form

1=4 Since J1=4 D C J (see Proposition 3.10), we have C

˝ 

1=4 ; C .

˛ ˝ / D 

; 1=4 . C

˛ / ;

so that k!

˝ ! k  

1=4
; 12 1=4 C C . C

˛ /  k

k2 ;

1=4 thanks to 12 .1=4 C C /  1. C Next let ;  2 P be arbitrary. Let  t0 be the modular automorphism group associated with .M 0 ; /, i.e.,  t0 .x 0 / D  i t x 0 i t , x 0 2 M 0 (see Remark 2.9). From Theorem 3.2(i) and Lemma 2.13 one can choose sequences xn0 ; yn0 2 MC0 such that xn0 ; yn0 are  t0 -analytic and kn k ! 0, kn k ! 0, where n WD  1=4 xn0  D  0 i=4 .xn0 / and n WD  1=4 yn0  D  0 i=4 .yn0 /. Here, by adding "n 1 ("n & 0) to xn0 , we may assume that xn0 C yn0  "n 1, so  0 i=4 .xn0 C yn0 / 2 M 0 are invertible. Hence n Cn D  1=4 .xn0 Cyn0 / D  0 i=4 .xn0 Cyn0 / 2 P is separating and also cyclic for M by Lemma 3.9(1). Therefore, from the first part it follows that k!n !n k  kn n k2 . Letting n ! 1 gives the asserted inequality. 

Lemma 3.16. The map  7! ! is a homeomorphism from P onto a closed subset E WD ¹! W  2 Pº of MC with respect to the norm topologies on P and MC . Proof. It follows from Lemma 3.15 that  2 P 7! ! 2 E is a homeomorphism. If ¹!n º is a Cauchy sequence in MC , then km n k2  k!m !n k ! 0 as m; n ! 1, so ¹n º is Cauchy in P. Hence n !  2 P and !n ! ! for some  2 P. Hence E is closed in MC .  Lemma 3.17. Let  2 P. If 2 MC and  ! , then there exists an  2 P such that    and .x/ D 12 .h; xi C h; xi/ for all x 2 M . Proof. First assume that  is cyclic and separating for M . As is well known (and easily shown), there exists a b 0 2 M 0 such that 0  b 0  1 and .x/ D h; b 0 xi for all x 2 M . Note that b 0 ; .1 b 0 / 2 P[ , where P[ is P [ for  D . Since  1=4 P[  P D P by Theorem 3.2(i) and Proposition 3.10, we have  WD  1=4 b 0  2 P and  

1=4

.1

 D

b / 2 P, i.e., 0    . Set 0


1 1=4   b  D 2 1 C 1=2  1=4 1=4
1 D 2  C   D f .log  /;

 WD 2 1 C 1=2 




1 0

where f .t / WD 2=.e t =4 C e t =4 / D 1= cosh.t =4/. Note that f .t / is the Fourier transform of 4= cosh.2s/, so f is a positive definite function on R. Hence by Theorem 3.2(v), 0   D f .log  /  f .log  / D :

37

3.2 Uniqueness theorem

/ D 12 . C F /; see Furthermore, since  D J  D J , we find that b 0  D 12 .1 C 1=2  Lemma 2.1(iii). Hence, for every x 2 M we have .x/ D hb 0 ; xi D 12 .h; xi C hF ; xi/ D 21 .h; xi C hS x; i/ D 12 .h; xi C hx  ; i/ D 21 .h; xi C h; xi/: Next let  2 P be arbitrary. Let e WD e./ D s.! / and q WD ej.e/. Then  2 qP is cyclic and separating for qM q whose standard form is .qM q; qH ; qJ q; qP/; see Proposition 3.8(2). By Proposition 3.8(1) one can define q 2 .qM q/C  by q .qxq/ D .exe/ D .x/ for x 2 M . Since  ! where ! is regarded as an element of .qM q/C  , it follows from the first part of the proof that there exists an  2 qP ( P by (d)) such that    and q .x/ D 12 .h; xi C h; xi/ for all x 2 qM q. Therefore, for every x 2 M , .x/ D q .qxq/ D 12 .h; xi C h; xi/:  Lemma 3.18. If 0 2 P,

2 MC and

D ! for some  2 P.

 !0 , then

Proof. Put 1 WD !0 . By Lemma 3.17 one can find an 1 2 P such that 1  12 0 and 1 .x/ D h1 ; x0 i C h0 ; x1 i for all x 2 M . Set 1 WD 0 1 2 P; then !1 .x/ D h0 ; x0 i D !0 .x/ so that !1

h0 ; x1 i

h1 ; x0 i C h1 ; x1 i

1 .x/

C !1 .x/ D

.x/ C !1 .x/;

1 .1/

D 2 Reh1 ; 0 i D 2h1 ; 0 i:

x 2 M;

D !1 and k

1k

D

Therefore, k!1

˝ ˛ k D k1 k2  1 ; 12 0 D 14 k

1k

D 14 k!0

k:

Apply the above argument to 1 in place of 0 ; then we have a 2 2 P such that k. Repeating the argument we find a sequence n 2 P such that k  14 k!1 k!2 k ! 0. By Lemma 3.16, D ! with  WD limn n 2 P.  k!n  ˛! for some ˛ > 0º is norm-dense in MC . P Proof. Recall P that each 2 MC has the form .x/ D 1 ; xn i for some sequence nD1 hnP n 2 H with n kn k2 < 1. Approximating by m .x/ WD m nD1 hn ; xn i and then n by xn0  with xn0 2 M 0 , one can approximate in norm by z 2 MC of the form P 0 0 z .x/ WD m nD1 hxn ; xxn i. Since ! m m X X 0 1=2 0 2 1=2 0 z .x/ D hx ; xn xn x i  kxn k h; xi; x 2 MC ; Lemma 3.19. The set ¹

2 MC W

nD1

it follows that z  ˛! for some ˛ > 0.

nD1



38

3 Standard form

End of Proof of Theorem 3.12. By Lemma 3.18 applied to 0 D , E D ¹! W  2 Pº  ¹

2 MC W

 ˛! for some ˛ > 0º:

By Lemmas 3.16 and 3.19 we have E D E D MC , which shows the first assertion of the theorem. Then the remaining follows from Lemmas 3.15 and 3.16. 

4  -Measurable operators

Non-commutative integration theory was created by Segal [84] (and followed by [88] and [63]), where measurable operators affiliated with a von Neumann algebra with a trace were discussed. Later in [66], Nelson proposed a more tractable approach to -measurable operators in a stricter connection with a given trace , which is more convenient in developing non-commutative integration. Throughout this chapter we assume that M is a semifinite von Neumann algebra on a Hilbert space H , having a faithful semifinite normal trace . The first section is a study of the theory of -measurable operators, mainly based on [97, Chap. I] whose exposition is considerably more readable than that in [66]. In the second section we present a brief exposition of the generalized s-numbers of -measurable operators. More detailed accounts are found in [24], which is the best literature on the topic. The third section gives a self-contained exposition of non-commutative Lp -spaces with respect to a trace. Non-commutative Lp -spaces Lp .M;  / on .M;  / were first developed in [21, 63, 84, 88] and were later discussed in [66, 109] in a more direct approach to -measurable operators.

4.1 -Measurable operators Let M be a von Neumann algebra on a Hilbert space H . Let aW D.a/ ! H be a linear operator with domain D.a/ a linear subspace of H . We say that a is affiliated with M , denoted by aM , if x 0 a  ax 0 for all x 0 2 M 0 or, equivalently, if u0 au0 D a for all unitaries u0 2 M 0 . The following facts are easy to verify (exercises): (a) If a, b are linear operators affiliated with M , then a C b with D.a C b/ D D.a/ \ D.b/ and ab with D.ab/ D ¹ 2 D.b/ W b 2 D.a/º are affiliated with M . (b) If a is densely defined and aM , then a M . (c) If a is closable and aM , then aM . (d) Assume that a is densely defined and closed, soRwe have the polar decomposition 1 a D wjaj and the spectral decomposition jaj D 0 e . Then aM if and only if w; e 2 M for all   0.

40

4 -Measurable operators

Hereafter, let M be a semifinite von Neumann algebra with a faithful semifinite normal trace , that is,  is a faithful semifinite normal weight on M (see Section 1.3) satisfying the trace condition that  .x  x/ D  .xx  / for all x 2 M . For each "; ı > 0 define ®
¯ O."; ı/ WD aM W eH  D.a/; kaek  " and  e ?  ı for some e 2 Proj.M / ; where Proj.M / is the set of projections in M , where k
k is the operator norm. Lemma 4.1. For any "1 ; "2 ; ı1 ; ı2 > 0 we have (1) O."1 ; ı1 / C O."2 ; ı2 /  O."1 C "2 ; ı1 C ı2 /, (2) O."1 ; ı1 /O."2 ; ı2 /  O."1 "2 ; ı1 C ı2 /. Proof. Let a 2 O."1 ; ı1 / and b 2 O."2 ; ı2 /, so there are e; f 2 Proj.M / such that
eH  D.a/; kaek  "1 ;  e ?  ı1 ;
f H  D.b/; kbf k  "1 ;  f ?  ı2 : (1) Letting p WD e ^ f 2 Proj.M / one has pH D eH \ f H  D.a/ \ D.b/ D D.a C b/; k.a C b/pk  kapk C kbpk  kaek C kbf k  "1 C "2 ;



 p ? D  e ? _ f ?   e ? C  f ?  ı1 C ı2 : Hence a C b 2 O."1 C "2 ; ı1 C ı2 /. (2) For every x 0 2 M 0 one has x 0 e ? bf D e ? x 0 bf  e ? bx 0 f D e ? bf x 0 ; which implies that e ? bf 2 M . Let g be the projection onto the kernel of e ? bf , so g 2 Proj.M / and bfg D ebfg. If  2 gH , then bf  2 eH and hence  2 D.abf /, so f  2 D.ab/. Let q WD f ^ g 2 Proj.M /. Then qH  D.ab/ and abq D abfgq D aebfgq D aebf q; so that kabqk  kaek kbf k  "1 "2 . Furthermore, one has (see [95, Props. V.1.5, V.1.6]) 
  g ? D the projection onto the closure of the range of e ? bf    the projection onto the closure of the range of e ? bf  e ? ; so that

 q ? D  f ? _ g? D  .f ? / C  ..f ? _ g ? /

f ? / D  .f ? / C  .g ?

.f ? ^ g ? //

  .f ? / C  .g ? /   .f ? / C  .e ? /  ı2 C ı1 : Hence ab 2 O."1 "2 ; ı1 C ı2 /.



41

4.1 -Measurable operators

Definition 4.2. A linear subspace L of H is said to be -dense if, for any ı > 0, there exists an e 2 Proj.M / such that eH  L and  .e ? /  ı. Lemma 4.3. A -dense linear subspace L of H is dense in H . Proof. Let L be -dense. For ı D 1=2k (k V 2 N) choose a qk 2 Proj.M / such that qk H  L and  .qk? /  1=2k . Let en WD 1 kDn qk 2 Proj.M /. Then en % and W ? en H  L . Since en? D 1 q , one has kDn k
?

 en D lim  m!1

m _

! qk?

kDn



1 X


 qk? 

kDn

1 2n 1

!0

.n ! 1/:

Hence en? & 0, i.e., en % 1. This implies that L is dense in H .



Lemma R 14.4. Let a be a densely defined closed operator with aM . Let a D wjaj and jaj D 0  de be as in (d) above. Then for any "; ı > 0,
a 2 O."; ı/ ”  e"?  ı: Proof. Assume that a 2 O."; ı/, so there is an e 2 Proj.M / such that k jajek D kaek  " and  .e ? /  ı. For any t > " and  2 e t? H n ¹0º, Z k jajk2 D 2 dke k2 > t 2 kk2 .t;1/

so that e ^ e t? D 0. So one has (see [95, Prop. V.1.6]) e t? D e t?

e ^ e t?  e _ e t?

e  e?;

which implies that  .e t? /   .e ? /  ı. Since e t? % e"? as t & ",  .e"? /  ı follows. The converse is obvious by taking e D e" .  Lemma 4.5. Let a and b be densely defined closed operator with a; bM . If there exists a -dense linear subspace L of H such that L  D.a/ \ D.b/ and ajL D bjL , then a D b. Proof. Consider the von Neumann algebra M .2/ WD M ˝ M2 .C/ D

² x11 x21

 ³ x12 W xij 2 M x22

on H .2/ WD H ˚ H with a faithful semifinite normal trace     x11 x12 x11 x12  .2/ WD  .x11 / C  .x22 / for 2 .M .2/ /C : x21 x22 x21 x22

42

4 -Measurable operators

Let pa , pb be the projections from H .2/ onto the graphs (closed subspaces) G.a/, G.b/ of a, b, respectively. Note that ² 0  ³ x 0 0 0 .M .2/ /0 D W x 2 M : 0 x0 For every x 0 2 M 0 and  2 D.a/ one has .x 0 ˚ x 0 /. ˚ a/ D x 0  ˚ ax 0  2 G.a/, so .x 0 ˚ x 0 /pa D pa .x 0 ˚ x 0 /pa . This implies that pa 2 .M .2/ /00 D M .2/ . Similarly, pb 2 M .2/ . For any ı > 0 there is an e 2 Proj.M / such that eH  L and  .e ? /  ı=2. Set e .2/ WD 0e 0e 2 Proj.M .2/ /; then  .2/ .e .2/? /  ı. Since ajL D bjL , one has G.a/ \ e .2/ H .2/ D ¹ ˚ a W  2 eH ; a 2 eH º D ¹ ˚ b W  2 eH ; b 2 eH º D G.b/ \ e .2/ H .2/ ; which means that pa ^ e .2/ D pb ^ e .2/ . Let p0 WD pa pa ^ pb ^ e .2/ , it follows that p0 ^ e .2/ D 0 so that p0 D p0

p0 ^ e .2/  p0 _ e .2/

pa ^ pb . Since pa ^ e .2/ D

e .2/  e .2/? :

Therefore,  .2/ .p0 /   .2/ .e .2/? /  ı. Since ı > 0 is arbitrary,  .2/ .p0 / D 0 so p0 D 0, i.e., pa D pa ^ pb . Similarly, pb D pa ^ pb , so pa D pb , i.e., G.a/ D G.b/ or a D b.  Definition 4.6. Let a be a densely defined closed operator such that aM . We say that a is -measurable if, for any ı > 0, there exists an e 2 Proj.M / such that eH  D.a/ and  .e ? /  ı. Since eH  D.a/ ” kaek < 1 due to the closed graph theorem, the condition is equivalent to there being, for any ı > 0, an " > 0 such that a 2 O."; ı/. We z the set of such -measurable operators. denote by M Proposition 4.7. LetR a be a densely defined closed operator affiliated with M with 1 a D wjaj and jaj D 0  de as above. Then the following conditions are equivalent: z; (i) a 2 M z; (ii) jaj 2 M (iii)  .e? / ! 0 as  ! 1; (iv)  .e? / < 1 for some  > 0. Proof. (i) ” (ii) is obvious and (iii) H) (iv) is trivial. (ii) ” (iii) immediately follows from Definition 4.6 and Lemma 4.4. (iv) H) (iii). Assume that  .e? / < 1 for some  > 0. Since e t e % 1 e as  < t ! 1, one has  .e t e / %  .1 e / < 1 so that
 e t? D  ..1 e / .e t e // D  .1 e /  .e t e / ! 0 as t ! 1. Hence (iii) follows.



4.1 -Measurable operators

43

For each "; ı > 0 define z \ O."; ı/ N ."; ı/ WD M
¯ ® z W kaek  " and  e ?  ı for some e 2 Proj.M / : D a2M z , then a 2 M z . Moreover, a 2 N ."; ı/ ” a 2 N ."; ı/. Lemma 4.8. If a 2 M R z with a D wjaj and jaj D 1  de . One can define a spectral Proof. Let a 2 M 0 R1 resolution ¹y e º0 by ey? WD we? w  2 Proj.M /. Then ja j D wjajw  D 0  d ey . z follows from Proposition 4.7. Moreover, this implies by Since  .y e? / D  .e? /, a 2 M Lemma 4.4 that a 2 O."; ı/ ” a 2 O."; ı/. Hence the latter assertion follows.  z , then a C b and ab are densely defined and closable, and Lemma 4.9. If a; b 2 M z a C b; ab 2 M . Moreover, if a 2 N ."1 ; ı1 / and b 2 N ."2 ; ı2 /, then a C b 2 N ."1 C "2 ; ı1 C ı2 / and ab 2 N ."1 "2 ; ı1 C ı2 /. z . For any ı1 ; ı2 > 0 there are "1 ; "2 > 0 such that a 2 O."1 ; ı1 / Proof. Let a; b 2 M and b 2 O."2 ; ı2 /. By Lemma 4.1 one has a C b 2 O."1 C "2 ; ı1 C ı2 / and ab 2 O."1 "2 ; ı1 C ı2 /. Since ı1 C ı2 is arbitrarily small, it follows that D.a C b/ and D.ab/ z are -dense. Hence a C b and ab are densely defined by Lemma 4.3. Since a ; b  2 M by Lemma 4.8, a C b  and b  a are also densely defined, so that .a C b  / and .b  a / exist. Note that a C b  .a C b  / and ab  .b  a / (exercises). Therefore, a C b and ab are closable, so a C b 2 O."1 C "2 ; ı1 C ı2 / and ab 2 O."1 "2 ; ı1 C ı2 /. Since z . Moreover, the latter assertion ı1 C ı2 is arbitrarily small, it follows that a C b; ab 2 M follows from the above proof.  z is a *-algebra with respect to the adjoint , the strong sum Proposition 4.10. The M a C b and the strong product ab. z is closed under the adjoint , the strong Proof. First note by Lemmas 4.8 and 4.9 that M z sum and product. Let a; b; c 2 M . Since a C b C c; a C b C c  a C b C c and D.a C b C c/ is -dense by Lemma 4.1(1), it follows from Lemma 4.5 that a C b C c D a C b C c: Since abc; abc  abc and D.abc/ is -dense by Lemma 4.1(2), it also follows that abc D abc: Since .a C b/c; ac C bc  .a C b/c and D..a C b/c/ is  -dense by Lemma 4.1, one has .a C b/c D ac C bc and similarly a.b C c/ D ab C ac. Also, since .a C b/ D .a C b/  a C b  , one has .a C b/ D a C b  . Since .ab/ D .ab/  b  a , one has .ab/ D b  a . Moreover, a D a holds. 

44

4 -Measurable operators

z we will use the convention that In view of Proposition 4.10, for every a; b 2 M a C b and ab mean the strong sum a C b and the strong product ab, respectively. A big advantage of -measurable operators is that we can freely take adjoint, sum and product z . So the domain problem never occurs, which is an annoying problem in the case of in M more general measurable operators as in [84]. Lemma 4.11. For any "; "1 ; "2 ; ı; ı1 ; ı2 > 0 we have (1) N ."; ı/ D N ."; ı/, (2) ˛N ."; ı/ D N .j˛j"; ı/ for all ˛ 2 C, ˛ ¤ 0, (3) "1  "2 , ı1  ı2 H) N ."1 ; ı1 /  N ."2 ; ı2 /, (4) N ." ^ "2 ; ı1 ^ ı2 /  N ."1 ; ı1 / \ N ."2 ; ı2 /, (5) N ."1 ; ı1 / C N ."2 ; ı2 /  N ."1 C "2 ; ı1 C ı2 /, (6) N ."1 ; ı1 /N ."2 ; ı2 /  N ."1 "2 ; ı1 C ı2 /. Proof. (1), (5) and (6) are in Lemmas 4.8 and 4.9, while (2)–(4) are obvious.



The main result of the section is the following. z is a complete metrizable Hausdorff topological *-algebra with Theorem 4.12. The M z. ¹N ."; ı/ W "; ı > 0º as a neighborhood basis of 0. Moreover, M is dense in M z Proof. From (2), (4) and (5) of Lemma 4.11 it follows that a linear topologyT is defined on M with ¹N ."; ı/ W "; ı > 0º as a neighborhood basis of 0. Assume that a 2 ";ı>0 N ."; ı/ R1 with the spectral decomposition jaj D 0  de . By Lemma 4.4,  .e"? /  ı for all "; ı > 0, which implies that e"? D 0 for all " > 0, so jaj D 0 or a D 0. Hence the defined topology is Hausdorff, which is metrizable since there is a countable neighborhood basis z . For any ¹N .1=n; 1=n/ W n 2 Nº of 0. By Lemma 4.11(1), a 7! a is continuous on M z a0 ; b0 2 M and any "; ı > 0, take r; s > 0 such that a0 2 N .r; ı=6/ and b0 2 N .s; ı=6/. Choose an "1 > 0 with "1 ."1 C r C s/  ". If a a0 ; b b0 2 N ."1 ; ı=6/, then ab

a0 b0 D .a

a0 /.b

b0 / C a0 .b

b0 / C .a

a0 /b0

2 N ."1 ; ı=6/N ."1 ; ı=6/ C N .r; ı=6/N ."1 ; ı=6/ C N ."1 ; ı=6/N .s; ı=6/  N ."; ı/; z z thanks to (5) and (6) of Lemma 4.11. R 1Hence .a; b/ 7! ab is continuous on M  M . z Let a D wjaj 2 M withRjaj D 0  de . For any "; ı > 0 choose an r > 0 such that  .er? /  ı, and let a1 WD w Œ0;r  de 2 M . Then Z  .a a1 /er D w  de er D 0; .r;1/

and so a

z. a1 2 N ."; ı/. Hence M is dense in M

45

4.1 -Measurable operators

z . By taking Finally, to prove the completeness, let ¹an º be a Cauchy sequence in M n n a subsequence we may assume that anC1 an 2 N .2 ; 2 / for all n 2 N. One can choose a sequence pn 2 Proj.M / such that for every n 2 N,
pn H  D.an /; k.anC1 an /pn k  2 n ;  pn?  2 n : V ? nC1 Let en WD 1 . When l > m  n, kDn pk 2 Proj.M /; then en % and  .en /  2 k.al

am /en k 

l 1 X

k.akC1

ak /pk k  2

nC1

:

kDm

S Hence one can define a0  WD limm!1 am  for  2 D.a0 / D n en H . Similarly, for  ¹an º choose a sequence qn 2 Proj.M / such that qn H  D.an /, k.anC1 an /qn k  2 n V1  ? n and  .qn /  2 . Let fn WD kDn qk and define b0  WD limm!1 am  for  2 D.b0 / D S n fn H . For every  2 D.a0 / and  2 D.b0 /,  ha0 ; i D lim ham ; i D lim h; am i D h; b0 i; m!1

m!1

b0 ,

which implies that a0  so a0 is closable. Now let a WD a0 . Since a0 M as easily verified, we have aM . Since en H  D.a0 /  D.a/ and  .en? /  2 nC1 z . Furthermore, for any "; ı > 0 choose an n0 with for all n 2 N, we have a 2 M n0 C1 ? 2  min¹"; ıº. Then  .en0 /  ı. When l > m  n0 , since k.al am /en0 k  ", we have k.al am /en0 k  " for all  2 H with kk  1. Letting l ! 1 gives k.a am /en0 k  ", so a am 2 N ."; ı/ for all m  n0 , showing that am ! a as m ! 1.  z given in Theorem 4.12 is called the measure topology. This The topology on M topology is not necessarily locally convex. Indeed, it is an exercise to show that if M is a finite non-atomic von Neumann algebra with a faithful normal finite trace , then a z is only the whole M z and there is no non-zero continuous non-empty open convex set in M z functional on M . z C be the set of positive self-adjoint a 2 M z (denoted by a  0). Note that M zC Let M z z z. is a closed convex cone in M . In fact, assume that an 2 MC (n 2 N) and an ! a 2 M Then in the last paragraph of the proof of Theorem 4.12, one can let en D fn , so a0 D b0 . Since h; a0 i  0 for all  2 D.a0 / and a D a0 , it follows that a is positive self-adjoint. z is an ordered topological space with the order a  b defined by b a  0 for Hence M self-adjoint a, b. z C , besides the order a  b mentioned above (i.e., b a 2 Remark 4.13. When a; b 2 M z MC ), we have another familiar notion of a  b in the form sense (i.e., D.b 1=2 /  D.a1=2 / and ka1=2 k2  kb 1=2 k2 for all  2 D.b 1=2 /); see Definition A.2 of Appendix A.1. But z C , a  b in we notice that both definitions of a  b are equivalent. Indeed, for a; b 2 M z z MC if and only if 1 C a  1 C b in MC , which is equivalent to .1 C b/

1=2

.1 C a/.1 C b/

1=2

 1 ” .1 C a/1=2 .1 C b/ ” .1 C b/

1

1

.a C a/1=2  1

 .1 C a/

1

;

46

4 -Measurable operators

that is, a  b in the form sense; see Lemma A.1. Furthermore, although the proof is z C (n 2 N) and an ! a in the omitted here, the following is worth noting: If a; an 2 M measure topology, then an ! a in the strong resolvent sense (i.e., .1Can / 1 ! .1Ca/ 1 strongly); see Definition A.6. (The details can be found in [41, §3].) z D B.H / and Examples 4.14. (1) When M D B.H / and  is the usual trace Tr, M z ; then there is the measure topology is the operator norm topology. Indeed, let a 2 M ? a projection e such that kaek < 1 and Tr.e / < 1. The latter implies that e ? D 0 or e D 1, so a 2 M . Moreover, when ı < 1, N ."; ı/ D ¹a 2 M W kak  "º, which means the equivalence of the measure and the operator norm topologies. z is the set of all densely (2) Let M be finite with a faithful normal finite trace . Then M R1 defined closed operators xM . Indeed, if a is in the latter set with jaj D 0  de , then  .e? / ! 0 as  ! 1 automatically. (3) Let .X; X ; / be a localizable measure space, where .X; X ; / is localizable if for every A 2 X there is a B 2 X such that B  A and .B/ < R1. For an abelian von Neumann algebra A D L1 .X; / D L1 .X; / with  .f / WD X f d for f 2 L1 .X; /C , e A is the space of measurable functions f on X such that f is bounded on X n A for some A 2 X with .A/ < 1. In e A, f D g means f .x/ D g.x/ -a.e. The following triangle inequality for -measurable operators is from [58], extending (partially) the inequality in [2] for bounded operators, which will be used in Section 4.3. z there exist partial isometries u; v 2 M such that Lemma 4.15. For every a; b 2 M ja C bj  ujaju C vjbjv  : Proof. Let Re.a/C denote the positive part of Re.a/ WD .a C a /=2. Let us first prove that there exists a partial isometry u 2 M such that Re.a/C  ujaju . Let a D wjaj be the polar decomposition ˇ and ˇe be the support projection of Re.a/C . Also let c WD e.1 C w/=2 and cjaj1=2 D uˇcjaj1=2 ˇ be the polar decomposition. Note that w, e, c, u are all in M and c  c  1. Moreover, note that ˇ ˇ2 cjajc  D uˇcjaj1=2 ˇ u D ujaj1=2 c  cjaj1=2 u  ujaju ; Re.a/C D e Re.a/e D 12 e.wjaj C jajw  /e: We hence have 4.ujaju

Re.a/C /  4cjajc 

2e.wjaj C jajw  /e

D e.1 C w/jaj.1 C w  /e D e.jaj D e.1

wjaj w/jaj.1

2e.wjaj C jajw  /e

jajw C wjajw  /e w/ e  0:

z let a C b D wja C bj Next let us prove the assertion of the lemma. For any a; b 2 M be the polar decomposition. Note that ja C bj D w  .a C b/ D 12 .w  .a C b/ C .a C b/ w/ D Re.w  a/ C Re.w  b/  Re.w  a/C C Re.w  b/C :

47

4.2 Generalized s-numbers

By the assertion proved first, there are partial isometries u; v 2 M such that ja C bj  ujw  aju C vjw  bjv   ujaju C vjbjv  :



4.2 Generalized s-numbers z . For an interval I of Œ0; 1/ let eI .jaj/ denote the spectral projection of Let a 2 M jaj corresponding to I . For example, eŒ0;s .jaj/ D es and e.s;1/ .jaj/ D es? when R1 jaj D 0  de is the spectral decomposition. z and t > 0 the (t th) generalized s-number  t .a/ is defined Definition 4.16. For a 2 M by  t .a/ WD inf¹s  0 W  .e.s;1/ .jaj//  tº: Note that  t .a/ < 1 for all t > 0 since  .e.s;1/ .jaj// ! 0 as s ! 1 by Proposition 4.7. It is an easy exercise that t 2 .0; 1/ 7!  t .a/ is non-increasing and continuous from the right. Examples 4.17. (1) Let M D B.H / with dim H D 1 and P  D Tr. Let a be a compact operator on H , and take the spectral decomposition jaj D 1 nD1 n jn ihn j into rank-one projections with 1  2 


! 0 and orthonormal vectors n . For each n 2 N with n 1  t < n, since Tr.e.s;1/ .jaj//  t ” s  n , we have  t .a/ D n

for t 2 Œn

1; n/;

(4.1)

which is the nth singular value of a. For a general a 2 B.H / let s0 WD sup¹s 2 R W Tr.e.s;1/ .jaj// D 1º P ( 0) and n0 WD Tr.e.s0 ;1/ .jaj// . 1). Then jxje.s0 ;1/ .jaj/ 0 n jn ihn j as above and  t .a/ is given as in (4.1), where is written in the form nnD1  t .a/ D s0 for all t  n0 if n0 < 1. be a localizable measure space, and M D L1 .X; / with  .f / D R (2) Let .X; X ; / 1 z X f d for f 2 L .X; /C . For f 2 M (see Example 4.14(3)), since e.s;1/ .jf j/ D ¹xWjf .x/j>sº (where ¹


º denotes the indicator function), we have  t .f / D inf¹s  0 W .¹x W jf .x/j > sº/  t º;

t > 0;

which is the decreasing rearrangement of jf j. z with the convention that In the following we use the operator norm kak for any a 2 M kak D 1 unless a 2 M . z for n 2 N. Lemma 4.18. Let a; an 2 M (1) For every s; t > 0,  t .a/  s ”  .e.s;1/ .jaj//  t ” a 2 N .s; t /: Hence,  t .a/ D inf¹s > 0 W a 2 N .s; t /º:

(4.2)

48

4 -Measurable operators

(2) For every t > 0, ®
¯  t .a/ D inf kaek W e 2 Proj.M /;  e ?  t :

(4.3)

Moreover, if A is any von Neumann subalgebra of M containing all spectral projections of jaj, then ®
¯  t .a/ D inf kaek W e 2 Proj.A/;  e ?  t : (4.4) z C , for every t > 0, (3) When a 2 M ®
¯  t .a/ D inf sup2eH ;kkD1 h; ai W e 2 Proj.M /;  e ?  t : (4.5) R1 R 1 Here, h; ai is defined to be 0  dke k2 , where a D 0  de is the spectral decomposition of a. (4) an ! a in the measure topology if and only if " .an

a/ ! 0 for any " > 0.

Proof. (1) The first equivalence is immediate from the definition of  t .a/. The second equivalence follows from Lemma 4.4. (2) For each t > 0 write r, r 0 for the right-hand sides of (4.3) and (4.4), respectively. Obviously, r  r 0 . For any s  0 let e WD eŒ0;s .jaj/ 2 A. Then kaek D kjajek  s and e ? D e.s;1/ .jaj/, from which one sees that  t .a/  r 0 . It remains to show that  t .a/  r. For any " > 0 there is an e 2 Proj.M / such that kaek  r C " and  .e ? /  t . Hence a 2 N .r C "; t /. So  t .a/  r C " by (1). Letting " & 0 gives  t .a/  r. (3) Similarly to the proof of (2) one has  t .a/  RHS of (4.5). Hence, by (2) it suffices to show that kaek D sup¹h; ai W  2 eH; kk D 1º for any e 2 Proj.A/, where A is the von Neumann subalgebra of M generated by all spectral projections of a, so A is commutative. If ae is bounded, then kaek D keaek D sup¹h; ai W  2 eH; kk D 1º. If ae is unbounded, then both sides are 1. (4) By (1), " .an a/ ! 0 for any " > 0 if and only if, for any "; ı > 0, there is an n0 such that an a 2 N ."; ı/ for all n  n0 , which means by Theorem 4.12 that an ! a in the measure topology.  z. Proposition 4.19. Let a; b; c 2 M (1) The function t 2 .0; 1/ 7!  t .a/ 2 Œ0; 1/ is non-increasing and right-continuous. (2)  t .a/ % kak (2 Œ0; 1) as t & 0. (3)  t .a/ D  t .jaj/ D  t .a / for all t > 0. (4)  t .˛a/ D j˛j t .a/ for all ˛ 2 C and t > 0. (5) If 0  a  b, then  t .a/   t .b/ for all t > 0. (6)  t .bac/  kbk kck t .a/ for all t > 0.

49

4.2 Generalized s-numbers

(7)  t Ct 0 .a C b/   t .a/ C  t 0 .b/ for all t; t 0 > 0. (8)  t Ct 0 .ab/   t .a/ t 0 .b/ for all t; t 0 > 0. (9) If f is a continuous non-decreasing function on Œ0; 1/ with f .0/ D 0, then z and  t .f .jaj// D f . t .a// for all t > 0. f .jaj/ 2 M (10) j t .a/

 t .b/j  ka

bk for all t > 0.

Proof. (1) is easy as already mentioned in Definition 4.16. (2) By (4.3),  t .a/  kak is clear. Let s0 WD lim t&0  t .a/; then  .e.s0 ;1/ .jaj// D 0 and so kak  s0 . Hence the assertion follows. (3) That  t .a/ D  t .jaj/ is obvious. By Lemma 4.11(1) and (4.2),  t .a/ D  t .a / follows. (4) is obvious by Lemma 4.11(2). show that  .e.s;1/ .a//   .e.s;1/ .b// for all s  0. Let a D R 1 (5) It suffices toR 1 0  de and b D  0  df be the spectral decompositions. Let 0 < s < s and  be in 0 the range of e.s 0 ;1/ .a/ ^ eŒ0;s .b/. Then one has Z Z skk2   dkf k2 D kb 1=2 k2  ka1=2 k2 D  dke k2  s 0 kk2 : .s 0 ;1/

Œ0;s

For the second inequality above, see Remark 4.13 and Lemma A.1. Hence  D 0, which shows that e.s 0 ;1/ .a/ ^ eŒ0;s .b/ D 0. Since e.s 0 ;1/ .a/ D e.s 0 1/ .a/

e.s 0 ;1/ .a/ ^ eŒ0;s .b/

 e.s 0 ;1/ .a/ _ eŒ0;s .b/

eŒ0;s .b/  e.s;1/ .b/;

one has  .e.s 0 ;1/ .a//   .e.s;1/ .b//. Letting s 0 & s gives the desired inequality. (6) By (4.3) and (3) one has  t .bac/  kbk t .ac/ D kbk t .c  a /  kbk kc  k t .a / D kbk kck t .a/: (7) By (4.2) and Lemma 4.11(5) one has  t .a/ C  t 0 .b/ D inf¹s C s 0 W s; s 0 > 0; a 2 N .s; t /; b 2 N .s 0 ; t 0 /º  inf¹s > 0 W a C b 2 N .s; t C t 0 /º D  tCt 0 .a C b/: (8) By (4.2) and Lemma 4.11(6) one has  t .a/ t 0 .b/ D inf¹ss 0 W s; s 0 > 0; a 2 N .s; t /; b 2 N .s 0 ; t 0 /º  inf¹s > 0 W ab 2 N .s; t C t 0 /º D  t Ct 0 .ab/: (9) Let A be the commutative vonR Neumann subalgebra of M generated by the 1 spectral projections of jaj. Let jaj D 0  de be the spectral decomposition. Then

50

4 -Measurable operators

R1 f .jaj/ D 0 f ./ de and hence e.s;1/ .f .jaj// D ef 1 ..s;1// .jaj/ 2 A. This implies z . By (4.4) we have that f .jaj/ 2 M ®
¯  t .f .jaj// D inf kf .jaj/ek W e 2 Proj.A/;  e ?  t : When e 2 Proj.A/, note that Z Z 1 f .jaj/e D f ./ d.e e/ D f 0

1

  d.e e/ D f .jaje/;

0

where we have used f .0/ D 0 for the second equality. Therefore, kf .jaj/ek D f .kjajek/ holds so that ®
¯  t .f .jaj// D inf f .kjajek/ W e 2 Proj.A/;  e ?  t D f . t .a//: (10) For every t > 0 and " > 0, by (7) and (2) one has  tC" .a/ D  tC" .b C .a

b//   t .b/ C " .a

By (1), letting " & 0 gives  t .a/   t .b/ C ka ka bk. Hence the assertion follows.

b/   t .b/ C ka

bk:

bk and similarly  t .b/   t .a/ C 

z C by We extend the trace  on MC to a 2 M Z 1  .a/ WD  d .e /;

(4.6)

0

R1 where a D 0  de is the spectral decomposition. In fact, when a 2 MC with a D R kak  de./, since 0 2n X kkak an WD e kkak .kC1/kak % a; 2n Π2n ; 2n / kD0

we have by the normality of , n

Z kak 2 X
kkak D  d .e /:  .a/ D lim  .an / D lim  e kkak .kC1/kak Π2n ; / n!1 n!1 2n 2n 0 kD0

zC It is immediate that the extended  is positive homogeneous ( .˛a/ D ˛ .a/ for a 2 M z and ˛  0), and the additivity ( .a C b/ D  .a/ C  .b/ for a; b 2 MC ) will be shown later (Corollary 4.24). zC, Proposition 4.20. For every a 2 M Z  .a/ D

1

 t .a/ dt:

(4.7)

0

Moreover, for any continuous non-decreasing function f on Œ0; 1/ with f .0/  0, Z 1  .f .a// D f . t .a// dt: (4.8) 0

51

4.2 Generalized s-numbers

Proof. For each n 2 N define fn ./ WD

1 X k  k kC1 ./; 2n Π2n ; 2n /

an WD fn .a/ D

kD0

1 X k e k kC1 .a/: 2n Π2n ; 2n /

kD0

Then we have 1 X
k  eΠk ; kC1 / .a/ n n n 2 2 2 kD0 Z 1 Z D fn ./ d .e / %

 .an / D

0

1

 d .e / D  .a/:

0

n Since ka an k  1=2n , we have j t .a/  Proposition 4.19(10), t .an /j  1=2 by R1 R1 so  t .an / %  t .a/ for all t > 0. Hence 0  t .an / dt % 0  t .a/ dt. Assume that  .e.s;1/ .a// D 1 for some s > 0; then  .eŒ1=2n ;1/ .a// D 1 for some n. Since n n a ;1/ .a/,  .a/   .an / D 1. Since  t .an /  1=2 for all t > 0, R n1 .1=2 /eŒ1=2Rn1  .a / dt D 1. Hence (4.7) holds. So we may assume that  .a/ dt  t n t 0 0  .e.s;1/ .a// < 1 for any s > 0; then it is immediate that

 t .an / D

k 2n

if


 eΠkC1 ;1/ .a/  t <  eΠ2n

k 2n

;1/ .a/




:

Hence we have 1

Z

 t .an / dt D 0

1 X
k  eΠk ; kC1 / .a/ D  .an /: n 2n 2n 2

kD0

Letting n ! 1 gives (4.7). Furthermore, (4.8) follows from Proposition 4.19(9).



Corollary 4.21. We have z C and a  b, then  .a/   .b/, (1) if a; b 2 M z, (2)  .a a/ D  .aa / for all a 2 M z C and x 2 M . (3)  .xax  /  kxk2  .a/ for all a 2 M Proof. (1) Assume that 0  a  b. Then  t .a/   t .b/ for all t > 0 by Proposition 4.19(5). Hence  .a/   .b/ follows from (4.7). z, (2) From (4.7) one has for any a 2 M Z 1 Z 1  .a a/ D  t .a/2 dt D  t .a /2 dt D  .aa /; 0

0

where (9) and (3) of Proposition 4.19 have been used. (3) is immediate from (4.7) and Proposition 4.19(6).



52

4 -Measurable operators

z (n 2 N) be such that an ! a in the measure topology. Then Lemma 4.22. Let a; an 2 M (1)  t .a/  lim inf n!1  t .an / for all t > 0, (2)  t .a/ D limn!1  t .an / if  t .a/ is continuous at t . Proof. (1) For any " > 0, Proposition 4.19(7) implies that  t C" .a/   t .an / C " .a an /. Since " .a an / ! 0 as n ! 1 by Lemma 4.18(4), one has  tC" .a/  lim inf  t .an /: n!1

Letting " & 0 gives the assertion. (2) If 0 < " < t , then  t .an /   t and so lim supn!1  t .an /   t

" .a/.

" .a/

C " .an

a/;

Letting " & 0 gives

lim sup  t .an /   t .a/; n!1



implying the assertion.

z C (n 2 N) with an ! a in the measure topology. Then Proposition 4.23. Let a; an 2 M (1) (Fatou’s lemma)  .a/  lim inf n!1  .an /, (2) (monotone convergence theorem) if an  a for all n (in particular, this is the case zC if an is increasing), then  .a1=2 ba1=2 / D limn!1  .an1=2 ban1=2 / for all b 2 M (in particular,  .a/ D limn!1  .an /). Proof. (1) One has Z

1

 .a/ D

 t .a/ dt

(by (4.7))

Z 01 

lim inf  t .an / dt n!1 Z 1  lim inf  t .an / dt

(by Lemma 4.22(1))

0

n!1

(by Fatou’s lemma)

0

D lim inf  .an /: n!1

(2) Since b 1=2 an b 1=2 ! b 1=2 ab 1=2 in the measure topology and b 1=2 an b 1=2  b ab 1=2 , one has  .b 1=2 ab 1=2 / D limn!1  .b 1=2 an b 1=2 / from Corollary 4.21(1) and (1) above. Hence the assertion follows from Corollary 4.21(2).  1=2

z C we have Corollary 4.24. For every a; b; c 2 M  ..a C b/1=2 c.a C b/1=2 / D  .a1=2 ca1=2 / C  .b 1=2 cb 1=2 / and in particular,  .a C b/ D  .a/ C  .b/.

53

4.2 Generalized s-numbers

R1 R1 Proof. Take the spectral decompositions a D 0  de , b D 0  df and c D R1 Rn Rn Rn 0  dg . Set an WD 0  de , bn D 0  df and cn WD 0  dg for each n 2 N. Then an % a, bn % b, cn % c and so an C bn % a C b in the measure topology. Hence by Proposition 4.23(2) and Corollary 4.21(2) one has  ..a C b/1=2 c.a C b/1=2 / D lim  ..an C bn /1=2 c.an C bn /1=2 / n!1

1=2 1=2 D lim lim  .cm .an C bn /cm / n!1 m!1

1=2 1=2 1=2 1=2 D lim lim Œ .cm an c m / C  .cm bn cm /

D

n!1 m!1 lim Œ .c 1=2 an c 1=2 / C  .c 1=2 bn c 1=2 / n!1 1=2 1=2 1=2 1=2

D  .a

ca

/ C  .b

cb

/:



z with an ! a in the Proposition 4.25 (Lebesgue’s convergence theorem). Let a; an 2 M 1 measure topology. Assume that there is an f 2 L ..0; 1/; dt / such that  t .an /  f .t / z C such that  .b/ < 1 a.e. for each n. (In particular, this is the case if there is a b 2 M and jan j  b for all n.) Then lim  .jan

n!1

aj/ D 0;

lim  .jan j/ D  .jaj/:

n!1

Proof. Note that  t .an

a/   t =2 .an / C  t =2 .a/  2f .t =2/ a.e.

R1 R1 and 0 f .t =2/ dt D 2 0 f .t / dt < 1. Since  t .an a/ ! 0 for all t > 0 by Lemma it follows from (4.7) and Lebesgue’s convergence theorem that  .jan aj/ D R4.18(4), 1  .a a/ dt R! 0. Since  t .anR/  f .t / a.e. and  t .an / !  t .a/ a.e. by Lemma t n 0 1 1 4.22(2),  .jan j/ D 0  t .an / dt ! 0  t .a/ dt D  .jaj/ similarly.  The next lemma will be used in Section 9.2 to prove Proposition 9.25. z C and v 2 M be a contraction. If f is a non-negative continuous Lemma 4.26. Let a 2 M and convex function on Œ0; 1/ with f .0/ D 0, then  t .f .vav  // D f . t .vav  //   t .vf .a/v  / for all t > 0. Proof. It is clear that the function f as stated is non-decreasing on Œ0; 1/. Hence z C and the first equality follow from Proposition 4.19(9). First assume that f .vav  / 2 M R kak a is bounded with the spectral decomposition a D 0  de . For any vector  2 H ,

54

4 -Measurable operators

kk D 1, since kv  k  1 and f .0/ D 0 one finds that Z kak  h; vf .a/v i D f ./ dke v  k2 0 kak

Z D

f ./ dke v  k2 C f .0/.1

kv  k2 /

0

Z

kak

f

 dke v  k2



D f .h; vav  i/:

0

In the above, vf .a/v   and vav   make sense and the inequality follows from the convexity of f . Hence by (4.5) one has ®
¯  t .vf .a/v  / D inf sup2eH ;kkD1 h; vf .a/v  i W e 2 Proj.M /;  e ?  t ®
¯
 f inf sup2eH ;kkD1 h; vav  i W e 2 Proj.M /;  e ?  t D f . t .vav  //: R R z be arbitrary with a D 1  de . Set an WD n  de for n 2 N; Next let a 2 M 0 0 Rn C R1 then f .an / D 0 f ./ de  0 f ./ de D f .a/ and vf .an /v   vf .a/v  . From the above case one has f . t .van v  //   t .vf .an /v  /   t .vf .a/v  /;

t > 0:

Since van v  ! vav  in the measure topology, it follows from Lemma 4.22(1) that  t .vav  /  lim inf n!1  t .van v  / so that f . t .vav  //  lim inf f . t .van v  //   t .vf .a/v  / n!1

thanks to f being non-decreasing and continuous on Œ0; 1/.  Rt Rt The integral quantities 0 s .a/ ds and exp 0 log s .a/ ds play a role in non-commutative integration theory on .M;  /. In the rest of the section we will show some variational formulas for those expressions from [23, 24], which will be used in the next section. The formula in the next proposition is familiar in matrix theory as well as in classical theory. The expression in the right-hand side of (4.9) is known as the K-functional in real interpolation theory (see [7]). z and t > 0 we have Proposition 4.27. For every a 2 M Z t ® ¯ z ; a D a1 C a2 : s .a/ ds D inf  .ja1 j/ C t ka2 k W a1 ; a2 2 M

(4.9)

0

z with a D a1 C a2 , by Proposition 4.19(10) one has Proof. For every a1 ; a2 2 M s .a/  s .a1 / C ka2 k for all s > 0, so that Z t Z t s .a/ ds  s .a1 / ds C t ka2 k   .ja1 j/ C tka2 k 0

0

55

4.2 Generalized s-numbers

Rt thanks to (4.7). Therefore, 0 s .a/ ds  RHS of (4.9). Conversely, assume that Rt 0 s .a/ ds < 1 and let r WD  t .a/ R21Œ0; 1/. Take the polar decomposition a D vjaj and the spectral decomposition jaj D 0  de . Define  Z Z  de C rer? : a1 WD v . r/ de ; a2 WD a a1 D v Œ0;r

.r;1/

R

Note that ka2 k  r and ja1 j D .r;1/ .   0. By Proposition 4.19(9) one has

r/ de D f .jaj/, where f ./ WD . ´

s .a1 / D f .s .a// D .s .a/ so that by (4.7), Z  .ja1 j/ D

r/C D

1

Z s .a1 / ds D

0

Therefore, Z

which implies that

s .a/ 0

r

r/C for

.0 < s < t /; .s  t /;

t

s .a/

rt:

0

t

s .a/ ds D  .ja1 j/ C rt 0 Rt 0 s .a/ ds  RHS of (4.9).

  .ja1 j/ C tka2 k; 

z is said to be -compact if  t .a/ ! 0 as t ! 1 Definition 4.28. An operator a 2 M e D S.M e or, equivalently,  .e.s;1/ .jaj// < 1 for all s > 0. We denote by S / the e z set of -compact operators in M . It is easy to see that S is the closure of the set z W  .s.jaj// < 1º in the measure topology, where s.jaj/ is the support projection ¹a 2 M e is a two-sided ideal of M e z . Also, define S WD S\M of jaj, and that S . When M D B.H / e with  D Tr, S D S is the set of compact operators on H . Proposition 4.29. Assume that M is non-atomic (i.e., M has no minimal projection). For e and t > 0 we have every a 2 S Z t ® ¯ s .a/ ds D sup  .ejaje/ W e 2 Proj.M /;  .e/ D t : (4.10) 0

Proof. For every e 2 Proj.M / with  .e/ D t, since  .e.r;1/ .ejaje//   .e/ D t for all r > 0, note that s .ejaje/ D 0 for all s  t. Moreover, since s .ejaje/  s .a/ for all s > 0 by Proposition 4.19(6), one has Z t Z 1 s .a/ ds  s .ejaje/ ds D  .ejaje/ 0

0

Rt

thanks to (4.7). Hence, 0 s .a/ ds  RHS of (4.10). Conversely, let s0 WD  t .a/ 2 Œ0; 1/. It is easy to see that  .e.s0 ;1/ .jaj//  t   .eŒs0 ;1/ .jaj//:

56

4 -Measurable operators

Indeed, the first inequality is clear. For the second, for any s 2 .0; s0 / one has t < e Letting s % s0 gives the second inequality. Since  .e.s;1/ .jaj// < 1 since a 2 S. M is non-atomic, there is an e 2 Proj.M / such that e.s0 ;1/ .jaj/  e  eŒs0 ;1/ .jaj/ and  .e/ D t . Then e.r;1/ .ejaje/ D e.r;1/ .jaj/ for all r  s0 , which implies that s .ejaje/ D s .a/ for all s 2 .0; t / and hence Z t Z 1 s .a/  s .ejaje/ ds D  .ejaje/:  0

0

Here, recall the notion of the determinant in a finite von Neumann algebra due to Fuglede and Kadison [25]. When .M;  / is a finite von Neumann algebra with a finite trace , the Fuglede–Kadison determinant of a 2 M is defined by kak

Z .a/ WD exp  .log jaj/ D exp

log  d .e /; 0

R kak where jaj D 0  de is the spectral decomposition. In the original paper [25], M is a factor of type II1 and  is a normalized trace, but everything carries over to the case where M is a general finite von Neumann algebra with a finite (non-normalized) trace. In fact, when 1 WD = .1/ and 1 .a/ is the determinant with respect to 1 , the above .a/ is 1 .A/.1/ . The most significant property of  is the multiplicativity .ab/ D .a/.b/;

a; b 2 M:

(4.11)

Furthermore, the following is well known but we give the proof for convenience. Lemma 4.30. Assume that M is a finite von Neumann algebra with  .1/ < 1. For every a 2 M we have Z  .1/

.a/ D exp

log  t .a/ dt: 0

Proof. For any n 2 N, applying Proposition 4.19(9) to the function f ./ D log.n C 1/,   0, we note that  t .log.njaj C 1// D log.n t .a/ C 1/ for t > 0. Hence we have  .log.jaj C n

1

1// D  .log.njaj C 1//  .1/ log n Z  .1/ D  t .log.njaj C 1// dt  .1/ log n 0  .1/

Z D

log. t .a/ C n

1

/ dt;

0

and also  .log.jaj C n

1

Z 1// D

kak

log. C n

1

/ d .e /:

0

By the monotone convergence theorem, letting n ! 1 gives the result.



4.3 Lp -spaces with respect to a trace

57

Returning to the semifinite von Neumann algebra setting, we give the following. z satisfies Definition 4.31. When a 2 M a2M

or

˛

 t .a/  C t

we define Z ƒ t .a/ WD exp

.t > 0/ for some C; ˛ > 0;

(4.12)

t

log s .a/ ds;

t > 0:

0

This is well defined (i.e., 1 1 does not occur) under (4.12). In particular, when  .1/ < 1 and a 2 M , .a/ D ƒ .1/ .a/ by Lemma 4.30. Proposition 4.32. Assume that M is non-atomic. For every a 2 S and t > 0 we have ® ¯ ƒ t .a/ D sup e .ejaje/ W e 2 Proj.M /;  .e/ D t ; (4.13) where e .ejaje/ is the Fuglede–Kadison determinant of ejaje in .eM e; jeM e /. Proof. For every e 2 Proj.M / with  .e/ D t one has Z t Z t log s .a/ ds  log s .ejaje/ ds D log e .ejaje/ 0

0

by Lemma 4.30. Hence, ƒ t .a/  RHS of (4.13). Conversely, let s0 WD  t .a/ and choose an eR 2 Proj.M / as in the R t second paragraph of the proof of Proposition 4.29. Then one t has 0 log s .a/ ds D 0 log s .ejaje/ ds D log e .ejaje/ by Lemma 4.30. 

4.3 Lp -spaces with respect to a trace As in Sections 4.1 and 4.2 let M be a semifinite von Neumann algebra with a faithful z be the space of -measurable operators. We begin semifinite normal trace  and let M p with the definitions of L -norm and Lp -space associated with .M;  /. z define Definition 4.33. For each a 2 M kakp WD  .jajp /1=p 2 Œ0; 1; 0 < p < 1: R1 By (4.8) note that kakp D . 0  t .a/p dt /1=p . Furthermore, define kak1 WD kak 2 z and 0 < p  1. Œ0; 1. By Proposition 4.19(3) note that kakp D ka kp for every a 2 M The non-commutative Lp -space on .M;  / is defined as ® ¯ z W kakp < 1 ; 0 < p  1: Lp .M / D Lp .M;  / WD a 2 M e for all (In particular, L1 .M / D L1 .M;  / D M .) It is clear that Lp .M /  S p 2 .0; 1/; see Definition 4.28.

58

4 -Measurable operators

Examples 4.34. (1) When M D B.H / and  D Tr, Lp .M / is the Schatten–von Neumann p-class Cp .H / WD ¹a 2 B.H / W Tr jajp < 1º with kakp WD .Tr jajp /1=p . In particular, the case p D 1 is the trace class and the case p D 2 is the Hilbert–Schmidt class. (2) When M D L1 .X; / on a localizable measure space .X; X ; /, Lp .M / is R the usual Lp -space Lp .X; / WD ¹f W measurable; X jf jp d < 1º with kf kp WD R . X jf jp d/1=p . Lemma 4.35. Let 0 < p  1. z. (1) Lp .M / is a linear subspace of M (2) If a 2 Lp .M / and x; y 2 M , then xay 2 Lp .M / and kxaykp  kxk kyk kakp . Proof. (1) Since the case p D 1 is obvious, let 0 < p < 1 and a; b 2 Lp .M /. For ˛ 2 C, since Z 1 1=p Z 1 1=p k˛akp D  t .˛a/p dt D .j˛j t .a//p dt D j˛j kakp < 1; 0

0

p

one has ˛a 2 L .M /. Since  t .a C b/   t =2 .a/ C  t =2 .b/  2 max¹ t =2 .a/;  t =2 .b/º; one has  t .a C b/p  2p Œ t =2 .a/p C  t =2 .b/p , so that Z 1 Z 1 Z p p p p ka C bkp D  t .a C b/ dt  2  t =2 .a/ dt C D2

0 pC1

0

1

p



 t =2 .b/ dt

0

.kakpp C kbkpp / < 1:

(4.14)

Hence a C b 2 Lp .M /. (2) The case p D 1 is obvious. For 0 < p < 1, since  t .xay/  kxk kyk t .a/ by Proposition 4.19(6), we may take the pth power of both sides and integrate to have the assertion.  Lemma 4.36. Set F WD ¹x 2 MC W  .x/ < 1º D MC \ L1 .M /; N WD ¹x 2 M W  .x  x/ < 1º D M \ L2 .M /; M WD span¹y  x W x; y 2 N º: Then (1) N is a two-sided ideal of M and is closed under the adjoint , (2) M D span F ( L1 .M /) and F D MC \ M ,

4.3 Lp -spaces with respect to a trace

59

(3) jF uniquely extends by linearity to a positive linear functional on M , and the extended  satisfies  .xy/ D  .yx/ for all x; y 2 N . Proof. (1) If x; y 2 N , then .x C y/ .x C y/  2.x  x C y  y/ and so x C y 2 N . Since  .x  x/ D  .xx  /, N is closed under . If x 2 N and y 2 M , then .yx/ .yx/  kyk2 x  x, so yx 2 N . Also, xy 2 N since xy D .y  x  / . Hence N is a two-sided ideal of M . (2) For every x; y 2 M recall the polarization yx D

3 1X k i .x C i k y/ .x C i k y/: 4

(4.15)

kD0

 If x; y 2 N , then x C i k y 2 N by (1), P so that y x 2 span F . Hence M  span F . The converse is obvious. Next, if a D jnD1 yj xj 2 MC \ M with xj ; yj 2 N , then from the polarization and a D a one has n

aD

1X ¹.xj C yj / .xj C yj / 4

.xj

yj / .xj

yj /º

j D1 n



1X .xj C yj / .xj C yj /; 4 j D1

which implies that a 2 F . Hence F D MC \ M . (3) From (2) we can uniquely extend jF to a linear functional W M ! C. For every x; y 2 N , using the polarization in (4.15) we have  .xy/ D

3 1 X  ..y C x  / .y C x  // 4

 ..y

x  / .y

x  //

kD0

C i  ..y C ix  / .y C ix  // D

3 1 X  ..y C x  /.y C x  / / 4

 ..y

i  ..y

ix  / .y

x  /.y

 ix  //

x  / /

kD0

C i  ..y C ix  /.y C ix  / / D

3 1 X  ..x C y  / .x C y  // 4

 ..x

i  ..y

ix  /.y

y  / .x

 ix  / /

y  //

kD0

C i  ..x C iy  / .x C iy  //

i  ..x

iy  / .x

 iy  //

D  .yx/; as asserted.



60

4 -Measurable operators

Proposition 4.37. The functional  on M (defined in Lemma 4.36(3)) uniquely extends to a positive linear functional on L1 .M / such that j .a/j  kak1 ;

a 2 L1 .M /:

(4.16)

Moreover, if a 2 L1 .M /C , then  .a/ coincides with the definition in (4.6). Proof. Since .y; x/ 2 N  N 7!  .y  x/ is an inner product, the Schwarz inequality says that j .y  x/j   .x  x/1=2  .y  y/1=2 D kxk2 kyk2 ; x; y 2 N : Let a 2 M \ L1 .M / with the polar decomposition a D wjaj. Set x WD jaj1=2 and y WD jaj1=2 w  ; then x; y 2 M . Since  .x  x/ D  .jaj/ and  .y  y/ D  .wjajw  / D  .jaj1=2 w  wjaj1=2 / D  .jaj/, one has x; y 2 N , kxk2 D kyk2 D kak1=2 and a D 1 y  x 2 M , so j .a/j  kxk2 kyk2 D kak1 . Therefore, M \ L1 .M /  M and a 2 M \ L1 .M /: (4.17) R 1 Next let a 2 L1 .M / with a D wjaj and jaj D 0  de . For n 2 N set an WD Rn Rn R w 0  de ; then jan j D 0  de and ja an j D .n;1/  de . Hence an ! a in the measure topology and  t .an /   t .a/ 2 L1 ..0; 1/; dt /, so we have an 2 M \ L1 .M / and kan ak1 ! 0 by Proposition 4.25. Moreover, by (4.17) and (4.14) for p D 1, j .a/j  kak1 ;

j .am /

 .an /j D j .am

an /j  kam

an k1 D k.am

 4.kam

ak1 C kan

ak1 / ! 0

a/ C .a

an /k1

as m; n ! 1:

So one can define  .a/ WD lim  .an /: n!1

Since j .an /j  kan k1 by (4.17) and kan k1 ! kak1 by Proposition 4.25, (4.16) holds. The linearity of  on L1 .M / and the uniqueness of  as a linear functional with (4.16) are easy to see, so the details are omitted. Rn Finally, let a 2 L1 .M /C and set an WD 0  de 2 MC \ L1 .M / as above. Then one has Z n Z 1  .a/ D lim  .an / D lim  d .e / D  d .e /; n!1

n!1 0

0



which is the definition in (4.6). z we have Lemma 4.38. For every a; b 2 M ka C bk1  kak1 C kbk1 :

Proof. By Lemma 4.15 there are partial isometries u; v 2 M such that ja C bj  ujaju C vjbjv  , from which we have by Corollaries 4.21 and 4.24, ka C bk1 D  .ja C bj/   .ujaju / C  .vjbjv  /   .jaj/ C  .jbj/ D kak1 C kbk1 :



4.3 Lp -spaces with respect to a trace

61

z the following hold: Theorem 4.39. For every a; b 2 M Rt Rt (1) 0 s .a C b/ ds  0 .s .a/ C s .b// ds for all t > 0, Rt Rt (2) 0 .s .a C b// ds  0 .s .a/ C s .b// ds for all t > 0 and for any continuous non-decreasing convex function  on Œ0; 1/. z (i D 1; 2) such that a D a1 C a2 and b D b1 C b2 , since Proof. (1) For every ai ; bi 2 M a C b D .a1 C b1 / C .a2 C b2 /, it follows from Proposition 4.27 applied to a C b that Z t s .a C b/ ds  ka1 C b1 k1 C t ka2 C b2 k 0

 .ka1 k1 C tka2 k/ C .kb1 k1 C t kb2 k/ by Lemma 4.38. By applying Proposition 4.27 again to a and b we have the asserted inequality. (2) Write f .s/ WD s .a C b/ and g.s/ WD s .a/ C s .b/, which are non-decreasing and continuous from the right on .0; 1/. First R t consider the function R t ./ WD . ˛/C on Œ0; 1/ with ˛  0. If ˛  f .0C /, then 0 .f .s// ds D 0  0 .g.s// ds. Assume that ˛ < f .0C /. When ˛  f .t /, let t0 WD t. When f .t / < ˛ < f .0C /, one can choose a t0 2 .0; t / such that f .t0 /  ˛  f .t0 /. Then it follows that Z

t

Z

t0

.f .s// ds D

Z .f .s/

0

Z

.g.s/

0 t0



t0

˛/ ds 

˛/ ds

0

Z .g.s/

0

˛/C ds 

t

.g.s// ds; 0

where the first inequality above is due to (1). Hence we see that the asserted inequality holds when  is a function of the form ./ D ˛0 C

k X

ˇi .

˛i /;

i D1

where ˛0 2 R, 0  ˛1 < ˛2 <


< ˛k < 1 and ˇi > 0 (1  i  k). For a general function  as stated in (2) one can choose a sequence n of the above form such that n ./ % ./ for all  2 Œ0; 1/. Hence the monotone convergence theorem gives Z t Z t Z t Z t .f .s// ds D lim n .f .s// ds  lim n .g.s// ds D .g.s// ds 0

n!1 0

n!1 0

0

for all t > 0.



z, Proposition 4.40 (Minkowski’s inequality). Let 1  p  1. For every a; b 2 M ka C bkp  kakp C kbkp :

(4.18)

62

4 -Measurable operators

z Proof. Since the case p D 1 is trivial, assume that 1  p < 1. For every a; b 2 M and t > 0, by Theorem 4.39(2) we have t

Z

p

1=p

s .a C b/ ds

Z

1=p

t p



.s .a/ C s .b// ds

0

0

Z 

t

s .a/p ds

1=p

t

Z

s .b/p ds

C

0

1=p ;

0

where the last inequality above is due to the classical Minkowski inequality. Now let t ! 1 and use Proposition 4.19(9) and (4.7); then the assertion follows.  Theorem 4.41. For every p 2 Œ1; 1, Lp .M / is a Banach space with the norm k
kp . In particular, L2 .M / is a Hilbert space with the inner product ha; bi D  .a b/. Moreover, M \ L1 .M / is dense in Lp .M / for any p 2 Œ1; 1/. Proof. We may assume that 1  p < 1. From Lemma 4.35(1) and (4.18) it follows that Lp .M / is a normed space, by noting that kakp D 0 H)  t .a/ D 0 for all t > 0 H) a D 0. So it remains to prove completeness of k
kp . Let ¹an º be a Cauchy sequence in Lp .M /. By (4.8) note that kam

an kpp

D  .jam

p

1

Z

an j / D

 t .am

an /p dt:

(4.19)

0

For any ı > 0, since ıı .am

an /p 

ı

Z

 t .am Z0 1



 t .am

an /p dt an /p dt ! 0

as m; n ! 1;

0

z , so by Theorem 4.12 there is an it follows from Lemma 4.18(4) that ¹an º is Cauchy in M z a 2 M such that an ! a in the measure topology. For any " > 0 there is an n0 2 N such that kam an kp  " for all m; n  n0 . Since am an ! a an as m ! 1 in the measure topology, we have  t .am an / !  t .a an / a.e. by Lemma 4.22(2). By Fatou’s lemma and (4.19) we have Z 1 Z 1  t .a an /p dt  lim inf  t .am an /p dt  "p ; n  n0 : 0

m!1

0

Therefore, if n  n0 , then a an 2 Lp .M / and ka an kp  ", which implies that a 2 Lp .M / and ka an kp ! 0. When p D 2, it is clear that ha; bi WD  .a b/ is an inner product on L2 .M /. Since kak2 D  .jaj2 /1=2 D ha; ai , it follows that L2 .M / is a Hilbert space.

4.3 Lp -spaces with respect to a trace

63

If a 2 M \ L1 .M /, then it is clear that jajp  kakp 1 jaj and so  .jajp / < 1. p Hence M \ L1 .M /  1 and a 2 Lp .M / with R 1L .M / for all p 2 Œ1; 1/. Let 1  pR < n a D wjaj and jaj D 0  de . For each n 2 N let an WD w 1=n  de ; then Z

n

 de  np

jan j D 1=n

1

Z

n

p de  np

1

jajp ;

1=n

so that  .jan j/ < 1 and hence an 2 M \ L1 .M /. Note that Z Z ja an jp D p de C p de  jajp 2 L1 .M /; .0;1=n/

Z



.0;1=n/

p  de

1

.n;1/

 .1=n/p ! 0;

Z

 p de en D 0

.n;1/

and  .en? / ! 0. Therefore, ja an jp ! 0 as n ! 1 in the measure topology, so ka an kpp D  .ja an jp / ! 0 by Proposition 4.25. Hence the last assertion follows.  The next proposition was shown in [23], which will be used to prove Hölder’s inequality. Proposition 4.42. For every a; b 2 S (see Definition 4.28) the following hold: (1) ƒ t .ab/  ƒ t .a/ƒ t .b/ for all t > 0, Rt Rt (2) 0 .s .ab// ds  0 .s .a/s .b// ds for all t > 0 and for any continuous non-decreasing function  on Œ0; 1/ such that  7! .e  / is convex on R. Proof. (1) To prove this, we may assume without loss of generality that M is non-atomic. R1 In fact, we can always embed M into M ˝L1 .Œ0; 1; dt / with the trace  ˝ 0
dt without changing generalized s-numbers, i.e.,  t .a/ D  t .a ˝ 1/ and so ƒ t .a/ D ƒ t .a ˝ 1/. Let a D ujaj and b D vjbj be the polar decompositions. For every e 2 Proj.M / with  .e/ D t let f be the support projection s.jeb  j/. Since f D s.jeb  j/  s.jbej/  e, there is a w 2 M such that w  w D f and ww   e. Note that ejabj2 e D eb  jaj2 be D .eb  w  e/.ewjaj2 w  e/.ewbe/

(4.20)

thanks to w  ew D w  w D f and f be D be. From the multiplicativity property (4.11) it follows that e .ejabj2 e/ D e .eb  w  e/e .ewjaj2 w  e/e .ewbe/ D e .ewjaj2 w  e/e .eb  w  ewbe/ D e .ewjaj2 w  e/e .ejbj2 e/ thanks to eb  w  ewbe D eb  f be D ejbj2 e. Proposition 4.32 implies that e .ejabj2 e/  ƒ t .wjaj2 w  /ƒ t .jbj2 /  ƒ t .jaj2 /ƒ t .jbj2 /:

64

4 -Measurable operators

Since this holds for all e 2 Proj.M / with  .e/ D t, by Proposition 4.32 again we have ƒ t .jabj2 /  ƒ t .jaj2 /ƒ t .jbj2 /. Since ƒ t .jaj2 / D ƒ t .a/2 etc., the desired inequality follows. (2) Write f .s/ WD s .ab/ and g.s/ WD s .a/s .b/. First consider the function Rt ./ WD . ˛/C on Œ 1; 1/ with ˛ 2 R. If e ˛  f .0C /, then 0 .log f .s// ds D Rt 0  0 .log g.s// ds. Assume that e ˛ < f .0C /. When e ˛  f .t /, let t0 WD t. When f .t / < e ˛ < f .0C /, let t0 2 .0; t / be such that f .t0 /  e ˛  f .t0 /. Then we have Z t Z t0 .log f .s// ds D .log f .s/ ˛/ ds 0 0 Z t0 Z t  .log g.s/ ˛/ ds  .log g.s// ds; 0

0

R t0

Rt where the first inequality follows since (1) gives 0 log f .s/ ds  00 log g.s/ ds. Hence the asserted inequality holds for any ./ WD .log /,  2 Œ0; 1/ when is a function of the form k X ./ D ˛0 C ˇi . ˛i /C ;  2 Œ 1; 1/; iD1

where ˛0 2 R, 1 < ˛1 < ˛2 <


< ˛k < 1 and ˇi > 0 (1  i  k). For a general function  as stated in (2) there is a sequence n of functions of the form above such that  n ./ % .e / for all  2 Œ 1; 1/. Hence we have Z t Z t .f .s// ds D lim n .log f .s// ds n!1 0 0 Z t Z t  lim .log g.s// ds D .g.s// ds n n!1 0

0

for all t > 0.



Proposition 4.43 (Hölder’s inequality). Let p; q; r 2 .0; 1 with 1=p C 1=q D 1=r. For z we have every a; b 2 M kabkr  kakp kbkq : (4.21) Proof. The inequality is trivial if a D 0 or b D 0. Assume that a; b ¤ 0, i.e., kakp ; kbkp > 0. If kakp D 1 or kbkq D 1, thenRthe inequality is trivial R 1 again. 1 So assume that kakp < 1 and kbkq < 1. Write a D u 0  de and b D v 0  df , Rn Rn and set an WD u 0  de and bn WD v 0  df for n 2 N. Then an ; bn 2 S. Hence by Proposition 4.42(2) applied to ./ WD r on Œ0; 1/ we have for every t > 0, Z

t r

s .an bn / ds

1=r

1=r

t

Z

r



.s .an /s .bn // ds

0

0 t

Z

s .an /p ds

 0

1=p Z

t

s .bn /q ds 0

1=q

4.3 Lp -spaces with respect to a trace

1=p Z

t

Z

p



65 1=q

t q

s .a/ ds

s .b/ ds

0

0

 kakp kbkq ; where the second inequality is due to the classical Hölder inequality. Since an ! a and bn ! b in the measure topology, it follows that an bn ! ab in the measure topology. By Lemma 4.22(1) and Fatou’s lemma we furthermore have Z

t

s .ab/r ds

1=r

t

Z

s .an bn /r ds

 lim inf n!1

0

1=r  kakp kbkq :

0

Letting t ! 1 gives the result.



A complex analytic method using the three-lines theorem is another standard way of showing Hölder’s inequality, though restricted to the case r D 1. Below we include the proof for convenience. Another proof in the case r D 1. Let p; q 2 Œ1; 1 with 1=p C1=q D 1. We may assume that kakp < 1 and kbkq < 1. Since the case p D 1 holds by Lemma 4.35(2), assume that 1 < p < 1. Let a D ujaj 2 Lp .M / and b D vjbj 2 Lq .M /. For each n 2 N define n

n2 X k ek ; xn WD 2n

where ek WD eŒ

k kC1 ; / 2n 2n

where fk WD eŒ

k kC1 ; / 2n 2n

kD0

.jaj/;

n

n2 X k yn WD fk ; 2n kD0

.jbj/:

Then xn  jaj, xn ! jaj in the measure topology and yn  jbj, yn ! jbj in the measure topology. Since n2n  X k p p ek  jajp ; xn D 2n kD0

one has n

1>

 .xnp /

n2  X k p D  .ek / 2n kD0

n

so that  .ek / < 1 for 1  k  n2 . Similarly,  .fk / < 1 for 1  k  n2n . For any n fixed, take the polar decomposition uxn vyn D wjuxn vyn j, and note that n

xnpz

n2  X k pz D ek ; 2n kD1

n

ynq.1 z/

n2  X k q.1 D 2n kD1

z/

fk

66

4 -Measurable operators

are in N for 0  Re z  1. Hence by Lemma 4.36 we can define f .z/ WD  w



uxnpz vynq.1 z/




n2n  X j pz  k q.1 D 2n 2n

z/

 .w  uej vfk /;

j;kD1

which is bounded continuous on 0  Re z  1 and analytic in 0 < Re z < 1. Moreover, we find that jf .i t /j  kw  uxnipt vynq yn i qt k1  kynq k1  k jbjq k1 D kbkqq ; jf .1 C i t /j  kw  uxnp xnipt vyn i qr k1  kxnp k1  k jajp k1 D kakpp ;

t 2 R:

From the three-lines theorem it follows that jf .1=p/j  .kbkqq /1

1=p

.kakpp /1=p D kakp kbkq :

Since f .1=p/ D  .w  uxn vyn / D kuxn vyn k1 , we have kuxn vyn k1  kakp kbkq . Note that uxn vyn ! ujajvjbj D ab in the measure topology and  t .uxn vyn /   t =2 .uxn / t =2 .vyn /   t =2 .xn / t =2 .yn /   t =2 .a/ t =2 .b/; Z 1 Z 1  t =2 .a/ t =2 .b/ dt D 2  t .a/ t .b/ dt 0

0

Z 2

1

1=p Z

p

1

 t .a/ dt

q

1=q

 t .b/ dt

0

0

D 2kakp kbkq < 1: By Proposition 4.25 we obtain kabk1 D limn!1 kuxn vyn k1  kakp kbkq .



Hölder’s inequality (Proposition 4.43) is a consequence of Proposition 4.42, but as shown in [24, Thm. 4.2] the latter can in turn be extended by use of the former. This extension will be used in Section 9.2. z satisfying (4.12), assertions (1) and (2) of Proposition Theorem 4.44. For every a; b 2 M 4.42 hold. Proof. The proof of (2) from (1) is the same as that of Proposition 4.42. So we need only z let us first show that to prove (1) here. For every a; b 2 M t

Z

s .ab/r ds 0

1=r

Z  0

t

s .a/2r ds

1=2r Z

t

s .b/2r ds

1=2r ;

t > 0:

(4.22)

0

We may assume that M is non-atomic, as remarked in the proof of Proposition 4.42(1). For every e 2 Proj.M / with  .e/ D t let f and w be taken as in the proof of

4.3 Lp -spaces with respect to a trace

67

Proposition 4.42(1). From (4.20) and Proposition 4.43, for any r > 0 one has kejabj2 ekr  keb  w  ek4r kewjaj2 w  ek2r kewbek4r 1=4r 1=2r Z t 1=4r Z t Z t 4r 4r 4r s .b/ s .a/ s .b/  D

0

0

0

Z

t

s .b/4r

1=2r Z

t

s .b/4r

1=2r :

0

0

Let s0 WD  t .ab/. For any " > 0, slightly modifying the proof of Proposition 4.29, one can choose an e 2 Proj.M / such that e commutes with all spectral projections of jabj, e.s0 ;1/ .jabj/  e  e.s0 ";1/ .jabj/ and  .e/ D t. Then it is readily seen that Rt s .ejabje/  s .ab/ " for all s 2 .0; t /. This implies that Œ 0 s .ab/2r 1=r is the Rt supremum of Œ 0 s .ejabje/2r ds1=r D kejabj2 ekr over e 2 Proj.M / with  .e/ D t. Therefore, by replacing 2r with r, (4.22) holds for any r > 0. Dividing both sides by t 1=r we have  Z t 1=r  Z t 1=2r  Z t 1=2r 1 1 1 s .ab/r ds  s .a/2r ds s .b/2r ds : t 0 t 0 t 0 Letting r & 0 gives ƒ t .ab/  ƒ t .a/ƒ t .b/ from the well-known fact (see, e.g., [80, p. 71]) that 1=r  Z t   Z t 1 1 r jf .s/j ds D exp log jf .s/j ds ; lim t 0 r&0 t 0 Rt whenever 1t 0 jf .s/jr ds < 1 for some r > 0.  Proposition 4.45. Let 1  p < 1 and 1=p C 1=q D 1. If a 2 Lp .M / and b 2 Lq .M /, then ab; ba 2 L1 .M / and  .ab/ D  .ba/. Proof. By linearity we R 1may assume that Ra;1b  0. When 1 < p < 1, with the spectral decompositions a D 0  de and b D 0  df define Z n Z n an WD  de D qeŒ1=n;n .a/; bn WD  df D beŒ1=n;n .b/: 1=n

1=n

Since eŒ1=n;n .a/  np

Z

n

p de  np ap ;

1=n

one has  .eŒ1=n;n .a// < 1, and similarly  .eŒ1=n;n .b// < 1. Let e WD eŒ1=n;n .a/ _ eŒ1=n;n .b/ 2 Proj.M /; then  .e/ < 1. Since an ; bn 2 eM e  N , one has  .an bn / D  .bn an / by Lemma 4.36(3). Since kan akp ! 0 and kbn bkq ! 0, we find by (4.21) that j.an bn /

 .ab/j  j ..an  kan

a/bn /j C j .a.bn akp kbn kq C kakp kbn

b//j bkq ! 0:

68

4 -Measurable operators

Hence we have  .an bn / !  .ab/, and similarly  .bn an / !  .ba/, so  .ab/ D  .ba/. When p D 1 and q D 1, since a1=2 ; a1=2 b; ba1=2 2 L2 .M /, we have  .ab/ D  .a1=2 a1=2 b/ D  .a1=2 ba1=2 / D  .ba1=2 a1=2 / D  .ba/:



Lemma 4.46. Let 1  p  1 and 1=p C 1=q D 1. For every a 2 Lp .M /, kakp D sup¹j .ab/j W b 2 Lq .M /; kbkq  1º:

(4.23)

Proof. It follows from (4.16) and (4.21) that kakp  RHS of (4.23). To prove the converse, we may assume that a ¤ 0. Let a D wjaj be the polar decomposition of a. When p D 1, let b WD w  2 M ; then kbk1  1 and  .ab/ D  .ba/ D  .jaj/ D kak1 . When p D 1, let ˛ WD kak1 and for any " 2 .0; ˛/ choose a non-zero e 2 Proj.M / such that e  e.˛ ";˛ .jaj/ and  .e/ < 1. Let b WD  .e/ 1 ew  . Then jb  j D  .e/ 1 .ew  we/1=2 D  .e/ 1 e so that kbk1 D kb  k1 D 1. Since ejaje  .˛ "/e, we have  .ab/ D  .ba/ D  .e/ 1  .ejaje/  ˛ ". Hence (4.23) follows. When 1 < p < 1, let b WD jajp 1 w  . Since jb  j2 D jajp 1 w  wjajp 1 D 2.p 1/ jaj , we have jb  j D jajp 1 and jb  jq D jajp , so kbkq D kb  kq D kakpp=q . Since ba D jajp , it follows that  .ab/ D  .ba/ D kakpp . We may assume that kakp > 0 (i.e., a ¤ 0), and let b1 WD kakp p=q b. We then find that kb1 kq D 1 and  .ab1 / D kakp . Hence (4.23) follows.  Remark 4.47. It is immediate to show (4.18) from (4.23). Since (4.23) is a consequence of (4.21), this is a standard way to prove Minkowski’s inequality via Hölder’s. But the majorizations for the generalized s-numbers for a C b and ab given in Theorem 4.39 and Proposition 4.42 (Theorem 4.44) provide a more effective approach to Minkowski’s and Hölder’s inequalities. In particular, when M D B.H /, this approach was fully adopted in [38]. Majorization theory is a major subject in matrix theory, which is worth discussing in the setting of (semi-)finite von Neumann algebras as well (see, e.g., [43, 44]). The following are famous inequalities for the k
kp -norms with 1 < p < 1; proofs can be found in [24] and are omitted here (also see [74] for these and related inequalities). But Clarkson’s inequality will be proved in Proposition 9.25 for more general Haagerup Lp -spaces. Proposition 4.48. When 2  p < 1, for every a; b 2 Lp .M /, ka C bkpp C ka

bkpp  2p

1

.kakpp C kbkpp /

(Clarkson’s inequality).

When 1 < p  2 and 1=p C 1=q D 1, for every a; b 2 Lp .M /, ka C bkpq C ka

bkpq  2.kakpp C kbkpp /q=p

(McCarthy’s inequality).

Recall that a Banach space X is said to be uniformly convex if, for any " 2 .0; 2/, ® ¯ ı."/ WD inf 1 k xCy k W x; y 2 X; kxk D kyk D 1; kx yk  " > 0: 2 As is well known, a uniformly convex Banach space X is reflexive, i.e., X  D X. The next result follows from Proposition 4.48.

4.3 Lp -spaces with respect to a trace

69

Corollary 4.49. When 1 < p < 1, Lp .M / is uniformly convex (hence reflexive). The last theorem of the section is the Lp –Lq -duality for Lp .M /, 1  p < 1. Theorem 4.50 (Lp –Lq -duality). Let 1  p < 1 and 1=p C 1=q D 1. Then the dual Banach space of Lp .M / is Lq .M / under the duality pairing .a; b/ 2 Lp .M /Lq .M / 7!  .ab/ 2 C. Proof. First assume that 1 < p < 1 and 1=p C 1=q D 1. Define ˆW Lq .M / ! Lp .M / by ˆ.b/.a/ WD  .ab/; a 2 Lp .M /; b 2 Lq .M /: In fact, from (4.23) (with p, q exchanged) it follows that ˆ.b/ 2 Lp .M / and kˆ.b/k D kbkq . Since ˆ is clearly linear, ˆ is a linear isometry. Now let us prove that ˆ is surjective. Note that Lp .M / , as well as Lp .M /, is reflexive by Corollary 4.49. Since ˆ.Lq .M // is norm-closed in Lp .M / , we have w

w

ˆ.Lq .M // D ˆ.Lq .M // D ˆ.Lq .M // w

;

w

where Œ


 and Œ


 denote the closures with respect to the w- and w*-topologies in the Banach space sense. Let a 2 Lp .M / and assume that ˆ.b/.a/ D  .ab/ D 0 for w all b 2 Lq .M /. Then a D 0 follows from (4.23). This implies that ˆ.Lq .M // D Lp .M / , so ˆ.Lq .M // D Lp .M / . Next assume that p D 1 and q D 1. Define ‰W L1 .M / ! M  by ‰.a/.x/ WD  .ax/;

a 2 L1 .M /; x 2 M D L1 .M /:

1 Then ‰ is a linear isometry R 1 since k‰.a/k D kak R n1 thanks to (4.23). Let a 2 L .M / with a  0. Let a D 0  de and an WD 0  de . For every net ¹x˛ º in MC such that x˛ % x 2 MC , note by Proposition 4.45 that  .ax˛ / D  .a1=2 x˛ a1=2 /   .a1=2 xa1=2 / D  .ax/. Moreover, one has

 .ax/

 .ax˛ /   ..a  2kan

an /x/ C  .an .x ak1 kxk1 C

x˛ // C  ..an

 .an1=2 xan1=2 /

a/x˛ /

 .an1=2 x˛ an1=2 /:

Since an1=2 x˛ an1=2 ; an1=2 xan1=2 2 MC and an1=2 x˛ an1=2 % an1=2 xan1=2 for any n fixed, the normality of  gives  .ax/

sup  .ax˛ /  2kan ˛

ak1 kxk1 ! 0

as n ! 1:

Therefore, ‰.a/.x˛ / D  .ax˛ / %  .ax/ D ‰.a/.x/, which implies that ‰.a/ 2 M . w Hence ‰ is a linear isometry from L1 .M / to M , so ‰.L1 .M // D ‰.L1 .M // . Let x 2 M and assume that ‰.a/.x/ D  .ax/ D 0 for all a 2 L1 .M /. For every e 2 Proj.M / with  .e/ < 1, since x  e 2 L1 .M /, it follows that  .x  ex/ D 0. We can let e % 1 to w have x D 0. This implies that ‰.L1 .M // D M , so ‰.L1 .M // D M . By taking the  dual map ˆ WD ‰ we have an isometric isomorphism ˆW M D .M / ! L1 .M / and ˆ.x/.a/ D ‰.a/.x/ D  .ax/ for a 2 L1 .M / and x 2 M . 

70

4 -Measurable operators

Corollary 4.51. We have M D L1 .M / under the correspondence ' 2 M $ a 2 L1 .M / given by '.x/ D  .ax/, x 2 M . Moreover, ' 2 MC ” a 2 L1 .M /C . Proof. The first assertion was shown in the proof of the case p D 1 of Theorem 4.50. For the latter assertion, if a  0, then '.x/ D  .a1=2 xa1=2 /  0 for all x 2 MC , so '  0. Conversely, if '  0, then for every x 2 MC ,  .ax/ D '.x/ D '.x/ D  .xa / D  .a x/; which implies that a D a . For each n 2 N let en be the spectral projection of a corresponding to Œ n; 1=n. Since aen  .1=n/en , one has 0   .aen /  .1=n/ .en /, so  .en / D 0 and hence en D 0 for all n. Hence a  0.  The a in L1 .M;  / given in Corollary 4.51 is called the Radon–Nikodym derivative of ' with respect to  and denoted by d'=d . Remark 4.52. Note that the Hilbert space L2 .M / is the completion of .N ; h
;
i /. For x 2 M consider the left multiplication .x/a WD xa for a 2 L2 .M /. Since kxak2  kxk kak2 and ha; xbi D  .a xb/ D  ..x  a/ b/ D hx  a; bi for a; b 2 L2 .M /, we have .x/ 2 B.L2 .M // and .x  / D .x/ . If .x/ D 0, then .x  x/ D .x/ .x/ D 0 and for every e 2 Proj.M / with  .e/ < 1 we have 0 D he; x  xei D kxek22 and hence xe D 0. Letting e % 1 gives x D 0. Thus,  is a faithful representation of M on L2 .M /. Furthermore, we note that
.M /; L2 .M /; J D  ; L2 .M /C is the standard form of M (in particular, see Example 3.6(2) for M D B.H /). For x 2 M , J .x/J in .M /0 D J .M /J acts as the right multiplications r .x/a WD ax  , a 2 L2 .M / (indeed, J .x/Ja D J xa D ax  ). By Corollary 4.51, for every ' 2 MC the Radon–Nikodym derivative a WD d'=d satisfies '.x/ D  .ax/ D ha1=2 ; xa1=2 i for x 2 M . Thus, a1=2 D .d'=d /1=2 2 L2 .M /C is the vector representative of '.

5 Conditional expectations and generalized conditional expectations

The notion of conditional expectations is essential in classical probability theory. Its noncommutative version in von Neumann algebras was first introduced by Umegaki [102] in finite von Neumann algebras, and also discussed by Tomiyama [99, 100] in a different approach. In the first part of this chapter we present Umegaki’s and Tomiyama’s results as well as Takesaki’s famous theorem in [92] on the existence of the conditional expectation in the general von Neumann algebra case. The second part is a concise account of the generalized (weaker) conditional expectation introduced by Accardi and Cecchini [1], which always exists even when the proper conditional expectation does not.

5.1 Conditional expectations Before entering the subject of this section let us recall different positivity notions of linear maps between general C  -algebras. Definition 5.1. Let A and B be C  -algebras and ˆW A ! B be a linear map. Define the following:  ˆ is positive if ˆ.a a/  0 for all a 2 A ,  ˆ is a Schwarz map if ˆ.a a/  ˆ.a/ ˆ.a/ for all a 2 A ,  for each n 2 N, ˆ is n-positive if ˆ.n/ D ˆ ˝ idn W Mn .A / ! Mn .B/ is positive, where Mn .A / D A ˝ Mn .C/ is the C  -algebra tensor product of A with the n  n matrix algebra Mn .C/ (whose elements are represented as n  n matrices Œaij ni;j D1 of aij 2 A ) and ˆ.n/ is defined by ˆ.n/ .Œaij / WD Œˆ.aij / (of course, 1-positivity means positivity),  ˆ is completely positive if it is n-positive for every n 2 N. The following is obvious: completely positive H) n-positive H) positive. A few basic properties of the positivity notions defined above are summarized in the next proposition. Proposition 5.2. Let A and B be unital C  -algebras and ˆW A ! B be a linear map.

72

5 Conditional expectations and generalized conditional expectations

(1) If ˆ is a Schwarz map, then it is positive. If ˆ is unital and 2-positive, then it is a Schwarz map. P (2) For each n 2 N, ˆ is n-positive if and only if ni;j D1 bi ˆ.ai aj /bj  0 for all ai 2 A and bi 2 B (i D 1; : : : ; n). (3) If A or B is commutative, then any positive ˆ is completely positive. (4) If ˆ is positive, then it is bounded with kˆk D kˆ.1/k. Hence, if ˆ is unital (i.e., ˆ.1/ D 1) and positive, then kˆk D 1. The next lemma will be useful in proving the above proposition.   Lemma 5.3. For any a; b 2 B.H / with a  0, ab b1  0 in B.H ˚ H / if and only if a  b  b. Proof. If a  b  b, then 

   b b b  1 b

a b

Conversely, assume that 0

 a

1=2

0

 a b  b 1

 0 a 1 b

  b b D 0 1

1 0

 

b 0

 1  0: 0

 0. If a is invertible, then b 1

 a

1=2

0

  1 0 D 1 ba 1=2

a

1=2 

b

1

 :

It is easy to verify that this is equivalent to w WD ba 1=2 being a contraction (an exercise), which implies that bb D .wa1=2 / .wa1=2 / D a1=2 w  wa1=2  a. When a is not  aC"1  b invertible, since  0 for every " > 0, we have b  b  a C "1. Letting " & 0 b 1  gives b b  a.  Proof of Proposition 5.2. We may assume that A and B are unital C  -subalgebras of B.H / and B.K / on Hilbert spaces H , K , respectively. (1) The first assertion is obvious. Assume that ˆ is unital and 2-positive. For      a/ ˆ.a/  any a 2 A , since aaa a1  0 by Lemma 5.3, we have ˆ.a  0 (here ˆ.a/ 1 ˆ.a / D ˆ.a/ for positive ˆ is standard), so that ˆ.a a/  ˆ.a/ ˆ.a/ by Lemma 5.3 again. P  (2) Since Œaij  Œaij  D nkD1 Œaki akj ni;j D1 for Œaij ni;j D1 2 Mn .A /, we see that ˆ  n is n-positive if and only if Œˆ.ai aj /i;j D1  0 for all ai 2 A (i D 1; : : : ; n). If ˆ is n-positive, then n X i;j D1

2 3 2 3 b1 b1 6 : 7 6 :: 7     bi ˆ.ai aj /bj D 4 : 5 ˆ.ai aj / 4 :: 5  0 bn

bn

73

5.1 Conditional expectations

for all ai 2 A and bi 2 B. Conversely, assume that the above inequality holds for all ai , bi . For any cyclic representation ¹0 ; K0 ; 0 º of B one has * ! + n n X X ˝ ˛    0 .bi /0 ; 0 .ˆ.ai aj //.bj /0 D 0 ; 0 bi ˆ.ai aj /bj 0  0 i;j D1

i;j D1

for all bi 2 B. This implies that .0 ˝ D Œ0 .ˆ.ai aL j //  0. Note that the representation B in B.K / is represented as the direct sum  D k L k of cyclic representations ¹k ; Kk ; k º of B. Then it is immediate to see that  z D k .k ˝ idn / is a faithful representation of Mn .B/. From the above discussion it follows that  z .Œˆ.ai aj //  0, and hence Œˆ.ai aj /  0. (3) Assume that A is commutative; then by the Gelfand–Naimark theorem we may write A D C.X /, the complex continuous functions on a compact Hausdorff space X. As in the proof of (2) it suffices to show that n X ˝ ˛ i ; ˆ.fi fj /j  0 idn /.Œˆ.ai aj //

i;j D1

for all fi 2 C.X / and i 2 K (i D 1; : : : ; n). By the Riesz–Markov theorem there are Radon measures ij (i; j D 1; : : : ; n) on the Borel space .X; BX / such that hi ; ˆ.f /j i D R f d ij for all f 2 C.X /. Choose a positive Radon measure  on .X; BX / such that X ij   (absolutely continuous) for all i; j , and let ij D dij =d 2 L1 .X; / be the Radon–Nikodym derivatives. For every c1 ; : : : ; cn 2 C, since * n !+ Z Z n n n X X X X ci cj ij d D ci cj f dij D ci i ; ˆ.f / cj j 0 f
X

i;j D1

X

i;j D1

iD1

j D1

Pn

for all f 2 C.X /C , we have i;j D1 ci cj ij .x/  0 for -a.e. x 2 X. This implies that Pn i;j D1 fi .x/fj .x/ij .x/  0 for -a.e. x 2 X, and hence Z X n n X ˝ ˛ i ; ˆ.fi fj /j D fi .x/fj .x/ij .x/ d.x/  0: X i;j D1

i;j D1

Next assume that B is commutative, so we write B D C.X / as above. For every ai 2 A and fi 2 C.X / (i D 1; : : : ; n) and every x 2 X one has ! n n X X  fi ˆ.ai aj /fj .x/ D fi .x/ˆ.ai aj /.x/fj .x/ i;j D1

i;j D1

D

n X


ˆ fi .x/fj .x/ai aj .x/

i;j D1

Dˆ

n X i D1

and hence ˆ is n-positive by (2).

! fi .x/ai

n X j D1

!! fj .x/aj

.x/  0;

74

5 Conditional expectations and generalized conditional expectations

(4) Assume first that ˆ is unital and positive. For each unitary u 2 A let A0 be the commutative C  -subalgebra of A generated by u, 1. Since ˆjA0 is completely positive    1 ˆ.u/  by (3) and u1 u1  0 by Lemma 5.3, one has ˆ.u/  0. Hence by Lemma 5.3 1  again one has ˆ.u/ ˆ.u/  1 and so kˆ.u/k  1. The Russo–Dye theorem says that the unit ball ¹a 2 A W kak  1º is the norm-closed convex hull of the unitaries of A ; see, e.g., [69, Prop. 1.1.12]. It thus follows that kˆ.a/k  1 for all a 2 A , kak  1, so that kˆk D 1. Next let ˆ be a positive map. When ˆ.1/ is invertible, define a unital map O O D 1 so ˆ.a/ WD ˆ.1/ 1=2 ˆ.a/ˆ.1/ 1=2 , a 2 A . The previous case implies that kˆk that 1=2 O O kˆ.a/k D kˆ.1/1=2 ˆ.a/ˆ.1/ k  kˆ.1/k kˆ.a/k  kˆ.1/k kak: Hence kˆk D kˆ.1/k follows. When ˆ.1/ is not invertible, define ˆ" .a/ WD ˆ.a/ C "!.a/1, where " > 0 and ! is a state of A . Since ˆ" .1/ is invertible, one has kˆ" .a/k  kˆ" .1/k kak. Letting " & 0 gives kˆ.a/k  kˆ.1/k kak, as desired.  Concerning completely positive maps, the most important result is the Stinespring representation theorem, that is, for any completely positive map ˆW A ! B.H /, there exist a representation ¹; K º of A and a bounded operator V W H ! K such that ˆ.a/ D V  .a/V for all a 2 A . Under the minimality condition .A /V H D K , the triplet ¹; K ; V º is unique up to a unitary conjugation. Moreover, ˆ is unital ” V is an isometry. See [68, Chap. 4] for the proof of the Stinespring theorem and other dilation theorems. Now we turn to the subject of the present section. Let us first recall the classical notion of conditional expectations, which is essential in probability theory. Let .X; X ; / be a probability space and Y be a sub--algebra of X . For every f 2 L1 .X; X ; / we R R have a unique fz 2 L1 .X; Y ; / such that B fz d D B f d for all B 2 Y , which is called the conditional expectation of f with respect to Y and denoted by E .f jY /. If f 2 L1 .X; X ; / and g 2 L1 .X; Y ; /, then E .fgjY / D E .f jY /g. In [102] Umegaki introduced the notion of conditional expectations in the finite von Neumann algebra setting as follows. Let M be a von Neumann algebra with a faithful normal tracial state . In this case, by Hölder’s inequality in (4.21) the following are easy to show (exercises): M  Lp .M;  /  L1 .M;  /;

1 < p < 1;

kak1  kakp  kak1 ;

z ; 1 < p < 1: a2M

Theorem 5.4 (Umegaki). Let .M;  / be as stated above and let N be a von Neumann subalgebra of M . (1) There exists a unique linear map E W M ! N such that (i) E .y/ D y for all y 2 N , (ii) E .y1 xy2 / D y1 E .x/y2 for all x 2 M and y1 ; y2 2 N , (iii)  .E .x// D  .x/ for all x 2 M .

75

5.1 Conditional expectations

Moreover, E also satisfies (iv) kE .x/k  kxk for all x 2 M , (v) E is completely positive, in particular, E .x  x/  E .x/ E .x/ for all x 2 M, (vi) E .x  x/ D 0 H) x D 0, (vii) E is normal, i.e., x˛ % x in MC H) E .x˛ / % E .x/. (2) The map E W M ! N uniquely extends to a linear map E W L1 .M;  / ! L1 .N;  / D L1 .N; jN / satisfying (i)0 (ii)0 (iii)0 (iv)0

E .b/ D b for all b 2 L1 .N;  /, E .y1 ay2 / D y1 E .a/y2 for all a 2 L1 .M;  / and y1 ; y2 2 N ,  .E .a// D  .a/ for all a 2 L1 .M;  /, kE .a/k1  kak1 for all a 2 L1 .M;  /.

(3) When restricted to the Hilbert space L2 .M;  /, E is the orthogonal projection from L2 .M;  / onto the closed subspace L2 .N;  / D L2 .N; jN /. Furthermore, when 1 < p < 1, kE .a/kp  kakp for all a 2 Lp .M;  /. (4) Let p; q1 ; q2 ; r 2 Œ1; 1 be such that 1=pC1=q1 C1=q2 D 1=r. Then E .b1 ab2 / D b1 E .a/b2 holds in Lr .N;  / for all a 2 Lp .M;  / and bi 2 Lqi .N;  / (i D 1; 2). Proof. (1) For every x 2 M define 'x .b/ WD  .xb/ for b 2 L1 .N;  /. Since j'x .b/j  kxbk1  kxkkbk1 , it follows that 'x 2 L1 .N;  / D N by Theorem 4.50 or Corollary 4.51, so there is a unique yx 2 N with kyx k  kxk such that 'x .b/ D  .yx b/ for all b 2 L1 .N;  /. By letting E .x/ WD yx we have a map E W M ! N , which is clearly linear and satisfies (i)–(iii). Indeed, (i) and (iii) are obvious and (ii) follows since, for x 2 M and y1 ; y2 2 N ,  .y1 xy2 b/ D  .xy2 by1 / D  .E .x/y2 by1 / D  .y1 E .x/y2 b/;

b 2 L1 .N;  /:

To show the uniqueness, assume that EW M ! N is a linear R 1map satisfying (i)–(iii). Then for every x 2 M and b 2 L1 .N;  / with b D wjbj D w 0  de , we have  .xb/ D lim  .xyn / D lim  .E.xyn // D lim  .E.x/yn / D  .E.x/b/; n!1

n!1

n!1

Rn

where yn WD w 0  de 2 N . Hence E D E follows. We show (iv)–(vii): (iv) is kyx k  kxk noted above. From Corollary 4.51 it follows that E is positive. Furthermore, for every xi 2 M and yi 2 N (1  i  n) we have n X i;j D1

yi E .xi xj /yj D

n X

E .yi xi xj yj /

i;j D1

D E

n X i D1

! xi yi

n X j D1

!! xj yj

 0;

(5.1)

76

5 Conditional expectations and generalized conditional expectations

which implies (v) thanks to Proposition 5.2(2). If E .x  x/ D 0, then  .x  x/ D  .E .x  x// D 0 so that x D 0. If x˛ % x in MC , then E .x˛ / % y  E .x/ for some y 2 NC . Since  .x˛ / D  .E .x˛ // %  .y/, we have  .y/ D  .x/ D  .E˛ .x//, so y D E .x/. Hence E .x˛ / % E.x/ follows. (2) We first prove that kE .x/k1  kxk1 for all x 2 M . For x 2 M let x D vjxj and E .x/ D wjE .x/j be the polar decompositions. We have kE .x/k1 D  .w  E .x// D  .E .w  x// D  .w  x/ D  .w  vjxj/  kw  vk kxk1  kxk1 thanks to (4.16) and Lemma 4.35(2). Since M is dense in L1 .M / by Theorem 4.41, we can uniquely extend E to E W L1 .M;  / ! L1 .N;  / by continuity. Then (i)0 –(iv)0 are verified by simple arguments taking limits, so we omit the details. (3) For every x 2 M and y 2 N note that kE .x/k2 D  .E .x/ E .x//1=2   .E .x  x//1=2 D  .x  x/1=2 D kxk2 ; hy; E .x/i D  .y  E .x// D  .y  x/ D hy; xi : Since M and N are dense in L2 .M;  / and L2 .N;  / respectively, we see by limit-taking arguments that the E extended to L1 .M;  / in (2) maps L2 .M;  / to L2 .N;  /, and the above inequality and equality are extended to all x 2 L2 .M;  / and y 2 L2 .N;  /. Hence the first assertion follows. Now let 1 < p < 1 and 1=p C 1=q D 1. For every a 2 M and b 2 N , by Lemma 4.46, j .E .a/b/j D j .E .ab//j D j .ab/j  kakp kbkq ; which can be extended to all b 2 Lq .M;  / by Theorem 4.41. Hence kE .a/kp  kakp for all a 2 M . Finally, using Theorem 4.41 again one can extend this to all a 2 Lp .M;  /. (4) follows by a limit-taking argument from (3), Theorem 4.41 and Proposition 4.43, whose details are left as an exercise for the reader.  The map E given in Theorem 5.4 is called the conditional expectation from M onto N with respect to . In [99, 100] Tomiyama characterized conditional expectations from a unital C  -algebra onto a C  -subalgebra in terms of norm-one projections. Theorem 5.5 (Tomiyama). Let A be a C  -algebra with a unit 1 and B be a C  subalgebra of A with 1 2 B. If EW A ! B is a norm-one projection, then it satisfies properties (ii) and (v) of Theorem 5.4. Proof. For simplicity let us prove the case where A is a von Neumann algebra and B is a unital von Neumann subalgebra of A . Consider the dual map E  W B  ! A  . If 2 B  and  0, then .E  /.1/ D .E.1// D .1/ D k k  kE  k, which  implies that E  0 (see, e.g., [69, Prop. 3.1.4]). Hence for every x 2 AC we have

77

5.1 Conditional expectations

.E.x// D .E  /.x/  0 for all positive 2 B  , so E.x/  0 follows. To prove (ii) it suffices to show that E.px/ D pE.x/ and E.xp/ D E.x/p for all x 2 AC and p 2 Proj.B/. We may and do assume that 0  x  1. Since 0  E.pxp/  E.p/ D p, one has

(5.2) E.pxp/ D pE.pxp/p; similarly E p ? xp ? D p ? E p ? xp ? p ? : Let y WD E.pxp ? /. For any  2 C and any  2 pH with kk D 1, one has


2 jh; yi C j2  ky C pk2 D E pxp ? C p

2

 pxp ? C p D pxp ? C p p ? xp C p

D pxp ? xp C jj2 p  1 C jj2 ; so that
jh; yij2 C 2 Re h; yi C jj2  1 C jj2 ; which implies that h; yi D 0 for all  2 pH . Therefore, pyp D 0. By a similar argument with  2 p ? H , one has p ? yp ? D 0 as well, so y D pyp ? Cp ? yp D pyCyp. Hence, for any  > 0 one has





. C 1/ p ? yp  pyp ? C . C 1/p ? yp D py C yp C p ? yp


D y C p ? yp D E pxp ? C p ? yp



¯ ®  pxp ? C p ? yp D max pxp ? ;  p ? yp ; which implies that p ? yp D 0. Since pyp D p ? yp ? D p ? yp D 0, we find that y D pyp ? . Since
E.x/ D E.pxp/ C y C y  C E p ? xp ? ; we have by (5.2), pE.x/p ? D pyp ? C p ? yp





D y D E pxp ? ;

pE.x/p D E.pxp/;

so that
E.px/ D E pxp ? C E.pxp/ D pE.x/p ? C pE.x/p D pE.x/: By taking adjoints we have E.xp/ D E.x/p too, and therefore E satisfies Theorem 5.4(ii). Then (v) is shown in the same way as in (5.1).  Takesaki [92] presented a necessary and sufficient condition for the existence of the conditional expectation onto a von Neumann subalgebra with respect to a given faithful normal state (or weight). Theorem 5.6 (Takesaki). Let M be a von Neumann algebra and N be a von Neumann subalgebra of M . Let ! be a faithful normal state on M . Then the following conditions are equivalent:

78

5 Conditional expectations and generalized conditional expectations

(a) N is globally invariant under the modular automorphism group  t! , i.e.,  t! .N / D N for all t 2 R; (b) there exists a conditional expectation (i.e., a normal norm-one projection) EW M ! N such that ! D ! ı E on M . The equivalence of (a) and (b) was more generally proved in [92] when ! is a faithful semifinite normal weight on M which is also semifinite on N . In this case, ! D ! ı E in (b) is understood to hold on M! , where M! is defined as in Lemma 4.36 with ! in place of  (also see Definition 6.1). Proof. By taking the GNS representation of M with respect to !, we may assume that M is represented on a Hilbert space H with a cyclic and separating vector  for M such that !.x/ D h; xi, x 2 M . Consider the modular operator  and the modular conjugation J with respect to !, so that S D J1=2 (see Section 2.1). Let HN WD N  and P be the projection onto HN ; then P 2 N 0 and  is a cyclic and separating vector for NP Š N jHN representing !jN . So the modular operator N and the modular conjugation JN with respect to !jN (or  2 HN ) are given on HN . Assume (a); then  t!jN D  t! jN is seen from the KMS condition (Theorem 2.14). Hence for every y 2 N and t 2 R, i t y  D i t y

it

 D  t! .y/ D  t!jN .y/ D iNt y ;

which implies that iNt D i t jHN , t 2 R, so N D jHN . Moreover, for every y 2 N ,  1=2 JN 1=2 y  D J1=2 N y  D y  D J N y ;

which implies that JN D J jHN and JP D PJ . Now let x 2 M and set x WD PJ x 2 HN . Since P 2 N 0 and J xJ 2 M 0 , one has for any y 2 N , kyx k D kP yJ xJ k D kPJ xJy k  kxk ky k: Hence one can define an x 0 2 B.HN / with kx 0 k  kxk so that x 0 .y / D yx for all y 2 N . Then it is easy to verify that x 0 2 .N jHN /0 . Moreover, one has P x D JN JP x D JN PJ x D JN x D JN x 0  D JN x 0 JN : Since JN x 0 JN 2 JN .N jHN /0 JN D N jHN by Tomita’s theorem (Theorem 2.2) for N jHN , there exists an E.x/ 2 N such that JN x 0 JN D E.x/jHN and so kE.x/k D kx 0 k  kxk and P x D E.x/. Since  is separating for N jHN , E.x/ 2 N is uniquely determined by the equality E.x/ D P x, so that EW M ! N is a linear map and E.y/ D y for all y 2 N . Furthermore, for every x 2 M , !.E.x// D h; E.x/i D h; P xi D h; xi D !.x/: Therefore, E is a norm-one projection onto N such that ! D ! ı E, which also verifies the normality of E. Thus (b) follows.

79

5.2 Generalized conditional expectations

Conversely, assume (b). For every x 2 M and y 2 N , hy ; E.x/i D !.y  E.x// D !.y  x/ D hy ; xi D hy ; P xi: Hence E.x/ D P x and moreover, PS x D P x   D E.x  / D SE.x/ D SP x;

x 2 M:

This implies that PS  SP and so .1 2P /S  S.1 2P /. Since 1 2P is a selfadjoint unitary, one has .1 2P /S D S.1 2P / and S  .1 2P / D .1 2P /S  . Since  D S  S , it follows that  D .1 2P /.1 2P / so that i t D .1 2P /i t .1 2P /. Therefore, one has .1 2P /i t D i t .1 2P / so that i t P D P i t for all t 2 R. For every y 2 N ,  t! .y/ D i t P y  D P i t y  D P  t! .y/ D E. t! .y//; implying that  t! .y/ D E. t! .y// 2 N , t 2 R. Hence (a) follows.



5.2 Generalized conditional expectations Theorem 5.6 says that the existence of the conditional expectation with respect to a nontracial normal state is rather restrictive. But there is a weaker and generalized notion of conditional expectations introduced by Accardi and Cecchini [1], which can be defined for any von Neumann subalgebra and for any faithful normal state. First we consider the general situation that M and N are von Neumann algebras and

W N ! M is a unital (i.e., .1/ D 1) positive linear map. Let a faithful ! 2 MC be given, and assume that !0 WD ! ı is normal and faithful on N . In this case, is automatically normal and faithful (i.e., .y  y/ D 0 H) y D 0). We may assume that M and N are represented on H and H0 with respective cyclic and separating vectors  and 0 satisfying !.x/ D h; xi

.x 2 M /;

!0 .y/ D h0 ; y 0 i .y 2 N /:

Theorem 5.7 (Accardi and Cecchini). In the above situation, there exists a unique positive linear map 0 W M 0 ! N 0 such that h .y/; x 0 i D hy 0 ; 0 .x 0 /0 i

for all y 2 N and x 0 2 M 0 :

Moreover, (1) 0 is unital, normal and faithful, (2) is completely positive if and only if so is 0 .

(5.3)

80

5 Conditional expectations and generalized conditional expectations

Proof. For every x 0 2 MC0 define and for every y 2 NC , x 0 .y/

x 0 .y/

WD hx 0 ; .y/i for y 2 N . Then

x0

2 NC

D kx 01=2 .y/1=2 k2  kx 0 k k .y/1=2 k2 D kx 0 k!. .y// D kx 0 k!0 .y/:

Hence there is a unique 0 .x 0 / 2 NC0 such that

x 0 .y/

D h0 ; 0 .x 0 /y 0 i, that is,

h .y/; x 0 i D hy 0 ; 0 .x 0 /0 i;

y 2 N:

(5.4)

By linearly extending 0 , for every x 0 2 M 0 there is a unique 0 .x 0 / 2 N 0 for which (5.4) holds. Then it is clear that 0 W M 0 ! N 0 is linear, positive and unital. Let ¹x˛0 º be a net in MC0 with x˛0 % x 0 2 MC0 . Then 0 .x˛0 / % y 0  0 .x 0 / for some y 0 2 NC0 . For every y 2 N, hy 0 ; y 0 0 i D limhy 0 ; 0 .x˛0 /0 i D limh .y/; x˛0 i ˛

˛ 0

0

D h .y/; x i D hy 0 ; .x 0 /0 i; which implies that y 0 0 D 0 .x 0 /0 and so y 0 D 0 .x 0 / since 0 is separating for N 0 . Hence 0 is normal. If x 0 2 MC0 and 0 .x 0 / D 0, then h; x 0 i D h .1/; x 0 i D h0 ; 0 .x 0 /0 i D 0, so x 0 D 0. Hence 0 is faithful. Finally, let xi0 2 M 0 and yi 2 N for 1  i  n. Note that n X

hyi 0 ; 0 .xi0 xj0 /yj 0 i D

i;j D1

n X

hyj yi 0 ; 0 .xi0 xj0 /0 i

i;j D1

D

n X

h .yj yi /; xi0 xj0 i

i;j D1

D

n X

h .yj yi /xi0 ; xj0 i;

i;j D1



which shows the assertion in (2).

In the situation of Theorem 5.7, let J and J0 be the modular conjugations with respect to  and 0 , respectively. One can transform 0 W M 0 ! N 0 into  W M ! N by defining

 WD j0 ı 0 ı j , where j WD J
J and j0 WD J0
J0 , so Theorem 5.7 is reformulated as follows. Corollary 5.8. In the situation of Theorem 5.7, there exists a unique unital normal positive map  W M ! N such that hJ x; .y/i D hJ0  .x/0 ; y 0 i

for all x 2 M and y 2 N :

We have !0 ı  D !, and is completely positive if and only if so is  .

(5.5)

81

5.2 Generalized conditional expectations

In fact, (5.5) is a restatement of (5.3), and !0 ı  D ! follows by letting y D 1 in (5.5). Definition 5.9. The map  (also denoted, more explicitly, by ! ) is called the !-dual map of , which is often also called the Petz recovery map because in [72, 73, 49] Petz successfully utilized the map  in the reversibility (or recovery) theorem for quantum operations in von Neumann algebras. Note that the correspondence between W N ! M and  W M ! N (determined by (5.5)) is completely dual so that is the !0 -dual of  . Now assume that W N ! M is a Schwarz map, i.e., .y  y/  .y/ .y/ for all y 2 N . Since ky 0 k2 D !0 .y  y/ D !. .y  y//  !. .y/ .y// D k .y/k2 ;

y 2 N;

(5.6)

we have a linear contraction V W H0 ! H by extending the operator given by Vy 0 D .y/;

y 2 N:

(5.7)

Then we have the following lemma. Lemma 5.10. Let W N ! M be a unital Schwarz map and V W H0 ! H be as given above. Let 0 W M 0 ! N 0 be as given in Theorem 5.7. Then the following conditions are equivalent: (i) V is an isometry; (ii) is a *-isomorphism from N into M ; (iii) 0 .x 0 / D V  x 0 V for all x 0 2 M 0 . Proof. (ii) ” (iii). Note that for every x 0 2 M 0 and y1 ; y2 2 N , hy1 0 ; 0 .x 0 /y2 0 i D hy2 y1 0 ; 0 .x 0 /0 i D h .y2 y1 /; x 0 i; hy1 0 ; V  x 0 Vy2 i D h .y1 /; x 0 .y2 /i D h .y2 / .y1 /; x 0 i: Hence (iii) holds if and only if .y2 y1 / D .y2 / .y1 / for all y1 ; y2 2 N , which is equivalent to (ii) since is faithful. (ii) H) (i) is obvious since the inequality in (5.6) becomes an equality. (i) H) (iii). For every x 0 2 MC0 and y 2 N one has hy 0 ; . 0 .x 0 /

V  x 0 V /y 0 i D hy  y 0 ; 0 .x 0 /0 i 

0

D h .y y/; x i D h. .y  y/

hVy 0 ; x 0 Vy 0 i

h .y/; x 0 .y/i

.y/ .y//; x 0 i  0;

which implies that ˆW M 0 ! B.H0 / defined by ˆ.x 0 / WD 0 .x 0 / V  x 0 V is a positive map. Assume (i); then ˆ.1/ D 0. Hence by Proposition 5.2(4) one has ˆ.x 0 / D 0 for all x 0 2 M 0 , i.e., (iii) holds. 

82

5 Conditional expectations and generalized conditional expectations

In particular, assume that N is a von Neumann subalgebra of M and W N ,! M is the injection. Let !0 WD !jN and we may take 0 D  and H0 D N   H . In this case, the isometry V given by (5.7) is the injection H0 ,! H , so that V  is the projection from H onto H0 . By specializing Corollary 5.8 with Lemma 5.10 we have the following. Corollary 5.11. Let M be a von Neumann algebra on H with a cyclic and separating vector , and !.x/ D h; xi, x 2 M . For every von Neumann subalgebra N of M , let P be the projection from H onto N . Let J and JN be the respective modular conjugations for M and N with respect to . Then the !-dual map E! WD ! W M ! N of the injection W N ,! M is explicitly given as E! .x/ D JN PJ xJPJN D JN PJ xJJN ;

x 2 M;

(5.8)

which satisfies ! ı E! D !. Definition 5.12. The map E! W M ! N given in Corollary 5.11 is called the !-conditional expectation or the generalized conditional expectation with respect to !, which is a weaker notion of conditional expectations due to Accardi and Cecchini [1] For further discussions we recall a few notions related to a Schwarz map between C  -algebras. Definition 5.13. Let A and B be unital C  -algebras. (1) For a unital Schwarz map W B ! A , the multiplicative domain of is defined as M WD ¹y 2 B W .y  y/ D .y/ .y/; .yy  / D .y/ .y/ º: (2) For a unital Schwarz map from A into itself, the fixed-point set of is defined as F WD ¹x 2 A W .x/ D xº: The next result was first shown by Choi [10] when is a unital 2-positive map. The proof below is from [1]. Lemma 5.14. Let be as in Definition 5.13(1). For any y 2 B,

.y  y/ D .y/ .y/ ” .by/ D .b/ .y/ for all b 2 B;

(5.9)

.yy / D .y/ .y / ” .yb/ D .y/ .b/ for all b 2 B:

(5.10)





Consequently, M D ¹y 2 B W .by/ D .b/ .y/; .yb/ D .y/ .b/ for all b 2 Bº; and it is a unital C  -subalgebra of B. If A , B are von Neumann algebras and is normal, then M is a von Neumann subalgebra of B.

5.2 Generalized conditional expectations

83

Proof. Let R WD ¹y 2 B W .y  y/ D .y/ .y/º and L WD ¹y 2 B W .yy  / D

.y/ .y/ º; then R  D L . Since (5.10) follows from (5.9) immediately, we will prove (5.9) only. Define D.b1 ; b2 / WD .b1 b2 / .b1 / .b2 / for b1 ; b2 2 B, which is sesquilinear on B  B and satisfies D.b; b/ D .b  b/ .b/ .b/  0 since is a Schwarz map. Assume that y 2 R , i.e., D.y; y/ D 0. For any 2 AC and b 2 B, since ı D is a positive sesquilinear form, the Schwarz inequality gives j .D.b; y//j 

.D.b; b//1=2 .D.y; y//1=2 D 0:

Hence .D.b; y// D 0 for all 2 AC and so D.b; y/ D 0 for all b 2 B, i.e., the right-hand side of (5.9) holds. The converse follows by taking b D y  in (5.9). From (5.9) and (5.10) it is easy to see that R and L are unital algebras, so M D R \ L is a unital C  -subalgebra of B. The last assertion is clear.  The set F is not generally a subalgebra of A , and there are no general inclusion relations between F and M ; see [42, App. B]. But we have the next result, which seems to have been first observed in [62] (see also [1], [6, Thm. 2.3]). Lemma 5.15. Let be as in Definition 5.13(2). Assume that there exists a faithful ! 2 AC such that ! ı D !. Then F D ¹x 2 A W .xa/ D x .a/; .ax/ D .a/x for all a 2 A º  M ;

(5.11)

and hence F is a unital C  -subalgebra of A . If A is a von Neumann algebra and is normal, then F is a von Neumann subalgebra of A . Proof. It is obvious that F  the set in the middle of (5.11). Conversely, assume that x 2 F . Then x  x D .x/ .x/  .x  x/ and !. .x  x/ x  x/ D !.x  x/ !.x  x/ D 0, implying .x  x/ D x  x D .x/ .x/. Similarly, .xx  / D xx  D

.x/ .x/ . Therefore, x 2 M and, by Lemma 5.14, we have .xa/ D .x/ .a/ D x .a/ and .ax/ D .a/ .x/ D .a/x for all a 2 A . Hence (5.11) has been shown. The remaining assertions are now clear.  Now we return to the situation of Corollary 5.11 and state the result concerning the fixed points of the !-conditional expectation. Theorem 5.16 (Accardi and Cecchini). Let N  M and ! be as in Corollary 5.11 and E! W M ! N be the !-conditional expectation. For any y 2 N the following conditions are equivalent: (i) E! .y/ D y; (ii) E! .yx/ D yE! .x/ and E! .xy/ D E! .x/y for all x 2 M ; (iii)  t! .y/ 2 N for all t 2 R; (iv)  t! .y/ D  t!jN .y/ for all t 2 R.

84

5 Conditional expectations and generalized conditional expectations

Proof. (i) ” (ii) holds by Lemma 5.15. p 2 (iii) H) (iv). Assume (iii). For each n 2 N let fn .z/ WD n e nz (z 2 C), which is an entire function. Define Z 1 Z 1 ! yn .z/ WD fn .s z/s .y/ ds; bn .z/ WD fn .t z/ t!jN .b/ dt 1

1

for any b 2 N . Then yn .z/ (resp., bn .z/) is an entire analytic M -valued (resp., N valued) function; see the proof of Lemma 2.13. For every r 2 R let y1 WD r! .y/ and b1 WD r!jN .b/. Note that Z 1 Z 1 yn .r/ D fn .s r/s! .y/ ds D fn .s/s! .y1 / ds; Z

1 1

yn .r C i / D

1

fn .s

r

1

i /s! .y/ ds

Z

1

D

fn .s 1

i /s! .y1 / ds;

and similarly for bn .r/ and bn .r C i /. From the KMS condition (see Definition 2.11 and [9, Prop. 5.3.12]) it follows that Z 1 Z 1 !.bn .r/yn .r// D fn .s/ fn .t /!. t!jN .b1 /s! .y1 // dt ds 1 1 Z 1 Z 1 D fn .s/ fn .t i /!.s! .y1 / t!jN .b1 // dt ds 1 1 Z 1 Z 1 D fn .t i / fn .s/!.s! .y1 / t!jN .b1 // ds dt 1 1 Z 1 Z 1 D fn .t i / fn .s i /!. t!jN .b1 /s! .y1 // ds dt 1

1

D !.bn .r C i /yn .r C i //;

r 2 R:

This implies that the entire function !.bn .z/yn .z// has period i , so it is bounded on C. The Liouville theorem yields that !.bn .z/yn .z//  !.bn .0/yn .0//, so that Z 1Z 1 fn .s/fn .t /!. t!jN .r!jN .b//s! .r! .y/// ds dt 1 1 Z 1Z 1 D fn .s/fn .t /!. t!jN .b/s! .y// ds dt; r 2 R: (5.12) 1

1

R1

Since limn!1 1 fn .s/.s/ ds D .0/ for any bounded continuous function  on R, it follows that Z 1 Z 1 fn .s/s! .r! .y// ds ! r! .y/; fn .t / t!jN .r!jN .b// dt ! r!jN .b/ 1

1

strongly. Therefore, letting n ! 1 for both sides of (5.12) we arrive at !.r!jN .b/r! .y// D !.by/;

i.e., hiNr b  ; i r y i D hb  ; y i

85

5.2 Generalized conditional expectations

for all r 2 R and b 2 N , where  and N are the respective modular operators for M and N with respect to . Since (iii) implies that i r y  D r! .y/ 2 N , it follows that hb  ; Ni r .i r y /i D hb  ; y i for all b 2 N , so that Ni r .i r y / D y . Hence r! .y/ D i r y  D iNr y  D r!jN .y/; implying that r! .y/ D r!jN .y/ for all r 2 R. (iv) H) (i). From (iv) it follows that i t y  D iNt y  for all t 2 R. Since 1=2 1=2 y  2 D.1=2 / \ D.1=2 y  D 1=2 N /, analytic continuation gives  N y  D N y  (see Theorem A.7 of Appendix A.1). Therefore, 1=2 y   D JN 1=2 y  D JN PJy   D E! .y  / N y  D JN P 

thanks to (5.8), implying that E! .y  / D y  so that E! .y/ D y. (i) H) (iii). Choose an invariant mean m on `1 .N/. For every x 2 M , note that 2 M 7! mŒ .E!n .x// is a bounded linear functional on M such that jmŒ .E!n .x//j  kxk k k. Hence there is an E.x/ 2 M such that kE.x/k  kxk and .E.x// D mŒ .E!n .x// for all 2 M . Since .E! .E.x/// D mŒ

ı E! .E!n .x// D mŒ .E!nC1 .x// D

.E.x//;

2 M ;

we have E! .E.x// D E.x/, i.e., E.x/ 2 FE! . If x 2 FE! , then .E.x// D .x/ for all 2 M , so E.x/ D x. Therefore, EW M ! FE! is a norm-one projection onto FE! . Moreover, since !.E.x// D mŒ!.E!n .x// D !.x/, it follows that E is the conditional expectation from M onto FE! with respect to !. By Theorem 5.6 we have  t! .FE! / D FE! for all t 2 R, which shows that (i) H) (iii).  Corollary 5.17. In the situation of Theorem 5.16, FE! is the largest von Neumann subalgebra of N onto which the conditional expectation from M with respect to ! exists. Proof. In the above proof of (i) H) (iii) we have shown the existence of the conditional expectation from M onto FE! with respect to !. Let N1 be a von Neumann subalgebra of N and assume that there is the conditional expectation from M onto N1 with respect to !. For every y 2 N1 , since Theorem 5.6 implies that  ! .y/ 2 N1  N for all t 2 R, we have y 2 FE! due to Theorem 5.16. Hence N1  FE! follows. 

6 Connes’ cocycle derivatives

Up to now, to make the presentations simpler we have avoided using the notion of normal weights on von Neumann algebras, except for the semifinite normal trace in Chapter 4, which may be rather harmless as long as von Neumann algebras are -finite. However, it seems that (faithful) semifinite normal weights are essential for studying Connes’ cocycle derivatives introduced in [13]. The notion of Connes’ cocycle derivatives plays a crucial role in the study of operator-valued weights and that of the structure theory of von Neumann algebras, presented in Chapters 7 and 8 below. Thus, in the first part of this chapter we give a minimal requirement for faithful semifinite normal weights on von Neumann algebras. Then in the second part we give a brief description of Connes’ cocycle derivatives for faithful semifinite normal weights. Further discussions on Connes’ cocycle derivatives are presented in Section 10.2.

6.1 Basics of faithful semifinite normal weights Let M be a von Neumann algebra on a Hilbert space H . We begin with the next definition. Definition 6.1. A functional 'W MC ! Œ0; 1 satisfying the following properties is called a weight on M : for any x; y 2 MC ,  '.x/ D '.x/,   0,  '.x C y/ D '.x/ C '.y/. Define F' WD ¹x 2 MC W '.x/ < 1º; N' WD ¹x 2 M W '.x  x/ < 1º; M' WD N' N' WD span¹y  x W x; y 2 N' º: Then N' is a left ideal of M , M' D span F' and F' D MC \ M' (similarly to Lemma 4.36). The weight ' is said to be  normal if, for any increasing net ¹x˛ º in MC with x˛ % x 2 MC , '.x˛ / % '.x/,  faithful if '.x  x/ D 0 H) x D 0 for any x 2 M ,  semifinite if the following equivalent conditions hold:

88

6 Connes’ cocycle derivatives

(a) N' is weakly dense in M ; (b) sup¹e 2 Proj.M / W '.e/ < 1º D 1; (c) there is an increasing net ¹u˛ º in F' such that u˛ % 1.1 Here, we record Haagerup’s theorem [27] characterizing normal weights on von Neumann algebras, where the implication (iv) H) (v) is due to [71, Thm. 7.2]. Theorem 6.2 (Haagerup). Let ' be a weight on M . Then the following properties are equivalent: (i) ' is normal; (ii) P ' is completely additive, i.e., ' i xi 2 MC ;

P

i


P xi D i '.xi / for any set ¹xi º in MC with

(iii) ' is -weakly lower semicontinuous; (iv) '.x/ D sup¹!.x/ W ! 2 MC ; !  'º for all x 2 MC ; P (v) '.x/ D i2I !i .x/ for all x 2 MC with some .!i /i2I  MC ; P (vi) '.x/ D i2I hi ; xi i for all x 2 MC with some .i /i 2I  H . Examples 6.3. (1) Let M D L1 .X; / be a commutative R von Neumann algebra over a localizable measure space .X; X ; /. Define '.f / WD X f d for f 2 L1 .X; /C . Then ' is a faithful semifinite normal weight on M . In this case, F' D L1 .X; /C \ L1 .X; /, N' D L1 .X; / \ L2 .X; / and M' D L1 .X; / \ L1 .X; /. (2) Let M be a semifinite von Neumann algebra with a faithful semifinite normal trace  (discussed in Chapter 4), in particular, M D B.H / with the usual trace Tr. Then  is a special case of faithful semifinite normal weights, F D MC \ L1 .M;  /, N D M \ L2 .M;  / and M D M \ L1 .M;  / as given in Lemma 4.36. Associated with a faithful semifinite normal weight ' on M , we can perform the GNS construction in the following way. A Hilbert space H' is the completion of N' with an inner product ha; bi' WD '.b  a/, a; b 2 N' . Let a 2 N' 7! a' 2 H' be the canonical injection of N' into H' . For any x 2 M , since N' is a left ideal of M and k.xa/' k D '.a x  xa/1=2  kxk ka' k, ' .x/ 2 B.H' / is defined by ' .x/a' WD .xa/' . Then it is easy to see that ' is a faithful representation of M on H' . So we may consider M as a von Neumann algebra on H' by identifying x with ' .x/. Note that N' \ N' has a *-algebra structure. Although we do not enter into the details here, the following fundamental results hold (see [96, 90] for details): w

1Generally, since N' is a left ideal of M , there is a (unique) projection e 2 M such that N' D M e, w where N' is the weak closure of N' . Also, there is an increasing net ¹u˛ º in ¹x 2 N' W 0  x  1º such that kx xu˛ k ! 0 for all x 2 N' . Such a net ¹u˛ º is called a (right) approximate identity of N' , and we have u˛ % e automatically. Moreover, it is known that an approximate identity of N' can be taken in ¹x 2 F' W 0  x  1º; hence there is a net ¹u˛ º in F' such that u˛ % e. See, e.g., [90, §§3.20, 3.21] for details on these facts.

6.1 Basics of faithful semifinite normal weights

89

(A) The subspace A' WD ¹a' W a 2 N' \ N' º of H' becomes a so-called left Hilbert algebra with the product a' b' D .ab/' and the involution .a' /] WD .a /' , and the associated left von Neumann algebra L.A' / WD ¹L W  2 A' º00 is ' .M / Š M , where L is the left multiplication (bounded operator) L  WD  for  2 A' . Moreover, the weight ' is recaptured from A' in such a way that for x 2 MC , ´ kk2 if ' .x/1=2 D L for some  2 A' ; '.x/ D 1 otherwise: (Note that if x 2 F' and so x 1=2 2 N' , then ' .x 1=2 / D L.x 1=2 /' and '.x/ D k.x 1=2 /' k2 .) (B) (Tomita’s theorem) Let S' be the closure of the conjugate-linear closable operator a' 2 A' 7! .a /' 2 A' , and S' D J' 1=2 be the polar decomposition of S' . ' Then J' and ' satisfy properties (i)–(iv) of Lemma 2.1, and Tomita’s fundamental theorem holds as J' ' .M /J' D ' .M /0 ;

i't ' .M /' i t D ' .M /; t 2 R:

(C) Define the modular automorphism group  t' (t 2 R) of M associated with ' by ' . t' .x// WD i't ' .x/' i t : Then ' ı  t' D ', t 2 R, and the KMS condition (i.e., Definition 2.11 with ˇ D 1 for x, y restricted to elements of N' \ N' ) holds. Furthermore,  t' is uniquely determined as a weakly continuous one-parameter automorphism group satisfying these properties. (D) More specific identities behind results (A)–(C) above are the following: The subspace J A' of H' is the right Hilbert algebra A0' WD ¹ 2 D.S' / W R is boundedº, where R is the right multiplication R x' WD ' .x/ for x 2 N' , and RJ  D JL J . 2 A' /;

J ' .x/Jy' D ' .y/J x' .x; y 2 N' /

hold. Furthermore, for every t 2 R, i t A' D A' and Lit  D i t L 

it

. 2 A' /;

. t' .x//' D i t x' .x 2 N' /

hold. (E) Although we do not give the explicit definition, there exists a subalgebra \ T'  A' \ A0' \ D.z' /; z2C

which is called the Tomita algebra and satisfies the nice properties J' T' D T' ; for all z 2 C.

z' T' D T' ;

T' is a core of z'

90

6 Connes’ cocycle derivatives

6.2 Connes’ cocycle derivatives Now consider the tensor product M ˝ M2 .C/ D M2 .M / of M and the 2  2 matrix algebra M2 .C/. Let ', be faithful semifinite normal weights on M . The balanced weight  D .'; / on M2 .M / is given by ! 2 2 X X  xij ˝ eij WD '.x11 / C .x22 /; xij ˝ eij 2 M2 .M /C ; (6.1) i;j D1

i;j D1

where eij (i; j D 1; 2) are the matrix units of M2 .C/. Then we have the following lemma. Lemma 6.4.

(1)  is a faithful semifinite normal weight on M2 .M /.

(2) N D N' ˝ e11 C N ˝ e12 C N' ˝ e21 C N ˝ e22 . (3) M D M' ˝ e11 C N' N ˝ e12 C N N' ˝ e21 C M ˝ e22 , where N N' WD span¹y  x W x 2 N' ; y 2 N º. (4) N \ N D .N' \ N' / ˝ e11 C .N \ N' / ˝ e12 C .N' \ N / ˝ e21 C .N \ N / ˝ e22 .   x12 Proof. (2) For X D xx11 2 M2 .M /, since 21 x22   .X  X / D '.x11 x11 C x21 x21 / C

  .x12 x12 C x22 x22 /;

(6.2)

it follows that .X  X / < 1 if and only if x11 ; x21 2 N' and x12 ; x22 2 N . (1) By (6.2), .X  X / D 0 ” X D 0, so  is faithful. From definition (6.1) it is easy to verify that  is normal. There are increasing nets ¹u˛ º  F' and ¹v˛ º  F such that u˛ % 1 and v˛ % 1. Then u˛ ˝ e11 C v˛ ˝ e22 2 F and u˛ ˝ e11 C v˛ ˝ e22 % 1, hence  is semifinite. (3) and (4) are immediate from (2).  By Lemma 6.4(2) the GNS Hilbert space associated with  is given as H D H' ˚ H ˚ H' ˚ H with the canonical injection 2

 a11 a21





a12 a 2 N 7 ! 11 a22 a21

a12 a22

 

3 .a11 /' 6.a12 / 7 7 D6 4 .a21 /' 5 ; .a22 /

and the GNS representation of M2 .M / associated with  is      x11 x12 a11 a12 x a C x12 a21  D 11 11 x21 x22 a21 a22  x21 a11 C x22 a21

x11 a12 C x12 a22 x21 a12 C x22 a22

 ; 

91

6.2 Connes’ cocycle derivatives

so that  is given in 4  4 form as 2 ' .x11 / 0   6 0 x11 x12  .x 11 /  D6 4' .x21 / x21 x22 0 0  .x21 /

3 ' .x12 / 0 0  .x12 /7 7: ' .x22 / 0 5 0  .x22 /

     a11   a21 a12 ( aa11 7 ! By Lemma 6.4(4) the closure S of aa11  a 21 21 a22   a12 22 given as 2 3 S' 0 0 0 60 0 S ;' 0 7 7; S D 6 4 0 S'; 0 05 0 0 0 S

a12 a22

 where S'; is the closure of a12 2 N \ N' 7! a12 2 N' \ N and S 

of a21 2 N' \ N ! 7 given as 2 J' 0 60 0 J D 6 4 0 J'; 0 0

 a21

2N \ 0

J

;'

0 0

N' .

3 0 07 7; 05 J



(6.3)

2 N \ N ) is

(6.4)

;'

is the closure

Thus, the polar decomposition S D J 1=2 is  2

' 6 0  D 6 4 0 0

0 '; 0 0

0 0 

;'

0

3 0 0 7 7; 0 5 

(6.5)

1=2 where S'; D J'; 1=2 '; and S ;' D J ;'  ;' . Concerning the modular automorphism group  t of M2 .M / we have the following lemma.

Lemma 6.5. The modular automorphism group  t is given as  t



x11 x21

x12 x22



"

 t' .x11 / D  t ;' .x21 /

#  t'; .x12 / ;  t .x22 /



x11 x21

 x12 2 M2 .M /; x22

(6.6)

where  t' ,  t are the modular automorphism groups of M associated with ', , respectively,  t ;' is a strongly continuous one-parameter group of isometries on M , and  t'; .x/ D  t ;' .x  / , x 2 M . Furthermore, ' . t

;'

.x// D i t;' ' .x/' i t ;

 . t

;'

.x// D i t  .x/';i t ;

x 2 M: (6.7)

  x12 Proof. Note that  . t .X // D it  .X / i t for X D xx11 2 M2 .M /. Hence 21 x22 the expressions (6.6) and (6.7) come from a direct computation based on (6.3) and (6.5). Since  t is a strongly continuous one-parameter automorphism group of M2 .M /, it is clear that  t ;' is a strongly continuous one-parameter group of isometries of M and  t'; .x/ D  t ;' .x  / . 

92

6 Connes’ cocycle derivatives

Theorem 6.6 ([13]). Define u t WD  t ;' .1/ 2 M for t 2 R. Then t 2 R 7! u t is a strongly* continuous map into the unitaries of M satisfying  t .x/ D u t  t' .x/ut ; usCt D

us s' .u t /;  0

t 2 R; x 2 M;

(6.8)

s; t 2 R (the cocycle identity): (6.9)  
D  t 01 00 for t 2 R, so t 7! u t is strongly*

Proof. By Lemma 6.5 note that u0t 0 continuous. Since            ut ut 0 0 0 0 0 0 0 0 0 D D  t 0 0 ut 0 ut 0 1 0 1 0    '    1 0 1 0  .1/ 0 D t D  t D 0 0 0 0 0 0    
  and similarly 00 u t0ut D  t 00 01 D 00 01 , we see that the u t are unitaries. Moreover,          0 0 0 0 0 0 x 0 0 0   D t D t 0 x 1 0 0 0 1 0 0  t .x/   '     0 0 0 0  t .x/ 0 0 0 D D ; ut 0 0 0 ut 0 0 u t  t' .x/ut 

so that (6.8) follows. Since        0 0 0 0 0 0 0  D sCt D s D s usCt 0 1 0 ut 0 1   '    0 0 0 0 s .u t / 0 D D ; us 0 0 0 us s' .u t / 0

 0 ut 0 0

 0 0



we have (6.9).

Definition 6.7. The map t ! 7 u t given in Theorem 6.6 is called Connes’ cocycle (Radon– Nikodym) derivative of with respect to ', and denoted by .D W D'/ t , i.e., u t D .D W D'/ t , t 2 R. By construction it is clear that  t';' D  t' and so .D' W D'/ t D 1 for all t 2 R. More properties of Connes’ cocycle derivatives .D W D'/ t will be given, for example, in Proposition 7.24, and will be discussed also for (not necessarily faithful) '; 2 MC in Section 10.2. We finish the section with Connes’ inverse theorem in [14, Thm. 1.2.4] without proof. Theorem 6.8 ([14]). Let ' be a faithful semifinite normal weight on M . Let t 2 R 7! u t 2 M be a strongly* continuous map satisfying usCt D us s' .u t /; u

t

D

'

 t .u t /;

t 2 R;

(6.10)

t 2 R:

(6.11)

6.2 Connes’ cocycle derivatives

Then there exists a unique semifinite normal weight for all t 2 R.

on M such that u t D .D

93 W D'/ t

In [14] the u t are assumed to be unitaries in M . In this case, (6.11) is redundant and given in the theorem is faithful. The above version without the assumption of the u t being unitaries is taken from [89, Thm. 5.1]. Here, we note that when is not necessarily faithful with the support projection s. / (though ' remains faithful), the modular automorphism group  t is defined on s. /M2 .M /s. /, where s. / D 1 ˝ e11 C s. / ˝ e22 is the support projection of  . Then Connes’ cocycle derivative u t D .D W D'/ t is defined    
as u0t 00 D  t s.0 / 00 or u t D  t ;' .s. // 2 s. /M (in fact,  t ;' acts on s. /M ). It is an easy exercise to show that (6.10) and (6.11) imply ´ the u t are partial isometries and (6.12) u t ut D u0 , ut u t D  t' .u0 / for t 2 R; and conversely (6.10) and (6.12) imply (6.11). Note that even when ' 2 MC , in the theorem cannot be in MC in general.2 When u t D .D W D'/ t with '; 2 MC , we will show properties (6.10)–(6.12) in Theorem 10.16(1)–(3).

2This fact suggests that von Neumann algebra theory cannot be self-completed when we stick to functionals in M , so the theory of (semifinite) normal weights is unavoidable.

7 Operator-valued weights

This chapter is aimed at giving an essential part of operator-valued weights in von Neumann algebras developed by Haagerup [32, 33]. Operator-valued weight is a notion generalizing both weights and conditional expectations, and plays a role in the structure theory of von Neumann algebras. In this chapter, after a brief account of generalized positive operators, we present in Section 7.2 a main result by Haagerup [32] on operator-valued weights. In Section 7.3 we present a concise description of the Pedersen–Takesaki construction [71] and related results, based on the results in Section 7.2.

7.1 Generalized positive operators Let M be a von Neumann algebra with the predual M as before. Definition 7.1. A functional mW MC ! Œ0; 1 satisfying the following properties is called a generalized positive operator affiliated with M : for any '; 2 MC ,  m.'/ D m.'/,   0,  m. C

/ D m.'/ C m. /,

 m is lower semicontinuous on MC . cC the set of generalized positive operators affiliated with M and call We denote by M cC by regarding x 2 MC as a it the extended positive part of M . Obviously, MC  M functional ' 2 MC 7! '.x/. cC , a 2 M and   0, define m, m C n, a ma 2 M cC by For any m; n 2 M .m/.'/ WD m.'/;

.m C n/.'/ WD m.'/ C n.'/;

.a ma/.'/ WD m.a'a /

for ' 2 MC , where a'a WD '.a
a/. Define m  n if m.'/  n.'/ for all ' 2 MC . cC , m 2 M cC is defined by m.'/ WD sup˛ m˛ .'/ for For an increasing net .m˛ / in M P C cC is ' 2 M . In this case, write m˛ % m. In particular, the sum m D i2I mi 2 M cC . defined for any family .mi /i 2I  M Examples 7.2. (1) Let A beR a positive self-adjoint operator affiliated with M with the 1 spectral decomposition A D 0  de . Define Z 1 mA .'/ WD  d'.e /; ' 2 MC : 0

96

7 Operator-valued weights

Then mA is lower semicontinuous on MC since mA .'/ D supn '.xn / where xn WD Rn c 0  de 2 MC , n 2 N. Hence we have mA 2 M C . For each  2 H write ! for the vector functional x 2 M 7! h; xi. Note that ´ Z 1 kA1=2 k2 if  2 D.A1=2 /; 2 mA .! / D  dke k D 1 otherwise: 0 For positive self-adjoint operators A, B affiliated with M , if mA D mB , then D.A1=2 / D D.B 1=2 / and kA1=2 k2 D kB 1=2 k2 for all  2 D.A1=2 /, which means that A D B (see Theorem A.11 of Appendix A.2). (2) Let M D L1 .X; / be a commutative von Neumann algebra over a -finite measure space .X; X ; /. Let f; gW X ! Œ0; 1 be measurable functions. Define Z mf .'/ WD f ' d; ' 2 L1 .X; /C Š MC : X

1

When '; 'n 2 L .X; /C with k'n 'k1 ! 0, a subsequence 'nk can be chosen so that mf .'nk / ! lim inf n mf .'n / and 'nk ! ' a.e. By Fatou’s lemma we have mf .'/  lim inf k mf .'nk / D lim inf n mf .'n /. Hence mf is lower semicontinuous on cC . It is clear that mf D mg ” f D g -a.e. Conversely, for L1 .X; /C , so mf 2 M cC , by Theorem 7.3 below, there is an increasing sequence fn 2 L1 .X; /C any m 2 M R such that X fn ' d % m.'/ for all ' 2 L1 .X; /C . By letting f WD supn fn we have cC is identified with ¹f W X ! Œ0; 1 measurableº, where f D g m D mf . Thus, M means f D g -a.e. cC in terms of a certain The next theorem gives an explicit description of m 2 M spectral decomposition in M . cC has a unique spectral decomposition of the form Theorem 7.3. Each m 2 M Z 1 m.'/ D  d'.e / C 1'.p/; ' 2 MC ;

(7.1)

0

where .e /0 is an increasing family of projections in M such that  7! e is strongly right-continuous and p D 1 lim!1 e . Moreover, e0 D 0 ” m.'/ > 0 for any ' 2 MC n ¹0º; p D 0 ” ¹' 2

MC

W m.'/ < 1º is dense in

(7.2) MC :

(7.3)

Consequently, there is an increasing sequence xn 2 MC such that '.xn / % m.'/ for all ' 2 MC . Proof. Define a function qW H ! Œ0; 1 by q./ WD m.! /,  2 H . Then it is immediate to see that q is a positive form in the sense of Definition A.14 of Appendix A.2, that is, the following hold:

7.1 Generalized positive operators

97

(a) q./ D jj2 q./ for all  2 H and  2 C, (b) q. C / C q.

/ D 2q./ C 2q./ for all ;  2 H .

From the lower semicontinuity of m on MC , it is clear that q is lower semicontinuous on H . Hence by Theorem A.16 there exists a unique positive self-adjoint operator A on K WD D.q/, where D.q/ WD ¹ 2 H W q./ < 1º, such that D.A1=2 / D D.q/ and ´ kA1=2 k2 if  2 D.A1=2 /; m.! / D q./ D (7.4) 1 otherwise: For every unitary u0 2 M 0 , since !u0  D ! on M , we have q.u0 / D q./. Hence we see that u0 D.q/ D D.q/ so that u0 K D K ; hence the orthogonal projection onto K 0 0 is in M . Moreover, from the uniqueness of A it follows that u R 1Au D A, that is, A is affiliated with M . Now take the spectral decomposition A D 0  de , where .e /0 is a spectral resolution in M such that e % e1 as  % 1, where e1 is the projection onto K . Let p WD 1 e1 , the projection onto K ? . It is clear that Z 1 Z 1 m.! / D  dke k2 C 1kpk2 D  de! .e / C 1! .p/;  2 H : 0

0

P Note that any ' 2 is written as ' D n !n for some .n /  H with n kk2 < 1. Therefore, (7.1) immediately follows. Conversely, assume that .eR /0 in M satisfies (7.1) with p D 1 lim!1 e . Let 1 K WD .1 p/H and A WD 0  de , a positive self-adjoint operator on K . Then we find in turn that (7.4) holds. Hence by Theorem A.16, A is uniquely determined by q (hence by m), which implies the uniqueness of .e /0 satisfying (7.1). Next, (7.2) is seen as MC

P

e0 D 0 ” m.! / > 0 for any  2 H n ¹0º ” m.'/ > 0 for any ' 2 MC n ¹0º: For (7.3) we have pD0 ” K DH ” ¹ 2 H W m.! / < 1º .D D.A1=2 // is dense in H : Since kn k ! 0 H) k!n ! k ! 0, it immediately follows from the above condition that ¹' 2 MC W m.'/ < 1º is dense in MC . Conversely, if p ¤ 0, then ¹' 2 MC W m.'/ < 1º is not dense in MC because ¹' 2 MC W m.'/ < 1/  ¹' 2 MC W '.p/ D 0º. Rn The last assertion follows by letting xn WD 0  de C np 2 MC for n 2 N.  Remark 7.4. Theorems 7.3 and A.16 (together with the above proof) say that there are one-to-one correspondences between the following four objects determined by (7.1) and (7.4):

98

7 Operator-valued weights

cC ,  generalized positive operators m 2 M  families .e /0 in M as stated in Theorem 7.3,  positive self-adjoint operators A on a closed subspace of H affiliated with M ,  lower semicontinuous positive forms q on H such that q ı u0 D q for all unitaries u0 2 M 0 .

2

The term “generalized positive operator” is well justified by these correspondences. When M D B.H /, an element of B .H /C means a positive self-adjoint operator on a closed subspace K (considered 1 on K ? ), whose idea goes back to [53]. Proposition 7.5. Any normal weight ' on M has a unique extension (denoted by the same cC satisfying ') to M cC ,   0, (1) '.m/ D '.m/ for all m 2 M cC , (2) '.m C n/ D '.m/ C '.n/ for all m; n 2 M cC , m˛ % m H) '.m˛ / % '.m/. (3) m; m˛ 2 M R cC with the spectral resolution m D 1  de C 1p, let xn WD Proof. For m 2 M 0 Rn 6.2 (due to [71]) note 0  de C np and define P '.m/ WD limn!1 '.xn /. By Theorem that ' is written as ' D i2I !i with some .!i /i 2I  MC . Then X X '.m/ D lim !i .xn / D m.!i /; n!1

i 2I

i 2I

from which it is easy to verify (1)–(3). To show the uniqueness of the extension, let cC satisfying (1)–(3). Since xn % m, one has another extension of ' to M .m/ D lim

n!1

Hence

cC . D ' on M

be

.xn / D lim '.xn / D '.m/: n!1



7.2 Operator-valued weights Throughout this section we assume that N is a von Neumann subalgebra of M . We begin b C -valued version of with the definition of operator-valued weights. The notion is the N weights on M discussed in Section 6.1 and is also considered as the unbounded version of conditional expectations from M onto N discussed in Section 5.1. b C satisfying the following properties is called an Definition 7.6. A map T W MC ! N operator-valued weight from M to N : for any x; y 2 MC ,

99

7.2 Operator-valued weights

 T .x/ D T .x/,   0,  T .x C y/ D T .x/ C T .y/,  T .b  xb/ D b  T .x/b for all b 2 N . For such T define FT WD ¹x 2 MC W T .x/ 2 NC º; NT WD ¹x 2 M W T .x  x/ 2 NC º; MT WD NT NT WD span¹y  x W x; y 2 NT º: It is easy to see that NT and MT are N -bimodules, i.e., linear subspaces of M closed under the left and right multiplication by any b 2 N . We have MT D span FT and FT D MC \ MT similarly to the weight case in Definition 6.1. The T is said to be  normal if, for any increasing net ¹x˛ º in MC with x˛ % x 2 MC , T .x˛ / % T .x/,  faithful if T .x  x/ D 0 H) x D 0 for any x 2 M ,  semifinite if NT is weakly dense in M or, equivalently, there is an increasing net ¹u˛ º in FT such that u˛ % 1. b C we can define a linear map TP W MT ! N For an operator-valued weight T W MC ! N in such a way that for any x D x1 x2 C i.x3 x4 / with xk 2 FT (k D 1; : : : ; 4), TP .x/ WD T .x1 /

T .x2 / C i.T .x3 /

T .x4 //:

Then we have TP .b1 xb2 / D b1 TP .x/b2 for all x 2 MT and b1 ; b2 2 N . Hence, if T .1/ D 1, then TP W M ! N is a conditional expectation. We write P .M / WD ¹' W faithful semifinite normal weight on M º; P .N / WD ¹' W faithful semifinite normal weight on N º; P .M; N / WD ¹T W faithful semifinite normal operator-valued weight from M to N º: Proposition 7.7. Let T be a normal operator-valued weight from M to N and ' be a normal weight on N . Then ' ıT is a normal weight on M . Furthermore, if T 2 P .M; N / and ' 2 P .N /, then ' ı T 2 P .M /. Proof. From Proposition 7.5 it is obvious that ' ı T is a well-defined normal weight on M . Let T 2 P .M; N / and ' 2 P .N /. Assume that x 2 M and '.T .x  x// D 0. By Theorem 7.3 choose a sequence yn 2 NC such that yn % T .x  x/. Then '.yn / D 0 and so yn D 0 for all n, implying T .x  x/ D 0, so x D 0 follows. Hence ' ı T is faithful. Since ' is semifinite, one can choose a net ¹b˛ º in N' such that b˛ ! 1 strongly. For any x 2 NT , since .' ı T /.b˛ x  xb˛ / D '.b˛ T .x  x/b˛ /  kT .x  x/k'.b˛ b˛ / < 1;

100

7 Operator-valued weights

it follows that xb˛ 2 N'ıT . Since xb˛ ! x strongly, N'ıT is strongly dense in NT . Since T is semifinite, N'ıT is strongly dense in N , so ' ı T is semifinite.  The rest of the section is devoted to the proof of the following theorem in [32]. Theorem 7.8 (Haagerup). Let T 2 P .M; N / and ';

2 P .N /. Then

(a)  t'ıT .y/ D  t' .y/ for all y 2 N and t 2 R, (b) .D

ı T W D' ı T / t D .D

W D'/ t for all t 2 R.

In particular, we have the following corollary. Corollary 7.9. Let EW M ! N be a faithful normal conditional expectation and '; 2 NC be faithful. Then  t'ıE .y/ D  t' .y/ and .D ı E W ' ı E/ t D .D W D'/ t for all y 2 N and t 2 R. Since .D W D'/ t D  t ;' .1/ and  t' D  t';' (see Definition 6.7 and a remark after that), statements (a) and (b) of Theorem 7.8 are unified into .?/  t

ıT;'ıT

.y/ D  t

;'

.y/ for all y 2 N and t 2 R.

Toward the proof of .?/, we start by recalling a result by Ciorˇanescu and Zsidó [11] on analytic generators for one-parameter groups of operators. Let  t (t 2 R) be a -weakly (equivalently, weakly) continuous one-parameter group of isometries on M (though [11] dealt with more general one-parameter groups of bounded linear operators on a Banach space). For ˛ 2 C with Im ˛ > 0, define D.˛ / WD ¹x 2 M W there exists a -weakly continuous M -valued function fx .z/ on 0  Im z  Im ˛, analytic in 0 < Im z < Im ˛, such that fx .t / D  t .x/, t 2 Rº; and similarly for ˛ 2 C with Im ˛ < 0. Define ˛ .x/ WD fx .˛/ for x 2 D.˛ /. We further write M. / for the set of  t -analytic elements, i.e., the set of x 2 M such that there is a  -weakly entire M -valued function fx .z/ such that fx .t / D  t .x/, t 2 R. The next theorem is from [11, Thms. 2.4, 4.4]. Theorem 7.10 (Ciorˇanescu and Zsidó). (1) In the above situation, for any ˛ 2 C, ˛ is a closed densely defined operator on M (with the  -weak topology). In fact, T M. / D ˛2C D.˛ / is -weakly dense in M . Moreover, ˛ ˇ D ˛Cˇ ; 

˛

D ˛ 1 ;

˛; ˇ 2 C; .Im ˛/.Im ˇ/  0;

(7.5)

˛ 2 C:

(7.6)

(2) Let zt (t 2 R) be another -weakly continuous one-parameter group of isometries on M . Then  t D zt for all t 2 R if and only if  i D z i .

101

7.2 Operator-valued weights

The operator  i is called the analytic generator of  t . A prototype of Theorem 7.10 is the familiar Stone representation theorem saying that a continuous one-parameter unitary group U t on a Hilbert space is uniquely determined by its generator (a positive self-adjoint operator) A as U t D Ai t (t 2 R); see Theorem A.7 of Appendix A.1. Theorem 7.10 explains the methodological idea of the following proof of Theorem 7.8; the main part (2) will not be directly utilized but its extension to the inclusion N  M setting will be given in Lemma 7.15. On the other hand, the easy part (1) will be used below, so we give its proof in the following: Proof of Theorem 7.10(1). The proof of the denseness of M. / is similar to that of Lemma 2.13, and the details are omitted. By analytic continuation it is immediate to see that ˛ D  t ir D i r  t for every ˛ D t C i r (t; r 2 R). Hence it suffices to prove (7.5) and (7.6) for ˛ D i r, ˇ D i s (r; s 2 R). These are obvious when r D 0 (or s D 0). Let r; s > 0 (the case r; s < 0 is similar). It is clear that i.rCs/  i r i s . Conversely, assume that x 2 D.i r i s /, i.e., x 2 D.i s / and i s .x/ 2 D.i r /. Then there is a -weakly continuous function fx .z/ on 0  Im z  s, analytic in 0 < Im z < s, such that fx .t / D  t .x/, t 2 R. Also, there is a  -weakly continuous function fi s .x/ .z/ on 0  Im z  r, analytic in 0 < Im z < r, such that fi s .x/ .t / D  t .i s .x//, t 2 R. Since fx .t C i s/ D  tCi s .x/ D  t .is .x// D fi s .x/ .t /;

t 2 R;

one can define gx .z/ WD

´ fx .z/ fis .x/ .z

if 0  Im z  s; i s/

if s  Im z  s C r;

which is -weakly continuous on 0  Im z  r C s and analytic in 0 < Im z < r C s. Hence x 2 D.i.rCs/ / and i.rCs/ .x/ D fi s .x/ .i r/ D i r .i s .x//. Therefore, (7.5) follows. Assume that x 2 D.i r / and y D i r .x/. With the above function fx .z/ define fy .z/ WD fx .z Ci r/ for r  Im z  0, which is -weakly continuous on r  Im z  0 and analytic in r < Im z < 0. Since fy .t / D fx .t C i r/ D  t .i r .x// D  t .y/;

t 2 R;

one has y 2 D. i r / and  i r .y/ D fy . i r/ D fx .0/ D x. Conversely, if y 2 D. i r / and x D  ir .y/, then one similarly has x 2 D.i r / and i r .x/ D y. Therefore, (7.6) is shown. 



Now let us start to prove .?/ above. To do so, we need to analyze the analytic generator for which the next theorem is the most essential.

;' i ,

102

7 Operator-valued weights

Theorem 7.11. Let '; equivalent: (i) a 2 D.

;' i /

2 P .M / and a; b 2 M . Then the following conditions are

and b D 

;' i .a/;

(ii) aN'  N , N b  N' and

.ax/ D '.xb/ for all x 2 N' N .

Corollary 7.12. Let '; 2 MC be faithful. Then a 2 D. only if .ax/ D '.xb/ for all x 2 M .

;' i /

and b D 

;' i .a/

if and

To prove Theorem 7.11, we first give the following lemma. Lemma 7.13. Let '; equivalent: (a) a a .D

2 P .M /, a 2 M and k > 0. Then the following conditions are

.a
a //  k 2 ';

(b) N' a  N and k.xa / k  kkx' k for all x 2 N' ; (c) a 2 D. If ' D

;' / i=2

and k

;' .a/k i=2

 k.

and (a)–(c) hold, then .xa /' D J' ' . 'i=2 .a//J' x' ;

x 2 N' :

(7.7)

Proof. (a) ” (b). (a) means that ..xa / .xa //  k 2 '.x  x/ for all x 2 M , which is equivalent to (b). (c) H) (a) [in the case ' D ]. Assume that a 2 D. 'i=2 / and k 'i=2 .a/k  k. Since i't ' .a/.x  /' D ' . t' .a//i't .x  /' ; x 2 N' ; we have, by analytic continuation,

so that That is,

'  1=2  1=2 ' ' .a/.x /' D ' . i=2 .a//' .x /' ;

x 2 N' \ N' ;

S' ' .a/.x  /' D J' ' . 'i=2 .a//J' S' .x  /' ;

x 2 N' \ N' :

.xa /' D J' ' . 'i=2 .a//J' x' ;

x 2 N' \ N' :

(7.8)

If x 2 MC and '.x/ < 1, then x 1=2 2 N' \ N' and so '.axa / D k.x 1=2 a /' k2  k' . 'i=2 .a//k2 k.x 1=2 /' k2  k 2 '.x/: Hence (a) holds. Moreover, since ¹x' W x 2 N' \ N' º is dense in H' and x' 7! .xa /' (x 2 N' ) is bounded thanks to (a) H) (b), the assertion in (7.7) follows from (7.8).

103

7.2 Operator-valued weights

(b) H) (c) [in the case ' D ]. By (b) there is a T 2 B.H' / with kT k  k such that T x' D .xa /' for all x 2 N' . For every x 2 N' \ N' , since ' .a/' 1=2 J' x' D ' .a/S' x' D ' .a/.x  /' D .ax  /' ; one has 1=2 1=2 J' x' D J' S' .ax  /' D J' .xa /' D J' T x' : ' ' .a/' 1=2 iz Let  2 D.1=2 /. Then the function z 7! hiz ' / and  2 D.' ' ; ' .a/' i is analytic in 1=2 < Im z < 0. When Im z D 0 so that z D t , one has

h' i t ; ' .a/' i t i D h; i't ' .a/' i t i D h; ' . t' .a//i; jh' i t ; ' .a/' i t ij  kak kk kk: When Im z D

1=2 so that z D t

ˇ˝ iz ˇ  ; ' .a/ '

iz '

i=2, one has

˛ˇ it 1=2  ˇ D jh1=2 ' i t ij ' ' ; ' .a/' 1=2 D jh' i t ; 1=2 ' i t ij ' ' .a/'

D jh' i t ; J' TJ' ' i t ij  kkk kk: Therefore, from the three-lines theorem it follows that ˇ˝ iz ˛ˇ ˇ  ; ' .a/ iz  ˇ  .kak _ k/kk kk; ' '

1=2  Im z  0;

so that there exists an f .z/ 2 B.H' / for 1=2  Im z  0 such that ˝ ˛ iz h; f .z/i D iz ' ; ' .a/'  ;

kf .z/k  kak _ k:

Then it is easy to see that f .z/ is -weakly continuous on 1=2  Im x  0 and analytic in 1=2 < Im z < 0. Moreover, we have f .t / D ' . t' .a//, t 2 R. It is easily seen that ' 1 .f .z// 2 M for 1=2  Im z  0, so that ' 1 .f .z// is the analytic continuation of  t' .a/ to 1=2  Im z  0. Hence we find that a 2 D. 'i=2 / and k 'i=2 .a/k  k. (a) ” (c) [in thegeneral case]. the balanced weight  WD / on   0.';  Consider x12 0  M2 .M /. Let az WD a0 00 and xz WD xx11 2 M .M / . Since a z x z a z D so  2 C 0 ax11 a 21 x22 that .z axzaz / D .ax11 a /, one has (a) ” .z axzaz /  k 2 .z x / for all xz 2 M2 .M /C


 ” az 2 D  i=2 and   i=2 .z a /  k ” (c) thanks to  t .z a/ D



0

t

;'

0 .a/ 0 ,

t 2 R, by Lemma 6.5.



104

7 Operator-valued weights

Proof of Theorem 7.11. (i) H) (ii) [in the case ' D ]. Assume that a 2 D. 'i / ' and b D  'i .a/. From Theorem 7.10(1) one can see that a 2 D. 'i=2 /, b 2 D.i=2 / ' and  'i=2 .a/ D i=2 .b/. For x 2 M , since  t' .x  / D  t' .x/ for t 2 R, note that x  2 D.z' / ” x 2 D.z' / for any z 2 C, so b  2 D. 'i=2 /. Hence by Lemma 7.13 one has N' a  N' , so aN'  N' , and N' b  N' . Therefore, aM'  M' and M' b  M' , so that '.ax/ and '.xb/ are well defined for any x 2 M' . If x 2 F' then, using (7.7) twice, one has '.ax/ D h.x 1=2 a /' ; .x 1=2 /' i D hJ' ' . 'i=2 .a//J' .x 1=2 /' ; .x 1=2 /' i ' D hJ' ' .i=2 .b//J' .x 1=2 /' ; .x 1=2 /' i ' D h.x 1=2 /' ; J' ' .i=2 .b/ /J' .x 1=2 /' i

D h.x 1=2 /' ; J' ' . 'i=2 .b  //J' .x 1=2 /' i D h.x 1=2 /' ; .x 1=2 b/' i D '.xb/: Since M' D span F' , it follows that '.ax/ D '.xb/ for all x 2 M' . (ii) H) (i) [in the case ' D ]. Assume (ii). Here we utilize the Tomita algebra T' associated with '; see (E) of Section 6.1. Let ;  2 T' and let 1 WD S'  and 1 WD F'  (F' WD S' ). Since 1 ; 1 2 T' , we write 1 D x' and 1 D y' with x; y 2 N' \ N' . Since (ii) implies that ay  2 N' and xb 2 N' , note that ya ; xb 2 N' \ N' . Since '  D F' S'  D F' x' ;

 D S' 1 D S' x' ;

' 1  D S' F'  D S' y' ;

 D F' 1 D F' y' ;

one has h' ; ' .a/' 1 i D hF' x' ; ' .a/S' y' i D hF' x' ; .ay  /' i D hF' x' ; S' .ya /' i D h.ya /' ; x' i D '.ay  x/ D '.y  xb/ D hy' ; .xb/' i D hS' .xb/' ; F' y' i D h.b  x  /' ; F' y' i D h' .b  /S' x' ; F' y' i D h; ' .b/i: For every ;  2 T' consider the entire function ˝ ˛ iz z 2 C 7 ! iz ' ; ' .a/'  : When Im z D 0, i.e., z D t , one has h' i t ; ' .a/' i t i D h; ' . t' .a//i; jh' i t ; ' .a/' i t ij  kak kk kk:

(7.9)

105

7.2 Operator-valued weights

When Im z D

1, i.e., z D t

i , one has, by applying (7.9) to ' i t ; ' i t  2 T' ,

h' ' i t ; ' .a/' 1 ' i t i D h' i t ; ' .b/' i t i D h; ' . t' .b//i; jh' ' i t ; ' .a/' 1 ' i t ij  kbk kk kk: Therefore, as in the proof of Lemma 7.13, there exists a B.H' /-valued -weakly continuous function f .z/ on 1  Im z  0, analytic in 1 < Im z < 0, such that f .t / D ' . t' .a// and f .t i / D ' . t' .b// for all t 2 R. Since ' 1 .f .z// 2 M for 1  Im z  0, we have a 2 D. 'i / and b D  'i .a/, i.e., (i) holds. (i) ” (ii) [in the general case]. Consider the balanced weight  WD .'; / on      0 0 M2 .M /. Let az WD a0 00 , bz WD b0 00 2 M2 .M /. Since  t .z a/ D  ;' .a/ 0 for t 2 R, t it follows that a 2 D.

;' i /

and b D 

Therefore, from the case ' D

;' i .a/

” az 2 D.  i / and bz D   i .z a/:

we find that

z for all xz 2 M : (i) ” azN  N ; N bz  N and .z axz/ D .z x b/ By Lemma 6.4 note that N bz  N ” N b  N' ;

azN  N ” aN'  N ; and for xz D

 x11

x12 x21 x22



2 M D



M' N 'N N N' M

z D .ax12 / and .z x b/

 , .z axz/ D

'.x12 b/. Combining these facts together, we have shown (i) ” (ii).



To prove Theorem 7.8, we need several lemmas. Let  t (t 2 R) be a -weakly continuous one-parameter group of isometries on M . An element x 2 M is said to be of -exponential type if x is -analytic and there is a c > 0 such that sup kz .x/ke

cj Im zj

z2C

< 1:

Lemma 7.14. Let Mexp . / be the set of x 2 M of -exponential type. Then Mexp . / is -weakly dense in M . Proof. Let ´ pR .x/ WD

R 0;

The Fourier transform of pR is Z 1 pyR .z/ D .2/ 1=2 pR .x/e

1=2

; jxj  R=2; jxj > R=2:

izx

dx D .2R/

1=2

he

Z

R=2

e

izx

dx

R=2

1

D .2R/

1=2

izx iR=2

iz

R=2

D

 2 1=2 sin.Rz=2/ ; R z

z 2 C:

106

7 Operator-valued weights

Define

2 sin2 .Rz=2/ 1 cos Rz D : 2 Rz Rz 2 We then notice that qR .s/  0 (s 2 R) and R1 (1) 1 qR .s/ ds D 1, R1 (2) limR!1 1 qR .s/f .s/ ds D f .0/ for all continuous bounded functions f on R, R1 Rjtj (3) for all t 2 R. 1 jqR .s C i t /j ds  e qR .z/ WD pyR .z/2 D

Indeed, (1) follows from Z

1 1

qR .s/ ds D kpyR k22 D kpR k22 D 1:

(2) is easy to check directly and (3) follows since Z 1 jqR .s C i t /j ds D kpyR .
C i t /k22 D kpR .x/e tx k22 1

D

1 R

Z

R=2

e 2tx dx D R=2

e Rt

e 2Rt

Rt

 e Rjtj :

R1 Now, for any x 2 M and R > 0, let xR WD 1 qR .s/s .x/ ds. Define Z 1 f .z/ WD qR .s z/s .x/ ds; z 2 C: 1

Then, by Lebesgue’s convergence theorem, one can see that f .z/ is an entire function. Since Z 1 f .t / D qR .s/sCt .x/ ds D  t .xR /; t 2 R; 1

it follows that xR is -analytic with f .z/ D z .xR /. Furthermore, by (3) one has Z 1 kf .z/k  kxk jqR .s z/j ds  kxke Rj Im zj ; z 2 C; 1

so that xR 2 Mexp . / for any R > 0. For every ' 2 M , note by (1) and (2) that Z 1 '.xR x/ D qR .s/'.s .x/ x/ ds ! 0 as R ! 1: 1

Hence, xR ! x  -weakly as R ! 1.



The next lemma extends Theorem 7.10(2) to the case of a von Neumann subalgebra N  M . When N D M , the lemma says that z i   i H) z i D  i , similarly to the fact that for self-adjoint operators A, B on a Hilbert space, A  B H) A D B.

107

7.2 Operator-valued weights

Lemma 7.15. Let N  M be a von Neumann subalgebra. Let  t and zt (t 2 R) be -weakly continuous one-parameter groups of isometries on M and N , respectively. If z i   i , then  t .y/ D zt .y/ for all y 2 N and all t 2 R. Proof. Assume that z i   i . First we prove that Nexp .z  /  Mexp . /. Let y 2 Nexp .z  /, cj Im zj i.e., y 2 N is z -analytic and kz z .y/k  Ke for all z 2 C with some K; c > 0. Hence, in particular, kyk  K. For every n 2 Z, since z i n D .z  i /n  . i /n D  i n , one has y 2 D. ni / and  i n .y/ D z i n .y/. Hence it is clear that y is -analytic. For every z 2 C write z D s C i t and n 1  t < n (or .n 1/  t > n) for some n 2 N. By the three-lines theorem, for Im z  0 we have     kz .y/k  sup ks .y/k _ sup ksCi n .y/k D kyk _ sup ks .i n .y//k s2R

s2R

s2R

D kyk _ ki n .y/k D kyk _ kz i n .y/k  Ke

cn

and the same holds also for Im z  0. Hence y 2 Mexp . /. Now, for any y 2 Nexp .z  / ( Mexp . /), set f .z/ WD z .y/ an entire function. Then the following hold:

 .Ke c /e cjtj ;

zz .y/, z 2 C, which is

 kf .z/k  kz .y/k C kz z .y/k  Ke cj Im zj , z 2 C, for some K; c > 0,  kf .t /k  2kyk, t 2 R,  f .i n/ D . i /n .y/

.z  i /n .y/ D 0 for all n 2 Z.

Carlson’s theorem1 implies that f .z/  0, so  t .y/ D zt .y/ for all t 2 R. Since Nexp .z / is -weakly dense in N by Lemma 7.14, the result follows.  Lemma 7.16. Let T 2 P .M; N / and TP W MT ! N be as defined after Definition 7.6. Define RW M2 .MT / ! M2 .N / by  R

x11 x21

x12 x22



 WD

TP .x11 / TP .x21 /

 TP .x12 / ; TP .x22 /

xij 2 MT :

Then (1) M2 .MT / is an M2 .N /-bimodule and R.axb/ D aR.x/b for all x 2 M2 .MT / and a; b 2 M2 .N /, (2) M2 .MT / D span M2 .MT /C and x 2 M2 .MT /C H) R.x/  0. 1Carlson’s theorem: Assume that f is a continuous function on Im z  0, analytic in Im z > 0, jf .z/j  0 Ke cjzj , Im z > 0, with some K; c > 0, jf .t /j  K 0 e c jtj , t 2 R, with some K 0 ; c 0 > 0, and f .i n/ D 0 for all non-negative integers n. Then f .z/  0.

108

7 Operator-valued weights

Proof. (1) Since MT is an N -bimodule, it is clear that M2 .MT / is an M2 .N /-bimodule. If x D Œxij  2 M2 .MT / and a D Œaij , b D Œbij  2 M2 .N /, then axb D P Œ 2k;lD1 aik xkl blj 2i;j D1 and " R.axb/ D

2 X

#2 ai k TP .xkl /blj

D aR.x/b:

k;lD1

i;j D1

(2) Since MT D NT NT , we have ²  x  y11 M2 .MT / D span x11  y

21 21

Furthermore, since      x11 y11 x12 y12 x D 11   x21 y21 x22 y22 0  0 C 0

 x12 y12  y x22 22

  0 y11 0 0   x21 0 0 0



³ W xij ; yij 2 NT :

  0 0 C 0 x12   y21 0 C 0 0

  0 0 0 y12   0 0 x22 0

 0 0  0 ; y22

we find that M2 .MT / D M2 .NT / M2 .NT /. Hence, as in the proof of Lemma 4.36(2) with use of polarization, one has M2 .MT / D span M2 .MT /C . For any ' 2 NC let  WD ' ˝ Tr and z WD .' ı T / ˝ Tr, where Tr is the usual trace on M2 , i.e., 
 a12 D '.a11 / C '.a22 /, aij 2 N . Since x 2 M2 .MT /C H) z.x/ < 1, one  aa11 21 a22 has Mz  span M2 .MT /C D M2 .MT /. Note that for any x D Œxij  2 M2 .MT /, z.x/ D ' ı TP .x11 / C ' ı TP .x22 / D  ı R.x/: With the cyclic representation . ; H ;  / of M2 .N / associated with , for any x 2 M2 .MT /C and a 2 M2 .N /, one has h .a/ ;  .R.x// .a/ i D .a R.x/a/ D  ı R.a xa/ D z.a xa/  0;

(by (1))

so that  .R.x//  0, i.e., s. /R.x/  0, where s. / is the support projection of  . It is easy to W see that ¹ D ' ˝ Tr W ' 2 NC º is a separating family of M2 .N /, which implies that  s. / D 1. Hence R.x/  0.  Lemma 7.17. Let T 2 P .M; N /, '; 2 P .N / and 'z WD ' ı T , z WD x 2 .N'z \ NT / .N z \ NT / H) TP x 2 N' N .

ı T . Then

Proof. First recall that TP x 2 N is well defined for x 2 NT NT D MT . We may assume that x D y  z with y 2 N'z \ NT and z 2 N z \ NT . Set x11 WD y  y, x12 WD y  z D x,      y z  x12 x21 WD z  y and x22 WD z  z; then xz WD xx11 2 M2 .MT / and xz D y0 z0 21 x22 0 0  0.

109

7.2 Operator-valued weights

P  P Hence R.z x / D TP .x11 / TP .x12 /  0 by Lemma 7.16(2). Consider the balanced weight T .x21 / T .x22 /  WD .'; / 2 P .M2 .N // and note that .R.z x // D '.T x11 / C

.T x22 / D '.y z  y/ C z .z  z/ < 1:

Therefore, we have R.z x / 2 M so that TP x D TP x12 2 N' N by Lemma 6.4(3).



We are now in a position to prove Theorem 7.8. Proof of Theorem 7.8. Let T 2 P .M; N / and '; 2 P .N /. We prove claim .?/. By Lemma 7.15 it suffices to prove that  i;'   iıT;'ıT . Let 'z WD ' ı T and z WD ı T . Assume that
(7.10) a 2 D  i;' and b D  i;' .a/ and prove that a2D 

z ;' z
i

and

bD

z ;' z i .a/:

(7.11)

;' /, i=2

;' By Theorem 7.10(1) we note that a 2 D. b 2 D.i=2;' / and  i=2 .a/ D i=2;' .b/. Since  t'; .b  / D  t ;' .b/ by Lemma 6.5, one has b  2 D. ';i=2 /. Hence by Lemma b C, 7.13, there is a k > 0 such that a a  k 2 ' and b'b   k 2 on NC . For any m 2 N

by Theorem 7.3 there is a sequence yn 2 NC such that yn % m. Then .ama / D .a a/.m/ D lim .a a/.yn /  k 2 lim '.yn / D k 2 '.m/ n!1

n!1

and similarly '.b  mb/  k 2 .m/. Therefore, for every x 2 MC one has z .axa / D

.aT .x/a /  k 2 '.T .x// D k 2 '.x/ z

and similarly '.b z  xb/  k 2 z .x/. By Lemma 7.13 again, one has N'z a  N z ;

i.e., aN'z  Nz ;

N z b  N'z ;

(7.12) (7.13)

k.ya / z k  kky'z k

for all y 2 N'z ;

(7.14)

k.zb/'z k  kkz z k

for all z 2 N z :

(7.15)

By Theorem 7.11 with (7.12) and (7.13), to show (7.11) it remains to prove that z .ax/ D '.xb/ z

for all x 2 N'z N z :

(7.16)

First assume that x0 D y0 z0 with y0 2 N'z \ NT and z0 2 N z \ NT . Since x0 2 MT and TP x0 2 N' N by Lemma 7.17, we have by (7.10) and Theorem 7.11, .TP .ax0 // D

.a.TP x0 // D '..TP x0 /b/ D '.TP .x0 b//:

110

7 Operator-valued weights

Moreover, note from (7.12) and (7.13) that ax0 D .y0 a / z0 2 .N z \ NT / .N z \ NT /  span.F z \ FT /; x0 b D y0 .z0 b/ 2 .N'z \ NT / .N'z \ NT /  span.F'z \ FT /; so that .TP .ax0 // D . ıT /.ax0 / D z .ax0 / and '.TP .x0 b// D .' ıT /.x0 b/ D '.x z 0 b/. Therefore, z .ax0 / D '.x z 0 b/: (7.17) Next assume that x D y  z with y 2 N'z and z 2 N z . Since '.T .y  y// < 1, R1 T .y  y/ has the spectral decomposition T .y  y/ D 0  de (see Theorem 7.3). For any s > 0 we have Z s T .es y  yes / D es T .y  y/es D

 de ; 0

' ı T .es y  yes / D '

s

Z



1

Z '

 de 0

  de

< 1;

0

so that yes 2 N'z \ NT . Moreover, k.y

yes /'z k2 D ' ı T ..y D ' ı T ..1

yes / .y

yes //



es /y y.1

es //



D '..1 Z D'

es /T .y y/.1 es //   de ! 0 as s ! 1:

1

s

R1 Similarly, T .z  z/ has the spectral decomposition T .z  z/ D 0  df , and for any s > 0, zfs 2 N z \ NT and k.z zfs / z k2 ! 0 as s ! 1. From (7.14) and (7.15) it follows that k.ya / z

.yes a / z k ! 0;

k.zb/'z

.zfs b/'z k ! 0

as s ! 1:

Therefore, we have z .ax/ D z ..ya / z/ D h.ya / z ; z z i D lim h.yes a / z ; .zfs / z i D lim z .aes y  zfs / D lim z .a.es xfs //; s!1

s!1

s!1



'.xb/ z D '.y z zb/ D hy'z ; .zb/'z i D lim h.yes /'z ; .zfs b/'z i D lim '..e z s xfs /b/: s!1

s!1

From (7.17) for x0 WD es xfs D .yes / .zfs / it follows that z .a.es xfs // D '..e z s xfs /b/; Letting s ! 1 gives (7.16).

s > 0: 

111

7.3 Pedersen–Takesaki construction

We finish the section with the next theorem without proof; it is a main result of [33], giving the converse of Theorem 7.8(a), and in a sense generalizes Takesaki’s theorem [92] (see Theorem 5.6 in the case of bounded '). Theorem 7.18 (Haagerup). Let N  M be von Neumann algebras and ' 2 P .M /, 2 P .N /. If  t' .x/ D  t .x/ for all t 2 R and all x 2 N , then there exists a unique T 2 P .M; N / such that ' D ı T .

7.3 Pedersen–Takesaki construction Define the centralizer M' of ' 2 P .M / as the fixed-point algebra of  t' (see Proposition 2.16 in the case of a faithful ' 2 MC ), i.e., M' WD ¹x 2 M W  t' .x/ D x; t 2 Rº: Of course, M' is a von Neumann subalgebra of M . Lemma 7.19. Let ' 2 P .M /. (1) For a 2 M , a 2 M' if and only if aN'  N' , N' a  N' and '.ax/ D '.xa/ for all x 2 M' . (2) For a unitary u 2 M , u 2 M' ” '.u
u / D '. (3) If a 2 .M' /C , then 'a .x/ WD '.a1=2 xa1=2 /;

x 2 MC

(7.18)

is a semifinite normal weight on M and 'a .x/ D '.ax/ D '.xa/ for all x 2 M' . Moreover, 'a .x  x/ D k' .a/1=2 J' x' k2 D hJ' x' ; ' .a/J' x' i;

x 2 N' :

(7.19)

(4) If a; b 2 .M' /C , then 'aCb D 'a C 'b . Hence, if a; b 2 M' and 0  a  b, then 'a  'b . (5) If a˛ is an increasing net in .M' /C and a˛ % a 2 .M' /C , then 'a˛ % 'a . Proof. (1) Assume that a 2 M' ; then it is obvious that a 2 D. 'i / and  'i .a/ D a. Hence the stated condition holds by Theorem 7.11 for ' D . Conversely, assume the stated condition. Then the same holds for a too since for y; z 2 N' , '.a y  z/ D '..ya/ z/ D '.z  ya/ D '.az  y/ D '..za / y/ D '.y  za /: Hence we may assume that a D a . By Theorem 7.11 for ' D again, we have a 2 D. 'i / and  'i .a/ D a, so that  t' i .a/ D  t' . 'i .a// D  t' .a/ for all t 2 R.

112

7 Operator-valued weights

Therefore, from the Schwarz reflection principle it follows that for any  2 H , f .t / WD h;  t' .a/i extends to an entire function f .z/ with period i . From the Liouville theorem, f .z/  f .0/ for all  2 H , so  t' .a/ D a for all t 2 R. Hence a 2 M' follows. (2) Assume that u 2 M' , so u 2 M' as well. Then by (1), uM' D M' u D M' and '.uxu / D '.xu u/ D '.x/ for all x 2 M' . Hence '.u
u / D ' holds. Conversely, assume that '.u
u / D '. Then '.u
u/ D ' holds as well. So N' u D N' u D N' and '.ux/ D '.u uxu/ D '.xu/ for all x 2 M' . Hence u 2 M' follows from (1). (3) Assume that a 2 .M' /C and so a1=2 2 M' . It is clear that 'a is a normal weight on M . If x 2 N' , then xa1=2 2 N' by (1), so that 'a .x  x/ D '..xa1=2 / .xa1=2 // < 1. This means that N'  N'a , so 'a is semifinite. For every x 2 M' , since a1=2 x; xa1=2 2 M' by (1), one has '.a1=2 xa1=2 / D '.ax/ D '.xa/. Moreover, for every x 2 N' , by (7.7) one has .xa1=2 /' D J' ' .a1=2 /J x' , which gives (7.19). (4) We have unique contractions v; w 2 M such that a1=2 D v.a C b/1=2 , b 1=2 D w.a C b/1=2 and v.1 s.a C b// D w.1 s.a C b// D 0, where s.a C b/ is the support projection of a C b. Since a C b D .a C b/1=2 .v  v C w  w/.a C b/1=2 , it immediately follows that v  v C w  w D s.a C b/. Moreover, applying  t' to a1=2 D v.a C b/1=2 gives  t' .v/ D v for all t 2 R. Hence v 2 M' and similarly w 2 M' . Let x 2 MC . First assume that 'aCb .x/ D 1. If 'a .x/ < 1 and 'b .x/ < 1, then x 1=2 a1=2 ; x 1=2 b 1=2 2 N' , so that x 1=2 .a Cb/1=2 v  v D x 1=2 a1=2 v 2 N' and x 1=2 .a Cb/1=2 w  w D x 1=2 b 1=2 w 2 N' thanks to (1). Hence x 1=2 .a C b/1=2 2 N' , contradicting 'aCb .x/ D 1. Therefore, 'a .x/ C 'v .x/ D 1 D 'aCb .x/ in this case. Next assume that 'aCb .x/ < 1 and so .a C b/1=2 x.a C b/1=2 2 M' . By (1) one has .a C b/1=2 x.a C b/1=2 v  2 M' and 'a .x/ D '.v.a C b/1=2 x.a C b/1=2 v  / D '..a C b/1=2 x.a C b/1=2 v  v/; and similarly 'b .x/ D '..a C b/1=2 x.a C b/1=2 w  w/. Hence 'a .x/ C 'b .x/ D 'aCb .x/ follows. The latter assertion of (4) is now obvious. (5) From (4) it is clear that 'a˛ % and 'a˛  'a . For any x 2 MC , since 1=2 an xan1=2 ! a1=2 xa1=2 strongly, by the lower semicontinuity of ' (see Theorem 6.2(iii)) one has 'a .x/ D '.a1=2 xa1=2 /  lim inf '.a˛1=2 xa˛1=2 /  sup 'a˛ .x/: ˛

˛

Hence 'a˛ % 'a follows.



Lemma 7.20. Let ' 2 P .M /. (1) If ˛ is an automorphism of M , then  t'ı˛ D ˛

1

ı  t' ı ˛;

t 2 R:

(2) If a 2 M' is positive invertible, then 'a 2 P .M / and  t'a .x/ D ai t  t' .x/a

it

;

x 2 M; t 2 R:

113

7.3 Pedersen–Takesaki construction

(3) If e 2 M' is a projection, then 'je WD 'jeM e 2 P .eM e/ and '

 t je .x/ D  t' .x/;

x 2 eM e; t 2 R:

(Note that 'je is essentially the same as 'e , although 'e is defined on M .) Proof. (1) It is immediate to see that N'ı˛ D ˛ 1 .N' / and so M'ı˛ D ˛ 1 .M' /. Define a -weakly continuous one-parameter automorphism group  t by  t WD ˛ 1 ı t' ı˛ (t 2 R). From Lemma 7.15 in the case of N D M (see also Theorem 7.10) it suffices to 1 .D. 'i // and  i .a/ D ˛ 1 ı 'i ı˛.a/ prove that  'ı˛ i D  i . Note that D. i / D ˛ for all a 2 D. i /. From Theorem 7.11 we find that 'ı˛ a 2 D. 'ı˛ i / and b D  i .a/

” aN'ı˛  N'ı˛ , N'ı˛ b  N'ı˛ and ' ı ˛.ax/ D ' ı ˛.xb/ for all x 2 M'ı˛ ” ˛.a/N'  N' , N' ˛.b/  N' and '.˛.a/x/ D '.x˛.b// for all x 2 M' ” ˛.a/ 2 D. 'i / and ˛.b/ D  'i .˛.a// ” a 2 D. i / and b D  i .a/: Therefore,  'ı˛ i D  i holds. (2) Assume that a 2 M' is positive invertible. It is clear that 'a 2 P .M /. Since ka 1 k 1  a  kak, it follows from Lemma 7.19(4) that ka 1 k 1 '  'a  kak' and so N' D N'a and M' D M'a . Define a -weakly continuous one-parameter automorphism group  t by  t .x/ WD ai t  t' .x/a i t (x 2 M , t 2 R). Note that D. i / D D. 'i / and  i .c/ D a 'i .c/a 1 for all c 2 D. i /. From Theorem 7.11 and Lemma 7.19(3), we find that c 2 D. 'ai / and b D  'ai .c/ ” cN'a  N'a , N'a b  N'a and 'a .cx/ D 'a .xb/ for all x 2 M'a ” cN'  N' , N' b  N' and '.cxa/ D '.xba/ for all x 2 M' ” cN'  N' , N' .a (since N' a '

1

1

ba/  N' and '.cx/ D '.xa

D N' and M' a

” c 2 D. i / and a

1

1

1

ba/ for all x 2 M'

D M' )

'

ba D  i .c/

” c 2 D. i / and b D  i .c/: Therefore,  'ai D  i holds, so  t'a D  t follows from Lemma 7.15 in the case of N D M. (3) It is clear that 'je is faithful and normal. Since N'je D eN' e, note that 'je is also semifinite. Moreover, M'je  eM' e is clear. If x 2 eF' e, then x 1=2 2 eN' e so that x 2 M'je . Hence eM' e  M'je , so one has M'je D eM' e. For every x 2 eM e, one

114

7 Operator-valued weights

has  t' .x/ D  t' .exe/ D e t' .x/e. Hence one can define a  -weakly continuous oneparameter automorphism group  t on eM e by  t .x/ D  t' .x/ D e t' .x/e (x 2 eM e, t 2 R). For a; b 2 eM e, from Theorem 7.11 we have a 2 D. i / and b D  i .a/ ” a 2 D. 'i / and b D  'i .a/ ” aN'  N' , N' b  N' and '.ax/ D '.xb/ for all x 2 M' H) aN'je  N'je , N'je b  N'je and '.ax/ D '.xb/ for all x 2 M'je ” a 2 D. Therefore, 

i



'je i

'je i /

and b D 

'je i .a/:

'

holds, so  t je D  t follows from Lemma 7.15.



Note that another standard way to prove Lemma 7.20 is to make use of the KMS condition characterizing the modular automorphism group (see (C) in Section 6.1). Let ' 2 P .M / and A be a positive self-adjoint operator affiliated with M' . We extend 'a (defined by (7.18) for a 2 .M' /C ) to 'A . To do so, set A" WD A.1 C "A/

1

2 .M' /C ;

" > 0:

(7.20)

Then Lemma 7.19(4) implies that 'A" % as " &, so we define 'A .x/ WD sup 'A" .x/ D lim 'A" .x/; ">0

"&0

x 2 MC :

(7.21)

Proposition 7.21. Let ' and A be as above. (1) 'A is a semifinite normal weight on M and 'A .x  x/ D k' .A/1=2 J' x' k2 ;

x 2 N' ;

where the quadratic form in the right-hand side is defined to be 1 unless J' x' 2 D.' .A/1=2 /. (2) 'A is faithful if and only if A is non-singular. (3) Let B be another positive self-adjoint operator affiliated with M' . Then A  B (in the sense of Definition A.2) ” 'A  'B . (4) Let A˛ be a net of positive self-adjoint operators affiliated with M' . Then A˛ % A (in the sense of Definitions A.2 and A.6) ” 'A˛ % 'A . Proof. (1) That 'A is normal is clear by definition (7.21). Let en be the spectral projection of A corresponding to Œ0; n, so en 2 M' . For every x 2 N' one has  1=2   'A" .en x  xen / D '.A1=2 " en x xen A" / D '.x xen A" /  n'.x x/ < 1

115

7.3 Pedersen–Takesaki construction

thanks to (1) and (4) of Lemma 7.19. Hence 'A .en x  xen / < 1 so that N' en  N'A . Since en % 1, 'A is semifinite. Moreover, for every x 2 N' , by (7.19) one has 'A .x  x/ D sup k' .A" /1=2 J' x' k2 D k' .A/1=2 J' x' k2 : ">0

(2) is easy to verify. (3) If A  B, then A"  B" for all " > 0 by Lemma A.1. By Lemma 7.19(4), for any x 2 MC one has 'A" .x/  'B" .x/  'B .x/. Hence 'A .x/  'B .x/. Conversely, assume that 'A  'B . Let fn be the spectral projection of B corresponding to Œ0; n. Then, for every x 2 N' one has hJ' x' ; ' .fn A" fn /J' x' i D h' .fn /J' x' ; ' .A" /' .fn /J' x' i D hJ' .xfn /' ; ' .A" /J' .xfn /' i

(by (7.7))



D 'A" .fn x xfn / (by (7.19))  'B .fn x  xfn / D sup '.B"1=2 fn x  xfn B"1=2 / ">0



 'Bfn .x x/ D hJ' x' ; ' .Bfn /J' x' i; which implies that fn A" fn  Bfn  B. Since fn % 1, A"  B for any " > 0, implying that A  B. (4) For any " > 0, since .A˛ /" % A" by Lemmas A.1 and A.4, for every x 2 MC one has '.A˛ /" .x/ % 'A" .x/ by Lemma 7.19(5). This implies that 'A˛ % and 'A .x/ D sup 'A" .x/ D sup sup '.A˛ /" .x/ D sup 'A˛ .x/; ">0 ˛

">0

˛

so that 'A˛ % 'A . Conversely, assume that 'A˛ % 'A . It follows from (3) that A˛ is increasing and A˛  A. Hence there is a positive self-adjoint operator B such that A˛ % B. Noting that BM' , one has 'A˛ % 'B by the first part of the proof. Hence 'A D 'B so that A D B holds by (3).  Theorem 7.22. Let ' 2 P .M / and A be a non-singular positive self-adjoint operator affiliated with M' . Then (1)  t'A .x/ D Ai t  t' .x/A

it

for all x 2 M and t 2 R,

(2) .D'A W D'/ t D Ai t for all t 2 R. For the proof we need one more lemma. Lemma 7.23. Let ' and A be as in Theorem 7.22. Let en be the spectral projection of A corresponding to Œ1=n; n for n 2 N. Then en 2 M'A . Proof. For every " > 0, since 'A"  'A by Proposition 7.21(3), one has N'A  N'A" . For any x 2 M one has  1=2   'A" .en x  xen / D '.A1=2 " en x xen A" / D 'A" en .x x/  'A .x x/:

116

7 Operator-valued weights

Letting " & 0 yields that 'A" .en x  xen /  'A .en x  xen /  'A .x  x/. Therefore, N'A en  N'A and N'A en  N'A" , so it follows that M'A en D N'A N'A en  N'A" N'A" D M'A" ; en M'A D en N'A N'A  N'A" N'A" D M'A" : 1=2   Furthermore, for every " > 0, since '.A1=2 " x xA" /  'A .x x/, we have

N'A A1=2  N' ; "

1=2 hence A1=2  M' : " M'A A"

(7.22)

For every x 2 M'A , since en x; xen 2 M'A" , we find that 1=2 1=2 1=2 'A" .en x/ D '.A1=2 " en xA" / D '.en A" xA" / 1=2 D '.A1=2 " xA" en /

D

1=2 '.A1=2 " xen A" /

(by (7.22) and Lemma 7.19(1)) D 'A" .xen /:

Letting " & 0 yields that 'A .en x/ D 'A .xen /, which, together with N'A en  N'A shown above, implies by Lemma 7.19(1) that en 2 M'A .  Proof of Theorem 7.22. (1) For each n 2 N let en be as given in Lemma 7.23. From ' Lemma 7.20(3) it follows that  t jen .Aen / D  t' .Aen / D Aen for all t 2 R, so that Aen 2 M'jen . On the other hand, en 2 M'A by Lemma 7.23. For every x 2 .en M en /C note that 1=2 .'A /jen .x/ D 'A .x/ D sup '.A1=2 " en xen A" / ">0

D sup.'jen /A" en .x/ D .'jen /Aen .x/; ">0

where the last equality follows from Proposition 7.21(4) since A" en % Aen . Therefore, .'A /jen D .'jen /Aen : Since Aen is invertible in en M en , for any x 2 en M en we have .'A /jen

 t'A .x/ D  t D

.'jen /Aen

.x/ D  t

Ai t en  t' .x/en A i t

D

'

.x/ D .Aen /i t  t jen .x/.Aen /

it

Ai t  t' .x/A i t ;

where the first and the fourth equalities S follow from Lemma 7.20(3) and the third equality follows from Lemma 7.20(2). Since n2N en M en is strongly dense in M due to en % 1, the result follows. (2) Consider the balanced weights  WD .'; '/ and z WD .'; 'A /. If a; b 2 M' ,  
  ' .a/ 0      then  t a0 b0 D t 0  ' .b/ D a0 b0 by Lemma 6.5, so a0 b0 2 M . Hence we have t

117

7.3 Pedersen–Takesaki construction

Az WD that

1

0 0A



 M . Then it is immediate to see that z D Az. Therefore, from (1) it follows

0 .D'A W D'/ t

    i t    0 0 0 1 0 0 0 1 z D  t D  t 0 1 0 0 A 1 0 0       1 0 0 0 0 0 0 1 D D ; 0 Ai t 1 0 0 A i t Ai t 0

0 A



so that .D'A W D'/ t D Ai t holds.

it



While (2) of Theorem 7.22 has been proved from (1), we note that (1) conversely follows from (2) in view of Theorem 6.6. Here we give some basic properties of Connes’ cocycle derivatives (introduced in Section 6.2) to use in the proof of Theorem 7.25. Proposition 7.24. Let ', (1) .D' W D / t D .D (2)

D ' ” .D

and 'i be faithful semifinite normal weights on M . W D'/t , t 2 R. W D'/ t D 1, t 2 R.

(3) .D'1 W D'3 / t D .D'1 W D'2 / t .D'2 W D'3 / t , t 2 R (the chain rule). (4) '1 D '2 ” .D'1 W D'/ t D .D'2 W D'/ t , t 2 R. Proof. (1) Let  WD .'; / and y WD . ; '/. Define an automorphism of M2 .M /       x12 by .x/ WD 01 10 x 01 10 for x D xx11 2 M2 .M /. Since y.x/ D . .x// for all 21 x22 x 2 M2 .M /, we find by Lemma 7.20(1) that    .x /  ;' .x12 / y  t .x/ D 1 ı  t ı .x/ D '; 11  .x21 /  ' .x22 / under the expression of  t in (6.6). Hence .D' W D / t D  t'; .1/ D  t ;' .1/ D .D W D'/t by Lemma 6.5. (2) That .D' W D'/ t D 1 is clear as noted just after Definition 6.7. Conversely, assume that .D 
all t 2 R. Let  WD .';  /. Then we have  
  W D'/ t D 1 for  t 01 00 D 01 00 and so  t 01 10 D 01 10 for all t 2 R. Hence 01 10 2 .M2 .M // follows. By Lemma 7.19(2) this implies that         x 0 0 1 x 0 0 1 0 0 '.x/ D  D D D .x/ 0 0 1 0 0 0 1 0 0 x for all x 2 MC . (3) It is convenient to use the triple balanced weight on M3 .M /: ! 3 3 3 X X X  xij ˝ eij WD 'i .xi i /; xij ˝ eij 2 M3 .M /C : i;j D1

iD1

i;j D1

118

7 Operator-valued weights

While the details are omitted here, as in the case of .'; / treated in Section 6.2, we have ' ;'j

 t .x ˝ eij / D  t i

.x/ ˝ eij ;

x 2 M; i; j D 1; 2; 3:

Therefore, .D'1 W D'3 / t ˝ e13 D  t'1 ;'3 .1/ ˝ e13 D  t .1 ˝ e13 / D  t .1 ˝ e12 / t .1 ˝ e23 / D .D'1 W D'2 / t .D'2 W D'3 / t ˝ e13 : (4) By (1), (3) and (2) we find that .D'1 W D'/ t D .D'2 W D'/ t ; t 2 R ” .D'1 W D'2 / t D 1; t 2 R ” '1 D '2 :



The main result of Pedersen and Takesaki [71] is the following. Theorem 7.25 (Pedersen and Takesaki). Let '; are equivalent: (i)

is  ' -invariant, i.e.,

ı  t' D

2 P .M /. Then the following conditions

for all t 2 R;

(ii) ' is  -invariant; (iii) .D

W D'/ t 2 M for all t 2 R;

(iv) .D

W D'/ t 2 M' for all t 2 R;

(v) .D

W D'/ t (t 2 R) is a strongly continuous one-parameter unitary group;

(vi) there exists a (unique) non-singular positive self-adjoint operator AM' such that D 'A . Proof. Set u t WD .D

W D'/ t , t 2 R.

(i) H) (iii). By Lemma 7.20(1) one has s D  't ı s ı  t' so that  t' ı s D s ı  t' ;

s; t 2 R:

If x 2 M , then  t' .x/ D  t' .s .x// D s . t' .x// for all s; t 2 R, so  t' .x/ 2 M . Hence  t' .M / D M , t 2 R. For every x 2 MC and t 2 R, .x/ D D

. t .x// D

.u t  t' .x/ut / (by (6.8))

. 't .u t  t' .x/ut // D

. 't .u t /x 't .u t / /:

Hence by Lemma 7.19(2) one has  't .u t / 2 M , so u t 2  t' .M / D M for all t 2 R.

7.3 Pedersen–Takesaki construction

119

(iii) ” (v). Since s .u t / D us s' .u t /us D usCt us by (6.8) and (6.9), we have u t 2 M ; t 2 R ” s .u t / D u t ; s; t 2 R ” u tCs D u t us ; s; t 2 R: (iv) ” (v). This is immediate from (iii) ” (v) since ut D .D' W D / t . (v) H) (vi). From (v) (also (iv)), Stone’s theorem says that there exists a non-singular positive self-adjoint operator AM' such that u t D Ai t for all t 2 R. Hence one has .D W D'/ t D Ai t D .D'A W D'/ t , t 2 R, by Theorem 7.22(2), so Proposition 7.24(4) implies that D 'A . The uniqueness of A in (vi) is immediate from Theorem 7.22(2). (vi) H) (i). Assume that

D 'A as stated in (vi). Then, for every x 2 MC one has

' ' 1=2 1=2 1=2 . t' .x// D sup '.A1=2 "  t .x/A" / D sup '. t .A" xA" // ">0

">0

1=2 D sup '.A1=2 " xA" / D 'A .x/ D

.x/;

">0

showing (i).



In this way, A 7! 'A is a bijective correspondence between the set of non-singular positive self-adjoint operators AM' and ¹ 2 P .M / W  ' -invariantº. When D 'A , A is called the Radon–Nikodym derivative of with respect to ', and 'A is often written as '.A
/. In particular, assume that ' D  is a faithful semifinite normal trace on M ; then  t D id (t 2 R) so that M D M . So Theorem 7.252 shows that any 2 P .M / is represented as D  .A
/ with a positive self-adjoint AM . This extends Corollary 4.51. In fact, Connes’ cocycle derivative .D W D'/ t can be extended to the case where is a (not necessarily faithful) semifinite normal weight on M , and the conditions (except (ii)) of Theorem 7.25 are still equivalent in this case. Since  t' .M' / D M' (t 2 R) trivially, Takesaki’s theorem [92] (see Theorem 5.6 when ' is a faithful normal state) implies that if ' is semifinite on M' , then there exists a faithful normal conditional expectation E' W M ! M' such that ' D ' ı E' . In fact, Combes [12] proved the next theorem characterizing this situation, which we record without proof in the following. Theorem 7.26. Let ' 2 P .M /. Then the following conditions are equivalent: (a) 'jM' is semifinite (so it is a faithful semifinite normal trace on M' ); (b) there exists a faithful normal conditional expectation E' W M ! M' with ' D 'ıE' (then E' is automatically  ' -invariant); P P (c) there exists a family ¹'i º  MC such that i s.'i / D 1 and ' D i 'i ; (d) M is  ' -finite in the sense that for any x 2 M , x ¤ 0, there exists a  ' -invariant ! 2 MC such that !.x/ ¤ 0. 2A point of Theorem 7.25 is that it holds without M' being semifinite.

120

7 Operator-valued weights

When the conditions of Theorem 7.26 hold, ' is said to be strictly semifinite.3 In this case, let  WD 'jM' . Then one can easily see that 2 P .M / is  ' -invariant if and only if is E' -invariant, and if this is the case, then D 'A D A ı E' with AM' as in (vi) of Theorem 7.25. Thus, by Theorem 7.8(b) one has .D'A W D'/ t D .DA ı E' W D ı E' / t D .DA W D / D Ai t ; so that Theorem 7.22(2) reduces to .DA W D / D Ai t . Furthermore, we note a result in [33] that M' is semifinite (a slightly weaker condition than (a)) if and only if there exists an  ' -invariant T 2 P .M; M' / (weaker than (b)). The next result is Takesaki’s theorem first proved in [91], saying that M is semifinite if and only if the modular automorphism group is inner. Theorem 7.27. The following conditions are equivalent: (i) M is semifinite; (ii)  ' is inner (i.e., there is a strongly continuous one-parameter unitary group ¹u t º t 2R in M such that  t' D u t
ut for all t 2 R) for any (equivalently, some) ' 2 P .M /. Proof. Assume that M is semifinite with a faithful semifinite normal trace  , so M D M . For any ' 2 P .M /, it follows from Theorem 7.25 that ' D A for some non-singular positive self-adjoint operator AM . Hence by Theorem 7.22(1),  t' .x/ D Ai t  t .x/A

it

D Ai t xA

it

;

x 2 M; t 2 R:

Conversely, assume that  ' is inner for some ' 2 P .M /, that is, there is a non-singular positive self-adjoint operator AM such that  t' D Ai t
A i t , t 2 R. Since  t' .Ai s / D Ais , we have Ais 2 M' for all s 2 R. Hence AM' , so we can define  WD 'A 1 2 P .M /. Then by Theorem 7.22(1),  t .x/ D A

it

 t' .x/Ai t D x;

x 2 M; t 2 R;

that is, M D M , which means that  is a trace.



We end the section with the next lemma, considered to be an extension of Lemma 6.5 to slightly more general weights than balanced weights, which will be used in Section 11.2. Lemma 7.28. Let e 2 M be a projection. Let ' 2 P .eM e/ and Define  2 P .M / by .x/ WD '.exe/ C

..1

e/x.1

e//;

2 P ..1 e/M.1 e//.

x 2 MC :

Then e; 1 e 2 M (the centralizer of ) and  t' ,  t are the restrictions of  t to eM e, .1 e/M.1 e/, respectively, for all t 2 R. 3Haagerup [29] gave an example of a faithful semifinite normal weight ' on the hyperfinite type II1 factor M such that M' is of type III (hence ' is not strictly semifinite).

7.3 Pedersen–Takesaki construction

121

Proof. One can modify the arguments around Lemmas 6.4 and 6.5 with the representation   exe  ex.1 e/  x12 of x 2 M in place of xx11 2 M2 .M /. Indeed, by arguing as 21 x22 .1 e/xe .1 e/x.1 e/ in Section 6.2 one can see that the GNS Hilbert space H and the modular operator  associated with .M;  / are given in the form H D H' ˚ H12 ˚ H21 ˚ H ;  D ' ˚ 12 ˚ 21 ˚  ; where ' is the modular operator on the GNS Hilbert space H' associated with .eM e; '/ and  is that associated with ..1 e/M.1 e/; /. This implies that  t' D  t jeM e and  t D  t j.1 e/M.1 e/ ; in particular,  t .e/ D e and  t .1 e/ D 1 e for all t 2 R.  Remark 7.29. Lemma 7.28 can also be shown by making use of Lemmas 7.19(1) and 7.20(3). In fact, note that x 2 M belongs to N if and only if exe; .1 e/xe 2 N' and ex.1 e/; .1 e/x.1 e/ 2 N . It is clear that N e  N . Moreover, for any x; y 2 N , using the polarization in (4.15) one can confirm that .ey  x/ D .y  xe/ and .y  x.1 e// D ..1 e/y  x/. The details are left as an exercise for the reader.

8 Takesaki duality and structure theory

In the first part of this chapter we present Takesaki’s duality theorem [93] for crossed products by locally compact abelian group actions, as well as a concise account of dual weights developed by Haagerup [31]. In the second part we give a short survey on Takesaki’s structure theorem in terms of the crossed product decomposition based on Takesaki’s duality, which will play an essential role in Chapter 9.

8.1 Takesaki’s duality theorem We begin with the definition of crossed products of von Neumann algebras. Let M be a von Neumann algebra on a Hilbert space H . Let G be a locally compact group and ˛ be a continuous action of G on M , i.e., t 2 G 7! ˛ t 2 Aut.M / (= the automorphism group of M ) is a weakly (equivalently, strongly) continuous homomorphism. The triplet .M; G; ˛/ is called a W  -dynamical system. We write ds for a left invariant Haar measure on G. Let L2 .G/ be the Hilbert space of square integral functions on G with respect to ds. Further, let L2 .G; H / be the H -valued square integral functions on R G with respect to ds, which becomes a Hilbert space with the inner product h; i WD G h.s/; .s/i ds for ;  2 L2 .G; H /. Note that L2 .G; H / Š H ˝ L2 .H /, the tensor product Hilbert space of H and L2 .G/, so we always identify L2 .G; H / and H ˝ L2 .G/. For every x 2 M and t 2 G define .˛ .x//.s/ WD ˛s

1

.x/.s/;

..t //.s/ WD .t

1

s/;

s 2 G;  2 L2 .G; H /:

Obviously, .t / is written as .t / D 1 ˝  t , where  t 2 B.L2 .G// is defined by . t f /.s/ WD f .t 1 s/, s 2 G, for f 2 L2 .G/. Lemma 8.1. Let ˛ and  be as above. Then (1) ˛ is a faithful normal representation of M on L2 .G; H /, (2)  is a strongly continuous unitary representation of G on L2 .G; H /, (3) ˛ .˛ t .x// D .t /˛ .x/.t / for all x 2 M and t 2 G (the covariance property). Proof. (1) and (2) are shown by direct computations (an exercise). As for (3) let x 2 M and t 2 G. For every  2 L2 .G; H / we have ..t /˛ .x/.t / /.s/ D .˛ .x/.t / /.t D ˛s

1

1

s/ D ˛s

1t

.x/..t / /.t

.˛ t .x//.s/ D .˛ .˛ t .x///.s/;

1

s/

s 2 G:



124

8 Takesaki duality and structure theory

Definition 8.2. The crossed product M o˛ G (or M ˝˛ G) of M by the action ˛ is the von Neumann algebra generated by ˛ .M / and .G/, i.e.,
00 M o˛ G WD ¹˛ .x/ W x 2 M º [ ¹.t / W t 2 Gº : By Lemma 8.1(3) note that span¹˛ .x/.t / W x 2 M; t 2 Gº is a *-subalgebra of M o˛ G and its strong closure is equal to M o˛ G. Examples 8.3. (1) When M D C is trivial and so G acts trivially on C, the crossed product C o G is nothing but the group von Neumann algebra L .G/ WD .G/00 generated by the left regular representation .t /, t 2 G, on L2 .G/. (2) Let n 2 N. Let M D Cn be an abelian von Neumann algebra on H D Cn , and G D Zn be the cyclic group of order n. Define the action ˛ of Zn on Cn by cyclic coordinate permutations. Then the crossed product Cn o˛ Zn is *-isomorphic to Mn .C/ D B.Cn /. This might be the simplest non-trivial example of crossed products. The proof is easy as follows: Let ei (1  i  n) be the natural basis of Cn , and w WD .1/, where 1 is the generator of Zn D ¹0; 1; : : : ; n 1º. Since ˛ .ei / D w i 1 ˛ .e1 /w .i 1/ , it follows that Cn o Zn is generated by Eij WD w i 1 ˛ .e1 /w .j 1/ (i; j D 1; : : : ; n). We find that Eij D Ej i and Eij Ekl D w i

1

˛ .e1 /w .j

D wi

1

w .j

D wi

j Ck 1

k/

k/

˛ .ej

˛ .ej

˛ .e1 /w .l

1/

kC1 /˛ .e1 /w

kC1 e1 /w

.l 1/

.l 1/

D ıj k Ei l ;

where j k C 1 is given in mod n. Hence ¹Eij ºni;j D1 constitutes a system of n  n matrix units. So Cn o˛ Zn Š Mn .C/. (3) Let M D `1 .Z/, an abelian von Neumann algebra on H D `2 .Z/, and G D Z act on `1 .Z/ by shift, i.e., .˛n .a//.i / D a.i C n/ for a 2 `1 .Z/ and n; i 2 Z. Then `1 .Z/ o˛ Z Š B.`2 .Z//. The proof is similar to that in (2). The crossed products in (2) and (3) are factors (of type I). This corresponds to the fact that the action ˛ of Z (or Zn ) on the space Z (or Zn ) is ergodic. b In the rest of this section we assume that G is a locally compact abelian group and G is the (Pontryagin) dual group of G. We write ht; pi WD p.t / for t 2 G and a character b For every p 2 G b define a unitary v.p/ 2 B.L2 .G; H // by p 2 G. .v.p//.s/ WD hs; pi.s/;

s 2 G;  2 L2 .G; H /:

b on L2 .G; H / and it is written Then v is a strongly continuous unitary representation of G as v.p/ D 1˝vp , where vp is defined by .vp f /.s/ WD hs; pif .s/, s 2 G, for f 2 L2 .G/. b Lemma 8.4. For every x 2 M , t 2 G and p 2 G, v.p/˛ .x/v.p/ D ˛ .x/;

v.p/.t /v.p/ D ht; pi.t /:

125

8.1 Takesaki’s duality theorem

The proof of the lemma is a straightforward computation (an exercise). b on M o˛ G can be defined by Definition 8.5. The continuous action ˛ y of G ˛ yp .y/ WD v.p/yv.p/ ;

b y 2 M o˛ G: p 2 G;

b is called the dual action. The action ˛ y of G The next proposition shows, in particular, that the isomorphism class of M o˛ G is determined by .M; G; ˛/ independently of the representing space of M (so one can take any representing Hilbert space of M to make M o˛ G). Proposition 8.6. Let .M; G; ˛/ and .N; G; ˇ/ be W  -dynamical systems. If ˛ and ˇ are conjugate, i.e., there exists a *-isomorphism W M ! N such that ı˛ t D ˇ t ı (t 2 G), then there exists a *-isomorphism zW M o˛ G ! N oˇ G such that z.˛ .x// D ˇ . .x// b where M , N are (x 2 M ), z.M .t // D N .t / (t 2 G) and z ı ˛ yp D ˇyp ı z (p 2 G), the unitary representations of G in constructing M o˛ G, N oˇ G, respectively. Proof. Note that z WD ˝ idW M ˝ B.L2 .G// ! N ˝ B.L2 .G// is a *-isomorphism. Consider a directed set consisting of finite Borel partitions of G, where, for finite Borel partitions ,  0 , the order    0 is defined if  0 is a refinement of . For every x 2 M P and any Borel partition  D ¹A1 ; : : : ; An º define xz WD niD1 ˛s 1 .x/ ˝ mAi with some i

si 2 Ai , where mAi is the multiplication operator on L2 .G/ by the indicator function of Ai . Then ¹z x º is a net in M ˝ B.L2 .G//. For  D  ˝ f with  2 H (where M  B.H /) and f 2 L2 .G/, k˛ .x/

n Z X

2

xz k D

i D1 Ai

k˛s

1

.x/

˛s

1 i

.x/k2 jf .s/j2 ds:

R For any " > 0 choose a compact K  G such that GnK jf .s/j2 ds < "2 . Moreover, choose a Borel partition ¹B1 ; : : : ; Bm º of K such that k˛s 1 .x/ ˛s 0 1 .x/k < " for all s; s 0 2 Bi , 1  i  m. If a Borel partition  refines ¹B1 ; : : : ; Bm ; G n Kº, then k˛ .x/

xz k2  kf k2 "2 C .2kxk kk/2 "2 :

Hence xz ! ˛ .x/ strongly. Since

z.z x / D

n X i D1

.˛s

1 i

.x// ˝ mAi D

m X

ˇs

1 i

. .x// ˝ mAi ;

iD1

one similarly has z.z x / ! ˇ . .x// strongly, so that z.˛ .x// D ˇ . .x// for all x 2 M . Also, one has z.M .t // D z.1M ˝ t / D 1N ˝ t D N .t / for all t 2 G. Therefore, it follows that z maps M o˛ G ( M ˝ B.L2 .G//) onto N oˇ G ( N ˝ B.L2 .G//). b and X 2 M o˛ G one has Furthermore, for every p 2 G

z.y ˛p .X // D z..1M ˝ vp /X.1M ˝ vp // D .1N ˝ vp /z

.X /.1N ˝ vp / D ˇyp .z

.X //: 

126

8 Takesaki duality and structure theory

We can introduce the second crossed product b .M o˛ G/ o˛y G; the crossed product of M o˛ G by the dual action ˛ y , which is a von Neumann algebra on b D H ˝ L2 .G  G/. b Moreover, we have the second dual action H ˝ L2 .G/ ˝ L2 .G/ b b dual to ˛ b Takesaki’s duality theorem [93] is stated as follows. b € ˛ of G D G, y of G. Theorem 8.7 (Takesaki). We have the *-isomorphism b Š M ˝ B.L2 .G//; .M o˛ G/ o˛y G under which the action b € ˛ is transformed to the action ˛ zt WD ˛ t ˝ Ad.t /, t 2 G, on 2   2 M ˝ B.L .G//, where Ad. t / WD  t
 t on B.L .G//. From Lemma 8.1 there is a faithful representation  of M on a Hilbert space H1 and a continuous unitary representation V of G on H1 such that .˛ t .x// D V t .x/V t for all x 2 M and t 2 G (in fact, we may take  D ˛ and V D  on H1 D L2 .G; H /). So by Proposition 8.6, to prove the theorem, we may assume that ˛ is given as ˛ t D Ad.V t / (D V t
V t ) with a continuous unitary representation V of G on H . We define a unitary U 2 B.L2 .G; H // by .U /.s/ WD Vs .s/;

s 2 G;  2 L2 .G; H /:

Lemma 8.8. By Ad.U / we have M o˛ G Š ¹x ˝ 1; V t ˝  t W x 2 M; t 2 Gº00 ; and ˛ yp D Ad.1 ˝ vp / is unchanged under this *-isomorphism. Proof. Since .U  /.s/ D Vs .s/ and ..x ˝ 1//.s/ D x.s/ for all  2 L2 .G; H /, one has .U  .x ˝ 1/U /.s/ D Vs xVs .s/ D ˛s

1

.x/.s/ D .˛ .x//.s/;

s 2 G;

so that U˛ .x/U  D x ˝ 1 for all x 2 M . Since ..V t ˝  t //.s/ D V t .t .U  .V t ˝  t /U /.s/ D Vs V t .U /.t D

Vs V t V t

1s

1

.t

1

(8.1)

s/, one has

s/ 1

s/ D ..t //.s/;

s 2 G;

(8.2)

so that U .t /U  D V t ˝  t for all t 2 G. Moreover, since U.1 ˝ vp /U  D 1 ˝ vp b obviously, one has Ad.U / ı ˛ yp ı Ad.U  / D ˛ yp for all p 2 G.  b 00 is the maximal abelian von Neumann algebra on L2 .G/ Lemma 8.9. (1) ¹vp W p 2 Gº consisting of multiplication operators by all f 2 L1 .G/. b 00 D B.L2 .G//. (2) We have ¹ t ; vp W t 2 G; p 2 Gº

8.1 Takesaki’s duality theorem

127

Proof. (1) Let A be the abelian von Neumann algebra consisting of all multiplication operators mf on L2 .G/ with f 2 L1 .G/. Since L1 .G/ D L1 .G/ (i.e., the Haar measure on G is localizable), it is well known that A is maximal abelian. Obviously, b 00  A. To prove equality, assume that ' 2 A and '.vp / D 0 for all p 2 G. b ¹vp W p 2 Gº P1 P1 2 2 2 There are sequences ¹un º; ¹vn º  L .G/ with nD1 kun k < 1, nD1 kvn k < 1 P P1 such that '.a/ D 1 nD1 hun ; avn i for all a 2 A. Set w.s/ WD nD1 un .s/vn .s/; then R b The injectivity of w 2 L1 .G/ and G hs; piw.s/ ds D '.vp / D 0 for all Rp 2 G. the Fourier transform yields w D 0. Hence '.mf / D G f .s/w.s/ ds D 0 for all b 00 follows. f 2 L1 .G/, so ' D 0. Thus, A D ¹vp W p 2 Gº b 0 ; then x 2 A0 D A by (1). Hence x D mf for (2) Let x 2 ¹ t ; vp W t 2 G; p 2 Gº 1 some f 2 L .G/. For any t 2 G, since mf D  t mf t , one has f .s/ D f .t 1 s/ a.e. This implies that f is constant (a.e.), so x 2 C1 follows.  Now we give a sketchy proof of Theorem 8.7. As well as the original paper [93], detailed expositions are given in [96, 105]. Sketch proof of Theorem 8.7. The proof is divided into several steps, which we sketch below. b is *-isomorphic to Step 1. .M o˛ G/ o˛y G ® ¯ b 00 N1 WD x ˝ 1 ˝ 1; V t ˝  t ˝ 1; 1 ˝ vp ˝  t W x 2 M; t 2 G; p 2 G b and b on H ˝L2 .G/˝L2 .G/, € ˛ t D Ad.1˝1˝v t / is unchanged under this *-isomorphism. In fact, applying Lemma 8.8 twice to ˛ y and then to ˛, we find that ® ¯ b Š y ˝ 1; 1 ˝ vp ˝ p W y 2 M o˛ G; p 2 G b 00 .M o˛ G/ o˛y G ® ¯ b 00 : Š x ˝ 1 ˝ 1; V t ˝  t ˝ 1; 1 ˝ vp ˝ vp W x 2 M; t 2 G; p 2 G Since 1˝1˝v t commutes with U ˝1, we see that b € ˛ is unchanged under the *-isomorphism above. Step 2. N1 is *-isomorphic to ® ¯ b 00 N2 WD x ˝ 1 ˝ 1; V t ˝  t ˝ 1; 1 ˝ vp ˝ vp W x 2 M; t 2 G; p 2 G on H ˝ L2 .G/ ˝ L2 .G/ and Ad.1 ˝ 1 ˝ v t / is transformed to Ad.1 ˝ 1 ˝ t / under b ! L2 .G/, this *-isomorphism. To see this, consider the Fourier transform F W L2 .G/  b which is a unitary operator. Then, since F p F D vp for all p 2 G, we have .1 ˝ 1 ˝ F /N1 .1 ˝ 1 ˝ F  / D N2 . Moreover, since F v t F  D t for all t 2 G, we see that Ad.1 ˝ 1 ˝ v t / is transformed by Ad.1 ˝ 1 ˝ F / to Ad.1 ˝ 1 ˝ t /. Step 3. N2 is *-isomorphic to ® ¯ b 00 N3 WD x ˝ 1; V t ˝  t ; 1 ˝ vp W x 2 M; t 2 G; p 2 G

128

8 Takesaki duality and structure theory

on H ˝L2 .G/, and Ad.1˝1˝t / is transformed to Ad.1˝t / under this *-isomorphism. To see this, consider a unitary operator W on L2 .G/ ˝ L2 .G/ D L2 .G  G/ defined by .Wf /.s; t / WD f .st; t / for f 2 L2 .G  G/. Then, since W  . t ˝ 1/W D  t ˝ 1;

W  .vp ˝ vp /W D vp ˝ 1;

we have .1˝W  /N2 .1˝W / D N3 ˝C1. Furthermore, since W  .1˝t /W D t ˝t , we see that Ad.1 ˝ 1 ˝ t / is transformed by Ad.W  / to Ad.1 ˝ t ˝ t /, which is Ad.1 ˝ t ˝ 1/ on N3 ˝ C1. Step 4. By Ad.U  / we have N3 Š M ˝ B.L2 .G//; and Ad.1 ˝ t / is transformed to ˛ zt under this *-isomorphism. To see this, let A WD ¹vp W b 00 . We have p 2 Gº U  .M ˝ C1/U D ˛ .M /  .C1 ˝ A/0 \ .M ˝ B.L2 .G/// D .C1 ˝ A/0 \ .M 0 \ C1/0 D .M 0 ˝ A/0 D M ˝ A; where we have used (8.1) for the first equality, Lemma 8.4 for the inclusion (also see the proof of Proposition 8.6) and Lemma 8.9(1) for the last equality. Since 1 ˝ vp commutes with U , we have U  .C1 ˝ A/U D C1 ˝ A, so U  .M ˝ A/U  M ˝ A. Furthermore, for every x 2 M , x 0 2 M 0 and  2 L2 .G; H / one has .U.x ˝ 1/U  .x 0 ˝ 1//.s/ D Vs xVs x 0 .s/ D ˛s .x/x 0 .s/ D x 0 ˛s .x/.s/ D ..x 0 ˝ 1/U.x ˝ 1/U  /.s/;

s 2 G;

.U.x ˝ 1/U  .1 ˝ vp //.s/ D hs; piVs xVs .s/ D ..1 ˝ vp /U.x ˝ 1/U  /.s/;

b s 2 G: p 2 G;

Therefore, U.M ˝ C1/U   .M 0 ˝ C1/0 \ .C1 ˝ A/0 D .M 0 ˝ A/0 D M ˝ A; so that U.M ˝ A/U   M ˝ A as well. Hence U  .M ˝ A/U D M ˝ A, from which we obtain U  N3 U D U  ..M ˝ A/ [ ¹V t ˝  t W t 2 Gº/00 U D ..M ˝ A/ [ ¹1 ˝  t W t 2 Gº/00 D M ˝ .A [ ¹ t W t 2 Gº/00 D M ˝ B.L2 .G//; where we have used (8.2) for the second equality and Lemma 8.9(2) for the last equality. Furthermore, since U  .1 ˝ t /U D V t ˝ t , we see that Ad.1 ˝ t / is transformed by Ad.U  / to Ad.V t ˝ t / D ˛ zt . 

129

8.1 Takesaki’s duality theorem

Remark 8.10. When M is properly infinite (i.e., any non-zero central projection in M is infinite) and G satisfies the second axiom of countability (so L2 .G/ is separable), since b is *-isomorphic to the original M . M ˝ B.L2 .G// Š M , it follows that .M o˛ G/ o˛y G The next theorem says that the original M (Š ˛ .M /) is captured as the fixed-point algebra of the dual action ˛ y. Theorem 8.11. We have b (1) ˛ .M / D ¹y 2 M o˛ G W ˛ yp .y/ D y; p 2 Gº, (2) M o˛ G D ¹x 2 M ˝ B.L2 .G// W ˛ zt .x/ D x; t 2 Gº. Proof. As in the proof of Theorem 8.7, we may assume that ˛ is given as ˛ t D Ad.V t / with a continuous unitary representation V of G on H . (1) That ˛ yp .˛ .x// D ˛ .x/ for all x 2 M is in Lemma 8.4. Since .˛ .x/.V t ˝ t //.s/ D ˛s

1

.x/V t .st / D V t ˛.st / 1 .x/.st /

D ..V t ˝ t /˛ .x//.s/;

 2 L2 .G; H /;

it follows that ˛ .x/ commutes with V t ˝ t . It is also clear that 1 ˝ s commutes with V t ˝ t . Hence M o˛ G  ¹V t ˝ t W t 2 Gº0 . Now assume that y 2 M o˛ G and b Then ˛ yp .y/ D y for all p 2 G. ® ¯ b 0: y 2 .M ˝ B.L2 .G/// \ V t ˝ t ; 1 ˝ vp W t 2 G; p 2 G Since V t ˝ t D U  .1 ˝ t /U (as already mentioned in Step 4 of the proof of Theorem 8.7) and 1 ˝ vp D U  .1 ˝ vp /U , we have  ¯  ® ¯ ® b 00 D U  C1 ˝ t ; vp W t 2 G; p 2 G b 00 U V t ˝ t ; 1 ˝ vp W t 2 G; p 2 G D U  .C1 ˝ B.L2 .G///U: Therefore, y 2 .M ˝ B.L2 .G// \ U  .B.H / ˝ C1/U; so that y D U  .x ˝ 1/U for some x 2 B.H /. Now let x 0 2 M 0 and W G ! H be a continuous function with compact support. From a direct computation we notice that ŒU  .x ˝ 1/U; x 0 ˝ 1 is continuous on G and
ŒU  .x ˝ 1/U; x 0 ˝ 1 .e/ D Œx; x 0 .e/; where Œx; x 0  WD xx 0 x 0 x and e is the identity of G. Since y commutes with x 0 ˝ 1, the above identity gives Œx; x 0  D 0 for any x 0 2 M 0 , so that we have x 2 M and y D U  .x ˝ 1/U D ˛ .x/ (see (8.1)), showing y 2 ˛ .M /.

130

8 Takesaki duality and structure theory

b ˛ (2) We can apply the above proof of (1) to .M o˛ G; G; y / in place of .M; G; ˛/. Then we obtain ® ¯ b Wb ˛y .M o˛ G/ D X 2 .M o˛ G/ o˛y G € ˛ t .X / D X; t 2 G : b onto M ˝ B.L2 .G// given in the Consider the *-isomorphism from .M o˛ G/ o˛y G b proof of Theorem 8.7, which transforms € ˛ to ˛ z . It is easy to check that this *-isomorphism maps ˛y .M o˛ G/ onto M o˛ G. Hence the assertion follows.  In the rest of the section we give a short account of the dual weights, which is an important notion in the theory of crossed products. The notion was introduced in Takesaki’s paper [93] for a special class of weights, and then developed by Digernes [20] and Haagerup [30, 31]. Let .M; G; ˛/ be a W  -dynamical system, where G is a general locally compact group, and let M o˛ G be the crossed product. Let P .M / be the set of faithful semifinite normal weights on M . The basic idea here is to construct a map ' 2 P .M / 7 ! 'z 2 P .M o˛ G/ in such a way that the modular automorphism groups  ' and  'z have a natural close relation. This is done by using a suitable construction of left Hilbert algebra whose left von Neumann algebra is N o˛ G. The approach to do this is to consider the set K.G; M / of  -strongly* continuous functions xW G ! M with compact support. The set K.G; M / becomes a *-algebra with the product Z .a ? b/.s/ WD ˛ t .a.st //b.t 1 / dt G

and the involution a] .s/ WD G .s/

1

˛s

1

.a.s

1 

/ /

for a; b 2 K.G; M /, where G is the modular function of G. With the covariant representation ¹˛ ; º of .M; G; ˛/ (see Lemma 8.1), define Z .s/˛ .a.s// ds; a 2 K.G; M /: .a/ WD G

Then it is not difficult to see that  is a *-representation of the *-algebra K.G; M / on L2 .G; H /, with range -weakly dense in M o˛ G. We may assume that M is represented in a standard form .M; H ; J; P/, so by the uniqueness of the standard form, for any ' 2 P .M /, we may identify the GNS Hilbert space H' with H . For a given ' 2 P .M / we set B' WD K.G; M /
N' D ¹a.
/x W a 2 K.G; M /; x 2 N' º; which is a left ideal in K.G; M /, and define ƒ' W B' ! L2 .G; H /;

.ƒ' b/.s/ WD .b.s//' :

131

8.1 Takesaki’s duality theorem

Then it is shown that A' WD ƒ' .B' \ B'] / has a left Hilbert algebra structure and its left von Neumann algebra is M o˛ G. By taking the weight 'z on M o˛ G associated with the left Hilbert algebra A' (see the last part of (A) in Section 6.1), the following theorem was proved by Haagerup [30]. The 'z is called the dual weight of '. The proof of this theorem is omitted here, because Theorem 8.13 below will be sufficient for our later discussions. Theorem 8.12 (Haagerup). For each ' 2 P .M / there exists a 'z 2 P .M o˛ G/ having the following properties: ] (1) '..a z ? a// D '..a] ? a/.e// for any a 2 B' ,

(2) the modular automorphism  t'z is given by ´  t'z .˛ .x// D ˛ . t' .x//; x 2 M; t 2 R; ' z it  t ..s// D G .s/ .s/˛ ..D' ı ˛s W D'/ t /; s 2 G; t 2 R; (3) for any ';

2 P .M /, .D z W D '/ z t D ˛ ..D

W D'/ t /;

t 2 R:

Moreover, 'z is determined as a unique element of P .M o˛ G/ satisfying (1) and (2). When G is a locally compact abelian group, Haagerup [31] gave an alternative construction of the dual weights by using an operator-valued weight. In the following let dp b taken under the normalization that the Fourier and the denote the dual Haar measure on G inverse Fourier transforms are formally given as Z Z y b fy.p/hs; pi dp .s 2 G/: f .p/ D f .s/hs; pi ds .p 2 G/; f .s/ D b G G Theorem 8.13 (Haagerup). Let M o˛ G be the crossed product of M by an action ˛ of a locally compact abelian group G. (a) The expression Z T x WD

˛ yp .x/ dp; x 2 .M o˛ G/C b G defines a faithful semifinite normal operator-valued weight from M o˛ G to ˛ .M /, where ˛ y is the dual action. (b) The T satisfies T  a] ? a









D ˛ a] ? a .e/ ; 

T ..s/x.s/ / D .s/T .x/.s/ ;

a 2 K.G; M /;

(8.3)

x 2 .M o˛ G/C ; s 2 G:

(8.4)

(c) For any ' 2 P .M / the dual weight 'z on M o˛ G is given by 'z D .' ı ˛ 1 / ı T:

(8.5)

132

8 Takesaki duality and structure theory

Proof. We write N for M o˛ G for brevity. b C is defined by (a) For any x 2 NC , a generalized positive operator T x 2 N Z .T x/.!/ D h!; T xi WD !.y ˛p .x// dp; ! 2 NC : G

b C by h!; ˛ b C , ! 2 NC . One can naturally extend ˛ yp to N yp .m/i WD h! ı ˛ yp ; mi for m 2 N Since Z h!; ˛ yp .T x/i D h! ı ˛ yp ; T xi D !.y ˛pq .x// dq D h!; T xi; ! 2 NC ; G

R ? b Let T x D 1  de C 1e1 one has ˛ yp .T x/ D T x for all p 2 G. (where e1 D 0 lim!1 e ) be the spectral decomposition of T x (see Theorem 7.3). Since the spectral decomposition of ˛ yp .T x/ D T x is Z 1
? ˛ yp .T x/ D  dy ˛p .e / C 1y ˛p e1 ; 0

2

b Hence Theorem it follows from Theorem 7.3 that ˛ yp .e / D e for all   0 and all p 2 G. 8.11(1) implies that e 2 ˛ .M / for all   0, which means that T x 2 ˛ .M /C . Hence T W NC ! ˛ .M /C is defined. For any x 2 NC and a 2 ˛ .M / one has Z Z h!; T .a xa/i D !.y ˛p .a xa// dp D !.a ˛ yp .x/a/ dp b b G G D ha!a ; T .x/i D h!; a T .x/ai; ! 2 NC ;

2

so that T .a xa/ D a T .x/a. Therefore, T is an operator-valued weight from N to ˛ .M /. It is straightforward to see that T is normal and faithful. The semifiniteness of T will be shown in the proof of (b). (b) To show this, let K.G/ be the set of continuous functions on G with compact support, and P .G/ be the set of continuous, positive definite functions on G. We show the R b then fy 2 L1 .G/ b fact that if f 2 K.G/ and fy.p/ D G f .s/hs; pi ds  0 for all p 2 G, and Z fy.p/ dp D f .e/: (8.6) b G R hs; pi d.p/ for some finite Indeed, for any  2 P .G/, since  is given as .s/ D b G b positive measure  on G (Bochner’s theorem; see [81, 1.4.3]), we have Z Z Z f .s/.s/ ds D f .s/hs; pi ds d.p/  0: b G G G R For any k 2 K.G/, since k   k 2 P .G/, one has G f .s/.k   k/.s/ ds  0, so f 2 R b and f .s/ D P .G/. Hence it follows from [81, 1.5.1] that fy 2 L1 .G/ fy.p/hs; pi dp b G for all s 2 G; in particular, (8.6) holds.

133

8.1 Takesaki’s duality theorem

Now let a 2 K.G; M / and ! 2 NC . Since Z ˝

˛ ˝

˛ ] 0  !; ˛ yp  a ? a D !; ˛ yp ..s//˛ a] ? a .s/ ds ZG ˝

˛ hs; pi !; .s/˛ a] ? a .s/ ds D

(by Lemma 8.4);

G

one can apply (8.6) to f .s/ WD h!; .s/˛ ..a] ?a/.s//i and fy.p/ WD h!; ˛ yp ..a] ?a//i; then Z ˝

˛ ˝

˛ !; ˛ yp  a] ? a dp D f .e/ D !; ˛ a] ? a .e/ ; ! 2 NC ; b G which means (8.3). In particular, we have T ..a/ .a// 2 ˛ .M /, i.e., .a/ 2 NT for all a 2 K.G; M /, so T is semifinite. For any x 2 N and s 2 G, since ˛ yp ..s/x.s/ / D b (8.4) is immediate. .s/y ˛p .x/.s/ for all p 2 G, (c) For each ' 2 P .M / we write 'z WD .' ı ˛ 1 / ı T ; then 'z 2 P .N / by Proposition 7.7. By Theorem 7.8, for every '; 2 P .M / we have 'ı˛ 1

 t'z .˛ .x// D  t

.D z W D '/ z t D .D

.˛ .x// D ˛ . t' .x//; 1

x 2 M; t 2 R; (8.7)

1

ı ˛ W D' ı ˛ / t D ˛ ..D

W D'/ t /;

t 2 R; (8.8)

where Lemma 8.14(1) given below has been used. From Lemma 8.1(3) and (8.4) it follows that

B ' ı ˛ .x/ D .' ı ˛ s

s

ı ˛ 1 /.T x/ D ' ı ˛ 1 ..s/.T x/.s/ /

 D .' ı ˛ 1 / ı T ..s/x.s/ / D '..s/x.s/ z /;

x 2 NC ; s 2 G:

Hence by (8.8) we have

B

˛ ..D' ı ˛s ; D'/ t / D .D ' ı ˛s W D '/ z t D .D 'z ı Ad..s// W D '/ z t D .s/  t'z ..s//;

s 2 G; t 2 R;

where Lemma 8.14(2) below has been used. Therefore,  t'z ..s// D .s/˛ ..D' ı ˛s W D'/ t /;

s 2 G; t 2 R:

Furthermore, for any a 2 K.G; M / it follows from (8.3) that





'z  a] ? a D .' ı ˛ 1 / ı T  a] ? a D ' a] ? a .e/ :

(8.9)

(8.10)

Thus, it follows from (8.7)–(8.10) that the map ' 2 P .M / 7! 'z 2 P .M o˛ G/ defined by (8.5) satisfies the properties in (1) and (2) (also (3)) of Theorem 8.12. So the uniqueness assertion in Theorem 8.12 shows that the present 'z coincides with the dual weight of '. 

134

8 Takesaki duality and structure theory

Note that (8.10) gives a slight extension of Theorem 8.12(1) to all a 2 K.G; M /. Also, the new definition (8.5) is applicable to all normal weights ' on M (see Proposition 7.7), for which

B ' C D 'z C z

for all normal weights ';

on M :

(8.11)

From Theorem 8.13 we may conveniently consider (8.5) as the definition of the dual weight '. z Lemma 8.14. Let ';

2 P .M /.

(1) Let ˛W M0 ! M be a *-isomorphism between von Neumann algebras. Then

.D

 t'ı˛ D ˛

1

ı ˛ W D' ı ˛/ t D ˛

1

ı  t' ı ˛;

t 2 R;

W D'/ t /;

..D

t 2 R:

(2) For every unitary u 2 M , .D' ı Ad.u/ W D'/ t D u  t' .u/;

t 2 R:

Proof. (1) The first formula is essentially the same as Lemma 7.20(1). For the second, apply the first to ˛ ˝ id2 W M2 .M0 / ! M2 .M / and the balanced weight  WD .'; /. Since  ı .˛ ˝ id2 / D .' ı ˛; ı ˛/, we have  t.'ı˛;

ı˛/

D .˛

1

˝ id2 / ı  t ı .˛ ˝ id2 /;

so that  .D

0 ı ˛ W D' ı ˛/ t

 0 D  t.'ı˛; 0 D .˛ D .˛  D ˛



0

 0 0

1M0   0 0 1 ˝ id2 / ı  t 1M 0   0 0 1 ˝ id2 / .D W D'/ t 0  0 0 : 1 ..D W D'/ t / 0

(2) Note that .'; ' ı Ad.u// D .'; '/ ı Ad  1  t.';'ıAd.u// D Ad 0

ı˛/

 1 0 
0u

. From (1) we have

   0 1 ı  t.';'/ ı Ad u 0

 0 ; u

135

8.2 Structure of von Neumann algebras of type III

so that 

0 .D' ı Ad.u/ W D'/ t

      0 1 0 1 0 0 0 .';'/ D Ad ı t ı Ad 0 0 u 0 u 1 0     1 0 0 0 D Ad ı t.';'/ 0 u u 0        0 0 0 0 1 0 : D  ' D Ad 0 u u  t .u/ 0  t' .u/ 0 

8.2 Structure of von Neumann algebras of type III Let M be a general von Neumann algebra on H . Choose a '0 2 P .M /, and construct the pair
N WD M o '0 R;  WD  '0 (8.12)

b

of the crossed product M o '0 R by the modular automorphism group  '0 and the dual action  '0 . Now choose another '1 2 P .M /, and let u t WD .D'1 W D'0 / t be Connes’ cocycle derivative and define a unitary U on L2 .R; H / by .U /.s/ WD u s .s/ for  2 L2 .R; H /. For any  2 L2 .R; H /, x 2 M and t; s 2 R, we compute

b

.U '0 .x/U  /.s/ D u s  '0s .x/u s .s/ D  '1s .x/.s/ D .

 '1

(by (6.8))

.x//.s/

and .U .t /U  /.s/ D u s u sCt .s D D

t / D u s .u s  '0s .u t // .s

u s  '0s .ut /u s .s . '1 .ut /.t //.s/:

t/ D

t / (by (6.9))

. '1s .ut /.t //.s/

Hence we have U '0 .x/U  D  '1 .x/ and U .t /U  D  '1 .ut /.t /, so that U.M o '0 R/U   M o '1 R. Since ut D .D'0 W D'1 / t by Proposition 7.24(1), the argument can be reversed, so that U  .M o '1 R/U  M o '0 R. Therefore, U.M o '0 R/U  D M o '1 R:

(8.13)

Moreover, since .U v.t /U  /.s/ D e i st .s/ D .v.t //.s/ for any  2 L2 .R; H / and t; s 2 R, we have Ad.U / ı  t'0 D  t'1 ı Ad.U /; t 2 R: (8.14)

b b

From (8.13) and (8.14) we see that the construction (8.12) is canonical in the sense that it is independent of the choice of '0 2 P .M /.

136

8 Takesaki duality and structure theory

Theorem 8.15. Let .N;  / be defined in (8.12) and 'z0 be the dual weight defined in (8.5). Then (1)  t'z0 D Ad..t // for all t 2 R, (2) the N is a semifinite von Neumann algebra with a faithful semifinite normal trace  (called the canonical trace) satisfying the trace scaling property s

 ı s D e

s 2 R;

;

(8.15)

(3) M ˝ B.L2 .R// Š N o R and M Š N  (the fixed-point algebra of ). Proof. (1) By (8.7) we have  t'z0 . '0 .x// D  '0 . t'0 .x// D .t / '0 .x/.t / ;

x 2 M; t 2 R:

Also, since '0 ı  '0 D '0 , by (8.9) we have  t'z0 ..s// D .s/ D .t /.s/.t / ;

s; t 2 R:

These show that  t'z0 D Ad..t // for all t 2 R. (2) By (1) and Theorem 7.27, N is semifinite. Furthermore, there is a non-singular positive self-adjoint operator A such that .t / D Ai t for all t 2 R. Then as in the proof of Theorem 7.27, it follows that AN'z0 and  WD .'z0 /A 1 2 P .N / is a faithful semifinite normal trace. Since s .A

1 it

/ D s .A

it

/ D s .. t//

De

i st

. t / (by Lemma 8.4 and Definition 8.5)

De

i st

A

it

D .e s A

1 it

t 2 R;

/ ;

one has s .A 1 / D e s A 1 for all s 2 R. Set B WD A 1 ; then  WD .'z0 /B . From definitions (7.20) and (7.21), for every x 2 NC it follows that  ı s .x/ D lim 'z0 .B"1=2 s .x/B"1=2 / D lim 'z0 . s .B"1=2 s .x/B"1=2 // "&0

"&0

1=2

D lim 'z0 . s .B" / "&0

De

s

 .x/;

where we have used 'z0 ı 

s

x s .B" /1=2 / D lim 'z0 .e "&0

s

Be1=2s " xBe1=2s " /

s 2 R; D 'z0 by definition (8.5) and

 s .B" / D  s .B/.1 C " s .B//

1

De

Hence (8.15) holds. (3) is seen from Theorems 8.7 and 8.11(1).

s

B.1 C e

s

"B/

1

De

s

Be

s"

:



8.2 Structure of von Neumann algebras of type III

137

Remark 8.16. In particular, when M is semifinite with a faithful semifinite normal trace 0 , we may let '0 D 0 ; then  0 D id and .N; s / D .M ˝ L .R/; id ˝ Ad.vs //. Via the Fourier transform F (a unitary on L2 .R/), note that F L .R/F  D L1 .R/ (the algebra of multiplication operators by f 2 L1 .R/), F s F  W  2 L2 .R/ 7! e i st .t / and F vs F  W  7! .
Cs/, so that Ad.vs / is transformed to the shift f 2 L1 .R/ 7! f .
Cs/. Furthermore, 'z0 and R(D .'z0 /A 1 ; see the proof of Theorem 8.15(2)) are transformed to R 0 ˝ R
dt and 0 ˝ R
e t dt respectively. Thus we see that .N; s ;  / is identified with   Z M ˝ L1 .R/; id ˝ .f 7! f .
C s//; 0 ˝
e t dt : (8.16) R

When M is of type III, we state the continuous crossed product decomposition or the structure theorem for von Neumann algebras of type III due to Takesaki [93] as follows. Theorem 8.17 (Takesaki). Let M be a von Neumann algebra of type III. Then there exist a von Neumann algebra N of type II1 , a faithful semifinite normal trace  on N and a continuous action W R ! Aut.N / such that the trace scaling property (8.15) holds and M D N o R: Furthermore, such .N;  / as above is unique up to conjugacy. When we have shown Theorem 8.15, what remain to prove for Theorem 8.17 are type II1 of N and the last uniqueness assertion, although their proofs are omitted here. For the details see [93] or [96] (a proof of type II1 of N is also in [105]). Remark 8.18. When M is a factor of type III, Connes and Takesaki [18] introduced an important concept called the flow of weights on M . Under the above continuous decomposition of M , the (smooth) flow of weights on M may be defined by .X; F tM /, where X WD N \ N 0 , the center of N , and F tM WD  t jX . Then .X; F tM / is an ergodic flow, and the type classification of factors of type III can be given in terms of .X; F tM / as follows:  M is of type III1 if X is a single point (i.e., N is a factor),  M is of type III , 0 <  < 1, if .X; F tM / has period

log ,

 M is of type III0 otherwise (i.e., .X; F tM / is aperiodic). Furthermore, the flow of weights is a complete invariant for the isomorphism classes of injective factors of type III. Finally we record the discrete crossed product decompositions of factors of type III (0   < 1) due to Connes [14]. Theorem 8.19 (Connes). Let M be a factor of type III where 0 <  < 1. Then there exist a factor N of type II1 and a  2 Aut.N / such that  ı  D  (where  is a faithful semifinite normal trace on N ) and M D N o Z. Furthermore, such an .N;  / is unique up to conjugacy.

138

8 Takesaki duality and structure theory

Theorem 8.20 (Connes). Let M be a factor of type III0 . Then there exist a von Neumann algebra N of type II1 with non-atomic center, a  2 .0; 1/, a  2 Aut.N / and a faithful semifinite normal trace  on N such that  ı   ,  is ergodic on the center of N and M D N o Z.

9 Haagerup’s Lp -spaces

In Sections 9.1 and 9.2 of this chapter, based on Terp’s thesis [97],1 we give a concise but self-contained exposition on Haagerup’s Lp -spaces associated with general von Neumann algebras, extending conventional non-commutative Lp -spaces Lp .M;  / on semifinite von Neumann algebras in Section 4.3. In [57] Kosaki proposed a new construction of Lp -spaces for general von Neumann algebras, based on the complex interpolation method. Section 9.3 is a brief survey of Kosaki’s construction. A survey article [74] is a good source for Banach space theoretic properties of non-commutative Lp -spaces, with a large number of references.

9.1 Description of L1 .M / Let M be a general von Neumann algebra, and choose a '0 2 P .M /. Consider the triplet .N; ;  / given in Theorem 8.15 associated with .M; '0 /, that is, N WD M o '0 R,  WD  '0 and  is the canonical trace on N satisfying (8.15). Here we may assume for simplicity that M  N by identifying x and  '0 .x/ for R x 2 M . Consider also the faithful semifinite normal operator-valued weight T WD R s ds from N to M (see Theorem 8.13(a)). We first give the following fact. This is rather obvious although a proof is given for completeness.

b

Lemma 9.1. The dual action s (s 2 R) on N uniquely extends to Nz (the -measurable operators affiliated with N ; see Section 4.1) as a one-parameter group of homeomorphic *-isomorphisms with respect to the measure topology. Proof. Recall Theorem 4.12 saying that Nz is a complete metrizable Hausdorff *-algebra and N is dense in Nz . Since the s are *-isomorphisms on N , we need only to show that if ¹yn º  N is Cauchy in the measure topology, then so is ¹s .yn /º for any s 2 R. The Cauchy-ness of ¹yn º means that for any "; ı > 0 there is an n0 2 N such that ym yn 2 N ."; ı/ for all m; n  n0 . So it suffices to prove that s .N ."; ı/ \ N / D N ."; e

s

ı/ \ N

for any "; ı > 0:

Assume that a 2 N ."; ı/ \ N , so kaek  " and  .e ? /  ı for some e 2 Proj.N /. Then k.a/s .e/k D kaek  " and  .s .e/? / D  .s .e ? // D e s  .e ? /  e s ı. Hence 1Unfortunately, the thesis [97] was not published as a paper, although it is well distributed. Haagerup [34] gives a brief survey of the topic.

9 Haagerup’s Lp -spaces

140

s .N ."; ı/ \ N /  N ."; e s ı/ \ N follows. The converse inclusion is similar. The uniqueness of the extension is clear.  More R 1 generally, let a be a positive self-adjoint operator affiliated with N , and let a D 0  de be the spectral decomposition of a. RThen one can define a positive self1 adjoint operator s .a/N for any s 2 R as s .a/ WD 0  ds .e /. It is immediate to see that this definition s .a/ for a 2 NzC coincides with that given in Lemma 9.1. In the following we will deal with general (not necessarily faithful) semifinite normal weights on M and N , so instead of P .M / and P .N / we write P .M / WD ¹' W semifinite normal weight on M º; P .N / WD ¹ P  .N / WD ¹

W semifinite normal weight on N º; 2 P .N / W

ı s D ; s 2 Rº;

N C WD ¹a W positive self-adjoint operator with aN º; N ;C WD ¹a 2 N C W s .a/ D e

s

Nz;C WD ¹a 2 NzC W s .a/ D e

s

a; s 2 Rº;

a; s 2 Rº;

where NzC is the positive part of the space Nz of -measurable operators affiliated with N (see Section 4.1). Obviously, P  .N /  P .N /;

Nz;C  N ;C  N C :

We first show the following correspondences. Lemma 9.2. (1) a 7! a D  .a
/ is a bijection between N C and P .N /, where a is defined by (7.20) and (7.21) for ' D  (see also [32, Prop. 1.11] for a ). Moreover, we have s. .a
// D s.a/ for support projections s.
/. (2) h 7!  .h
/ is a bijection between N ;C and P  .N /. Moreover, we have s. .h
// D s.h/. (3) ' 7! 'z is a bijection between P .M / and P  .N /, where 'z is defined by 'z WD ' ı T , extending (8.5) to P .M / (omitting the notation  '10 ). Moreover, we have s.'/ z D s.'/. Proof. (1) We have already mentioned this assertion after Theorem 7.25. But we give a brief proof for completeness. It is clear that  .a
/ 2 P .N / for any a 2 N C . The bijection between ¹a 2 N C W non-singularº and P .N / is really a special case of (i) ” (vi) of Theorem 7.25 when ' D . For any 2 P .N / we can apply the faithful case to .s. /N s. /; js. /N s. / /. Hence a 7!  .a
/ is surjective. To see injectivity, assume that  .a
/ D  .b
/ for a; b 2 N C . Then s.a/ D s.b/ follows immediately. Hence the question reduces to the faithful case as above.

9.1 Description of L1 .M /

141

(2) From (1) it suffices to prove that for a 2 N C ,  .a
/ 2 P  .N / if and only if a 2 N ;C . If  .a
/ 2 P  .N /, then  .ax/ D  .a s .x// D e s  .s .a s .x/// D e s  .s .a/x/;

x 2 N;

so that a D e s s .a/ for all s 2 R, i.e., a 2 N ;C . The argument can be reversed to see the converse implication. (3) We see from the proof of Proposition 7.7 with definition (8.5) that 'z is a semifinite normal weight on N if ' 2 P .M /. First prove the equality for supports. For any ' 2 P .M / let p0 WD 1 s.'/ 2 M and q0 WD 1 s.'/ z 2 N . Note that Mp0 D ¹x 2 M W '.x  x/ D 0º and Np0 D ¹y 2 N W '.y z  y/ D 0º. For any y 2 NT and any x 2 M , '.p z 0 x  y  yxp0 / D '.T .p0 x  y  yxp0 // D '.p0 x  T .y  y/xp0 / D 0; so that NT Mp0  N q0 . Since NT is -weakly dense in M , one has Mp0  N q0 , which implies that p0  q0 . Since 'z is  -invariant, so is q0 and hence q0 2 M . So, to prove that p0 D q0 , it suffices to show that '.q0 / D 0. There is an increasing net ¹u˛ º  FT such that u˛ % 1. For any u˛ and any n 2 N one has '.q0 .T .nu˛ / ^ 1/q0 /  n'.T .q0 u˛ q0 // D n'.q z 0 u˛ q0 / D 0: Letting n ! 1 gives '.q0 s.T .u˛ //q0 / D 0. Since T .u˛ / is increasing, s.T .u˛ // % e0 for some projection e0 2 M . Let f0 WD 1 e0 . Then 0 D f0 T .u˛ /f0 D T .f0 u˛ f0 / % T .f0 / so that one has T .f0 / D 0. Hence f0 D 0 and e0 D 1, so that '.q0 / D 0 follows. To see injectivity, assume first that '1 ; '2 2 P .M / and 'z1 D 'z2 . By (8.8) one has .D'1 W D'2 / t D .D 'z1 W D 'z2 / t D 1 for all t 2 R. Hence '1 D '2 by Proposition 7.24(2). Now assume that '1 ; '2 2 P .M / and 'z1 D 'z2 . Then s.'1 / D s.'2 / follows from the fact already proved. Choose an ! 2 P .M / such that s.!/ D 1 s.'1 /. Then one has '1 C ! D '2 C ! by (8.11), which implies that '1 C ! D '2 C !. Hence '1 D '2 follows. Next, to prove surjectivity, we first assume that 2 P  .N / is faithful. Choose a '1 2 P .M /. Since both and 'z1 are -invariant, by Lemma 8.14(1) we have that the .D W D 'z1 / t are -invariant, so that u t WD .D W D 'z1 / t 2 M . Then

B

B

usCt D us s'z1 .u t / D us s'1 .u t /;

s; t 2 R;

thanks to (8.7). Hence, by Theorem 6.8 there exists a ' 2 P .M / such that .D' W D'1 / t D u t for all t 2 R. By (8.8), .D 'z W D 'z1 / t D .D' W D'1 / t D u t D .D

W D 'z1 / t ;

t 2 R:

Hence 'z D holds by Proposition 7.24(4). For general 2 P  .N / set e0 WD 1 s. /. Since is -invariant, one has e0 2 M . Choose a semifinite normal weight ! on M

9 Haagerup’s Lp -spaces

142

such that s.!/ D e0 . Then ! z 2 P  .N / and s.!/ z D s.!/ D e0 as already proved. Since C! z is faithful, it follows from the faithful case that C ! z D 'z for some ' 2 P .M /. Then, D s. /
. C !/ z
s. / D s. /
'z
s. / D .s. /
'
s. //;

G

where the last equality is easily verified from definition (8.5) (extended to ' 2 P .M /).



Combining the correspondences given in (2) and (3) of Lemma 9.2, we immediately see that a bijective map (9.1) ' 2 P .M / 7 ! h' 2 N ;C is determined by the equality 'z D  .h'
/ for each ' 2 P .M /. Moreover, we have s.h' / D s.'/ for the support projections. The next lemma plays a key role in defining Haagerup’s Lp -spaces. Lemma 9.3. In the correspondence (9.1) between P .M / and N ;C , ' 2 MC ” h' 2 Nz;C ; i.e., h' is -measurable: Hence we have the bijection ' 2 MC 7! h' 2 Nz;C by restricting (9.1) to MC . R1 Proof. Let ' 2 P .M / and h' 2 N ;C be as given in (9.1). Let h' D 0  de be the spectral decomposition. Since Z 1 Z 1 Z 1 s s  ds .e / D s .h' / D e h' D e  de D  dees  ; (9.2) 0

0

0

it follows that s .e / D ees  for all   0 and s 2 R. Hence we compute Z
s h' 1 e1? ds R   Z Z Z Z  1 dees  ds  1 de ds D D s .1;1/ R .1;1/ R  Z Z e s  1 de ds D R .e s ;1/ Z  Z s D e ds  1 de (note  > e s ” s < log ) .0;1/ . 1;log / Z D  1 de D s.h' / D s.'/; .0;1/

so that in view of  .h'
/ D 'z D ' ı T we have

 e1? D 'z h' 1 e1? D '

Z R


s h' 1 e1? ds D '.s.'// D '.1/:

9.1 Description of L1 .M /

143

Therefore, for every  > 0,



 e? D  log  .e1 /? D  log  e1? D e

log 


1  e1? D '.1/: 

(9.3) 

From this and Proposition 4.7 the assertion follows. Lemma 9.4. For every ';

2 MC and x 2 M we have

(1) h'C D h' C h , (2) hx'x  D xh' x  , where .x'x  /.y/ WD '.x  yx/, y 2 M . Proof. (1) One has

B ' C D .' C

/ıT D'ıT C

ı T D 'z C z D  .h'
/ C  .h
/:

Since h' ; h 2 Nz;C , note that h' C h (WD h' C h ) 2 NzC is well defined and by Lemma 9.1, s .h' C h / D s .h' / C s .h / D e s .h' C h /: Hence h' C h 2 Nz;C , and so it suffices to show that  ..h' C h /
/ D  .h'
/ C  .h
/: For this, choose sequences ¹an º; ¹bn º  NC such that an % h' and bn % h in the measure topology. From Proposition 4.23(2) one has  .an
/ %  .h'
/,  .bn
/ %  .h
/ and  ..an Cbn /
/ %  ..h' Ch /
/.2 Since  ..an Cbn /
/ D  .an
/C .bn
/ by Corollary 4.24, the desired equality follows. (2) For every y 2 NC one has

B

.x'x  /.y/ D .x'x  / ı T .y/ D '.x  T .y/x/ D '.T .x  yx// (see Definition 7.6) 1=2  1=2 xh' x  y 1=2 / D '.x z  yx/ D  .h1=2 ' x yxh' / D  .y

D  ..xh' x  /1=2 y.xh' x  /1=2 /; where Corollary 4.21(2) has been used twice. Note that xh' x  2 NzC and by Lemma 9.1, s .xh' x  / D s .x/s .h' /s .x/ D e

s

xh' x  :

Hence one has xh' x  2 Nz;C , so that hx'x  D xh' x  follows.



We are now in a position to prove the following: 2These follow from Proposition 7.21(4) as well, by noting that if an ; a  0 are  -measurable operators and an % a in the measure topology, then an % a in the strong resolvent sense; see Remark 4.13.

9 Haagerup’s Lp -spaces

144

Theorem 9.5. (a) The bijection ' 2 MC 7! h' 2 Nz;C extends to a linear bijection, still denoted by ' 7! h' , from M onto the subspace Nz WD ¹a 2 Nz W s .a/ D e

s

a; s 2 Rº

(9.4)

of Nz . (b) For every ' 2 M and x; y 2 M , hx'y  D xh' y  ;

h'  D h' :

(c) If ' D uj'j is the polar decomposition of ' 2 M (see [95, §III.4]), then h' D uhj'j is the polar decomposition of h' . Hence jh' j D hj'j and the partial isometry part of h' is u 2 M . Proof. First note that Nz is an M -bimodule and closed under a 7! a and a 7! jaj. For the closedness under a 7! jaj, we may just note that js .a/j D s .jaj/. (a) Each ' 2 M is written as ' D '1 '2 C i.'3 '4 / with 'k 2 MC . So define h' WD h'1

h'2 C i.h'3

h'4 /:

By Lemma 9.4(1) it is easy to check that this definition is well defined independently of the expression of ' as above and ' 7! h' is a linear map from M to Nz . On the other hand,   D a1 a2 and a 2ia D a3 a4 be the Jordan decompositions. for each a 2 Nz let aCa 2 Since  a C a  a C a D e s s D e s .s .a1 / s .a2 //; 2 2 it follows from the uniqueness of the Jordan decomposition that e s s .a1 / D a1 and e s s .a2 / D a2 for all s 2 R, so a1 ; a2 2 Nz;C . Similarly, a3 ; a4 2 Nz;C . Hence there are 'k 2 MC such that h'k D ak for k D 1; : : : ; 4. Putting ' WD '1 '2 C i.'3 '4 / 2 M we have h' D a. Hence ' 7! h' is a linear surjection from M onto Nz . The injectivity of the map will be shown in (c). (b) The property h'  D h' is obvious from the definition of ' 7! h' in (a). Consider the polarization 3 1X k x'y  D i .x C i k y/'.x C i k y/ 4 kD0



and that for xh' y . Then the property hx'y  D xh' y  follows from Lemma 9.4(2). (c) Let ' D uj'j be the polar decomposition of ' 2 M , so u is a partial isometry in M with u u D s.j'j/. By (b) we have h' D uhj'j . Hence it suffices to show that h' D uhj'j is indeed the polar decomposition of h' . But this is clear since u u D s.j'j/ D s.hj'j /. In particular, jh' j D hj'j . Finally, if ' 2 M and h' D 0, then hj'j D 0 so that j'j D 0, i.e., ' D 0 follows. Hence ' 7! h' is injective. 

9.2 Haagerup’s Lp -spaces

145

Definition 9.6. We rewrite (9.4) as Haagerup’s L1 -space L1 .M / WD ¹a 2 Nz W s .a/ D e

s

a; s 2 Rº:

(9.5)

Due to the linear bijection given in Theorem 9.5(a), define a linear functional tr on L1 .M / by tr.h' / WD '.1/; ' 2 M : Then tr.jh' j/ D tr.hj'j / D j'j.1/ D k'k;

' 2 M :

1

This means that kak1 WD tr.jaj/ for a 2 L .M / is the norm on L1 .M / copied from the norm on M by the linear bijection. In this way, .L1 .M /; k
k1 / becomes a Banach space identified with M . Since j'.1/j  j'j.1/ in the above, note that jtr.a/j  kak1 ;

a 2 L1 .M /:

(9.6)

The following rewriting of (9.3) for j'j with a D h' is useful:  .e.;1/ .jaj// D

1 1 tr.jaj/ D kak1 ;  

a 2 L1 .M /;  > 0;

(9.7)

where e.;1/ .jaj/ is the spectral projection of jaj corresponding to the interval .; 1/.

9.2 Haagerup’s Lp -spaces After the description of Haagerup’s L1 .M / in Section 9.1 we are now in a position to define the following. Definition 9.7. Including L1 .M / in (9.5), for each p 2 .0; 1 we define Haagerup’s Lp -space Lp .M / WD ¹a 2 Nz W s .a/ D e s=p a; s 2 Rº: Clearly, the Lp .M / are (closed) linear subspaces of Nz , which are M -bimodules and closed under a 7! a and a 7! jaj. Moreover, similarly to the proof of Theorem 9.5(a), they are linearly spanned by their positive part Lp .M /C WD Lp .M / \ NzC . By Theorem 9.5 with Definition 9.6 we have .L1 .M /; k
k1 / Š M

(isometric):

Note that the Lp .M / are disjointly realized in Nz for different p, i.e., Lp1 .M /\Lp2 .M / D ¹0º if p1 ¤ p2 . This situation is quite different from that of the Lp .M;  / in Section 4.3. Proposition 9.8. We have L1 .M / D M .

9 Haagerup’s Lp -spaces

146

1 Proof. In view of Theorem 8.15(3) it suffices R 1 to show that every a 2 L .M / is bounded. 1 Assume that a 2 L .M / with jaj D 0  de . Since s .jaj/ D jaj, it follows that s .e / D e for all   0 and s 2 R. Since




 e? D  s e? D e

s


 e? ;

s 2 R;

one has  .e? / D 0 or 1. But  .e? / < 1 for some  > 0 by Proposition 4.7. Hence one has  .e? / D 0 and so e? D 0 for some  > 0. This means that jaj is bounded.  Remark 9.9. In contrast to Proposition 9.8, any non-zero element of Lp .M / for 0 < p < 1 assume that 0 ¤ a 2 Lp .M /, 0 < p < 1, with R 1is unbounded. Indeed, ? jaj D 0  de . Then  .e / > 0 for some  > 0. Similarly to the argument in (9.2) one has s .e / D ees=p  for any   0, so that


 ee?s=p  D  s e? D e

s


 e? ;

s 2 R;

which implies that jaj is unbounded. Remark 9.10. To construct Haagerup’s Lp -spaces Lp .M /, we have started with .M; '0 / where '0 2 P .M /. But we remark that the construction is canonical independently of the choice of '0 . To see this, choose another '1 2 P .M /. Take a unitary U on L2 .G; H / defined at the beginning of Section 8.2. Setting  WD Ad.U / we have W N0 WD M o '0 R ! N1 WD M o '1 R;  ı s0 ı 

1

D s1 ;

s 2 R;

where  0 ,  1 are the corresponding dual actions on N0 , N1 ; see (8.13) and (8.14). Hence  ı T0 ı 

1

D T1

(9.8)

for the corresponding operator-valued weights T0 , T1 . Under identification M D  '0 .M / D  '1 .M / (as we did at the beginning of Section 9.1), since jM D idM , (9.8) may be written as T0 ı 

1

D T1 ;

so that for any ' 2 P .M / we furthermore have 'z.0/ ı 

1

D 'z.1/

for 'z.0/ WD ' ı T0 , 'z.1/ WD ' ı T1 :

From this and the definition of  in the proof of Theorem 8.15(2) it follows that 0 ı 

1

D 1

9.2 Haagerup’s Lp -spaces

147

for the corresponding canonical traces 0 , 1 on N0 ; N1 ; indeed, this is seen from .D 'z1.1/ W D.0 ı  D D

1

// t

.D.'z1.0/ ı  1 / W D.0 ı  1 // t D ..D 'z1.0/ ..D 'z1.0/ W D 'z0.0/ / t .D 'z0.0/ W D0 / t /

W D0 / t /

D ..D'1 W D'0 / t .t //

(by Lemma 8.14(1)) (by Proposition 7.24(3))

(by (8.8) and Theorem 7.22(2))

D .D'1 W D'0 / t ..t // D .D'1 W D'0 / t .D'1 W D'0 /t .t / D .t / D .D 'z1.1/ W D1 / t : Extending  to zW Nz0 ! Nz1 , we find that z transforms Lp .M / with respect to '0 to Lp .M / with respect to '1 . Example 9.11. Assume that M is semifinite with a faithful semifinite normal trace 0 . Then .N; s ;  / is identified with (8.16) in Remark 8.16. In this case, for each ' 2 MC we find that Z d' 'z D ' ı T D ' ˝
dt; h' D ˝ e t; d R where d'=d is the Radon–Nikodym derivative of ' with respect to ; see Corollary 4.51. Hence, for each p 2 .0; 1, Lp .M / D Lp .M; 0 / ˝ e

t =p

and ka ˝ e t=p kLp .M / D kakLp .M;0 / for all a 2 Lp .M; 0 /. By neglecting the superfluous tensor factor e t =p we may identify Lp .M / with Lp .M; 0 /. Lemma 9.12. Let a 2 Nz with the polar decomposition a D ujaj. Then for every p 2 .0; 1/, a 2 Lp .M / ” u 2 M and jajp 2 L1 .M /: Proof. Assume that a 2 Lp .M /. Since a D e s=p s .a/ D e s=p s .u/s .jaj/ D e s=p s .u/js .a/j; it follows from the uniqueness of the polar decomposition that u D s .u/;

jaj D e s=p s .jaj/; p

s 2 R: s

Since the latter equality above is equivalent to jaj D e s .jajp /, we have u 2 M and jajp 2 L1 .M /. Conversely, assume that u 2 M and jajp 2 L1 .M /. Then s .a/ D us .jajp /1=p D e s=p a for all s 2 R. Hence a 2 Lp .M / follows.  Definition 9.13. In view of Lemma 9.12, for every a 2 Lp .M / where 0 < p  1, define kakp 2 Œ0; 1/ by kakp WD tr.jajp /1=p

if 0 < p < 1;

kak1 WD kak

if p D 1:

In the case p D 1, k
k1 is the same as given in Definition 9.6.

9 Haagerup’s Lp -spaces

148

Lemma 9.14. For every p 2 .0; 1/ and "; ı > 0, N ."; ı/ \ Lp .M / D ¹a 2 Lp .M / W kakp  "ı 1=p º: Furthermore,  t .a/ D t

1=p

kakp ;

a 2 Lp .M /; 0 < p < 1;

(9.9)

where  t .a/ is the (tth) generalized s-number of a (as an element in Nz with respect to ); see Definition 4.16. Proof. Let a 2 Lp .M /; then jajp 2 L1 .M /C and so jajp D h' for some ' 2 MC . Since, by (9.7),  .e.";1/ .jaj// D  .e."p ;1/ .jajp // D

1 1 tr.jajp / D p kakpp ; "p "

by Lemma 4.18(1) we have a 2 N ."; ı/ ”  .e.";1/ .jaj//  ı 1 ” p kakpp  ı ” kakp  "ı 1=p ; "

(9.10)

implying the first assertion. For every t > 0 it follows from (9.10) that  t .a/ D inf¹s > 0 W  .e.s;1/ .jaj//  t º D inf¹s > 0 W s

p

kakpp  t º; 

which implies (9.9).

The formula given in (9.9) was first explicitly pointed out in [58] and is quite useful in treating Haagerup’s Lp -norm. Proposition 9.15 (Minkowski’s inequality). Let 1  p  1. For every a; b 2 Lp .M /, ka C bkp  kakp C kbkp :

(9.11)

Proof. Let a; b 2 Lp .M /. The case p D 1 is obvious. The case p D 1 is actually contained in Definition 9.6. In fact, for a; b 2 L1 .M /, writing a D h' and b D h we have ka C bk1 D tr.jh'C j/ D k' C k  k'k C k k D kak1 C kbk1 : Assume now that 1 < p < 1. From (9.9) and Theorem 4.39(1) (for Nz ) we have Z

1

s

1=p

1

Z ka C bkp ds 

0

s

1=p

.kakp C kbkp / ds

0

so that

p p

1

ka C bkp 

p p

1

.kakp C kbkp /:



9.2 Haagerup’s Lp -spaces

149

Lemma 9.16. Let 0 < p  1. For every a 2 Lp .M / and x; y 2 M , kxaykp  kxk kyk kakp :

(9.12)

Proof. The case p D 1 is obvious. Let 0 < p < 1, a 2 Lp .M / and x; y 2 M . Note that Lp .M / is an M -bimodule, as mentioned right after Definition 9.7. Hence xay 2 Lp .M /. By (9.9) and Proposition 4.19(6) one has kxaykp D 1 .xay/  kxk kyk1 .a/ D kxk kyk kakp ; 

as asserted.

Proposition 9.17 (Hölder’s inequality). Let p; q; r 2 .0; 1 with 1=p C 1=q D 1=r. If a 2 Lp .M / and b 2 Lq .M /, then ab 2 Lr .M / and kabkr  kakp kbkq :

(9.13)

Proof. When p D 1 or q D 1, (9.13) follows from (9.12). Assume that 0 < p; q < 1, and let a 2 Lp .M /, b 2 Lq .M /. Since s .ab/ D s .a/s .b/ D .e s=p a/.e s=q b/ D e s=r ab for all s 2 R, it follows that ab 2 Lr .M /. Hence by (9.9) we have  t .a/ D t 1=p kakp ,  t .b/ D t 1=q kbkq and  t .ab/ D t 1=r kabkr , so that each of a, b and ab satisfy condition (4.12). Hence by Theorem 4.44(1) we have Z 1 Z 1 Z 1 1=r 1=p log.s kabkr / ds  log.s kakp / ds C log.s 1=q kbkq / ds; 0

so that

0

1  1  1 C log kabkr  C log kakp C C log kbkq ; r p q

that is, kabkr  kakp kbkq . Theorem 9.18. k
kp .

0



(a) For every p 2 Œ1; 1, Lp .M / is a Banach space with respect to

(b) In particular, L2 .M / is a Hilbert space with respect to the inner product ha; bi WD tr.a b/ (D tr.ba /) for a; b 2 L2 .M /. (c) For any p 2 Œ1; 1/, the norm topology on Lp .M / coincides with the relative topology induced from Nz with the measure topology. More precisely, the uniform structure on Lp .M / by k
kp coincides with that induced from Nz . Proof. (a) Let a; b 2 Lp .M /. Proposition 9.15 gives the triangle inequality of k
kp . It is obvious that kakp D 0 H) a D 0. The completeness will be shown after proving (c). (b) It is straightforward to check that .a; b/ 7! ha; bi is an inner product on L2 .M / and kak2 D ha; ai1=2 . (c) For every p 2 Œ1; 1/, since k
kp is a norm on Lp .M /, the result is immediate from Lemma 9.14.

9 Haagerup’s Lp -spaces

150

Finally, we show the completeness of .Lp .M /; k
kp /. The case p D 1 is obvious. When 1  p < 1, it suffices by (c) to show that Lp .M / is complete with respect to the uniform topology on Nz . But this is immediate since Nz is a complete metrizable space (see Theorem 4.12) and Lp .M / is a closed subspace of Nz .  Lemma 9.19. Let a 2 NzC . The function z 2 ¹z 2 C W Re z > 0º 7 ! az 2 Nz d z a D az log a. is differentiable in the measure topology and dz R1 Proof. Let a D 0  de 2 NzC . Then, for any z 2 C, Re z > 0, note that az D R1 z z Re z , so az 2 Nz follows. 0  de and ja j D a First assume that a is bounded, and prove that z 7! az is differentiable in the operator d z norm on Re z > 0. Note that for any  > 0, dz  D z log  on Re z > 0 and z z 7!  log  is continuous on Re z  0. Hence one can easily show that for any z0 2 C, Re z0 > 0 and " > 0, there exists an r > 0 with r < Re z0 such that ˇ z z0 ˇ ˇ ˇ jz z0 j  r H) sup ˇ z0 log ˇ  ": z0 0kak z

This implies that if Re z > 0 and jz

az

z

a z0 z0

z0 j  r, then

az0 log a 

ˇ z ˇ sup ˇ 0kak z

z 0 z0

ˇ ˇ z0 log ˇ  ";

showing the assertion. Next let a 2 NzC and "; ı > 0 be arbitrary. Choose a  > 0 such that  .e? / < ı. Let p WD e . Since ap is bounded, the first case implies that one can choose an r > 0 with r < Re z0 such that

 az az0

 .ap/z .ap/z0



az0 log a p D .ap/z0 log.ap/  "

z z0 z z0 if jz

z0 j  r. Hence az z

az0 z0

az0 log a 2 N ."; ı/ if jz

z0 j  r: 

This shows the result.

Lemma 9.20. Let a; b 2 L1 .N /C . Then for any z 2 C with 0 < Re z < 1, we have az b 1 z 2 L1 .N / and the function z 2 ¹0 < Re z < 1º 7 ! az b 1 is analytic in the norm k
k1 .

z

2 L1 .N /

9.2 Haagerup’s Lp -spaces

Proof. For 0 < Re z < 1, since az ; b 1 Lemma 9.19), by Lemma 9.1 we have s .az b 1

z

z

/ D s .az /s .b 1 De

zs z

a
e

; az b 1 z

z

151

2 Nz (see the beginning of the proof of

/ D s .a/z s .b/1

.1 z/s 1 z

b

De

z

s z 1 z

a b

;

s 2 R;

so that az b 1 z 2 L1 .M /. By Theorem 9.18(c) we may prove the differentiability of z 7! az b 1 z on 0 < Re z < 1 as a function of Nz . For any z0 , 0 < Re z0 < 1, we have az b 1

z

az0 b 1 z0

z

z0

b 1 z b 1 z0 az az0 1 z b C a z0 z z0 z z0 ! az0 log a
b 1 z0 az0
b 1 z0 log b;

D



thanks to Lemma 9.19. Lemma 9.21. For any t 2 R set Nz

1 2


® C i t WD a 2 Nz W s .a/ D e

s. 1 2 Ci t /

¯ a; s 2 R :

If a; b 2 Nz . 21 C i t /, then ab  ; b  a 2 L1 .M / and tr.ab  / D tr.b  a/. Proof. That ab  ; b  a 2 L1 .M / is shown similarly to the proof of Lemma 9.20. To prove that tr.b  a/ D tr.ab  /, assume first that a D b. From (9.7) it follows that tr.a a/ D  .e.1;1/ .a a// D  .e.1;1/ .aa // D tr.aa /: For the general case, since a C i k b 2 Nz . 12 C i t / (k D 0; : : : ; 3), the result immediately follows from the polarizations ba D

3 1X k i .a C i k b/ .a C i k b/; 4

ab  D

kD0

3 1X k i .a C i k b/.a C i k b/ 4 kD0



and the linearity of tr.

Proposition 9.22. Let 1  p; q  1 with 1=p C 1=q D 1. If a 2 Lp .M / and b 2 Lq .M /, then ab; ba 2 L1 .M / and tr.ab/ D tr.ba/: Proof. If p D 1 (so q D 1), then we have a D h' for some ' 2 M . Hence by Theorem 9.5(b) one has tr.h' b/ D tr.h'b / D .'b/.1/ D '.b/ D .b'/.1/ D tr.hb' / D tr.bh' /; and the case q D 1 is similar.

9 Haagerup’s Lp -spaces

152

Now assume that 1 < p; q < 1. Note that a can be written as a D a1 a2 Ci.a3 a4 / with ak 2 Lp .M /C and similarly for b. By the linearity of tr we may assume that a 2 Lp .M /C and b 2 Lq .M /C , so ap ; b q 2 L1 .M /C . Then it follows from Lemma 9.20 that ab; ba 2 L1 .M / (also shown in Proposition 9.17) and the functions

F .z/ WD tr apz b q.1 z/ ; G.z/ WD tr b q.1 z/ apz 1 1 are analytic in 0 < Re z < 1. For any t 2 R, since ap. 2 Ci t / ; b q. 2 Ci t / 2 Nz . 12 C i t /, by Lemma 9.21 we have



1 1 1 1 F 12 C i t D tr ap. 2 Ci t / b q. 2 i t / D tr ap. 2 Ci t / b q. 2 Ci t /



1 1 1 1 D tr b q. 2 Ci t / ap. 2 Ci t / D tr b q. 2 i t / ap. 2 Ci t / D G 12 C i t :

This implies that F D G on 0 < Re z < 1; in particular, tr.ab/ D F .1=p/ D G.1=p/ D tr.ba/:



Lemma 9.23. Let 1  p; q  1 with 1=p C 1=q D 1. Then for every a 2 Lp .M /, kakp D sup¹jtr.ab/j W b 2 Lq .M /; kbkq  1º:

(9.14)

Proof. If p D 1, then a D h' for some ' 2 M and kh' k1 D k'k D sup¹j'.b/j W b 2 M; kbk1  1º D sup¹jtr.h' b/j W b 2 M; kbk1  1º: If p D 1, then a 2 M and kak1 D sup¹j'.a/j W ' 2 M ; k'k  1º D sup¹jtr.ah' /j W h' 2 L1 .M /; kh' k1  1º: Now assume that 1 < p; q < 1. We may assume that kakp D 1. Let a D ujaj be the polar decomposition. Put b WD jajp=q u ; then b 2 Lq .M / and jbjq D ujajp u . Hence one has kbkqq D tr.ujajp u / D tr.jajp / D 1 by Proposition 9.22, so kbkq D 1. By Proposition 9.22 again one has tr.ab/ D tr.ba/ D tr.jajp=q jaj/ D tr.jajp / D 1; so that kakp  RHS of (9.14). Since jtr.ab/j  kabk1  kakp kbkq by (9.6) and (9.13), the reverse inequality holds as well.  Remark 9.24. It is immediate to show Minkowski’s inequality (9.11) from (9.14), and (9.14) is a consequence of Hölder’s inequality. The proof of the latter inequality in [97] is complex analytic based on the three-lines theorem, though restricted to r D 1. Our proofs of Propositions 9.15 and 9.17 from [24] are real analytic based on Theorems 4.39 and 4.44.

9.2 Haagerup’s Lp -spaces

153

Proposition 9.25 (Clarkson’s inequality). Let 2  p < 1. Then for every a; b 2 Lp .M /, ka C bkpp C ka

bkpp  2p

1

.kakpp C kbkpp /:

Proof. An elegant complex analytic proof of this based on the three-lines theorem is in [97]. Here we give a real analytic proof from [24]. Set p 0 WD p=2, so 1  p 0 < 1. For any a; b 2 Lp .M /, from Lemma 9.26 given below we have ka C bkpp C ka

0

bkpp D k ja C bj2 kpp0 C k ja

0

bj2 kpp0 0

bj2 kpp0

 k ja C bj2 C ja

0

0

0

D k2.jaj2 C jbj2 /kpp0 D 2p k jaj2 C jbj2 kpp0 0 0
0 0  2p 2p 1 k jaj2 kpp0 C k jbj2 kpp0 D 2p

1

.kakpp C kbkpp /:



Lemma 9.26. Let 1  p < 1. For every a; b 2 Lp .M /C , 21

p

ka C bkpp  kakpp C kbkpp  ka C bkpp :

Proof. The first inequality follows from 2

p

ka C bkpp 

 kak C kbk p kakpp C kbkpp p p :  2 2

For the second, there are unique contractions v and w in N such that a1=2 D v.a C b/1=2 and b 1=2 D w.a C b/1=2 with v.1 s.a C b// D w.1 s.a C b// D 0. Note that v  v C w  w D s.a C b/, as in the proof of Lemma 7.19(4). Moreover, applying s to a1=2 D v.a C b/1=2 gives e s=2p a1=2 D e s=2p s .v/.a C b/1=2 , so that s .v/ D v for all s 2 R. Hence v 2 M and similarly w 2 M . We now have kakpp C kbkpp D kv.a C b/v  kpp C kw.a C b/w  kpp D 1 ..v.a C b/v  /p / C 1 ..w.a C b/w  /p / (by (9.9) and Proposition 4.19(9)) p 

p

 1 .v.a C b/ v / C 1 .w.a C b/ w  / D tr.v.a C b/p v  / C tr.w.a C b/p w  /

(by Lemma 4.26) (by (9.9))

D tr..a C b/p=2 v  v.a C b/p=2 / C tr..a C b/p=2 w  w.a C b/p=2 / (by Proposition 9.22) D tr..a C b/p=2 s.a C b/.a C b/p=2 / D tr..a C b/p / D ka C bkpp :  p

q

Theorem 9.27 (L –L -duality). Let 1  p < 1 and 1=p C 1=q D 1. Then the dual Banach space of Lp .M / is Lq .M / under the duality pairing .a; b/ 2 Lp .M /Lq .M / 7! tr.ab/ 2 C.

154

9 Haagerup’s Lp -spaces

Proof. Let 1  p < 1 and write ˆ.b/.a/ WD tr.ab/ for a 2 Lp .M / and b 2 Lq .M /. From (9.14) it follows that ˆ.b/ 2 Lp .M / and kˆ.b/k D kbkq . Hence ˆW Lq .M / ! Lp .M / is a linear isometry so that ˆ.Lq .M // is a norm-closed (hence w-closed) subspace of Lp .M / , since Lq .M / is complete and hence so is ˆ.Lq .M //. If 2  p < 1, then Lp .M / is uniformly convex by Clarkson’s inequality (Proposition 9.25) and so Lp .M / is reflexive. Hence Lp .M / is also reflexive so that ˆ.Lq .M // is w*-closed. But it follows from (9.14) that ˆ.Lq .M // is w*-dense in Lp .M / . Hence ˆ.Lq .M // D Lp .M / follows. If 1 < p < 2, then 2 < q < 1. Hence from the above case it follows that Lp .M / D Lq .M / under the duality .b; a/ 7! tr.ba/ D tr.ab/ thanks to Proposition 9.22. Hence Lp .M / D Lq .M / D Lq .M / under the duality .a; b/ 7! tr.ab/. Finally, the result holds for p D 1 since L1 .M / D .M / D M .  Proposition 9.28. Let 1  p; q  1 with 1=p C 1=q D 1. Let a 2 Lq .M /. Then a  0 ” tr.ab/  0 for all b 2 Lp .M /C : Proof. When p D 1, the result is trivial. When p D 1, the result is well known. Now assume that 1 < p < 1, so 1 < q < 1. If a 2 Lq .M /C and b 2 Lp .M /C , then a1=2 ba1=2 2 L1 .M /C and hence tr.ab/ D tr.a1=2 ba1=2 /  0. Conversely, assume that a 2 Lq .M / satisfies tr.ab/  0 for all b 2 Lp .M /C . Since tr.ab/ D tr.ab/ D tr..ab/ / D tr.ba / D tr.a b/ for all b 2 Lp .M /C , we have a D a . Let a D aC aC a D 0. For b WD aq=p 2 Lp .M /C we have 0  tr.ab/ D tr.aC b/

tr.a b/ D

a be the Jordan decomposition, so tr.a b/ D

tr.aq /;

which implies that a D 0. Therefore, we have a D aC 2 Lq .M /C .



The last result in the section is the following. Theorem 9.29. For each x 2 M we define the left action .x/ and the right action .x/ on the Hilbert space L2 .M / by .x/a WD xa;

.x/a WD ax;

a 2 L2 .M /;

and the involution J on L2 .M / by Ja WD a . Then (1) the  (resp., ) is a normal faithful representation (resp., anti-representation) of M on L2 .M /, (2) the von Neumann algebras .M / and .M / are the commutants of each other and .M / D J .M /J; (3) ..M /; L2 .M /; J; L2 .M /C / is a standard form of M .

9.2 Haagerup’s Lp -spaces

155

Proof. (1) Since ha ; bi D tr.ab/ D tr.ba/ D hb  ; ai for a; b 2 L2 .M / by Proposition 9.22, it is clear that J is an involution on L2 .M /. Since .ax/ D x  a for x 2 M and a 2 L2 .M /, we have .x/ D J .x  /J; x 2 M: (9.15) Note that x 2 M; a; b 2 L2 .M /;

ha; .x/bi D tr.a xb/ D tr..x  a/ b/ D h.x  /a; bi; and if x˛ % x 2 MC , then

ha; .x˛ /ai D tr.a x˛ a/ D tr.x˛ aa / % tr.xaa / D tr.a xa/ D ha; .x/ai for all a 2 L2 .M /. Assume that x 2 M and .x/ D 0; then 0 D ha; .x/bi D tr.a xb/ D tr.xba /;

a; b 2 L2 .M /:

Since L1 .M / D ¹ba W a; b 2 L2 .M /º, we have x D 0. Therefore,  is a normal faithful representation of M on L2 .M /. The assertion for  is immediate from (9.15). (2) By (1), .M / and .M / are von Neumann algebras, and .M / D J .M /J follows from (9.15). Moreover, .M /  .M /0 is obvious by definition. To prove the converse inclusion, let T 2 .M /0 , and we prove that there is a bounded linear map QW L1 .M / ! L1 .M / such that ! n n X X Q bi ai D bi T .ai / iD1

i D1

P for any ai ; bi 2 L2 .M /. To show that Q is well defined, assume that niD1 bi ai D 0.
Pn 1=2  Put a WD 2 L2 .M /C . For 1  i  n, since ai ai  a2 , one can iD1 ai ai choose a unique xi 2 N such that xi .1 s.a// D 0 and ai D xi a. Applying s gives e s=2 ai D s .xi /.e s=2 a/, implying that s .xi / D xi for all s 2 R. Hence we have xi 2 M . Since ! ! n n n X X X bi xi a D bi ai D 0 and bi xi .1 s.a// D 0; i D1

we have

Pn

iD1

n X i D1

i D1

iD1

bi xi D 0 so that

bi T .ai / D

n X i D1

bi T .xi a/ D

n X i D1

bi T ..xi /a/ D

n X

! bi xi T .a/ D 0;

iD1

as desired. For any c 2 L1 .M / one can choose a; b 2 L2 .M / such that c D ab and kck1 D kak2 kbk2 . Since kQ.c/k1 D kbT .a/k1  kbk2 kT .a/k2  kbk2 kT k kak2 D kT k kck1 ;

156

9 Haagerup’s Lp -spaces

it follows that QW L1 .M / ! L1 .M / is a bounded linear map. Now set x WD Q .1/ 2 M , where Q W M ! M is the dual map of Q. For every a; b 2 L2 .M / we find that hb  ; T .a/i D tr.bT .a// D tr.Q.ba// D tr.Q .1/ba/ D tr.xba/ D tr.bax/ D hb  ; axi D hb  ; .x/ai; which implies that T D .x/ 2 .M /. Therefore, .M /0  .M / has been obtained, so that .M / D .M /0 . (3) That L2 .M /C is a self-dual cone follows from Proposition 9.28. The conditions of Definitions 3.5 are verified as follows: (a) J .M /J D .M / D .M /0 . (b) Any element in the center of .M / is given as .z/ with z 2 M \ M 0 . Since J .z/J D .z  / D .z/ , we need to show that .z/ D .z/. For every a 2 L1 .M / and y 2 M we have tr.zay/ D tr.ayz/ D tr.azy/. This implies that za D az for all a 2 L1 .M /C . By considering the spectral decomposition of a, we find that za1=2 D a1=2 z for all a 2 L1 .M /C , which means that zb D bz for all b 2 L2 .M /C . Hence .z/ D .z/ holds. (c) Ja D a D a for all a 2 L2 .M /C . (d) For every x 2 M and a 2 L2 .M /C we have .x/j..x//a D .x/.x  /a D xax  2 L2 .M /C .  Remark 9.30. For any projection e 2 M , Haagerup’s Lp -space Lp .eM e/ is identified with eLp .M /e. To see this, we may take a '0 2 P .M / as '0 D e'1 e C .1 e/'2 .1 e/ with '1 2 P .eM e/ and '2 2 P ..1 e/M.1 e//. Then from Lemma 7.28 it follows that e 2 M' and  t'1 D  t'0 jeM e (t 2 R). Hence one can easily see that corresponding to .N; s ;  / for .M; '0 /, the triplet associated with .eM e; '1 / is given as .eNe; s jeNe ; jeNe /, showing the assertion. Furthermore, note that ej.e/L2 .M / D eL2 .M /e and
eM e; eL2 .M /e; J D  ; eL2 .M /C e is a standard form of eM e (see Proposition 3.8), where eM e acts on eL2 .M /e as the left multiplication. Remark 9.31. Let N be a von Neumann subalgebra of M and ! be a faithful normal state on M . Assume that there exists a conditional expectation EW M ! N with respect to !. Since  t! .N / D N (t 2 R) by Theorem 5.6, one can easily see that Lp .N / can be naturally isometrically identified with a subspace of Lp .M / for any p 2 .0; 1/. Junge and Xu [51] extended E to a contractive projection from Lp .M / onto Lp .N / for any p 2 Œ1; 1/ and showed that similar properties to those in Theorem 5.4 (in the tracial case) hold. See [51] for details.

9.3 Kosaki’s interpolation Lp -spaces

157

9.3 Kosaki’s interpolation Lp -spaces This section is a brief survey of Kosaki’s interpolation Lp -spaces [57], without proofs. A merit of Kosaki’s Lp -spaces is that the construction itself allows us to apply complex interpolation techniques of the Riesz–Thorin type to them. We begin with a general description of the complex interpolation method (see, e.g., [7, 101] for details). Let X D .X0 ; X1 / be a compatible couple of Banach spaces, that is, there is a z Then Hausdorff topological linear space Xz so that both of X0 and X1 are subspaces of X. z becomes a Banach space with the norm the sum †.X / WD X0 C X1 (in X) kxk† WD inf¹kx0 kX0 C kx1 kX1 W x D x0 C x1 ; xj 2 Xj º; and the inclusions Xj ,! †.X / (j D 0; 1) are injective (so we may take Xz D †.X /). We denote by F .X / the space of all functions f W ¹z 2 C W 0  Re z  1º ! †.X / satisfying the following three conditions: (i) f is bounded continuous on 0  Re z  1 and analytic in 0 < Re z < 1, (ii) f .j C i t / 2 Xj for all t 2 R and j D 0; 1, (iii) for j D 0; 1, the function t 2 Xj 7! f .j C i t / is k
kXj -continuous and kf .j C i t /kXj ! 0 (t ! ˙1). For each f 2 F .X / define ® ¯ kf kF WD max sup t2R kf .i t /kX0 ; sup t2R kf .1 C i t /kX1 : Then it follows that .F .X /; k
kF / becomes a Banach space. Definition 9.32. For each  2 .0; 1/ define C .X / D C .X0 ; X1 / WD ¹x 2 †.X / W x D f . / for some f 2 F .X /º; ® ¯ kxk WD inf kf kF W f . / D x; f 2 F .X / ; x 2 C .X /: Then .C .X /; k
k / becomes a Banach space, called the complex interpolation space. The functor C for making C .X / from X D .X0 ; X1 / is called the complex interpolation method, which is an exact interpolation functor of exponent  in the sense that the following abstract version of the Riesz–Thorin theorem holds. Theorem 9.33. Let X D .X0 ; X1 / and Y D .Y0 ; Y1 / be two compatible couples of Banach spaces. If a linear mapping T W †.X / ! †.Y / satisfies, for j D 0; 1, T .Xj /  Yj

with kT xj kYj  `j kxj kXj

.xj 2 Xj /

for some `j < 1, then for every  2 .0; 1/, T .C .X //  C .Y / and kT xk  `10

  `1 kxk ;

x 2 C .X /:

158

9 Haagerup’s Lp -spaces

A slightly modified interpolation method was also considered in [57], where F .X / is replaced with the slightly larger space F 0 .X / of all functions f W ¹z W 0  Re z  1º ! †.X / satisfying conditions (i), (ii) above and ® ¯ (iii)0 kf kF 0 WD max sup t2R kf .i t /kX0 ; sup t 2R kf .1 C i t /kX1 < 1, and then C0 .X / and k
k0 are defined from F 0 .X / in the same way as above (see [101, §1.9]). It is obvious that C .X /  C0 .X / and kxk0  kxk , x 2 C .X /. We now turn to Kosaki’s construction. Let M be a von Neumann algebra and '0 be a distinguished faithful normal state on M (so M is assumed to be -finite). We have an embedding M ,! M via x 2 M 7 ! x'0 .WD '0 .
x// 2 M ;

(9.16)

so that we have a compatible couple .M; M / D .M '0 ; M / under the identification x D x'0 in M . For this couple, since kx'0 k1  kxk1 k'0 k1 D kxk1 .D kx'0 k1 /;

(9.17)

we have †.M; M / D M ;

k
k † D k
k1 :

Definition 9.34. For each p 2 .1; 1/ we define Lp .M; '0 / WD C D1=p .M; M / .D CD1=p .M '0 ; M //; which is called Kosaki’s (interpolation) Lp -space associated with M with respect to '0 . For a 2 Lp .M; '0 / let kakp denote the complex interpolation norm kakD1=p . Moreover, we write L1 .M; '0 / WD M and L1 .M; '0 / WD M '0 (identified with M ). From (9.17) and general properties of the complex interpolation method (see [7]) we have, for 1 < p 0 < p < 1, 0

.M D/ M '0  Lp .M; '0 /  Lp .M; '0 /  M ; .kxk1 D/ kx'0 k1  kx'0 kp  kx'0 kp0  kx'0 k1 ;

(9.18)

x 2 M:

Note also that M '0 (D M ) is dense in each Lp .M; '0 /, 1  p < 1. Let .M; H ; J; P/ be a standard form of M , and 0 2 P be the vector representative of '0 , i.e., '0 D h0 ;
0 i (see Theorem 3.12), so 0 is a cyclic and separating vector for M . We have the embeddings M ,! H ,! M by x 2 M 7 ! x0 2 H ;

 2 H 7 ! h0 ;
i 2 M ;

whose composition is x 7! h0 ;
x0 i D x'0 . The first main result of [57] is the following. Theorem 9.35. Kosaki’s L2 -space L2 .M; '0 / is the standard Hilbert space H with equal norms, where H is being embedded into M via  7! h0 ;
i.

9.3 Kosaki’s interpolation Lp -spaces

159

Thus, the standard Hilbert space H is captured as the “centered” interpolation space C D1=2 .M; M /. The reiteration theorem (see [7]) gives Lp .M; '0 / D C2=p .M; L2 .M; '0 //; Lp .M; '0 / D C2=p

1 .L

2

2 < p < 1;

.M; '0 /; M / D C2.1

2 1=p/ .M ; L .M; '0 //;

1 < p < 2:

Furthermore, it was proved in [57] that 0 Lp .M; '0 / D C1=p .M; M / D C1=p .M; M /

so that either C or C0 can be used to construct Kosaki’s interpolation Lp -spaces. Apart from the “left” injection (9.16) we may consider the “right” injection x 2 M 7! '0 x 2 M ('0 x WD '0 .x
/) and the resulting interpolation Lp -space Lp .M; '0 /R D C D1=p .'0 M; M /, and we write Lp .M; '0 /L D C D1=p .M '0 ; M / to distinguish both Lp -spaces. Under the standard form of M let '00 be the state on M 0 D JMJ defined by '00 WD ' ı j , where j.x 0 / WD J x 0 J , x 0 2 M 0 . Since '0 .j.x 0 // D .x 0 '00 / ı j for x 0 2 M 0 , it is straightforward to see that Lp .M; '0 /R Š Lp .M 0 ; '00 /L

(isometric):

In the second half of [57] Kosaki gave a more explicit description of his Lp -spaces in terms of Haagerup’s Lp -spaces Lp .M / introduced in the previous section. Now let '0 and 0 be faithful normal states on M , whose corresponding elements in L1 .M /C are denoted by h0 and k0 , respectively, i.e., h0 D h'0 , k0 D h 0 ; see Theorem 9.5 and Definition 9.6. For each  2 Œ0; 1 consider the embedding x 2 M 7 ! k0 xh01



2 L1 .M / .Š M /:

The following is the second main result of [57] concerned with the interpolation space in this setting. Theorem 9.36. For each  2 Œ0; 1 and p 2 .1; 1/, the complex interpolation space CD1=p .k0 M h10  ; L1 .M // is exactly k0=q Lp .M /h0.1 /=q . L1 .M // with the norm

=q .1

k ah 0 0

/=q p

a 2 Lp .M /;

D kakp ;

where 1=p C 1=q D 1. Hence, CD1=p .k0 M h10 isometrically isomorphic to Lp .M / via a 2 Lp .M / 7 ! k0=q ah.1 0

/=q



(9.19)

; L1 .M // D k0=q Lp .M /h.1 0

2 k0=q Lp .M /h0.1

/=q

/=q

is

:

In particular, when  D 0; 1, the above theorem tells us that Lp .M; '0 /L D Lp .M /h1=q 0 ;

Lp .M;

0 /R

D k01=q Lp .M /

(9.20)

9 Haagerup’s Lp -spaces

160 with the norms

kah1=q 0 kp D kakp ;

kk01=q akp D kakp ;

a 2 Lp .M /;

(9.21)

respectively. When 0 D '0 , i.e., k0 D h0 , we write Lp .M; '0 / for the interpolation space in Theorem 9.36. In particular, in the case  D 1=2, this is called the “symmetric” Kosaki Lp -space associated with M with respect to '0 , which is sometimes denoted simply as Lp .M; '0 / (though different from that in Definition 9.34). Thus, all of Kosaki’s interpolation Lp -spaces constructed above are isometrically isomorphic to Haagerup’s Lp .M /. Note that Kosaki’s Lp -spaces are realized in the space 0 L1 .M / and satisfy the natural inclusion property as in (9.18), while Lp .M / \ Lp .M / D ¹0º if p ¤ p 0 . Also, note that the above theorem indirectly shows the unicity theorem in the sense that all of the interpolation spaces C1=p .k0 M h10  ; L1 .M //, 0    1, do not depend (up to an isometric isomorphism) on the choice of h0 ; k0 (or '0 ; 0 ). But the proof of the unicity theorem for Lp .M; '0 /L , i.e., the independence of the choice of '0 , was given in [57] based on complex interpolation theory. The following is the last main theorem of [57], called the non-commutative Stein–Weiss interpolation theorem. Theorem 9.37. For each 0 <  < 1 and 1 < p < 1, the complex interpolation space 1=q p p p C .Lp .M /h1=q 0 ; k0 L .M // (D C .L .M; '0 /L ; L .M; 0 /R /; see (9.20), (9.21)) is  1  =q =q p 1 k0 L .M /h0 (D C1=p .k0 M h0 ; L .M /) with the norm (9.19). We end the section with a few remarks. Remark 9.38. When 1  p < 1 and 1=p C 1=q D 1, the Lp –Lq -duality of Kosaki’s Lp -spaces can be given by transforming that of Haagerup’s Lp -spaces in view of Theorem 9.36. For instance, for ; 0 2 Œ0; 1 the duality pairing between Lp .M; '0 / and Lq .M; '0 /0 is written as ˝

.1 h=q 0 ah0

/=q

0

; h0 =p bh0.1

 0 /=p ˛ p;q

D tr.ab/;

a 2 Lp .M /; b 2 Lq .M /:

When 0 D 1 , this duality is somewhat convenient in the sense that for every x; y 2 M , ˝  1  1  ˛ h0 xh0 ; h0 yh0 p;q ˝
=p ˛ .1 /=q
=p D h=q h=p h0 ; h0.1 /=p h0.1 /=q yh=q h0 p;q 0 0 xh0 0 =p .1 /=p .1 /=q =q
 1 
D tr h0 xh0 h0 yh0 D tr h0 xh0 y ; where the last expression is independent of p 2 Œ1; 1/. Here note that h0 M h10 in Lp .M; '0 / for every p 2 Œ1; 1/.



is dense

Remark 9.39. When ' 2 MC is not necessarily faithful with the support e WD s.'/ 2 M , Kosaki’s Lp -space with respect to ' is defined with restriction to eM e. More specifically,

9.3 Kosaki’s interpolation Lp -spaces

161

for 0    1 and 1 < p < 1, with the embedding x 2 eM e 7! h' xh1' eL1 .M /e D L1 .eM e/, from Remark 9.30 and Theorem 9.36 we have p .1 Lp .M; '/ D h=q ' eL .M /eh'

/=q

p .1 D h=q ' L .M /h'

/=q



2

. eL1 .M /e/

.1 /=q with the norm kh=q kp D kakp for a 2 eLp .M /e, where 1=p C 1=q D 1. In ' ah' p particular, we have L .M; '/ D h1=2q Lp .M /h1=2q for the symmetric case  D 1=2. ' '

10 Relative modular operators and Connes’ cocycle derivatives (continued)

The notion of relative modular operators was introduced by Araki [4] to extend relative entropy, the most important quantum divergence in quantum physics and quantum information, to the general von Neumann algebra setting. The notion is a kind of Radon–Nikodym derivative for two functionals ; ' 2 MC , in a similar vein to Connes’ cocycle derivatives introduced in Section 6.2. In the first part of this chapter we give a concise account of relative modular operators. In the second we give a more detailed description of Connes’ cocycle derivatives for functionals in MC (not weights, while not necessarily faithful) in connection with Araki’s relative modular operators.

10.1 Relative modular operators Let M be a von Neumann algebra represented in a standard form .M; H ; J; P/; see Section 3.1. Let ; ' 2 MC be given with their vector representatives  ; ' 2 P, so that .x/ D h ; x i and '.x/ D h' ; x' i for all x 2 M . The support projection s.'/ (2 M ) of ' is the projection onto M 0 ' , which is more explicitly written as sM .'/, the M -support of '. Also, let sM 0 .'/ (2 M 0 ) be the projection onto M ' , the M 0 -support of '. Note that JsM .'/J D sM 0 .'/; indeed, JsM .'/J H D JsM .'/H D JM 0 ' D JM 0 J ' D M ' D sM 0 .'/H thanks to J ' D ' and JM 0 J D M . Definition 10.1. For every ; ' 2 MC define the operators S 0 ;' and F 0;' by S 0 ;' .x' C / WD sM .'/x   ;

x 2 M;  2 .1

F 0;' .x 0 ' C / WD sM 0 .'/x 0  ;

x 0 2 M 0 ;  2 .1

sM 0 .'//H ; sM .'//H ;

which are closable conjugate-linear operators, as shown in the next lemma. Let S F ;' be the closures of S 0 ;' , F 0;' , respectively. Define 

;'

WD S ;' S

;' ,

;'

and call it the relative modular operator with respect to ; '. When D ', we simply write S' , F' and ' for S';' , F';' and ';' , respectively, and call ' the modular

164

10 Relative modular operators and Connes’ cocycle derivatives

operator with respect to '. Note that when ' is faithful, ' here coincides with the modular operator in Section 2.1. Lemma 10.2. The operators S 0 ;' and F 0;' are well defined, and they are densely defined and closable conjugate-linear operators. Proof. Assume that x1 ' C 1 D x2 ' C 2 for xi 2 M and i 2 .1 sM 0 .'//H . Since .x1 x2 /' D 2 1 is in sM 0 .'/H \ .1 sM 0 .'//H D ¹0º, one has .x1 x2 /' D 0, which implies that x1 sM .'/ D x2 sM .'/ so that sM .'/x1  D sM .'/x2  . Hence S 0 ;' is well defined and similarly for F 0;' . It is clear that S 0 ;' and F 0;' are conjugate linear and densely defined. For every x 2 M ,  2 .1 sM 0 .'//H and x 0 2 M 0 ,  2 .1 sM .'//H , one has hS 0 ;' .x' C /; x 0 ' C i D hsM .'/x   ; x 0 ' C i D hx   ; x 0 ' i D hx 0  ; x' i D hsM 0 .'/x 0  ; x' C i D hF 0;' .x 0 ' C /; x' C i: Since S 0 ;' and F 0;' are densely defined, the above equality implies that so are S 0;' and F 0;' . Hence S 0 ;' and F 0;' are closable.  Proposition 10.3. For every ; ' 2 MC the following hold: (1) the support projection of 

;'

WD S ;' S

(2) S

;'

D F ;' and F

(3) S

;'

D J1=2 ;' (the polar decomposition),

(4) ';1 D J

;' J ,

;'

;'

is sM . /sM 0 .'/ D s. /Js.'/J ,

D S ;' ,

where 

1 ;'

is defined with restriction to the support.

Proof. First assume that D '. Let e WD sM .'/, e 0 WD sM 0 .'/ and q WD ee 0 D eJeJ . By Proposition 3.8, .qM q; qH ; qJ q; qP/ is a standard form of qM q Š eM e. Define ' 2 .qM q/C  by '.qxq/ WD '.exe/ for x 2 M , whose vector representative is ' D q' 2 qP. Note that ' is cyclic and separating for qM q on qH . Let ' and J' be the modular operator and the modular conjugation with respect to '. Since S' ..1 e/x' / D ex  .1 e/' D 0 for all x 2 M , note that S' j.1 e/e0 H D 0 as well as S' j.1 e0 /H D 0. Since .1 e 0 / C .1 e/e 0 D 1 q, we find that S' j.1 q/H D 0, and similarly F' j.1 q/H D 0. For every x 2 M , qxq' D ex' and S' .qxq' / D qx  q' D ex  ' D S' .x' / D S' .qxq' /: Also, for every x 0 2 M 0 , qx 0 q' D e 0 x 0 ' and F' .qx 0 q'/ D qx 0 q' D e 0 x 0 ' D F' .x 0 ' / D F' .qx 0 q' /:

165

10.1 Relative modular operators

Therefore, we find that S' D S' ˚ 0;

F' D F' ˚ 0;

so ' D ' ˚ 0

on the decomposition H D qH ˚ .1 q/H . Hence the support projection of ' is q. Since J' D qJ q by Proposition 3.10, we have S' D J' '1=2 ˚ 0 D .J q ˚ J.1


q// '1=2 ˚ 0 D J1=2 ' :

So (1) and (3) hold in this case. Furthermore, (2) and (4) follow from the properties of S' , F' and ' ; see Lemma 2.1 and Remark 2.9. Next prove the case of general , '. Let M .2/ WD M ˝ M2 , for which the standard form .M .2/ ; H .2/ ; J .2/ ; P .2/ / was described in Example 3.11. 
Define  2 .M .2/ /C  to  x11 x12 be the balanced functional of ', similar to (6.1), i.e.,  x21 x22 WD '.x11 / C .x22 /, with vector representative in P .2/ given by 2 3 '   607 ' 0 7  D D6 4 0 5: 0   It is clear that sM .2/ . / D

 sM .'/ 0

 0 sM . /

or, in 4  4 form,

2

3 0 0 7 7 0 5 sM . /

(10.1)

3 0 0 0 sM 0 . / 0 0 7 7: 0 0 sM .'/ 0 5 0 0 sM 0 . /

(10.2)

sM .'/ 0 0 6 0 s .'/ 0 M sM .2/ . / D 6 4 0 0 sM . / 0 0 0 and also, by (3.6),

s.M .2/ /0 . / D J .2/ sM .2/ . /J .2/

2 sM 0 .'/ 6 0 D6 4 0 0

Furthermore, in view of (3.4), S0 is defined as 02 3 2 3 2 31 x11 0 x12 0 ' 11 B6 6 0 7 612 7C x11 0 x12 7 0 B6 0 7 6 7 6 7C S @4 C x21 0 x22 0 5 4 0 5 421 5A 0 x21 0 x22  22 2  32 3  x11 0 x21 0 '  76 6 0 x 0 x21 07 11 6 7 6 7; D sM .2/ . / 4   x12 0 x22 0 54 0 5   0 x12 0 x22 

166

10 Relative modular operators and Connes’ cocycle derivatives

that is, 2

x11 ' 6 x 12  S0 6 4 x21 ' x22  T

3 3 2  C 11 ' sM .'/x11  6 C 12 7  7 7 7 D 6 sM .'/x21  ' 5 C 21 5 4 sM . /x12  sM . /x22  C 22

 for xij 2 M and 11 12 21 22 2 .1 s.M .2/ /0 . //H .2/ . Therefore, extending (3.5) (also (6.4)) we find that the closure of S0 is 2 3 S' 0 0 0 60 0 S ;' 0 7 7 S D 6 (10.3) 4 0 S'; 0 05 0 0 0 S and so  is written as 2  S' S' 0 0  6 0 S'; S 0 ';  D 6 4 0 0 S ;' S 0 0 0

3 2 0 ' 6 0 0 7 7D6 0 5 4 0 ;' 0 SS

3 0 '; 0 7 7 : (10.4) 0  ;' 0 5 0 0    x12 On the other hand, in view of (3.4) and (3.6) note that J .2/ XJ .2/ for X D xx11 2 21 x22 .2/ M is represented in 4  4 form as 3 2 J x11 J J x12 J 0 0 6J x21 J J x22 J 0 0 7 7: J .2/ XJ .2/ D 6 4 0 0 J x11 J J x12 J 5 0 0 J x21 J J x22 J Hence, F0 is defined by 02

0 x11 B6x 0 6 21 F0 B @4 0 0

for xij0 2 M 0 and closure of F0 is



11

0

0 0

0 x12 0 x22 0 0

3 2 3 2 31 0 0 ' 11 6 0 7 612 7C 0 0 7 7 6 7 C 6 7C 0 0 54 x11 x12 0 5 421 5A 0 0 x21 x22  22 2 0 32 3 0 x11 x21 0 0 ' 6x 0 x 0 76 0 7 0 0 12 22 76 7 D s.M .2/ /0 . / 6 0 0 5 4 4 0 0 x11 x21 05 0 0 0 0 x12 x22  T 12 21 22 2 .1 sM .2/ . //H .2/ , from which we find that the 2 F' 60 6 F D 4 0 0

0 0 F

;'

0

0 F'; 0 0

3 0 0 7 7: 0 5 F

(10.5)

167

10.1 Relative modular operators

Now assertions (1)–(4) of the proposition follow from those (applied to ) in the previous case D '. That is, (1) is seen by (10.4), (10.1) and (10.2), (2) is seen by (10.3) and (10.5), (3) follows from (10.3), (3.6) and (10.4), and (4) follows from (10.4) and (3.6).  Examples 10.4. (1) Let M D L1 .X; / as in Example 3.6(1). For every ; ' 2 L1 .X; /C Š MC , it is immediate to verify that  ;' is the multiplication of ¹'>0º . ='/, which is the Radon–Nikodym derivative of d with respect to ' d (restricted on the support of ') in the classical sense. (2) Let M D B.H / as in Example 3.6(2). For every ; ' 2 B.H /C  we have the density operators (positive trace-class operators) D , D such that .x/ ' P P D Tr D x for x 2 B.H / and similarly for D' . Let D D a>0 aPa and D' D b>0 bQb be the spectral decompositions of D , D' , where Pa and Qb are finite-dimensional projections. Then the relative modular operator  ;' on C2 .H / is given as X ab 1 LPa RQb ; (10.6)  ;' D LD RD' 1 D a>0;b>0

where LŒ  and RŒ  denote the left and right multiplications respectively and D' 1 is the generalized inverse of D' . In the rest of the section we further discuss relative modular operators in standard form on Haagerup’s L2 .M /; see Theorem 9.29. The next lemma (due to Kosaki [57]) gives the description of the modular automorphism group  t' in terms of the element h' in L1 .M / corresponding to ' 2 MC . Lemma 10.5. For every ' 2 MC let  t' be the modular automorphism group with respect to 'js.'/M s.'/ , where s.'/ is the M -support of '. Then  t' .x/ D hi't xh' i t ;

x 2 s.'/M s.'/; t 2 R;

(10.7)

where hit ' is defined with restriction to the support of h' (note that s.h' / D s.'/). Proof. First assume that ' is faithful. Define ˛ t .x/ WD hi't xh' i t for x 2 M and t 2 R. Since s .˛ t .x// D s .hi't /s .x/s .h' i t / D .e

i st

h' /x.e i st h' / D ˛ t .x/;

s 2 R;

it follows that ˛ t .x/ 2 M and hence ˛ t is a strongly continuous one-parameter automorphism group of M . Let x; y 2 M and assume that x is entire ˛-analytic with the analytic extension ˛z .x/ (z 2 C); see Lemma 2.13. By analytic continuation one can see that hi's ˛z .x/ D ˛zCs .x/hi's for all s 2 R and z 2 C. For every  2 D.h' / ( K WD L2 .R; H /, the representing Hilbert space for N D M o '0 R; see Section 8.2), by Theorem A.7 of Appendix A.1 there exists a K -valued bounded continuous function f ./ on 1  Im   0, analytic in 1 < Im  < 0, such that f .s/ D hi's  (s 2 R). Then, for each z 2 C, ˛zC .x/f ./ is a bounded continuous function on 1  Im   0,

168

10 Relative modular operators and Connes’ cocycle derivatives

analytic in 1 < Im  < 0, such that ˛zCs .x/f .s/ D hi's ˛z .x/ (s 2 R). By Theorem A.7 again, this gives ˛z .x/ 2 D.h' / and h' ˛z .x/ D ˛z i .x/h' . Hence, in view of Lemma 4.5, one has h' ˛z .x/ D ˛z i .x/h' (z 2 C). Since '.˛ t .x/y/ D tr.h' ˛ t .x/y/ D tr.˛ t i .x/h' y/ D '.y˛ t i .x//;

t 2 R;

it follows that ' satisfies the . t' ; 1/-KMS condition for x, y. By a convergence argument based on Lemma 2.13 one can show that the KMS condition holds for all x; y 2 M (for further details see, e.g., [9, p. 82]). Hence Theorem 2.14 implies that  t' D ˛ t . For general ' 2 MC let e WD s.'/. Since h' 2 eL1 .M /e corresponds to 'jeM e (see Remark 9.30), the result follows from the case above.  Before further discussions we here examine Haagerup’s Lp -spaces for M .2/ WD M ˝ M2 . Take the tensor product '0 ˝ Tr of a faithful semifinite normal weight '0 on M and the trace functional Tr on M2 . Then it is immediate to see that  t'0 ˝Tr D  t'0 ˝ id2 , where id2 is the identity map on M2 . By looking at the construction of the crossed products N WD M o '0 R and N .2/ WD M .2/ o '0 ˝id2 R (see the first paragraph of Section 8.2), the following are easily seen: (a) N .2/ D N ˝ M2 , (b) the canonical trace on N .2/ is  ˝ Tr, where  is the canonical trace on N , (c) the dual action on N .2/ is s ˝ id2 (s 2 R), where s is the dual action on N .

e

e

Based on these facts we see that N .2/ D Nz ˝ M2 for the spaces Nz and N .2/ of -measurable and  ˝ Tr-measurable operators affiliated with N and N .2/ , respectively. Therefore, for 0 < p  1, Haagerup’s Lp -space

e

® Lp .M .2/ / WD a 2 N .2/ D Nz ˝ M2 W .s ˝ id2 /.a/ D e

s=p

a; s 2 R

¯

is written as ®  Lp .M .2/ / D Lp .M / ˝ M2 D a D aa11 21

e

a12 a22



¯ W aij 2 Lp .M /; i; j D 1; 2 ;

and its positive part is .Lp .M / ˝ M2 / \ N .2/ C . In particular, L2 .M .2/ / D L2 .M / ˝ M2 is viewed as the Hilbert space tensor product of L2 .M / and M2 with the Hilbert–Schmidt inner product. Moreover, by looking closely at the construction of the functional tr, we notice the following: (d) The tr-functional on L1 .M .2/ / D L1 .M /˝M2 is tr˝Tr, where tr is the tr-functional on L1 .M /. In this way, the standard form of M .2/ D M ˝ M2 is given in terms of Haagerup’s L -space (more specifically than Example 3.11) as
M ˝ M2 ; L2 .M / ˝ M2 ; J D  ; .L2 .M / ˝ M2 /C ; 2

169

10.1 Relative modular operators

where Œxij 2i;j D1 2 M ˝ M2 acts on L2 .M / ˝ M2 as the 2  2 matrix left multiplica    11 12      21 12  x12 tion xx11 for , and J D  is the matrix -operation 11 D 11   21 x22 21 22 21 22 12

Œij 2i;j D1 2 L2 .M / ˝ M2 .

22

Proposition 10.6. Let ; ' 2 MC with corresponding h ; h' 2 L1 .M /. Then i t;'  D hi t h' i t ; 1

p 2 p ;' .xh1=2 ' / D h xh'

p

;

 2 L2 .M /; t 2 R;

(10.8)

x 2 M; 0  p  1=2;

(10.9)

with the convention that h0 D s. /, h0' D s.'/ and 0 ;' D s. /Js.'/J . Proof. Since s. /j.s.'//L2 .M / D s. /L2 .M /s.'/, we may assume that  2 s. /L2 .M /s.'/ and x 2 s. /M s.'/; see Proposition 10.3(1). First assume that D '. Since ' h1=2 D h1=2 and so i't h1=2 D h1=2 ' ' ' ' , for every x 2 s.'/M s.'/ and t 2 R one has it i t 1=2 i't .xh1=2 D  t' .x/h1=2 ' / D ' x' h' ' it D hi't xh' i t h1=2 D hi't .xh1=2 ' ' /h'

(10.10)

thanks to Lemma 10.5. Since s.'/M h1=2 is dense in s.'/L2 .M /s.'/, the above implies ' 1=2 (10.8) in the case D '. Since xh' 2 D.1=2 ' /, Theorem A.7 implies that there is an s.'/L2 .H /s.'/-valued bounded continuous function f on 0  Im z  1=2, analytic in 0 < Re z < 1=2, such that f .i t / D i't .xh1=2 ' /, t 2 R. On the other hand, consider 1

Cz

the function g.z/ WD h'2 1=2 < r < 1=2 we write

1

xh'2 1

g.r C i t / D h'2

z

for

1=2 < Re z < 1=2. For z D r C i t with

1 Cr i t r h' xh' i t h'2

1

D h'2

Cr

1

 t' .x/h'2

r

thanks to Lemma 10.5 again. From Lemma 9.19 and Theorem 9.18(c) it follows that g.z/ is an s.'/L1 .M /s.'/-valued analytic function in 1=2 < Re z < 1=2. For every y 2 s.'/M s.'/ note that ' 1=2 tr.yg.i t // D tr.yh1=2 '  t .x/h' / it 1=2 D tr.yh1=2 ' ' .xh' //

D

tr.yh1=2 ' f .i t //;

(by (10.10))

t 2 R:

By analytic continuation this implies that tr.yg.z// D tr.yh1=2 ' f .z// for all z with 0  Re z < 1=2. Taking z D p with 0  p < 1=2 one has 1

p 2 tr.yh1=2 ' h' xh'

p

p 1=2 / D tr.yh1=2 ' ' .xh' //;

0  p < 1=2:

170

10 Relative modular operators and Connes’ cocycle derivatives 1

p

1=2 2 Noting that hp' xh'2 and p' .xh1=2 ' / are in s.'/L .M /s.'/ and s.'/M h' is dense in 2 s.'/L .M /s.'/, we see that (10.9) holds for 0  p < 1=2 in the case D '. When 1=2 1=2 p D 1=2, note that 1=2 D J.s.'/x  h1=2 ' xh' ' / D h' xs.'/. Next prove the case of general , '. Define the balanced functional  2 .M .2/ /C  as in the proof of Proposition 10.3. From the previous case D ' applied to  we have

it „ D hit „h i t ; 1

p=2 Xh1=2 D hp Xh2  

p

„ 2 L2 .M .2/ /; t 2 R; ;

X 2 M .2/ ; 0  p  1=2:

In view of the above description of Lp .M .2/ / before the proposition, note that the element   h 0  of L1 .M .2/ / corresponding to  is h D 0' h . Apply the above formulas to „ D 0 00   with  2 L2 .M / and X D x0 00 with x 2 M ; then (10.8) and (10.9) are seen in view of (10.4).  We end the section with the next theorem, giving an explicit description of the positive powers p ;' of  ;' represented on L2 .M /. This was first observed by Kosaki [55] when 0  p  1=2 (see also [39, Lem. A.3]), and the extension to the whole p  0 is due to Jenčová [48].1 Theorem 10.7. For every ; ' 2 MC and every p  0, the domain of p ;' defined on L2 .M / coincides with Dp . ; '/ WD ¹ 2 L2 .M / W hp s.'/ D hp' for some  2 L2 .M /s.'/º;

(10.11)

where hp s.'/ D hp' means the equality as elements of Nz (or L2=.1C2p/ .M /). Moreover, if ,  are given as in the right-hand side of (10.11), then p ;'  D p ;' .s.'// D :

(10.12)

Proof. First note that  2 L2 .M /s.'/ in the right-hand side of (10.11) is uniquely determined for each  2 Dp . ; '/. By Proposition 10.3(1) the first equality in (10.12) is clear. Assume first that 0  p  1=2 . Let  2 L2 .M / and  2 L2 .M /s.'/ be given with hp s.'/ D hp' and so s.'/  hp D hp'  . Due to (10.9), for every x 2 M one has ˝ p 21 h; p ;' .xh1=2 ' /i D ; h xh'

1

D tr hp'  xh'2 D tr. xh1=2 ' / D

1

p˛

D tr s.'/  hp xh'2 p


p


(by Proposition 9.22)

h; xh1=2 ' i:

This immediately extends to h; p ;' i D h; i;

2  2 M h1=2 ' C L .M /.1

s.'//:

1The author is grateful to A. Jenčová for permitting her proof to be included here.

171

10.1 Relative modular operators

p 2 Since M h1=2 s.'// is a core of 1=2 ' C L .M /.1 ;' , it is also a core of  ;' for 0  p  p 1=2; see [3, Lem. 4]. Hence we find that  2 D. ;' / and (10.12) holds. Conversely, assume that  2 D.p ;' / and p ;'  D ; then p ;' .s.'// D  2 2 L2 .M /s.'/. Since M h1=2 ' C L .M /.1 ¹xn º in M such that

kxn h1=2 '

s.'/k2 ! 0;

s.'// is a core of p ;' , there exists a sequence kp ;' .xn h1=2 ' / 1

p 2 Let n WD p ;' .xn h1=2 ' /; then n D h xn h'

p

k2 ! 0:

thanks to (10.9). Hence one has

n hp' D hp xn h1=2 ' :

(10.13)

By Hölder’s inequality (Proposition 9.17) one has kn hp' khp xn h1=2 '

hp' k2=.1C2p/  kn

k2 khp' k1=p ! 0;

hp s.'/k2=.1C2p/  khp k1=p kxn h1=2 '

(10.14)

s.'/k2 ! 0:

(10.15)

Combining (10.13)–(10.15) gives hp s.'/ D hp' . Thus, D.p ;' / D Dp . ; '/ has been shown in the case 0  p  1=2. Now we define ® ¯ D1 . ; '/ WD  2 L2 .M / W t 2 R 7! i t;'  2 L2 .M / extends to an entire function : By a familiar regularization technique with Gaussian kernels (as in the proof of Lemma 2.13 and also used in the last part of the proof here), it is seen that D1 . ; '/ is dense in L2 .M / and is a core of p ;' for any p  0. Let  2 D1 . ; '/, and show that for every p  0,  2 Dp . ; '/ and .p ;' /hp' D hp s.'/: (10.16) The case proved above implies that (10.16) holds when 0  p  1=2. For every p > 1=2 write p D k=2 C p0 with k 2 N and 0  p0 < 1=2. In view of Proposition A.8 of Appendix A.1 one has p 1=2 p .p ;' /hp' D .1=2 //h1=2 ' h' ;' . ;'

1=2

p 1 1=2 p D h1=2 .1=2 ;' . ;' //h' h'

1

D h1=2 .p ;'1=2 /hp' D h .p ;'1 /hp'

1=2

1

D


D hk=2 .p0;' /hp' 0 D hp s.'/: Therefore, (10.16) has been shown for all p  0. Since (10.16) means that hp s.'/ D hp' for  D p ;' , it follows that D1 . ; '/  Dp . ; '/ and (10.12) hold. Assume that p > 1=2, and let Tp be the operator with domain Dp . ; '/ defined by Tp  WD , which is clearly linear. By the previous paragraph note that Tp  D p ;'  for all  2 D1 . ; '/. Let us show that Tp is a closed operator with a core D1 . ; '/. Then the result follows since D1 . ; '/ is also a core of p ;' .

172

10 Relative modular operators and Connes’ cocycle derivatives

So let ¹n º be a sequence in Dp . ; '/ such that n !  and Tp n !  in L2 .M /. Then by Theorem 9.18(c), n !  and Tp n !  in Nz . Hence, from Theorem 4.12 it follows that hp s.'/ D lim hp n s.'/ D lim.Tp n /hp' D hp' n

n

in Nz with the measure topology, so that  2 Dp . ; '/ and Tp  D . Hence Tp is closed. To show that D1 . ; '/ is a core of Tp , let  2 Dp . ; '/ and  D Tp . For n 2 N set r Z 1 n 2 e nt i t;'  dt C .1 s.'//; n WD  1 r Z 1 n 2 n WD e nt i t;'  dt:  1 Then n 2 D1 . ; '/ and n ! , n !  in L2 .M /. By (10.8) note that hp it;'  D hi t hp h' i t D hi t hp' h' i t D .i t;' /hp' ;

t 2 R:

(10.17)

For each n, to see that Tp n D n , one can take sequences ¹n;k º1 , ¹n;k º1 of kD1 kD1 Riemann sums r mk .k/
.k/ 2 n X .k/ it tl tl.k/1 e n.tl /  l;'  C .1 s.'//; n;k WD  lD1

r n;k WD

mk nX tl.k/ 


tl.k/1 e

.k/ 2 /

n.tl

.k/



i tl ;'

;

lD1

.k/ < 1, such that kn;k n k2 ! 0 and kn;k n k2 ! with 1 < t0.k/ < t1.k/ <


< tm k 0 as k ! 1. Then it follows from (10.17) that

hp n s.'/ D lim hp n;k s.'/ D lim n;k hp' D n hp' k

k

in Nz , so that Tp n D n for all n. Hence D1 . ; '/ is a core of Tp , as desired.



Remark 10.8. For the case p D 1, Theorem 10.7 says that  ;'  D  when ;  2 L2 .M / satisfy h s.'/ D h' with s.'/ D . The condition might be formally written as  D h h' 1 , so we may write  ;' D Lh Rh' 1 in a formal sense. This is the same expression as in (10.6).

10.2 Connes’ cocycle derivatives (continued) In this section, as a continuation of Section 6.2, we continue to discuss Connes’ cocycle derivatives, here restricted to bounded functionals '; 2 MC , but not necessarily faithful unlike Section 6.2.

173

10.2 Connes’ cocycle derivatives (continued)

For each '; 2 MC consider the balanced functional  D .'; / on M .2/ WD
P2 M ˝ M2 defined by  .x22 /, xij 2 M , where eij i;j D1 xij ˝ eij WD '.x11 / C (i; j D 1; 2) are the matrix units of M2 .C/. The  has already been treated in the proof of Proposition 10.3. Since Pthe support projection of  is s. / D s.'/ ˝ e11 C s. / ˝ e22 , note that Œxij 2i;j D1 D 2i;j D1 xij ˝ eij 2 M .2/ belongs to s. /M .2/ s. / if and only if x11 2 s.'/M s.'/;

x12 2 s.'/M s. /;

x21 2 s. /M s.'/;

x22 2 s. /M s. /:

(10.18)

We then define the modular automorphism group  t on s. /M .2/ s. / as well as  t' on s.'/M s.'/ and  t on s. /M s. /. Lemma 10.9. Let ',

and  be as above. Then

(1) s.'/ ˝ e11 ; s. / ˝ e22 2 .s. /M .2/ s. // (the centralizer of js. /M .2/ s. / ; see Definition 2.15), (2)  t .x ˝ e11 / D  t' .x/ ˝ e11 for all x 2 s.'/M s.'/ and t 2 R, (3)  t .x ˝ e22 / D  t .x/ ˝ e22 for all x 2 s. /M s. / and t 2 R, (4)  t .s. /M s.'/ ˝ e21 /  s. /M s.'/ ˝ e21 . Proof. (1) Note that for every Œxij  2 s. /N s. /,    s.'/ 0 x11 x12  D '.s.'/x11 / D '.x11 s.'// 0 0 x21 x22    x11 x12 s.'/ 0 D x21 x22 0 0 and similarly 

 0 0

0 s. /

 x11 x21

x12 x22

 D

 x11 x21

x12 x22

 0 0

0 s. /

 :

Hence (1) follows. (2) By (1) with Proposition 2.16 there is a strongly* continuous automorphism group ˛ t on s.'/M s.'/ such that  t .x ˝ e11 / D ˛ t .x/ ˝ e11 ;

x 2 s.'/M s.'/; t 2 R:

When x; y 2 s.'/M s.'/, the KMS condition of  for  t with x ˝ e11 and y ˝ e11 induces that of ' for ˛ t with x and y. Hence Theorem 2.14 implies that ˛ t D  t' on s.'/M s.'/. (3) is similar to (2).

174

10 Relative modular operators and Connes’ cocycle derivatives

(4) For every x 2 s. /M s.'/, by (1) with Proposition 2.16 we have       0 0 0 0 0 0 s.'/ 0  t D  t x 0 0 s. / x 0 0 0       0 0 0 0 s.'/ 0  D  2 s. /M s.'/ ˝ e21 : 0 s. / t x 0 0 0



Definition 10.10. Let '; 2 MC and  D .'; /. By Lemma 10.9(4) there exists a strongly* continuous map t 2 R 7! u t 2 s. /M s.'/ such that  t .s. /s.'/ ˝ e21 / D u t ˝ e21 ;

t 2 R:

The map t 7! u t is called Connes’ cocycle (Radon–Nikodym) derivative of with respect to ', and denoted by .D W D'/ t . Note that the definition here extends Definition 6.7 where ', are faithful. .D

The next proposition specifies the relationship between Connes’ cocycle derivative W D'/ t and Araki’s relative modular operator  ;' .

Proposition 10.11. For every ';

2 MC ,

.D

W D'/ t Js.'/J D i t;' ' i t ;

(10.19)

.D

W D'/ t Js. /J D i t ';i t ;

(10.20)

.D

W

D'/ t i't

D

it ;'

(10.21)

for all t 2 R. Proof. Consider the balanced functional  D .'; / on M .2/ D M ˝ M2 and the associated  represented in the standard form of M .2/ as described in Example 3.11 and in the proof of Proposition 10.3. From (10.1) and (10.2) the support projection of  is 2 3 s.'/Js.'/J 0 0 0 6 7 0 s.'/Js. /J 0 0 7: s. / D 6 (10.22) 4 5 0 0 s. /Js.'/J 0 0 0 0 s. /Js. /J For every X D Œxij  2 s. /M .2/ s. / (see (10.18)) we write 2 3 x11 0 x12 0 6 0 x11 0 x12 7 6 7  it  t .X / D it  4x 0 x22 0 5  21 0 x21 0 x22 2 it it ' x11 ' i t 0 i't x12  ;' it it 6 0 '; x11 '; 0 D6 it it 4i t x21 ' i t 0  x ;' ;' 22  ;' it it 0  x21 '; 0

3 0 i';t x12  i t 7 7 : (10.23) 5 0 it it  x22 

175

10.2 Connes’ cocycle derivatives (continued)

In particular, letting X D s. /s.'/ ˝ e21 we find that .D

W D'/ t js.'/Js.'/J H D i t;' s. /s.'/' i t js.'/Js.'/J H ;

.D

W D'/ t js.'/Js.

/J H

D i t s. /s.'/';i t js.'/Js.

/J H

;

where s.'/Js.'/J and s.'/Js. /J arise from (10.22). By taking account of the supports of  ;' , ' ,  and '; we have (10.19) and (10.20). It is immediate to see that (10.19) gives (10.21) as well.  Remark 10.12. For any projection e 2 M , recall that x 2 eM e 7! xe 0 2 eM ee 0 (e 0 WD JeJ ) is a *-isomorphism (see Proposition 3.8(1)). This may justify writing (10.19) and (10.20) in a simpler way as follows: .D

W D'/ t D i t;' ' i t

if s. /  s.'/;

.D

W D'/ t D i t ';i t

if s.'/  s. /:

Similarly, from (10.23) we may also write  t .x/ D i t;' x

it ;' ;

x 2 s. /M s. /;

if s. /  s.'/:

We next show the expression of .D W D'/ t in terms of elements of Haagerup’s L1 .M /, which is quite convenient for deriving properties of Connes’ cocycle derivatives. Lemma 10.13. Let ; ' 2 MC with corresponding h ; h' 2 L1 .M /. Then .D

W D'/ t D hi t h' i t ;

t 2 R;

(10.24)

with the same convention as in Lemma 10.5. Proof. In view of the description of Lp .M ˝ M2 / before Proposition 10.6, since the h 0  element of L1 .N / corresponding to .'; / is h D 0' h , it follows from (10.7) that for Œxij  2 s. /N s. /,  t

 x11 x21

x12 x22



 i t    it h' 0 x11 x12 h' 0 0 h x21 x22 0 h  it it  it it h x11 h' h' x12 h D i't ; t 2 R: h x21 h' i t hi t x22 h i t D

(10.25)

Therefore,  .D

0 W D'/ t

so that (10.24) follows.

  0 0 D  t 0 s. /s.'/

  0 0 D it it h h' 0

 0 0 

176

10 Relative modular operators and Connes’ cocycle derivatives

Remark 10.14. Indeed, the assertions of Lemma 10.9 are immediately seen from the expression in (10.25) and Lemma 10.5. Example 10.15. Assume that M is a semifinite von Neumann algebra with a faithful semifinite normal trace  . As explained in Example 9.11, Haagerup’s L1 -space L1 .M / in this case is identified with the conventional L1 -space L1 .M;  / with respect to . More precisely, for each ' 2 M , h' in L1 .M / and the Radon–Nikodym derivative d'=d 2 L1 .M;  / are related as h' D .d =d / ˝ e t . Hence, for every '; 2 MC we have i t   i t  d' d : .D W D'/ t D d d In particular, when B D B.H / with the usual trace Tr, we have .D W D'/ t D D i t D' i t , where D' , D are the density (trace-class) operators representing '; 2 B.H /C . In the following theorems we present important properties of Connes’ cocycle derivative .D W D'/ t , which were first given by Connes [13, 14] for the case of faithful semifinite normal weights. Properties (2)–(4) appeared in Theorems 6.6 and 6.8 for (faithful) semifinite normal weights ', . Theorem 10.16. Let '; 2 MC and assume that s. /  s.'/. Then u t WD .D satisfies the following properties:

W D'/ t

(1) u t ut D s. / D u0 and ut u t D  t' .s. // for all t 2 R; in particular, the u t are partial isometries with the final projection s. /, (2) usCt D us s' .u t / for all s; t 2 R (the cocycle identity), (3) u

t

D  't .ut / for all t 2 R,

(4)  t .x/ D u t  t' .x/ut for all t 2 R and x 2 s. /M s. /, (5) for every x 2 s.'/M s. / and y 2 s. /M s.'/, there exists a bounded continuous function F on 0  Im z  1, analytic in 0 < Im z < 1, such that F .t / D

.u t  t' .y/x/;

F .t C i / D '.xu t  t' .y//;

t 2 R:

Furthermore, u t (t 2 R) is uniquely determined by a strongly* continuous map t 2 R 7! u t 2 M satisfying (1), (2), (4) and (5) above. (Note that (3) follows from (1) and (2).) Proof. First note that s.h / D s. /  s.'/ D s.h' /. In the proof below we repeatedly

177

10.2 Connes’ cocycle derivatives (continued)

use (10.24) as well as (10.7). We compute u t ut D .hi t h' i t /.hi t h' i t / D hi t h' i t hi't h

it

D hi t s.'/h

it

D hi t h

it

D s. /;

u0 D s. /s.'/ D s. /; ut u t D hi't h

h h' i t D hi't s. /h' i t D  t' .s. //;

it it

us s' .u t / D hi s h' i s hi's hi t h' i t hi's D hi.sCt / h' i.sCt / D usCt ;  't .ut / D h' i t hi't h

it it h'

Dh

it it h'

D u t;

and for any x 2 s. /M s. /, u t  t' .x/ut D hi t h' i t hi't xh' i t hi't h

it

D hi t xh

it

D  t .x/:

Hence all the properties in (1)–(4) have been shown. It is an easy exercise to verify that (3) follows from (1) and (2). (5) Let x 2 s.'/M s. / and y 2 s. /M s.'/. When X D x ˝ e12 and Y D y ˝ e21 , by (10.25) we write     0 0 0 0 D .u t  t' .y/x/; D D 0 hi t yh' i t x 0 u t  t' .y/x     i t xh yh' i t 0 xu t  t' .y/ 0 D '.xu t  t' .y//: D .X t .Y // D  0 0 0 0

. t .Y /X /

Hence the assertion follows from the KMS condition of  for   . To prove the uniqueness assertion for u t , assume that t 2 R 7! u t 2 M is a strongly* continuous map satisfying (1), (2), (4) and (5). For each t 2 R define a map  t from s. /M .2/ s. / into itself, where M .2/ WD M ˝ M2 , by  x11 t x21

x12 x22



  t' .x11 /  t' .x12 /ut WD u t  t' .x21 /  t .x22 /    1 0  t' .x11 /  t' .x12 / 1 D 0 u t  t' .x21 /  t' .x22 / 0 

0 ut



for Œxij  2 s. /M .2/ s. /, where the last equality is due to (4). For X D Œxij , Y D Œyij  2 s. /M .2/ s. / we find by (1) and (2) that  t .X / t .Y / D  t .X Y /,  t .X / D  t .X  /, 0 .X / D X and   s' . t' .x11 // s' . t' .x12 /ut /us s . t .X // D ' ' ' ' us s .u t  t .x21 // us s .u t  t .x22 /ut /us   ' ' sCt .x11 / sCt .x12 /usCt D D sCt .X /: ' ' usCt sCt .x21 / usCt sCt .x22 /usCt

178

10 Relative modular operators and Connes’ cocycle derivatives

Hence  t (t 2 R) is a strongly* continuous one-parameter automorphism group on s. /M .2/ s. /. Furthermore, we write  '    t .y11 /  t' .y12 /ut x11 x12 . t .Y /X / D  x21 x22 u t  t' .y21 /  t .y12 / D '. t' .y11 /x11 / C '. t' .y12 /ut x21 / C

.X t .Y //

.u t  t' .y21 /x12 / C

. t .y22 /x22 /; ' D '.x11  t .y11 // C .x21  t' .y12 /ut / C '.x12 u t  t' .y21 // C .x22  t .y22 //:

By property (5) there are bounded continuous functions F , G on 0  Im z  1, analytic in 0 < Im z < 1, such that F .t / D

.u t  t' .y21 /x12 /;

F .t C i / D '.x12 u t  t' .y21 //;

G.t / D

  /x21 /; .u t  t' .y12

  G.t C i / D '.x21 u t  t' .y12 //:

z Set G.z/ WD G.z C i / for 0  Im z  1, which is bounded continuous on 0  Im z  1 and analytic in 0 < Im z < 1. Then z / D G.t C i / D '. t' .y12 /ut x21 /; G.t

z C i / D G.t / D G.t

.x21  t' .y12 /ut /:

z as well as the KMS conditions of ' for  t' From these boundary conditions of F and G and of for  t , it follows that  satisfies the KMS condition for  t . Therefore, Theorem 2.14 implies that  t D  t , from which u t ˝ e21 D u t  t' .s. // ˝ e21 D  t .s. / ˝ e21 / D  t .s. / ˝ e21 / D .D so that u t D .D

W D'/ t ˝ e21

W D'/ t for all t 2 R.



Proposition 10.17. Let '; ;  2 MC . Then we have (1) .D

W D'/t D .D' W D / t for all t 2 R,

(2) if either s. /  s.'/ or s./  s.'/, then .D D/ t for all t 2 R (the chain rule), (3) if s. /  s.'/ and s./  s.'/, then .D and only if D ,

W D'/ t .D' W D/ t D .D

W D'/ t D .D W D'/ t for all t 2 R if

(4) for every ˛ 2 Aut.M /, .D. ı ˛/ W D.' ı ˛// t D ˛

1

..D

W D'/ t / for all t 2 R.

Proof. (1) By Lemma 10.13 one has .D

W

W D'/t D .hi t h' i t / D hi't h

it

D .D' W D / t :

179

10.2 Connes’ cocycle derivatives (continued)

(2) Since s.h / D s. /  s.'/ or s.h / D s./  s.'/, one has .D

W D'/ t .D' W D/ t D hi t h' i t hi't h i t D hi t s.'/h i t D hi t h i t D .D

W D/ t :

(3) Since s. /; s./  s.'/, one has .D

W D'/ t D .D W D'/ t for all t 2 R ” hi t h' i t D hit h' i t for all t 2 R ” hi t D hit for all t 2 R ” h D h ”

D :

(4) Let  WD .'; /, the balanced functional on M .2/ WD M ˝ M2 .C/. Note that ˛ ˝ id2 2 Aut.M .2/ /,  ı .˛ ˝ id2 / D .' ı ˛; ı ˛/ and s. ı .˛ ˝ id2 // D .˛ 1 ˝ id2 /.s. //. Hence ˛ 1 ˝ id2 is a *-isomorphism from s. /M .2/ s. / onto s. ı .˛ ˝ id2 //M .2/ s. ı .˛ ˝ id2 //. As in Lemma 7.20(1) (or more directly by using the KMS condition with Theorem 2.14) we have  tı.˛˝id2 / D .˛

1

˝ id2 / ı  t ı .˛ ˝ id2 /;

t 2 R:

Therefore, for every t 2 R we have .D.

ı ˛/ W D.' ı ˛// t ˝ e21 D  t ı.˛˝id2 / .s. D .˛

1

˝

D .˛

1

˝

ı ˛/s.' ı ˛/ ˝ e21 /

id2 /. t ..˛ id2 /. t .s.

˝ id2 /.˛

1

.s. /s.'// ˝ e21 ///

/s.'/ ˝ e21 // D ˛

1

..D

W D'/ t / ˝ e21 :

The next proposition is a modification of Theorem 7.25 in the setting of '; with additional characterizations in terms of h' , h . Proposition 10.18. Let '; equivalent: (i)

ı  t' D

 2 MC

2 MC with s. /  s.'/. The following conditions are

for all t 2 R;

(ii) .D

W D'/ t 2 .s.'/M s.'//' (the centralizer of 'js.'/M s.'/ ) for all t 2 R;

(iii) .D

W D'/ t 2 .s. /M s. // for all t 2 R;

(iv) t 2 R 7! .D

W D'/ t is a one-parameter group of unitaries in s. /M s. /;

it is (v) his hit ' D h' h for all s; t 2 R;

(vi) h h' D h' h as elements of Nz (the -measurable operators affiliated with N ).

180

10 Relative modular operators and Connes’ cocycle derivatives

Proof. First, in view of [77, Thm. VIII.13] one can easily verify that (v) ” (vi) and they are also equivalent to the property that all spectral projections of h and h' commute. (i) ” (v). By (10.7), (i) implies that tr.h' i t h hi't x/ D tr.h hi't xh' i t / D tr.h x/;

x 2 M; t 2 R;

so that h' it h hi't D h and hence .s.'/ C h / 1 hi't D hi't .s.'/ C h / 1 in s.'/M s.'/ for all t 2 R. This implies (v). The argument can be reversed to see that (i) and (v) are equivalent. (ii) ” (v). By (10.7) and Lemma 10.13, one can rewrite (ii) as hi's .hi t h' i t /h' i s D hi t h' i t ;

s; t 2 R;

/ which is equivalent to (v) by multiplying hi.sCt from the right. '

(iii) ” (v). As in the proof of (ii) ” (v), condition (iii) means that hi t h' i t 2 s. /M s. / and hi s .hi t h' i t /h i s D hi t h' i t for all s; t 2 R. This equality is equivalent to s. /h' it h is D h i s h' i t , so (iii) implies that s. /hi't hi s D hi s hi't for all s, t . Letting s D 0 gives s. /hi't s. / D s. /hi't , so that s. /hi't D hi't s. / for all t . Hence (iii) H) (v) follows, and the converse is similar. (ii) H) (iv). Let u t WD .D W D'/ t . By Theorem 10.16(2), condition (ii) implies that usCt D us s' .u t / D us u t for all s, t . Letting t D 0 gives us D us u0 D us s. / by Theorem 10.16(1). Hence us us D us us s. / so that us us  s. /. Since (ii) implies that '.us us / D '.us us / D '.s. //, we have '.s. / us us / D 0 and hence us us D s. / D us us . Therefore, t 2 R 7! u t is a one-parameter unitary group in s. /M s. /. (iv) H) (ii). Assume (iv). For every s; t 2 R, by (1) and (2) of Theorem 10.16 we have ut u t D u t ut D u0 D s. / and s' .u t / D s' .s. /u t / D us us s' .u t / D us usCt D u t ; which implies (ii).



Definition 10.19. We say that commutes with ' if the equivalent conditions of Theorem 10.18 hold. Condition (i) is often used to define the commutativity for normal positive functionals (also semifinite normal weights), but (v) and (vi) are quite natural definitions of commutativity that might be available for any '; 2 MC without s. /  s.'/.

11 Spatial derivatives and spatial Lp -spaces

In Sections 11.1 and 11.2 we give a concise account of the notion of spatial derivatives due to Connes [17], following not only the original approach in [17] but also a slightly different (but equivalent) approach described in [97]. The notion is a kind of Radon– Nikodym derivative like the relative modular operator in Section 10.1. But the spatial derivative is defined for a functional in MC (or a semifinite normal weight on M ) with respect to a faithful semifinite normal weight on the commutant M 0 , unlike the relative modular operator defined for two functionals in MC . A merit of the spatial derivative is that it is defined in any representing Hilbert space for M , the reason for the term “spatial”, while the relative modular operator is given in a standard form of M . Section 11.3 is a description of the spatial Lp -spaces due to Connes [17] and Hilsum [46].

11.1 Spatial derivatives Let M be a von Neumann algebra on a Hilbert space H . Let ‰ be a faithful semifinite normal weight on the commutant N WD M 0 (on H ). Let .H‰ ; ‰ ; ‰ / be the GNS representation of N with respect to ‰, that is, H‰ is the Hilbert space defined by completing N‰ WD ¹y 2 N W ‰.y  y/ < 1º with respect to the inner product hx; yi‰ WD ‰.x  y/ for x; y 2 N‰ , ‰ W N‰ ! H‰ is the canonical injection and ‰ W N ! B.H‰ / is the representation given by ‰ .x/‰ .y/ D ‰ .xy/ for x 2 N , y 2 N‰ . Definition 11.1. A vector  2 H is said to be ‰-bounded if there is a constant C < 1 such that kyk  C k‰ .y/k for all y 2 N‰ . Define D.H ; ‰/ WD ¹ 2 H W ‰-boundedº: For  2 D.H ; ‰/ a bounded operator R‰ ./W H‰ ! H is defined as R‰ ./‰ .y/ WD y;

y 2 N‰ :

(11.1)

Define  ‰ .; / WD R‰ ./R‰ ./ , a bounded operator on H . Lemma 11.2. With the above notation the following hold: (1) for every  2 D.H ; ‰/ and x 2 M , x 2 D.H ; ‰/ and R‰ .x/ D xR‰ ./, (2) D.H ; ‰/ is an M -invariant dense subspace of H ,

11 Spatial derivatives and spatial Lp -spaces

182

(3) for every  2 D.H ; ‰/, R‰ ./ is N -linear, i.e., R‰ ./‰ .x/ D xR‰ ./ for all x 2 N, (4) for every  2 D.H ; ‰/,  ‰ .; / belongs to MC D NC0 , (5) if  2 D.H ; ‰/ and R‰ ./ D 0, then  D 0. Proof. (1) is easy. (2) It is clear by (1) that D.H ; ‰/ is an M -invariant subspace of H . To prove the denseness of D.H ; ‰/, let e be the projection onto D.H ; ‰/. Since exe D xe for all x 2 M , one hasPe 2 M 0 D N . Suppose that e ¤ 1; then ‰.1 e/ > 0. Since ‰ is written as ‰ D i !i for some .i /  H by Theorem 6.2 (here ! WD h;
i, a vector functional), there is a  2 H such that !  ‰ and .1 e/ ¤ 0. Since kyk2 D ! .y  y/  ‰.y  y/ D k‰ .y/k2 ;

y 2 N‰ ;

it follows that  2 D.H ; ‰/, so .1 e/ D 0, a contradiction. Hence e D 1 follows. (3) is immediate since for every  2 D.H ; ‰/ one has, for every x 2 N and y 2 N‰ , R‰ ./‰ .x/‰ .y/ D R‰ ./‰ .xy/ D xy D xR‰ ./‰ .y/: (4) Since (3) gives ‰ .x/R‰ ./ D R‰ ./ x for all x 2 N , one has xR‰ ./R‰ ./ D R‰ ./‰ .x/R‰ ./ D R‰ ./R‰ ./ x;

x 2 N:

(5) If  2 D.H ; ‰/ and R‰ ./ D 0, then y D 0 for all y 2 N‰ . This gives  D 0 since ‰ is semifinite; see Definition 6.1.  The next lemma is essential to define the spatial derivative. Lemma 11.3. Let H and ‰ be as stated above. Let ' be a (not necessarily faithful) semifinite normal weight on M . Define the function q' W D.H ; ‰/ ! Œ0; 1 by q' ./ WD '. ‰ .; //;

 2 D.H ; ‰/:

Then we have (1) D.q' / WD ¹ 2 D.H ; ‰/ W q' ./ < 1º is dense in H , (2) q' W D.q' / ! Œ0; 1/ is a positive quadratic form in the sense of Definition A.9(1), (3) q' is lower semicontinuous on D.H ; ‰/. Proof. (2) is easy, so we prove (1) and (3) only. (1) Since D.H ; ‰/ is dense in H and there is a net ¹u º  N' such that 0  u % 1, it suffices to prove that if y 2 N' and  2 D.H ; ‰/, then q' .y/ < 1. For such y, , since R‰ .y/R‰ .y/ D yR‰ ./R‰ ./ y   kR‰ ./R‰ ./ kyy  ; we have q' .y/  kR‰ ./R‰ ./ k'.yy  / < 1, as desired.

183

11.1 Spatial derivatives

(3) Theorem 6.2 says that '.x/ D sup¹!.x/ W ! 2 MC ; !  'º forPx 2 MC . P C Moreover, each ! 2 M is written as ! D i !i for some ¹i º  H with i ki k2 < 1. So it suffices to show the result in the case where '.x/ D h0 ; x0 i, x 2 MC , for some 0 2 H . Then for every  2 D.H ; ‰/, we have q' ./1=2 D kR‰ ./ 0 k D sup¹jhR‰ ./ 0 ; ‰ .y/ij W y 2 N; ‰.y  y/  1º; hR‰ ./ 0 ; ‰ .y/i D h0 ; R‰ ./‰ .y/i D h0 ; yi;

y 2 N‰ :

Since the last term is continuous in , the result has been shown.



Definition 11.4. Let H , ‰ be as above. For every semifinite normal weight ' on M let q' W D.q' / ! Œ0; 1/ be a positive quadratic form given in Lemma 11.3. By Lemma 11.3 and Theorem A.12 of Appendix A.2 we have the closure q' of q' , which is a closed positive quadratic form that is the smallest closed extension of q' . Then by Theorem A.11 there exists a unique positive self-adjoint operator A on H such that D.A1=2 / D D.q' / and q' ./ D kA1=2 k2 ;  2 D.q' /: This A is denoted by d'=d‰ and called the spatial derivative of ' with respect to ‰. Note (see Remark A.13) that d'=d‰ is the largest positive self-adjoint operator A on H satisfying D.A1=2 /  D.q' /

and q' ./ D kA1=2 k2 ;

 2 D.q' /:

(11.2)

Also, note that D.q' / is a core of .d'=d‰/1=2 . In particular, when ' 2 MC , D.H ; ‰/ D D.q' /  D..d'=d‰/1=2 /. Example 11.5. Assume that M D B.H /, a type I factor, so that N D M 0 D C1. Consider the trivial state ‰.1/ D  on C1. Write ‰ D 1. Any ' 2 MC is given as '.x/ D Tr.D' x/ with the density (positive trace-class) operator D' on H ; D' is also denoted by d'=dTr, the Radon–Nikodym derivative of ' with respect to Tr. It is clear that H‰ D C and D.H ; ‰/ D H . For any  2 H , R‰ ./1 D  and hence  ‰ .; / D jihj (i.e., the rank-one projection onto C). Therefore, we find that '. ‰ .; // D Tr.D' jihj/ D kD'1=2 k2 ;

2H;

which means that d'=d1 (D d'=d‰) is D' (D d'=dTr). Furthermore, for every semifinite normal weight ' on B.H /, by Theorem 7.25 (applied to eB.H /e D B.eH / with e D s.'/), there exists a unique positive self-adjoint operator A such that ' D TrA . Then we have d'=d1 D A (D d'=dTr). In the special case where M is represented in its standard form, for '; 2 MC with faithful, the next proposition says that the spatial derivative d'=d 0 (where 0 2 .M 0 /C  is defined by the same representing vector as ) is nothing but the relative modular operator '; . From the proposition we may consider that spatial derivatives properly extend relative modular operators to the setting of an arbitrary representation M  B.H /.

11 Spatial derivatives and spatial Lp -spaces

184

Proposition 11.6. Let .M; H ; J; P/ be a standard form of a von Neumann algebra M 0 0 and '; 2 MC with faithful. Define a faithful 0 2 .M 0 /C .x / WD .J x 0 J /  by 0 0 0 0 0 for x 2 M , i.e., .x / D h0 ; x 0 i with the vector representative 0 2 P of . Then d' D '; : d 0 Proof. By Theorems 3.13 and 9.29 we may assume that
M; H ; J; P/ D .M; L2 .M /; J D  ; L2 .M /C ; where M is represented by the left multiplication on Haagerup’s L2 -space L2 .M /. We use the linear bijection ! 2 M 7! h! 2 L1 .M / and the functional tr on L1 .M /; see Definition 9.6. Note that M 0 D JMJ is identified with the opposite von Neumann algebra M o D ¹x o W x 2 M º with the reverse product x o y o D .yx/o , which is represented by the right multiplication r .x o / WD x,  2 L2 .M /. In fact, we can write r .x o / D J x  J for x 2 M , which is an isomorphism from M o onto M 0 . By this isomorphism, 0 on M 0 corresponds to 0 .x o / D .x/ on M o (here we use the same notation 0 on M o as 0 on M 0 ). Define  0 W M o ! L2 .M / by  0 .x o / WD h1=2 x, x 2 M . Then for every x; y 2 M we write h 0 .x o /;  0 .y o /i D hh1=2 x; h1=2 yi D tr.x  h y/ D tr.h yx  / D

.yx  / D

0

..yx  /o / D

0

..x o / y o /;

 0 .x o y o / D  0 ..yx/o / D h1=2 yx D  0 .y o /x D r .x o / 0 .y o /: 1=2 M D L2 .M / and Since 2 MC and so 0 2 .M o /C  are faithful, note that h o 2 N 0 D M . Hence .L .M /; r ;  0 / is identified with the GNS representation of M o associated with 0 . 0 For every  2 D.L2 .M /; 0 / we have a bounded operator R ./W L2 .M / ! L2 .M / such that 0 (11.3) R ./ 0 .y o / D r .y o / D y; y 2 M:

Furthermore, for every x; y 2 M , 0

0

R ./r .x o / 0 .y o / D R ./ 0 ..yx/o / D yx 0

0

D .R ./ 0 .y o //x D r .x o /R ./ 0 .y o /; 0

0

so that R ./ 2 r .M o /0 D .JMJ /0 D M 00 D M . Hence we put a WD R ./ 2 M . Since (11.3) is rewritten as y D ah1=2 y for all y 2 M , we have  D ah1=2 . On the other hand, if  D ah1=2 with a 2 M , then we have kr .y o /k2 D kyk2 D kah1=2 yk2  kak k 0 .y o /k2 ;

y 2 M;

185

11.1 Spatial derivatives

implying that  2 D.L2 .M /;

0

/. Therefore, we obtain

D.L2 .M /;

0

/ D M h1=2 :

Now, for every  D ah1=2 2 D.L2 .M /; 0 argument that R ./ D a and so 0

0

(11.4)

/ with a 2 M , it follows from the above

0

2 '.R ./R ./ / D '.aa / D tr.h' aa / D ka h1=2 ' k2 :

(11.5)

From the definition of the relative modular operator '; it also follows that (11.4) is a core of 1=2 '; and 1=2 1=2 2 2 2 k1=2 k2 D ka h1=2 ' k2 : '; k2 D kJ'; ah

(11.6)

By (11.5) and (11.6) we have 0

0

2 '.R ./R ./ / D k1=2 '; k2 ;

which implies due to Definition 11.4 that d'=d

0

 2 D.L2 .M /; D '; .

0

/; 

The following three theorems are the main results shown in [17]. The first two give significant properties of the spatial derivative d'=d‰ in connection with the modular automorphism group and Connes’ cocycle derivative. The third is the characterization theorem for d'=d‰. Their proofs are quite involved, so we defer them to the next section. Theorem 11.7 (Connes). Let ‰ be a faithful semifinite normal weight on N D M 0 . Let ', '1 and '2 be semifinite normal weights on M , and assume that ' and '2 are faithful. Then    it  d' d' i t ' (1)  t .x/ D x for all x 2 M and t 2 R, d‰ d‰   it   d' d' i t ‰ (2)  t .y/ D y for all y 2 N D M 0 and t 2 R, d‰ d‰ (3) the support projection of d'1 =d‰ coincides with s.'1 /,     d'1 i t d'2 i t (4) .D'1 W D'2 / t D for all t 2 R, where .D'1 W D'2 / t is d‰ d‰ it Connes’ cocycle derivative and .d'1 =d‰/ is defined with restriction to the support. The expression in (4) above has a resemblance to (10.24). Theorem 11.8 (Connes). Let ' and ‰ be faithful semifinite normal weights on M and N D M 0 , respectively. Then  d'  1 d‰ D : (11.7) d‰ d'

11 Spatial derivatives and spatial Lp -spaces

186

Expression (11.7) is the counterpart of Proposition 7.24(1) for Connes’ cocycle derivatives and Proposition 10.3(4) for the relative modular operators. Theorem 11.9 (Connes). Let M , N D M 0 and ‰ be as above. Let T be a positive self-adjoint operator on H . Then the following conditions are equivalent: (i) there exists a semifinite normal weight ' on M such that T D d'=d‰; (ii) T it  t‰ .y/ D yT i t for all t 2 R and y 2 N , where T i t is defined with restriction to the support of T . Furthermore, in this case, ' is uniquely determined by T . In the rest of the section, following the approach in [97, Chap. III], we extend the notion of spatial derivatives d'=d‰ to general (not necessarily semifinite) normal weights ' on M . This approach is apparently more tractable than Connes’ original approach in [17]. The idea in [97] is to define d'=d‰ as a generalized positive operator on H , i.e., an element of B .H /C ; see Section 7.1. We first extend the notion  ‰ .; / D R‰ ./R‰ ./ to general  2 H as an element c of M C . To do this, for each  2 H we consider R‰ ./ to be a densely defined operator defined by (11.1) with D.R‰ .// D N‰  H‰ . Then we have the adjoint operator R‰ ./ with domain D.R‰ ./ / a (not necessarily dense) subspace of H .

2

cC such that Lemma 11.10. For each  2 H there exists a unique element  ‰ .; / 2 M for every  2 H , ´ kR‰ ./ k2 if  2 D.R‰ ./ /; ‰  .; /.! / D (11.8) 1 otherwise; where ! .x/ WD h; xi, x 2 M . Proof. Let  2 H and p be the projection from H onto D.R‰ ./ /. In the present situation, the proof of Lemma 11.2(3) shows that xR‰ ./  R‰ ./‰ .x/;

x 2 M 0 .D N /;

(11.9)

which implies that ‰ .x/R‰ ./  R‰ ./ x;

x 2 M 0:

(11.10)

Hence one has xD.R‰ ./ /  D.R‰ ./ / so that xp D pxp for all x 2 M 0 . Therefore, p 2 M follows. Consider R‰ ./ as a densely defined closed operator from pH to H‰ . Then jR‰ ./ j2 can be defined as a positive self-adjoint operator on pH .1 From (11.9) 1This is seen from the following well-known theorem: Let T W K ! H be a densely defined closed operator between Hilbert spaces K , H , and define an operator T  T on K with D.T  T / WD ¹ 2 K W T  2 D.T  /º. Then D.T  T / is dense in K (more strongly, D.T  T / is a core of T ) and T  T is a positive self-adjoint operator on K . Hence the absolute value jT j D .T  T /1=2 of T can be defined. (Unlike the bounded operator case, the proof is rather complicated; see, e.g., [54, Chap. 5, §3.7], [70, Chap. 5].)

187

11.1 Spatial derivatives

and (11.10) one can further see that u jR‰ ./ j2 u D jR‰ ./ j2 for all unitaries u 2 M 0 (an exercise). Hence it follows that jR‰R./ j2 is affiliated with pMp. So one can take the 1 spectral decomposition jR‰ ./ j2 D 0  de with e 2 M and p D lim!1 e , for which Z 1 kR‰ ./ k2 D k jR‰ ./ jk2 D  dke k2 ;  2 D.R‰ ./ /: 0

Then, in view of the proof of Theorem 7.3 (also see Example 7.2(1)), an element  ‰ .; / 2 cC is defined as M Z 1 ‰  .; /.'/ D  d'.e / C 1'.1 p/; ' 2 MC ; 0

or equivalently, for every  2 H ,  ‰ .; /.! / D

Z

1

 dke k2 C 1k.1

´0 kR‰ ./ k2 D 1

p/k2

if  2 D.R‰ ./ /; otherwise;

cC satisfying (11.8) is immediate. showing (11.8). The uniqueness of  ‰ .; / 2 M



Note that if  2 D.H ; ‰/, then  ‰ .; / D R‰ ./R‰ ./ 2 MC is the same as that in Definition 11.1. Lemma 11.11. For every  2 H and x 2 M ,  ‰ .x; x/ D x ‰ .; /x  ; where .x ‰ .; /x  /.'/ WD  ‰ .; /.x  'x/ for ' 2 MC . Proof. Let  2 H and x 2 M . As in Lemma 11.2(1) it follows that R‰ .x/ D xR‰ ./ and so R‰ .x/ D R‰ ./ x  . Hence for every  2 H , by expression (11.8) we have  ‰ .x; x/.! / D  ‰ .; /.!x   / D  ‰ .; /.x  ! x/ D .x ‰ .; /x  /.! /; implying the assertion.



Lemma 11.12. Let ' be a (not necessarily semifinite) normal weight on M . Define qO ' W H ! Œ0; 1 by qO ' ./ WD '. ‰ .; //;  2 H : (See Proposition 7.5 for this definition.) Then qO ' is a lower semicontinuous positive form in the sense of Definition A.14 of Appendix A.2. Proof. We need to show the following:

188

11 Spatial derivatives and spatial Lp -spaces

(1) qO ' ./ D jj2 qO ' ./ for all  2 H and  2 C, (2) qO ' .1 C 2 / C qO ' .1

2 / D 2qO ' .1 / C 2qO ' .2 / for all 1 ; 2 2 H ,

(3) qO ' is lower semicontinuous on H . (1) is easy. We first prove (2) and (3) in the case ' D ! with  2 H . For (2) let us prove that qO ' .1 C 2 / C qO ' .1 2 /  2qO ' .1 / C 2qO ' .2 /: (11.11) If  62 D.R‰ .1 / / or  62 D.R‰ .2 / /, then RHS of (11.11) D 1, so (11.11) holds trivially. So assume that  2 D.R‰ .1 / / and  2 D.R‰ .2 / /. Since R‰ .1 ˙ 2 / D R‰ .1 /˙R‰ .2 /, one has R‰ .1 / ˙R‰ .2 /  R‰ .1 ˙2 / , so that  2 D.R‰ .1 ˙ 2 / /. Therefore, one has kR‰ .1 C 2 / k2 C kR‰ .1

2 / k2

D kR‰ .1 /  C R‰ .2 / k2 C kR‰ .1 / 

R‰ .2 / k2

D 2kR‰ .1 / k2 C 2kR‰ .1 / k2 ; implying (11.11). Replacing 1 , 2 with 1 C 2 , 1

2 in (11.11) we also have

qO ' .21 / C qO ' .22 /  2qO ' .1 C 2 / C 2qO ' .1

2 /;

which is the reverse inequality of (11.11). Thus (2) has been shown for ' D ! . Next, from (11.8) it follows that for every  2 H , qO ' ./ D  ‰ .; /.! / D sup¹jh; R‰ ./‰ .y/ij2 W y 2 N‰ ; k‰ .y/k  1º D sup¹jh; yij2 W y 2 N‰ ; k‰ .y/k  1º: Since  2 H 7! jh; yij2 is continuous, (3) follows in the case ' D ! . Now P let ' be an arbitrary normal weight on M . By Theorem 6.2 we can write ' D i !i for some .i /  H . Since qO ' ./ D '. ‰ .; // D

X

 ‰ .; /.!i /;

2H;

i

properties (2) and (3) follow from those in the case ' D ! proved above.



Definition 11.13. Let H , ‰ be as above. For every normal weight ' on M define qO ' W H ! Œ0; 1 as in Lemma 11.12. Then by Lemma 11.12 and Theorem A.16 there exists a unique positive self-adjoint operator A on K WD D.qO ' /, where D.qO ' / WD ¹ 2 H W qO ' ./ < 1º, such that ´ kA1=2 k2 if  2 D.A1=2 /; qO ' ./ D 1 otherwise:

189

11.1 Spatial derivatives

We denote A by d'=d‰ and call Rit the spatial derivative of ' with respect to ‰. Taking 1 the spectral decomposition A D 0  dE with P D lim!1 E , the projection onto K , and setting 1 on K ? , we can consider d'=d‰ as an element of B .H /C as Z 1 d' .! / D  dkE k2 C 1! .1 P /;  2 H I d‰ 0

2

2

see Theorem 7.3. Thus, d'=d‰ is uniquely determined as an element of B .H /C such that d' .! / D '. ‰ .; //;  2 H : d‰ Remark 11.14. Assume that ' is a semifinite normal weight on M . Then for every x 2 N‰ and  2 D.H ; ‰/ we have qO ' .x  / D '. ‰ .x  ; x  // D '.x   ‰ .; /x/ (by Lemma 11.11)  k ‰ .; /k'.x  x/ < 1: Since ¹x   W x 2 N‰ ;  2 D.H ; ‰/º is dense in H , it follows that D.qO ' / is dense in H and so d'=d‰ is a positive self-adjoint operator on H . Indeed, the next theorem holds true. Theorem 11.15. Assume that ' is a semifinite normal weight on M . Then the d'=d‰ in Definitions 11.4 and 11.13 coincide. To prove the theorem, we need a lemma. Lemma 11.16. For every  2 H there exists a sequence n 2 D.H ; ‰/ such that n !  and qO ' .n / ! qO ' ./ as n ! 1 for all normal weights ' on M . Moreover, q' .n / D qO ' .n / for all n if ' is semifinite. Proof. As in the proof of Lemma 11.10 write the spectral decomposition of  ‰ .; / as R1  ‰ .; / D 0  de C 1.1 p/, where p is the projection onto D.R‰ ./ /. For each n 2 N, since R‰ .en / D R‰ ./ en (see the proof of Lemma 11.11), note that R‰ .en / is bounded, i.e., en  2 D.H ; ‰/. By Lemma 11.2(2) choose a sequence n 2 D.H ; ‰/ such that n ! . Then one has .1 p/n 2 D.H ; ‰/ by Lemma 11.2(1). For each n 2 N define n WD en  C .1 p/n 2 D.H ; ‰/I then n ! p C .1 p/ D  as n ! 1. The latter assertion is clear from the definitions in Lemmas 11.3 and 11.12. Let ' be a normal weight on M , and prove that qO ' .n / ! qO ' ./. If qO ' ./ D 1, then this is clear by the lower semicontinuity P of qO ' (Lemma 11.12). Now assume that q OP ./ < 1. By Theorem 6.2 write ' D O ' ./ D ' i !i for some .i /  H . Since q ‰ ‰  .; /.! / < 1, one has  .; /.! / < 1 and so  2 pH for all i . This i i i i P P implies that p'p D i p!i p D i !i D '. Furthermore, by Lemma 11.11, p ‰ .n ; n /p D  ‰ .pn ; pn / D  ‰ .en ; en / D en  ‰ .; /en % p ‰ .; /p:

11 Spatial derivatives and spatial Lp -spaces

190 Therefore, one has

qO ' .n / D '. ‰ .n ; n // D '.p ‰ .n ; n /p/ % '.p ‰ .; /p/ D '. ‰ .; // D qO ' ./:



Proof of Theorem 11.15. Let d =d‰ denote the spatial derivative in Definition 11.13, which is a positive self-adjoint operator as noted in Remark 11.14. In view of the description in Definition 11.4, it suffices to show that d'=d‰ is the largest positive self-adjoint operator on H satisfying (11.2). It is clear that d'=d‰ satisfies (11.2). Now let A be any positive self-adjoint operator on H satisfying (11.2). Let us prove that A  d'=d‰ (in the sense of Definition A.2). By Lemma 11.16, for every  2 H there exists a sequence n 2 D.H ; ‰/ such that n !  and q' .n / ! qO ' ./. Define a positive form qA on H by ´ kA1=2 k2 if  2 D.A1=2 /; qA ./ WD 1 otherwise; which is lower semicontinuous on H (see Theorem A.16). If  2 D..d'=d‰/1=2 /, i.e., qO ' ./ < 1, then for all n large enough, we have n 2 D.q' / so that qA .n / D kA1=2 n k2 D q' .n / by (11.2). Hence we find that

 d' 1=2 2

 ; qA ./  lim inf qA .n / D lim q' .n / D qO ' ./ D n!1 n!1 d‰ which implies that A  d'=d‰.



The following are basic properties of d'=d‰.

2

Proposition 11.17. Let ', '1 and '2 be normal weights on M and a 2 M . Then the following hold as elements of B .H /C : d'1 d'2 d.'1 C '2 / D C . d‰ d‰ d‰  d'  d.a'a / Da (2) a . d‰ d‰

(1)

(3) If '1  '2 , then d'1 =d‰  d'2 =d‰. (4) Let .'i / be an increasing net of normal weights on M . If 'i % ', then d'i =d‰ % d'=d‰. Proof. The assertions in (1)–(3) result from the following computations for every  2 H . d.'1 C '2 / (1) .! / D .'1 C '2 /. ‰ .; // D '1 . ‰ .; // C '2 . ‰ .; // d‰  d' d'1 d'2 d'2  1 D .! / C .! / D C .! /: d‰ d‰ d‰ d‰

191

11.1 Spatial derivatives

d.a'a / .! / D '.a  ‰ .; /a/ D '. ‰ .a ; a // (by Lemma 11.11) d‰   d'   d' d'  D .!a  / D .a ! a/ D a a .! /: d‰ d‰ d‰ d'1 d'2 (3) .! / D '1 . ‰ .; //  '2 . ‰ .; // D .! /: d‰ d‰ (4) That d'i =d‰ % follows from (3). For every  2 H we have (2)

d'i d' .! / D '. ‰ .; // D sup 'i . ‰ .; // D sup .! /; d‰ i i d‰ where the second R 1 equality above is seen as follows: With the spectral decomposition  ‰ .; / D 0  de C 1p in Theorem 7.3 we have Z n  '. ‰ .; // D sup '  d C 1'.p/ n 0 Z n   D sup sup 'i  de C 1'i .p/ n i 0    Z n  de C 1'i .p/ D sup 'i . ‰ .; //:  D sup sup 'i i

n

0

i

2

Remark 11.18. As mentioned in Remark 7.4, elements of B .H /C are described in terms of (not necessarily densely defined) positive self-adjoint operators as well as lower semicontinuous positive forms on H . Accordingly, the sum d'1 =d‰ C d'2 =d‰ in (1) above is interpreted as the form sum in Example A.17, and d'1 =d‰  d'2 =d‰ in (3) is defined in terms of either operators or positive forms equivalently; see Proposition A.18. As for d'i =d‰ % d'=d‰ in (4), see Proposition A.19. Proposition 11.19. Let ' be a semifinite normal weight on M . Then the support projection of d'=d‰ coincides with s.'/. Proof. Let e D s.'/ 2 M . For each  2 H note that  is in the kernel R 1 of d'=d‰ if and only if .d =d‰/.! / D 0, i.e., '. ‰ .; // D 0. Write  ‰ .; / D 0  de C1.1 p/ as in the proof of Lemma 11.10; then we find that Z n  '. ‰ .; // D sup '  de C 1'.1 p/ D 0 n 0 Z n  ” '  de D 0 .n 2 N/ and '.1 p/ D 0 Z 0n  ” e  de e D 0 .n 2 N/ and e.1 p/e D 0 0 ‰

” e .; /e D 0 ”  ‰ .e; e/ D 0

(by Lemma 11.11)

‰

” R .e/ D 0 ” e D 0: The last equivalence above is seen similarly to Lemma 11.2(5). Hence the assertion follows. 

192

11 Spatial derivatives and spatial Lp -spaces

11.2 Proofs of theorems To prove Theorem 11.7, we consider the direct sum representation W N ! B.H ˚ H‰ /;

.y/ WD y ˚ ‰ .y/;

y 2 N;

and the commutant algebra .N /0  B.H ˚H‰ /. Let e be the projection from H ˚H‰ onto H . We write HomN .H‰ ; H / WD ¹T 2 B.H‰ ; H / W T ‰ .y/ D yT; y 2 N º: We have the identifications M D N 0 D e.N /0 e; ‰ .N /0 D .1

e/.N /0 .1

HomN .H‰ ; H / D e.N / .1 0

e/;

e/:

Define a faithful semifinite normal weight  on .N /0 (like the balanced weight; see Section 6.2) by .T / WD '.eT e/ C ‰ 0 ..1

e/T .1

e//;

T 2 .N /0C ;

(11.12)

where ‰ 0 .y 0 / WD ‰.J‰ y 0 J‰ / for y 0 2 ‰ .N /0C ; see (B) in Section 6.1. Let  t (t 2 R) be the modular automorphism group of .N /0 associated with  ; see (C) in Section 6.1. Then  t jHomN .H‰ ;H / defines a one-parameter group of isometries of HomN .H‰ ; H /. By Lemma 7.28 note that  t .e/ D e and  t .1 e/ D 1 e for all t 2 R. Hence for every T 2 HomN .H‰ ; H / one has  t .T /e D  t .T e/ D 0 and .1 e/ t .T / D  t ..1 e/T / D 0, so that  t .T / 2 e.N /0 .1 e/ D HomN .H‰ ; H /. In particular, when  2 D.H ; ‰/, note by Lemma 11.2(3) that R‰ ./ 2 HomN .H‰ ; H / so that  t .R‰ .// 2 HomN .H‰ ; H / for all t 2 R. The next lemma is a significant part of the proof of Theorem 11.7. Lemma 11.20. There exists a unique one-parameter unitary group v t' 2 B.H /, t 2 R, such that (a)  t' .x/ D v t' xv ' t for all x 2 M , t 2 R, (b)  t‰ .y/ D v ' t yv t' for all y 2 N , t 2 R, (c) for every  2 D.H ; ‰/ and t 2 R, v t'  2 D.H ; ‰/ and R‰ .v t' / D  t .R‰ .//. Proof. First, the uniqueness of v t' is clear from condition (c) by Lemma 11.2(2) and (5). The rest of the proof is divided into the three steps below. Step 1. Assume that for a faithful semifinite normal weight '0 on M there is a oneparameter unitary group v t'0 satisfying conditions (a)–(c) (for '0 in place of '). For

193

11.2 Proofs of theorems

every faithful semifinite normal weight ' on M define v t' WD u t v t'0 for t 2 R, where u t WD .D' W D'0 / t . Since '0 '0 '0 D us s'0 .u t /vsCt D us vs'0 u t v '0s vsCt usCt vsCt D us vs'0 u t v t'0 ;

it follows that v t' is a one-parameter unitary group and so v ' t D .v t' / D v '0t ut . Let us prove that v t' satisfies (a)–(c). (a) For every x 2 M , v t' xv ' t D u t v t'0 xv '0t ut D u t  t'0 .x/ut D  t' .x/

(by (6.8)):

(b) For every y 2 N , v t' yv ' t D u t v t'0 yv '0t ut D u t  ‰t .y/ut D u t ut  ‰t .y/ D  ‰t .y/: (c) Let 0 be defined for '0 similarly to  for ' in (11.12). For every  2 D.H ; ‰/, note by Lemma 11.2(1) that v t'  D u t v t'0  2 D.H ; ‰/ and R‰ .v t' / D u t R‰ .v t'0 / D u t  t0 .R‰ .//: Hence we may show that u t  t0 .R‰ .// D  t .R‰ .//. To do this, consider the balanced weights ‚ on M2 ..N /0 / and ˆ on M2 .M / given by     X11 X12 X11 X12 ‚ D 0 .X11 / C .X22 /; 2 M2 ..N /0 /C ; X21 X22 X21 X22     x11 x12 x11 x12 ˆ D '0 .x11 / C '.x22 /; 2 M2 .M /C : x21 x22 x21 x22 For every X D

 X11

X12 X21 X22



2 M2 ..N /0 /C note that

‚.X / D 0 .X11 / C .X22 / D '0 .eX11 e/ C ‰ 0 ..1 e/X11 .1 e// C '.eX22 e/ C ‰ 0 ..1 y 0 ..1 E/X.1 E//; D ˆ.EXE/ C ‰

e/X22 .1

e//

  y 0 is the balanced weight of ‰ 0 on ‰ .N /0 and itself. Since where E WD 0e 0e and ‰ 0 EM2 ..N / /E D M2 .M /, Lemma 7.28 implies that  tˆ D  t‚ jM2 .M / and E is in the centralizer M2 ..N /0 /‚ of ‚. Therefore,       0 0 0 0 0 0 D  t‰ D  t‚ ut 0 1M 0 e 0     0 0 0 0 D E t‚ ED : 1 0 e.D W D0 / t e 0

11 Spatial derivatives and spatial Lp -spaces

194

On the other hand, since .1 E/M2 ..N /0 /.1 E/ D M2 .‰ .N /0 /, we similarly have         0 0 0 0 0 0 0 0 b ‰0 ‚ D D t D t : 1 e 0 1‰ .N /0 0 1‰ .N /0 0 1 e 0 Since



0 .D W D0 / t

    0 0 0 0 D  t‚ C  t‚ 0 e 0 1 e

it follows that .D W D0 / t D u t C .1 e.N /0 .1 e/, we have

 0 ; 0

e/. Since R‰ ./ 2 HomN .H‰ ; H / D

u t  t0 .R‰ .// D .D W D0 / t  t0 .R‰ .// D  t .R‰ .//.D W D0 / t D  t .R‰ .//: Step 2. Assume that the assertion of the lemma holds under some realization of N and M D N 0 on H . Then it holds true for any isomorphic representation of N . To see this, recall (see [95, Thm. IV.5.5]) that any von Neumann algebra isomorphism is the composition of an amplification (from N  B.H /), an injective induction and a spatial isomorphism. So we may consider the following cases separately:  amplification: N ˝ C1  B.H / ˝ B.K / D B.H ˝ K / and M ˝ B.K / D .N ˝ C1/0  B.H / ˝ B.K /, where K is any Hilbert space,  injective induction: N q  B.qH / and qM q D .N q/0  B.qH /, where q is a projection in M with the central support cM .q/ D 1,  spatial isomorphism: UN U  and UM U  D .UN U  /0 in B.K / for a unitary UWH ! K . Note that this step was skipped in [17], although the details are given in [89, §7.6]. We omit the details here. Step 3. From Steps 1 and 2 it suffices to prove the lemma in the case where H D H‰ and ' D ‰ 0 (WD ‰.J‰
J‰ /). Let ‰ be the modular operator for ‰; then ‰1 D J‰ ‰ J‰ is the modular operator for ‰ 0 ; see (B) in Section 6.1. Set v t' WD ‰i t . Then we have (a) and (b); see (C) in Section 6.1. Since  D .‰ 0 ; ‰ 0 / in the present case,  
  ‰0 .x /  ‰0 .x /  x12 note that  t xx11 D t‰0 11 t‰0 12 for xij 2 M . Let  2 D.H‰ ; ‰/; then 21 x22  t .x21 /  t .x22 /

R‰ ./ 2 N 0 D M . For every y 2 N‰ we have R‰ .‰i t /‰ .y/ D y‰i t  D ‰i t  t‰ .y/ D ‰i t R‰ ./‰ . t‰ .y// 0

D ‰i t R‰ ./i‰t ‰ .y/ D  t‰ .R‰ .//‰ .y/ D  t .R‰ .//‰ .y/; so that (c) follows.



195

11.2 Proofs of theorems

Lemma 11.21. Let ‰ 0 on ‰ .N /0 be as in (11.12), i.e., ‰ 0 .x 0 / WD ‰.J‰ x 0 J‰ /, x 0 2 ‰ .N /0C . Then for every  2 D.H ; ‰/, ‰ 0 .R‰ ./ R‰ .// D kk2 : Hence by polarization, ‰ 0 .R‰ ./ R‰ .// D h; i for all ;  2 D.H ; ‰/. Proof. By Lemma 11.2(3) note that R‰ ./ R‰ ./ 2 ‰ .N /0C for any  2 D.H ; ‰/. First consider the particular case where H D H‰ and N D ‰ .N /  B.H‰ /. In this case, for  2 H‰ write R in place of R‰ ./, i.e., R ‰ .y/ D ‰ .y/ for y 2 N‰ , so D.H‰ ; ‰/ WD ¹ 2 H‰ W R is boundedº. Then, related to the construction of the right Hilbert algebra (see [96, 90]), it is known that ‰ 0 .R R / D kk2 for all  2 D.H‰ ; ‰/; the details are omitted here.2 Now let H be general and  2 D.H ; ‰/. Take the polar decomposition R‰ ./ D V jR‰ ./ j with a partial isometry V W H ! H‰ . From Lemma 11.2(3) it easily follows that jR‰ ./ j 2 N 0 and ‰ .y/V D Vy for all y 2 N . Since ‰ .y/V  D Vy D VR‰ ./‰ .y/ for all y 2 N‰ , we find that V  2 D.H‰ ; ‰/ and RV  D VR‰ ./, so that RV  RV  D R‰ ./ V  VR‰ ./ D R‰ ./ R‰ ./. Therefore, ‰ 0 .R‰ ./ R‰ .// D ‰ 0 .RV  RV  / D kV k2 by the first case. Moreover, choose yi 2 N‰ with yi ! 1 strongly to have R‰ ./‰ .yi / D ‰ .yi / ! , so that  is in the closure of the range of R‰ ./. Hence V  V  D  and kV k2 D kk2 .  Lemma 11.22. For v t' given in Lemma 11.20 we have v t' D .d'=d‰/i t , t 2 R. Proof. Let  on .N /0 be given in (11.12), and  be the modular operator for . On the other hand, by Stone’s theorem there is a non-singular positive self-adjointRoperator B' on 1 H such that v t' D B'i t , t 2 R. Take the spectral decompositions  D 1 e s dEs and R1 s B' D 1 e dFs . For every  2 D.H ; ‰/ let T WD R‰ ./. Note that T D eT .1 e/ 2 .N /0 and .T  T / D kk2 < 1 by Lemma 11.21, so that T 2 N . One has .T   t .T // D .R‰ ./ R‰ .v t' // ‰

0



‰

.v t' //

D ‰ .R ./ R Z 1 D e i t s dkFs k2 ;

(by Lemma 11.20(c)) D

h; v t' i

t 2 R:

(by Lemma 11.21) (11.13)

1

2The result is obvious when ‰ 2 NC (a bounded functional). Indeed, in this case, ‰.y/ D h0 ; ‰ .y/0 i and ‰ .y/ D ‰ .y/0 for y 2 N with a cyclic and separating vector 0 for N . Hence ‰ 0 .R R / D kR 0 k2 D kR ‰ .1/k2 D kk2 for any  2 D.H‰ ; ‰/.

196

11 Spatial derivatives and spatial Lp -spaces

On the other hand, one has .T   t .T // D h .T /;  . t .T //i (see (D) in Section 6.1) D h .T /; it  .T /i Z 1 D e i t s dkEs  .T /k2 ; t 2 R:

(11.14)

1

From (11.13) and (11.14) it follows that dkFs k2 D dkEs  .T /k2 . Note that .T T  / D '. ‰ .; // D q' ./; see Lemma 11.3. Assume that  2 D.q' /, i.e., q' ./ < 1, so T 2 N \N (i.e.,  .T / 2 A ; see (A) in Section 6.1). Then we have q' ./ D .T T  / D k .T  /k2 D k1=2  .T /k2  Z Z D e s dkEs  .T /k2 D e s dkFs k2 D kB'1=2 k2 : Since this holds for all  2 D.q' /, we find by Definition 11.4 that B'  d'=d‰. To prove that B' D d'=d‰, it now suffices to show that B' and d'=d‰ commute.3 For every  2 D.H ; ‰/ note that q' .v t' / D '.R‰ .v t' /R‰ .v t' / / D . t .R‰ ./R‰ ./ // (by Lemma 11.20(c)) D '.R‰ ./R‰ ./ / D q' ./: Therefore, v ' t .d'=d‰/v t' D d'=d‰, which implies the desired commutativity (see [77, Thm. VIII.13]).  To prove Theorem 11.9, we give two more lemmas. Lemma 11.23. There exists a net ¹u˛ º in MC such that each u˛ is of the form P n ‰ i D1  .i ; i / with i 2 D.H ; ‰/ and u˛ % 1. Proof. Write  ‰ .; / WD R‰ ./R‰ ./ for ;  2 D.H ; ‰/, and let I‰ be the linear span of ¹ .; / W ;  2 D.H ; /º. Similarly to the proof of Lemma 11.2 we easily see that I‰ is a two-sided ideal of M . For every x 2 M , x ¤ 0, by (2) and (5) of Lemma 11.2 there is a  2 D.H ; ‰/ such that x ¤ 0 and hence R‰ .x/ ¤ 0, so 0 ¤  .x; x/ D x ‰ .; /x  by Lemma 11.11. This implies that I‰ is a -weakly 3Let A, B be positive self-adjoint operators on H such that A, B commute and A  B (in the sense of Definition A.2). If there is a subspace D dense in H such that D  D.B 1=2 / and kA1=2 k2 D kB 1=2 k2 for all  2 D, then A D B. This can be seen, for instance, as follows. Let A be the abelian von Neumann algebra generated by the spectral projections of A, B. Decompose H into the direct sum of cyclic (hence separable) subspaces for A . Then one can use the measure space representation theorem (see, e.g., [19, Thm. IX.7.8]) for each direct summand. The details are left as an exercise for the reader.

197

11.2 Proofs of theorems

dense two-sided ideal of M , so there is a net ¹u˛ º in PI‰ \ MC such that u˛ % 1 (see, e.g., [90, §3.20]). Let a 2 I‰ \ MC and write a D niD1  ‰ .i ; i /. Note that n

aD

1X 1 ŒR .i /R‰ .i / C R‰ .i /R‰ .i /  .a C a / D 2 2 iD1 n

1X ‰  R .i C i /R‰ .i C i / .DW b/: 2 i D1

Since a1=2 D xb 1=2 for some x 2 M , one has n

a D xbx  D

1X ‰  .x.i C i /; x.i C i // 2 i D1



by Lemma 11.11. Hence the result follows.

Lemma 11.24. Let e 2 M be a projection, and consider eM e and Ne D .eM e/0 on eH . z on Ne for which the following There there exists a faithful semifinite normal weight ‰ hold: z

(a)  t‰ .ye/ D  t‰ .y/e for all t 2 R and y 2 N ; (b) if ' is a semifinite normal weight on M with s.'/ D e, then the support projection z D .d'=d‰/jeH , where 'z WD 'jeM e . of d'=d‰ is e and d'=d z ‰ Proof. Let eQ WD cM .e/, the central support of e, which is also a central projection in N . Since ye 2 Ne 7! y eQ 2 N eQ (y 2 N ) is a *-isomorphism, one can define a faithful z on Ne by ‰.ye/ z semifinite normal weight ‰ WD ‰.y e/ Q for y 2 N . Since  z ‰..ye/ .ye// D ‰.y  y e/ Q D ‰..y e/ Q  .y e//; Q

y 2 N;

(11.15)

note that ye 2 N‰z ” y eQ 2 N‰ . Let us prove (a) and (b). (a) One can define a strongly continuous one-parameter automorphism group ˛ t of Ne by ˛ t .ye/ WD  t‰ .y/e (t 2 R). Then for every y 2 NC one has z t .ye// D ‰.˛ t‰ .y/e/ z ‰.˛ Q D ‰.˛ t‰ .y e// Q D ‰.y e/ Q D ‰.ye/: For every ye; ze 2 N‰z \ Nz , equivalently y e; Q z eQ 2 N‰ \ N‰ thanks to (11.15), since ‰

z t .ye/.ze// D ‰. z t‰ .y/ze/ D ‰. t‰ .y/z e/ ‰.˛ Q D ‰. t‰ .y e/.z Q e//; Q ‰ ‰ z z ‰..ze/˛ Q D ‰..z e/ Q t‰ .y e//; Q t .ye// D ‰.z t .y/e/ D ‰.z t .y/e/ z

z satisfies the KMS condition with respect to ˛ t . Hence ˛ t D  t‰ holds; it follows that ‰ see (C) in Section 6.1.

11 Spatial derivatives and spatial Lp -spaces

198

(b) First, it is easy to see that s.d'=d‰/  s.'/ D e. Indeed, for every  2 D.H ; ‰/ with e D 0 we have k.d'=d‰/1=2 k2 D '. ‰ .; // D '.e ‰ .; /e/ D '. ‰ .e; e// D 0 (see Lemma 11.11). From this with Lemma 11.2(2) we have s.d'=d‰/  e. From (11.15) one can define an isometry W W H‰z ! H‰ by W ‰z .ye/ D ‰ .y e/ Q D z ‰ .e/ Q ‰ .y/ for y 2 N , with range ‰ .e/H Q ‰ . Now let  2 D.eH ; ‰/. Since kyk D kyek  C k‰z .ye/k D C k‰ .y e/k Q  C k‰ .y/k;

y 2 N;

for some C > 0, we have  2 D.H ; ‰/ and z

Q D eR‰ ./‰ .y e/ Q D eR‰ ./W ‰z .ye/; R‰ ./‰z .ye/ D ye D ey e z

so that R‰ ./ D eR ./W . Therefore, z

 ‰ .; / D eR‰ ./W W  R‰ ./ e D eR‰ ./‰ .e/R Q ‰ ./ e:

(11.16)

Since R‰ ./‰ .e/ Q ‰ .y/ D R‰ ./ .ey/ Q D ey Q D eR Q ‰ ./‰ .y/; we have R‰ ./‰ .e/ Q D eR Q ‰ ./ so that R‰ ./‰ .e/R Q ‰ ./ D e Q ‰ .; /e. Q From this z ‰ ‰ and (11.16) it follows that  .; / D e .; /e, so that z

'. z ‰ .; // D '.e ‰ .; /e/ D '. ‰ .; // z Hence, noting that s.d'=d‰/  e, we find that d'=d z D for every  2 D.eH ; ‰/. z ‰ z .d'=d‰/jeH , so that s.d'=d‰/ D e since s.d'=d z ‰/ D e.  Note that the assertion s.d'=d‰/ D s.'/ in (b) above was shown in Proposition 11.19 in a more general setting. We are now in a position to prove Theorems 11.7–11.9. Proof of Theorem 11.7. (1) and (2) follow from Lemmas 11.20 and 11.22; (3) has been shown in Lemma 11.24(b) (also Proposition 11.19). (4) Assume that '1 , as well as '2 , is faithful. Then v t'2 D .d'2 =d‰/i t satisfies (a)–(c) of Lemma 11.20 for '2 . By Step 1 of the proof of Lemma 11.20 we see that v t'1 WD .D'1 W D'2 / t v t'2 satisfies the same properties for '1 . Hence by Lemma 11.22 we have v t'1 D .d'1 =d‰/i t , i.e., .d'1 =d‰/i t D .D'1 W D'2 / t .d'2 =d‰/i t , t 2 R. Next assume that '1 is not faithful, and let e WD s.'1 /. Choose a semifinite normal weight '0 on M with s.'0 / D 1 e, and take a faithful ' WD '1 C '0 . By the first case we have .D' W D'2 / t D .d'=d‰/i t .d'2 =d‰/ i t , t 2 R. From Proposition 11.17(1) together with (3) of the theorem it follows that e

 d' i t d‰

D

 d' i t 1

d‰

;

t 2 R:

199

11.2 Proofs of theorems

So we may prove that e.D' W D'2 / t D .D'1 W D'2 / t ;

t 2 R:

To do this, let  D .'2 ; '/ and 1 D .'2 ; '1 / be the weights on M2 .M / D  balanced  M ˝ M2 .C/; see (6.1). Note that s.1 / D E WD 10 0e and for every X D Œxij  2 M2 .M /C , .X / D '2 .x11 / C '.x22 / D '2 .x11 / C '1 .ex22 e/ C '0 ..1 D 1 .EXE/ C '0 ..1 Therefore, it follows from Lemma 7.28 that and t 2 R. We hence have     0 0 0  D t E 1 .D'1 W D'2 / t 0  0 D E t 1

E/X.1

e/x22 .1

e//

E//:

 t1 .X / D  t .X / for all X 2 EM2 .M /E   0 E 0   0 0 ED 0 e.D' W D'2 / t

 0 : 0 

Proof of Theorem 11.8. The proof is along similar lines to that of Lemma 11.20. Step 1. Assume that (11.7) holds for a faithful semifinite normal weight ‰0 on N (and ' on M ). We prove that it holds true for every faithful semifinite normal weight ‰ on M (and ' on M ). Consider M ˝ C1
and N ˝ M2 D .M ˝ C1/0 on H ˝ C2 D H ˚ H . Define ' on M ˝ C1 by ' x0 x0 WD
'.x/ for x 2 MC (here we use the same notation y12 WD ‰.y11 / C ‰0 .y22 / for Œyij  2 .N ˝ M2 /C . ') and ‚ WD .‰; ‰0 / by ‚ yy11 21 y22 Apply Theorem 11.7(2) to obtain     0 0 0 0 D  t‚ .D‰0 W D‰/ t 0 1 0  d'  i t 0 0  d' i t ; t 2 R: (11.17) D 1 0 d‚ d‚ Now let us show that d' D d‚

"

d' d‰

0

0 d' d‰0

# :

(11.18)

Note that H‚ D H‰ ˚ H‰0 ˚ H‰ ˚ H‰0 and for every Y D Œyij  2 N‚ (see Lemma 6.4(2)), ‚ .Y / D ‰ .y11 / ˚ ‰0 .y12 / ˚ ‰ .y21 / ˚ ‰0 .y22 /; R‚ .1 ˚ 2 /‚ .Y / D Y .1 ˚ 2 / D .y11 1 C y12 2 / ˚ .y21 1 C y22 2 / D .R‰ .1 /‰ .y11 / C R‰0 .2 /‰0 .y12 // ˚ .R‰ .1 /‰ .y21 / C R‰0 .2 /‰0 .y22 //:

200

11 Spatial derivatives and spatial Lp -spaces

Hence we see that 1 ˚2 2 D.H ˚H ; ‚/ ” 1 2 D.H ; ‰/ and 2 2 D.H ; ‰0 /, and then for 1 ˚ 2 2 H ˚ H , R‚ .1 ˚ 2 / .1 ˚ 2 / D R‰ .1 / 1 ˚ R‰0 .2 / 1 ˚ R‰ .1 / 2 ˚ R‰0 .2 / 2 : Therefore, for every 1 ˚ 2 2 D.H ˚ H ; ‚/ we have  ‰   .1 ; 1 / C  ‰0 .2 ; 2 / 0  ‚ .1 ˚ 2 ; 1 ˚ 2 / D ; 0  ‰ .1 ; 1 / C  ‰0 .2 ; 2 / so that '. ‚ .1 ˚ 2 ; 1 ˚ 2 // D '. ‰ .1 ; 1 / C  ‰0 .2 ; 2 //; from which (11.18) follows. Combining (11.17) and (11.18) gives .D‰0 W D‰/ t D

 d'  d‰0

it 

d' i t ; d‰

t 2 R:

(11.19)

On the other hand, we apply Theorem 11.7(4), exchanging the roles of M , N , to obtain .D‰0 W D‰/ t D

 d‰ i t  d‰  0

d'

d'

it

;

t 2 R:

(11.20)

From (11.19) and (11.20) it follows that  d' i t  d‰ i t d‰

d'

D

 d' i t  d‰ i t 0 : d‰0 d'

This gives the desired assertion. Step 2. The validity of (11.7) is independent of the choice of a representation of N; M on H . This is seen similarly to Step 2 of the proof of Lemma 11.20; again, the details are omitted here (see [89, §7.6]). Step 3. From Steps 1 and 2 it suffices to prove (11.7) in the case where H D H‰ and ' D ‰ 0 . Then from Lemma 11.22 and the proof of Step 3 of the proof of Lemma 11.20, it follows that .d‰ 0 =d‰/i t D ‰i t . Since ‰0 D J‰ ‰ J‰ D ‰1 , exchanging the roles of M , N gives .d‰=d‰ 0 /i t D ‰i0 t D i‰t . Hence .d‰ 0 =d‰/ 1 D d‰=d‰ 0 follows.  Proof of Theorem 11.9. First we prove the theorem when T is non-singular. (i) H) (ii). Assume (i); then ' is faithful by Theorem 11.7(3). Hence (ii) follows from Theorem 11.7(2). (ii) H) (1). Take a faithful semifinite normal weight '1 on M and set T1 WD d'1 =d‰, u t WD T it T1 it (t 2 R). By Theorem 11.7(2), for every y 2 N we have T i t T1 i t yT1i t T

it

D T i t  t‰ .y/T

it

D yT i t T

it

D y:

201

11.2 Proofs of theorems

Hence u t y D yu t for all y 2 N so that u t 2 M , which is a strongly continuous one-parameter family of unitaries. By Theorem 11.7(1) note that us s'1 .u t / D T i s T1

is

T1i s .T i t T1 i t /T1

is

D T i.sCt / T1

i.sCt /

D usCt ;

that is, u t is a  '1 -cocycle. Thus, by Theorem 6.8 (and its remark) there exists a faithful semifinite normal weight ' on M such that u t D .D' W D'1 / t for all t 2 R. By Theorem 11.7(4) we have .d'=d‰/i t D u t T1i t D T i t for all t 2 R, so d'=d‰ D T . Next we prove the general case. Let e WD s.T /, the support projection. If (i) holds, then e D s.'/ 2 M by Theorem 11.7(3). If (ii) holds, then the case t D 0 gives ey D ye for all y 2 N , so e 2 N 0 D M . So, to prove that (i) ” (ii), we may assume that e 2 M . z on Ne be as given in Lemma 11.24. By Lemma 11.24 we see that Let Tz WD T jeH and ‰ (i) and (ii) are equivalent to the following (i)0 and (ii)0 , respectively: z (i)0 there exists a faithful semifinite normal weight 'z on eM e such that Tz D d'=d z ‰; z (ii)0 Tz it  t‰ .y/ Q D yQ Tz i t for all t 2 R and yQ 2 Ne.

The first case says that (i)0 ” (ii)0 holds, so that the result follows. Finally, the uniqueness of ' in (i) is immediately seen from Definition 11.13 together with Lemmas 11.23 and 11.11.  We end the section with a corollary (given in [17, Cor. 16]) of the theorems proved above. Corollary 11.25. Let M  B.H / and N D M 0 be as above. Let P .N / WD ¹‰ W faithful semifinite normal weight on N º; P .B.H /; M / WD ¹T W faithful semifinite normal operator-valued weight from B.H / to M º

(see Section 7.2):

Then there is a bijection ‰ $ T between P .N / and P .B.H /; M / determined by ' ı T D Trd'=d‰ ;

i.e.;

d.' ı T / d' D d1 d‰

(11.21)

(see Example 11.5) for any fixed ' 2 P .M / or, equivalently, for all ' 2 P .M /. Furthermore, (11.21) holds for all semifinite normal weights ' on M . Proof. Let ‰ 2 P .N /. Choose a '0 2 P .M / and set 'z0 WD Trd'0 =d‰ 2 P .B.H //; see Proposition 7.21. By Theorems 11.7(1) and 7.22(1) we have  d' i t  d'  0 0 x d‰ d‰  d' i t  d'  0 0 ' z0  t .x/ D x d‰ d‰  t'0 .x/ D

it

;

t 2 R; x 2 M;

;

t 2 R; x 2 B.H /;

it

11 Spatial derivatives and spatial Lp -spaces

202

so that  t'0 D  t'z0 jM . Hence by Theorem 7.18 there exists a T 2 P .B.H /; M / such that 'z0 D '0 ı T , i.e., (11.21) holds for ' D '0 . Let us prove (11.21) for general ' 2 P .M /. For every ' 2 P .M / let 'z WD Trd'=d‰ . We then have  d' i t  d'  i t 0 D .D' W D'0 / t D .D.' ı T / W D 'z0 / t : .D 'z W D 'z0 / t D d‰ d‰ In the above, the first equality follows from Proposition 7.24 and Theorem 7.22(2), the second from Theorem 11.7(4) and the third from Theorem 7.8(b). Therefore, 'z D ' ı T follows from Proposition 7.24(4), so we have (11.21) for all ' 2 P .M /. This can be further extended to non-faithful ' in a similar way to the proof of Theorem 11.7(4), although the details are left as an exercise for the reader. In this way, we have shown that (11.21) holds for all semifinite normal weights ' on M once it holds for some '0 2 P .M /. In particular, for any rank-one projection jihj we have '.T .jihj// D Trd'=d‰ .jihj/ D '. ‰ .; //;

' 2 MC

(see Definition 11.13), implying that T .jihj/ D  ‰ .; / for all  2 H . This says that ‰ 7! T is injective. To prove surjectivity, let T 2 P .B.H /; M /. Choose a '0 2 P .M / and let 'z0 D '0 ı T 2 P .B.H //. There exists a non-singular positive self-adjoint operator A on H such that 'z0 D TrA ; see Theorem 7.25. By Theorems 7.8(a) and 7.22(1) we have  t'0 D  t'z0 .x/ D Ai t xA it

 t'0 .x/

it

;

x 2 M;

it

so that A D xA for all x 2 M . Hence, by Theorem 11.9 applied to .N; M; '0 / in place of .M; N; ‰/, there exists a ‰ 2 P .N / such that A 1 D d‰=d'0 . Thus we have A D d'0 =d‰ by Theorem 11.8, so that (11.21) is satisfied for ' D '0 .  Remark 11.26. The T 2 P .B.H /; M / corresponding to ‰ 2 P .N / in Corollary 11.25 is denoted by ‰ 1 . It is worth noting the following more general result in [33, Thms. 5.9, 6.13]: Let N  M  B.H / be von Neumann algebras. There exists a bijection T 2 P .M; N / $ T 1 2 P .N 0 ; M 0 / determined by d.' ı T d

1

/

D

d.

d' ıT/

for ' 2 P .N 0 / and 2 P .N /. In fact, Corollary 11.25 is a particular case of this, where .N  M; T; T 1 / is .C1  N D M 0 ; ‰; T D ‰ 1 /. When N  M is a pair of factors and EW M ! N is a faithful normal conditional expectation (see Section 5.1), Kosaki [59] introduced the index of E (the extension of Jones’ index [50]; see Section 1.10(1)) by applying the above correspondence T 7! T 1 to E.

11.3 Spatial Lp -spaces Let M be a von Neumann algebra on H and ‰ be a faithful semifinite normal weight on M 0 . The spatial Lp -spaces are non-commutative Lp -spaces associated with .M; H ; ‰/

11.3 Spatial Lp -spaces

203

introduced by Connes [17] and Hilsum [46] based on the spatial theory [17] discussed in the previous sections. In this section we present a description of spatial Lp -spaces based on [17, 46, 97]. Definition 11.27. Let 2 R. A densely defined closed operator a on H with the polar decomposition a D ujaj is said to be -homogeneous with respect to ‰ if u2M

jaji t y D  ‰t .y/jaji t ;

and

t 2 R; y 2 M 0 ;

(11.22)

where jajit is defined with restriction to the support of jaj. Remark 11.28. It is known (see [56, Lem. 2.1], [98, Def. 19]) that for a densely defined closed operator a on H , the above condition (11.22) is equivalent to ya  ai‰ .y/ for all  t‰ -analytic elements y 2 M 0 . The proof of the equivalence is based on Carlson’s theorem for analytic functions. Let a D ujaj be a densely defined closed operator on H . For p 2 .0; 1/, a is . 1=p/-homogeneous if and only if u 2 M and jajp is . 1/-homogeneous. Note that a is 0-homogeneous if and only if u 2 M and jaj is affiliated with M . Theorem 11.9 says that a positive self-adjoint operator a on H is . 1/-homogeneous if and only if there is a unique semifinite normal weight ' on M such that a D d'=d‰. In this case, we define the ‰-integral of a by Z a d‰ WD '.1/: R R Hence a d‰ < 1 if and only if a D d'=d‰ with ' 2 MC . The integral
d‰ is a notion similar to the tr-functional on Haagerup’s L1 -space (Definition 9.6). Definition 11.29. For each p 2 .0; 1/ let M

1=p

WD ¹a W densely defined closed . 1=p/-homogeneous operators on H º:

Define the spatial Lp -space by ® Lp .‰/ D Lp .M; H ; ‰/ WD a 2 M

1=p

W

R

¯ jajp d‰ < 1 ;

and for a 2 Lp .‰/ define kakp 2 Œ0; 1/ by Z kakp WD

jajp d‰

1=p :

Moreover, for p D 1 define L1 .‰/ WD M , and kak1 denotes the usual operator norm of a 2 M .

11 Spatial derivatives and spatial Lp -spaces

204

Now let '0 be a distinguished faithful semifinite normal weight on M and d0 WD d'0 =d‰. By Theorem 11.7(1) note that  t'0 .x/ D d0i t xd0 i t ;

t 2 R; x 2 M:

Recall the crossed product N WD M o '0 R generated by  '0 .M / and .R/ on L2 .R; H / D H ˝ L2 .R/, having the dual action s (s 2 R) and the canonical trace ; see Section 8.2. (The symbol N is different from N D M 0 in Sections 11.1 and 11.2.) As in Section 9.1 we simply let M  N by identifying x and  '0 .x/ for x 2 M . Also, recall the unitary representation  t (t 2 R) on L2 .R/ given by . t f /.s/ WD f .s t / for s 2 R and f 2 L2 .R/, so that .t / D 1 ˝  t . Let A be a positive self-adjoint operator on L2 .R/ such that  t D Ai t ; t 2 R: Furthermore, define a unitary U0 on L2 .R; H / by t 2 R;  2 L2 .R; H /:

.U0 /.t / WD d0i t .t /; Lemma 11.30. We have

U0 xU0 D x ˝ 1; U0 .t /U0

d0i t

D

˝ t ;

x 2 M;

(11.23)

t 2 R:

(11.24)

Proof. For every  D v ˝ f 2 L2 .R; H / (v 2 H , f 2 L2 .R/) and s 2 R we have .U0 xU0 /.s/ D d0i s  '0s .x/d0 i s .s/ D x.s/ D f .s/xv D ..x ˝ 1/.v ˝ f //.s/; t / D d0i s d0 i.s

.U0 .t /U0 /.s/ D d0i s .U0 /.s D f .s

t /d0i t v

D

..d0i t

t/

.s

t / D d0i t .s

t/

˝  t /.v ˝ f //.s/:



Proposition 11.31. Let p 2 .0; 1/. Let N ; 1=p denote the set of densely defined closed operators h affiliated with N satisfying s .h/ D e s=p h (s 2 R). Then the mapping a 7 ! U0 .a ˝ A1=p /U0 is a bijection from M Proof. Let a 2 M and, by (11.23),

1=p

1=p

onto N ;

1=p .

with the polar decomposition a D ujaj. Then one has jajp 2 M

1

U0 .a ˝ A1=p /U0 D U0 .u ˝ 1/U0 U0 .jaj ˝ A1=p /U0 D u.U0 .jajp ˝ A/U0 /1=p : On the other hand, let h 2 N ; 1=p with the polar decomposition h D wjhj. Then w is s -invariant so that w 2 M , and one has jhjp 2 N ; 1 . Note that h D U0 .a ˝ A1=p /U0 if and only if w D u and jhjp D U0 .jajp ˝ A/U0 . Hence it suffices to show

11.3 Spatial Lp -spaces

205

that a 7! U0 .a ˝ A/U0 is a bijection from .M 1 /C WD ¹a 2 M 1 W positiveº onto .N ; 1 /C WD ¹h 2 N ; 1 W positiveº. By Lemma 9.2 (see also (9.1)) we have a bijection ' 7! h' from P .M / WD ¹' W semifinite normal weight on M º onto .N ; 1 /C (denoted by N ;C in Section 9.1). Since ' 2 P .M / 7! d'=d‰ 2 .M 1 /C is a bijection by Theorem 11.9, it remains to prove that   d' ˝ A U0 ; ' 2 P .M /: (11.25) h' D U0 d‰ In view of Lemma 9.2 the mapping ' 2 P .M / 7! h' 2 .N ;

1 /C

is determined by

'z D ' ı T D  .h'
/ 2 P  .N / (see Section 9.1); R

where T D R s ds, an operator-valued weight from N to M . First assume that ' 2 P .M / is faithful. We find that hi't h'0i t D .D 'z W D / t .D W D 'z0 / t

(by Theorem 7.22(2))

D .D 'z W D 'z0 / t

(by Proposition 7.24(3))

D .D' W D'0 / t  d' i t  d'  0 D d‰ d‰

(by Theorem 7.8(b))

Since .d'=d‰/i t .d'0 =d‰/ (11.23) we have

it

it

(by Theorem 11.7(4)):

2 M is seen from condition (ii) of Theorem 11.9, by

U0 hi't h'0i t U0 D

 d' i t  d'  0

it

˝ 1: d‰ d‰ Furthermore, 'z0 D '0 ı T D  ..1 ˝ A/
/ is seen from the proof of Theorem 8.15(2), where the previous A is 1˝A in the present situation. Hence h'0 D 1˝A and hi't0 D .t /, t 2 R, so that by (11.24) we have  d' i t 0 ˝ Ai t : U0 hi't0 U0 D U0 .t /U0 D d0it ˝  t D d‰ Therefore, U0 hi't U0 D .U0 hi't h'0i t U0 /.U0 hi't0 U0 / D

 d' i t d‰

˝ Ai t ;

which gives (11.25). Next, when ' is not faithful with e WD s.'/, choose a '1 2 P .M / with s.'1 / D 1 e. The first case gives  d' d.' C '1 / d'1  ˝AD C U0 h'C'1 U0 D ˝A d‰ d‰ d‰ by Proposition 11.17(1). By Lemma 9.2 one has s.h' / D s.'/ z D s.'/ D e and s.h'1 / D 1 e, from which h' D eh'C'1 e is easily verified. Also, by Proposition 11.19

11 Spatial derivatives and spatial Lp -spaces

206

(or Lemma 11.24) one has s.d'=d‰/ D e and s.d'1 =d‰/ D 1 e. Therefore, by (11.23) we have U0 h' U0 D U0 eh'C'1 eU0 D .e ˝ 1/U0 h'C'1 U0 .e ˝ 1/   d' d'1  d' C ˝ A .e ˝ 1/ D ˝ A; D .e ˝ 1/ d‰ d‰ d‰ 

as desired.

Lemma 11.32. Let p 2 .0; 1 and a 2 M 1=p . Then a 2 Lp .‰/, i.e., jajp d‰ < 1 if and only if U0 .a ˝ A1=p /U0 2 Lp .M / (Haagerup’s Lp -space). Moreover, in this case we have a 2 Lp .‰/, a ˝ A1=p D .a ˝ A1=p / and R

kakp D ka kp D kU0 .a ˝ A1=p /U0 kp : Proof. Assume that 0 < p < 1, since the case p D 1 is obvious from (11.23). Let a D ujaj 2 M 1=p . Since ja j D ujaju , it is clear that a 2 M 1=p . Note that jajp 2 M 1 and jajp D d'=d‰ for some ' 2 P .M /. From Proposition 11.31 (in particular, see (11.25)) we find that Z jajp d‰ D '.1/ < 1 ” h' 2 L1 .M / (by Theorem 9.5(a) and Definition 9.6) ” U0 .jaj ˝ A1=p /U0 2 Lp .M / ” U0 .a ˝ A1=p /U0 2 Lp .M / and, in this case, Z kakp D

p

jaj d‰

1=p

D tr.h' /1=p D kU0 .a ˝ A1=p /U0 kp :

By (11.23) note that U0 .ja j ˝ A1=p /U0 D u.U0 .jaj ˝ A1=p /U0 /u ; U0 .a ˝ A1=p /U0 D U0 .jaj ˝ A1=p /U0 u D .uU0 .jaj ˝ A1=p /U0 / D .U0 .a ˝ A1=p /U0 / D U0 .a ˝ A1=p / U0 : Hence we have a 2 Lp .‰/, ka kp D kakp and a ˝ A1=p D .a ˝ A1=p / .



Remark 11.33. Here we give an explicit description of the tensor product operator T ˝ A1=p for a densely defined closed operator T on H , which will be useful in the rest of the section. (The description was given in [97, Chap. IV, Lem. 9] in a slightly more general setting.) In view of Remark 8.16 note that F AF  is the operator for multiplication by e t , where F is the Fourier transform on L2 .R/, so F A1=p F  is multiplication by

11.3 Spatial Lp -spaces

207

e t=p . Hence we may assume that A1=p itself by e t =p . Then we can R 1is multiplication 1=p 1=p t =p write A as the spectral integral A D 1t dm t , where m t is multiplication by the indicator function of . 1; t . Let T be a densely defined closed operator on H T D U jT j and the spectral decomposition jT j D R 1. Take the polar decomposition1=p  dE . We notice that jT j ˝ A is represented as the two-variable spectral integral  R0 t=p e d.E ˝ m /. Hence  2 L2 .R; H / is in the domain D.jT j ˝ A1=p / if t  Œ0;1/R and only if  Z Z 1 Z 1 2 2t =p 2 2 2  e dk.E ˝ m t /k D  dkE .t /k e 2t =p dt < 1; Œ0;1/R

0

1

R1 that is, .t / 2 D.jT j/ D D.T / for a.e. t 2 R and 0 e 2t =p kT .t /k2 dt < 1. Thus, we conclude that ® ¯ R1 D.T ˝ A1=p / D  2 L2 .R; H / W .t / 2 D.T / a.e. t 2 R; 0 e 2t =p kT .t /k2 dt < 1 and for  2 D.T ˝ A1=p /, ..T ˝ A1=p //.t / D e

t =p

T .t /;

t 2 R:

From this we easily see (an exercise) that T  ˝ A1=p  .T ˝ A1=p / :

(11.26)

Lemma 11.34. Let p; q; r 2 .0; 1 with 1=p C 1=q D 1=r. (1) If a; b 2 Lp .‰/, then a C b is densely defined and closable, and a C b 2 Lp .‰/. (2) If a 2 Lp .‰/ and b 2 Lq .‰/, then ab is densely defined and closable, and ab 2 Lr .‰/. Proof. (1) The case p D 1 is obvious; so assume that 0 < p < 1. Let e be the projection from H onto D.a/ \ D.b/. Since a ˝ A1=p , b ˝ A1=p 2 U0 Lp .M /U0 by Lemma 11.32, it follows that D.a ˝ A1=p / \ D.b ˝ A1=p / is dense in L2 .R; H /. By Remark 11.33, if  2 D.a ˝ A1=p / \ D.b ˝ A1=p /, then e.t / D .t / for a.e. t 2 R, so that  2 .e ˝ 1/L2 .R; H /. This implies that e D 1, so D.a/ \ D.b/ is dense in H . Hence a C b is densely defined. The argument above can be also applied to a , b  by Lemma 11.32, so that a C b  is densely defined. Since a C b  .a C b  / , a C b is closable. Next let us show that .a C b/ ˝ A1=p D a ˝ A1=p C b ˝ A1=p :

(11.27)

From the description of a ˝ A1=p in Remark 11.33 it is immediate to see that a ˝ A1=p C b ˝ A1=p  .a C b/ ˝ A1=p ;

(11.28)

208

11 Spatial derivatives and spatial Lp -spaces

and similarly a ˝ A1=p C b  ˝ A1=p  .a C b  / ˝ A1=p . Therefore, we have .a C b/ ˝ A1=p  .a C b  / ˝ A1=p  ..a C b  / ˝ A1=p /

(by (11.26) for T D a C b  )

 .a ˝ A1=p C b  ˝ A1=p / D a ˝ A1=p C b ˝ A1=p ; where the last equality follows by considering the *-operation in Lp .M / with Lemma 11.32. Hence (11.27) holds since (11.28) gives a ˝ A1=p C b ˝ A1=p  .a C b/˝A1=p . Moreover, a C b 2 Lp .‰/ by Lemma 11.32. (2) The case r D 1 (so p D q D 1) is obvious, so assume that 0 < r < 1. Let e be the projection from H onto D.ab/. As in the proof of (1) it follows that D..a˝A1=p /.b˝ A1=q // is dense in L2 .R; H /. By Remark 11.33, if  2 D..a ˝ A1=p /.b ˝ A1=q //, then .t / 2 D.b/ for a.e. t 2 R and b.t / 2 D.a/ for a.e. t 2 R, so that .t / 2 D.ab/ for a.e. t 2 R and hence e.t / D .t / for a.e. t 2 R, i.e.,  2 .e ˝ 1/L2 .R; H /. Therefore, e D 1 and ab is densely defined. Similarly, b  a is densely defined. Since ab  .b  a / , ab is closable. Next let us show that ab ˝ A1=r D .a ˝ A1=p /.b ˝ A1=q /:

(11.29)

By Remark 11.33, if  2 D..a ˝ A1=p /.b ˝ A1=q //, then ..a ˝ A1=p /.b ˝ A1=q //.t / D e

t =p

a.e

t =q

b.t // D e

t =r

ab.t /;

t 2 R:

Hence .a ˝ A1=p /.b ˝ A1=q /  ab ˝ A1=r , so that .a ˝ A1=p /.b ˝ A1=q /  ab ˝ A1=r . Similarly .b  ˝ A1=p /.a ˝ A1=q /  b  a ˝ A1=r . Therefore, we have ab ˝ A1=r  .b  a / ˝ A1=r  .b  a ˝ A1=r /  ..b  ˝ A1=q /.a ˝ A1=p // D .a ˝ A1=p /.b ˝ A1=q / as in the proof of (1). Hence (11.29) follows. Since the right-hand side of (11.29) is in  U0 Lr .M /U0 , we have ab 2 Lr .‰/ by Lemma 11.32. We simply write a C b and ab for the strong sum a C b and the strong product ab in (1) and (2) of Lemma 11.34, respectively. Definition 11.35. Let a D ujaj 2 L1 .‰/. By Lemma 11.32 (and its proof) we have jaj D d'=d‰ with ' 2 MC and h' D U0 .jaj ˝ A/U0 . We define the ‰-integral of a by Z

a d‰ D tr.U0 .a ˝ A/U0 / D tr.uh' /:

11.3 Spatial Lp -spaces

209

Proposition 11.36. Let p; q 2 Œ1; 1 and 1=p C 1=q D 1. Then for every a 2 Lp .‰/ and b 2 Lq .‰/ we have Z Z ab d‰ D ba d‰: Proof. Lemma 11.34(2) implies that ab; ba 2 L1 .‰/. Thanks to (11.29), in the case r D 1 one has Z ab d‰ D tr.U0 .a ˝ A1=p /.b ˝ A1=q /U0 / D tr.U0 .a ˝ A1=p /U0
U0 .b ˝ A1=q /U0 / D tr.U0 .b ˝ A1=q /U0
U0 .a ˝ A1=p /U0 / (by Proposition 9.22) Z D ba d‰:



Now the following main results on Lp .‰/ are obtained immediately by transforming the corresponding results on Lp .M / in Section 9.2 in view of Lemmas 11.32 and 11.34, together with Proposition 11.36. Theorem 11.37. (a) For every p 2 Œ1; 1, Lp .‰/ is a Banach space with respect to the strong sum a C b and the norm k
kp . Moreover, Lp .‰/ Š Lp .M / (isometric) via the mapping a 2 Lp .‰/ 7! U0 .a ˝ A1=p /U0 2 Lp .M /. 2 (b) RIn particular, L is a Hilbert space with respect to the inner product ha; bi WD R .‰/  a b d‰ (D ba d‰) for a; b 2 L2 .‰/.

(c) (Hölder’s inequality). Let p; q; r 2 .0; 1 with 1=p C 1=q D 1=r. If a 2 Lp .‰/ and b 2 Lq .‰/, then ab 2 Lr .‰/ and kabkr  kakp kbkq : (d) (Lp –Lq -duality). Let 1  p < 1 and 1=p C 1=q D 1. Then the dual Banach space of Lp .‰/ is Lq .‰/ under the duality pairing .a; b/ 2 Lp .‰/  Lq .‰/ 7! R ab d‰ 2 C. (e) ..M /; L2 .‰/; J D  ; L2 .‰/C / is a standard form of M , where .x/ (x 2 M ) is the left action .x/a WD xa for a 2 L2 .‰/ and L2 .‰/C is the positive part of L2 .‰/.

Appendix Positive self-adjoint operators and positive quadratic forms

In this appendix we summarize the basic facts of positive self-adjoint operators and positive quadratic forms, some of which are used in the main body of this book. The appendix might be of use as a concise self-contained exposition on the topic of the title.

A.1 Positive self-adjoint operators Let H be a Hilbert space. Let A be a positive self-adjoint operator on H , which is represented by the spectral decomposition 1

Z AD

 dE 0

with a unique spectral measure dE on Œ0; 1/. The domain of A is given as ® ¯ R1 D.A/ D  2 H W 0 2 dkE k2 < 1 and for  2 D.A/ we have Z

1

A D

n!1 0

0 2

Z  dE  D lim

Z

kAk D

1

n

 dE  (strongly);

2 dkE k2 :

0

For a complex-valued Borel function f on Œ0; 1/, the Borel functional calculus f .A/ is defined by Z 1 f .A/ WD f ./ dE ; 0

with domain ® ¯ R1 D.f .A// D  2 H W 0 jf ./j2 dkE k2 < 1 :

212

Appendix Positive self-adjoint operators and positive quadratic forms

The operator f .A/ is self-adjoint if f is real-valued, positive self-adjoint if f is nonnegative, and bounded if f is bounded. For instance, Z 1 Ap D p dE .p > 0/; 0

.˛1 C A/

1

1

Z D 0

1 dE ; ˛C

A.1 C ˛A/

1

1

Z D 0

 dE 1 C ˛

.˛ > 0/:

Moreover, when A is non-singular (i.e., the support s.A/ D 1), Z 1 Z 1 A 1D t 1 dE t ; Ai t D i t dE .t 2 R/: 0

0

In fact, A 1 and Ai t are defined for singular A as well R with restriction to s.A/H (or in the sense of the generalized inverse), that is, A 1 D .0;1/ t 1 dE t and similarly for Ai t . (Note that A0 is used to mean either A0 D 1 or A0 D s.A/ according to the situation.) The next lemma summarizes equivalent conditions for order between two positive self-adjoint operators. Lemma A.1. Let A, B be positive self-adjoint operators on H . Then the following conditions are equivalent: (i) D.B 1=2 /  D.A1=2 / and kA1=2 k2  kB 1=2 k2 for all  2 D.B 1=2 /; (ii) there exists a core D of B 1=2 such that D  D.A1=2 / and kA1=2 k2  kB 1=2 k2 for all  2 D; (iii) .˛1 C B/

1

(iv) A.1 C ˛A/

 .˛1 C A/ 1

1

for some (equivalently, any) ˛ > 0;

 B.1 C ˛B/

1

for some (equivalently, any) ˛ > 0.

Proof. (i) ” (ii). (i) H) (ii) is trivial. Conversely, let D0 be a core of B 1=2 for which condition (ii) holds. For any  2 D.B 1=2 / there exists a sequence ¹n º in D0 such that n !  and B 1=2 n ! B 1=2 . Since kA1=2 n

A1=2 m k D kA1=2 .n

m /k  kB 1=2 .n

m /k ! 0

as n; m ! 1, it follows that A1=2 n converges to some  2 H , and hence we have  2 D.A1=2 / and A1=2  D . Therefore, kA1=2 k D lim kA1=2 n k  lim kB 1=2 n k D kB 1=2 k: n

n

(i) H) (iii). Assume (i) and prove (iii) for any ˛ > 0. For every  2 D.B 1=2 /, k.1 C A/1=2 k2 D kk2 C kA1=2 k2  kk2 C kB 1=2 k2 D k.1 C B/1=2 k2 : Hence there is a contraction C on H such that .1 C A/1=2  D C.1 C B/1=2 ;

 2 D.B 1=2 /:

213

A.1 Positive self-adjoint operators

For any  2 H let  WD .1 C B/  D .1 C B/1=2 , we have .1 C A/ so that .1 C A/

1=2

1=2

1=2

1=2

C  D .1 C A/

C D .1 C B/

.1 C B/

1

1=2

D Œ.1 C A/

D˛

1

.1 C ˛

1

1=2 1=2 1

For every ˛ > 0, since (i) holds for ˛ 1

C.1 C B/1=2  D  D .1 C B/

1=2

;

. Therefore,

D .1 C A/

.˛1 C A/

 2 D..1 C B/1=2 / D D.B 1=2 /. Then, since

C Œ.1 C A/

1

1

C 

1=2

C C  .1 C A/

A, ˛

A/

1=2

 .1 C A/

1

:

B, we have

˛

1

.1 C ˛

1

B/

1

D .˛1 C B/

1

:

(iii) H) (i). Assume (iii) for some ˛ > 0 and prove (i). Similarly to the last part of the proof of (i) H) (iii), we may assume (iii) for ˛ D 1. This means that k.1 C B/ 1=2 k  k.1 C A/ 1=2 k for all  2 H . Hence there is a contraction C on H such that .1 C B/ 1=2 D C.1 C A/ 1=2 D .1 C A/ 1=2 C  . Since D.B 1=2 / D D..1 C B/1=2 / is equal to the range R..1CB/ 1=2 / of .1CB/ 1=2 , for any  2 D.B 1=2 / there is an  2 H such that  D .1 C B/ 1=2  D .1 C A/ 1=2 C  . Then  2 D..1 C A/1=2 / D D.A1=2 / and .1 C A/1=2  D C  . Therefore, kk2 C kA1=2 k2 D k.1 C A/1=2 k2 D kC  k2  kk2 D k.1 C B/1=2 k2 D kk2 C kB 1=2 k2 ; so that kA1=2 k2  kB 1=2 k2 . (iii) ” (iv). Since t .1 C ˛t / A.1 C ˛A/

1

D˛

1 1

D˛ 1

˛

1

˛ 2

.˛

2 1

.˛

1

C t/

1 C A/

1

;

from which (iii) ” (iv) is seen immediately.

1

, one has ˛ > 0; 

Definition A.2. For positive self-adjoint operators A, B we write A  B (in the form sense) if the conditions given in Lemma A.1 hold. By Lemma A.1 and Definition A.2 we have the following. Proposition A.3. For densely defined closed operators A, B on H the following are equivalent: (i) A A  B  B; (ii) there exists a core D of B such that D  D.A/ and kAk  kBk for all  2 D. The next lemma gives equivalent conditions for convergence of positive self-adjoint operators.

214

Appendix Positive self-adjoint operators and positive quadratic forms

Lemma A.4. Let An (n 2 N) and A be positive self-adjoint operators on H . Then the following conditions are equivalent: 1

(i) .˛1 C An / (ii) .i C An /

1

! .i C A/

(iii) An .1 C ˛An /

1

! .˛1 C A/

1

1

strongly for some (equivalently, any) ˛ > 0;

strongly; 1

! A.1 C ˛A/

strongly for some (equivalently, any) ˛ > 0.

Moreover, assume that all An and A are non-singular, so we write An D e Hn and A D e H , where Hn WD log An and H WD log A. Then the above conditions are also equivalent to the following: (iv) .i C Hn /

1

! .i C H /

1

strongly;

it (v) Ait n ! A strongly for all t 2 R.

To prove the above lemma, we state the next result due to Kadison [52] without proof. For details, see [52] or [89, Thm. A.2]. Lemma A.5 (Kadison). Let f be a complex-valued continuous function on a subset S  C such that .S n S / \ S D ; and sup z2S

jf .z/j < 1: 1 C jzj

Then f is strong-operator continuous, i.e., the functional calculus f .A/ from ¹A 2 B.H / W normal;  .A/  S º to B.H / is strongly continuous, where  .A/ is the spectrum of A. Proof of Lemma A.4. (i) ” (ii). For each ˛ > 0 consider a homeomorphism ® 1 ¯ ® 1 ¯ 1 W Œ0; ˛ 1  D ˛Ct W 0  t  1 ! i Ct W0t 1 ;  1  1 1 WD (where 1 .0/ WD 0 for t D 1): ˛Ct i Ct Since .i C A/ 1 D 1 ..˛1 C A/ seen from Lemma A.5.

1

/ and .˛1 C A/

1

D 1 1 ..i C A/

1

/, the result is

(i) ” (iii). For each ˛; ˇ > 0 consider a homeomorphism ® t ¯ 2 W Œ0; ˛ 1  ! Œ0; ˇ 1  D 1Cˇ W0t 1 ; t  1  t 2 WD (where 2 .0/ WD ˇ 1 for t D 1): ˛Ct 1 C ˇt Since A.1 C ˇA/ 1 D 2 ..˛1 C A/ follows from Lemma A.5.

1

/ and .˛1 C A/

1

D 2 1 .A.1 C ˇA/

1

/, the result

215

A.1 Positive self-adjoint operators

Next assume that all An and A are non-singular. Let Hn WD log An and H WD log A. (i) H) (iv) follows as above by considering a continuous function ® 1 ¯ ® 1 ¯ W 0  t  1 ! iClog W0t 1 ; 3 W Œ0; ˛ 1  D ˛Ct t  1  1 3 WD (where 3 .˛ 1 / D 3 .0/ WD 0 for t D 0; 1): ˛Ct i C log t (iv) H) (v). By considering a homeomorphism ® 1 ¯ ® ¯ 4 W iCt W 1  t  1 ! i 1t W 1  t  1 ;  1  1 4 WD (where 4 .0/ WD 0 for t D ˙1); i Ct i t it follows from (iv) that .i Hn / 1 ! .i H / 1 strongly as well. Let C0 .R/ denote the Banach space of complex continuous functions  on R vanishing at infinity (i.e., lim t!˙1 f .t / D 0) with the sup-norm. By the Stone–Weierstrass theorem, the set of polynomials of .i ˙ t / 1 is dense in C0 .R/. Hence for any  2 C0 .R/ and " > 0 one can choose a polynomial p..i C t / 1 ; .i t / 1 / such that k.t /

p..i C t /

1

t/

1

H/

1

/k < ";

Hn /

1

/k < ";

; .i

/k1 < ":

Hence k.H / k.Hn /

p..i C H / p..i C Hn /

1 1

; .i

; .i

n 2 N:

For any  2 H with kk D 1, there is an n0 such that kp..i C Hn /

1

; .i

Hn /

1

p..i C H /

/

1

; .i

H/

1

/k < ";

n  n0 :

Therefore, k.Hn /

.H /k < 3";

n  n0 ;

which implies that .Hn / ! .H / strongly for every  2 C0 .R/. Now, for each fixed s 2 R, set f .t / WD e i st ; For k 2 N set

8 ˆ 0. Then, for every  2 H , the following conditions are equivalent:

A.1 Positive self-adjoint operators

217

(i)  2 D.A˛ /; R1 (ii) 0 2˛ dkE k2 < 1; (iii) there exists an H -valued bounded, weakly continuous function f on ˛  Im z  0, weakly analytic in ˛ < Im z < 0, such that f .t / D Ai t  for all t 2 R; (iv) there exists an H -valued bounded, strongly continuous function f on ˛  Im z  0, strongly analytic in ˛ < Im z < 0, such that f .t / D Ai t  for all t 2 R. Furthermore, in either condition (iii) or (iv), the function f .z/ is unique and A˛  D f . i ˛/ holds. Proof. Define finite positive measures  on .0; 1/ and  on R by d./ WD dkE k2 for  > 0, and d.s/ WD d.e s / for s 2 R. Without loss of generality we may assume that ˛ D 1, by replacing A with A˛ . (i) ” (ii) is a well-known fact and (iv) H) (iii) is trivial. (i) H) (iv). From (i) (and (ii)) one can define Z 1 iz iz dE ; f .z/ WD A  D

1  Im z  0;

0

which is bounded on 1  Im z  0, since Z 1 Z kf .z/k2 D jiz j2 d./  0

1

.1 C 2 / d./ < 1:

0

Assume that 1  Im zn  0 (n 2 N) and zn ! z0 . Since jizn

iz0 j2  2.jizn j2 C jiz0 j2 /  4.1 C 2 /

and izn ! iz0 for all  2 .0; 1/, Lebesgue’s convergence theorem gives Z 1 kf .zn / f .z0 /k2 D jizn iz0 j2 d./ ! 0: 0

Hence f .z/ is strongly continuous on 1  Im z  0. Next assume that 1 < Im z0 < 0, and choose a ı > 0 such that 1 C 2ı  Im z0  2ı. For each  2 .0; 1/ and any  2 C with 0 < jj  ı, by the mean value theorem (applied to Re i.z0 Ct  / and Im i.z0 Ct  / in t 2 Œ0; 1), there are 1 , 2 in the segment joining z0 and z0 C  such that i.z0 C / 

iz0

D Re.iz /0 .1 / C i Im.iz /0 .2 / D Re.i i1 log / C i Im.i i2 log /;

218

Appendix Positive self-adjoint operators and positive quadratic forms

so that ˇ i.z0 C / ˇ ˇ 

iz0

ˇ  ˇ i iz0 log ˇ  2 max jiz jz z0 jı

 iz0 j j log j:

Note that if jz z0 j  ı, then Im z  Im z0 C ı  ı and Im z  Im z0 ı  Hence, when 0 <   1, one has     max jiz iz0 j j log j  max . Im z C  Im z0 / j log j jz z0 jı

(A.1) 1 C ı.

Im z ı

 2ı j log j; and when   1, one has    max jiz iz0 j j log j  jz z0 jı

max

Im z 1Cı

 21

ı

.

(A.2)

Im z

log  D 2

C

Im z0

 / log 

log  : ı

(A.3)

Set K0 WD sup ı j log j < 1;

log  < 1: ı 1 

K1 WD sup

0 0:

./2 d./ < 1 by condition (ii) and by (A.1)–(A.3) we have ˇ i.z0 C / ˇ ˇ 

iz0

ˇ ˇ i iz0 log ˇ  4./;

 > 0:

since jiz0 log j  ./ for  > 0Rsimilarly to (A.2) and (A.3), it follows that RMoreover, 1 1 iz0 log j2 d./ < 1, so that 0 WD 0 i iz0 log  dE  2 H is defined. We 0 j then have

f .z C / f .z /

2 Z 1 ˇ i.z0 C / iz0 ˇ2

ˇ ˇ 0 0 0 D i iz0 log ˇ d./ ! 0

ˇ   0 as ı  jj ! 0 by Lebesgue’s convergence theorem, which implies that f 0 .z0 / D 0 . Hence f .z/ is strongly analytic in 1 < Im z < 0. (iii) H) (i). Let f .z/ be as given in (iii), and set '.z/ WD h; f .z/i, 1  Im z  0, which is bounded continuous on 1  Im z  0 and analytic in 1 < Im z < 0. Note that Z Z 1

'.t / D 0

i t d./ D

1

1

e i st d.s/;

t 2 R;

(A.4)

219

A.1 Positive self-adjoint operators

that is, we have '.t / D b  . t /, where b  is the Fourier transform of . For n 2 N 2 2 consider an entire function Gn .z/ WD pn e n z =2 , z 2 C. Then it is well known that 2 R1 1 Gn .t / dt D 1 and Z 1 s2 b G n .s/ WD e i st Gn .s/ ds D e 2n2 ; s 2 R: (A.5) 1

For any ı 2 .0; 1=2/ take the contour integral of Gn .z/'. i z/ along the rectangle joining R i ı, R i ı, R i.1 ı/ and R i.1 ı/, and then take the limit as R ! 1 to obtain Z 1 Z 1 Gn .t i ı/'. t i.1 ı// dt D Gn .t i.1 ı//'. t i ı/ dt: 1

1

By Lebesgue’s convergence theorem, letting ı & 0 gives Z 1 Z 1 Gn .t /'. t i / dt D Gn .t i /'. t / dt 1 1 Z 1  Z 1 i st D Gn .t i / e d.s/ dt 1 1  Z 1 Z 1 i st D Gn .t i /e dt d.s/ 1 1  Z 1 Z 1 i s.t Ci / D Gn .t /e dt d.s/ 1 1  Z 1 Z 1 D Gn .t /e i st dt e s d.s/ 1 1 Z 1 b n .s/e s d.s/: D G

(by (A.4))

1

By (A.5), letting n ! 1 yields Z

1

'. i / D

e s d.s/ D

1

Z

 d./; 0

1

R1 which implies that 0  d./ < 1 and so  2 D.A1=2 /. Hence, from the above proof of (i) H) (iv) and the uniqueness of analytic continuation it follows that f .z/ D Aiz  for 1=2  Im z  0. Next, for any  2 D.A/ let g.z/ WD Aiz  for 1  Im z  0. From (i) H) (iv), g.z/ is bounded and strongly continuous on 1  Im z  0 and strongly analytic in 1 < Im z < 0. Define F .z/ WD hg.z

i /; f .z/i;

1  Im z  0:

It is easy to verify that F .z/ is bounded continuous on 1  Im z  0 and analytic in 1 < Im z < 0 (because g is strongly analytic and f is weakly analytic in 1 < Im z < 0).

220

Appendix Positive self-adjoint operators and positive quadratic forms

For any r 2 .0; 1=2/ we have F . i r/ D hg. i.1

r//; f . i r/i D hA1 r ; Ar i D hA; i D F .0/;

which implies that F .z/  F .0/ for 1  Im z  0. Hence we have hA; i D F .0/ D F . i / D h; f . i /i: Since this holds for all  2 D.A/, we have  2 D.A/ and A D f . i /. Therefore, (i) follows and the last statement has been shown as well.  From the analytic continuation characterization in (iii) and (iv) above, one can show the following (similarly to the proof of Theorem 7.10(1) of Section 7.2). Proposition A.8. Let A be a positive self-adjoint operator on H . Let ˛; ˇ  0. Then we have A˛Cˇ D A˛ Aˇ ; more precisely, for  2 H ,  2 D.A˛Cˇ / if and only if  2 D.Aˇ / and Aˇ  2 D.A˛ /, and in this case, A˛Cˇ  D A˛ .Aˇ /. (Either convention A0 D 1 or A0 D s.A/ is available here.) Moreover, we have .A 1 /˛ D .A˛ / 1 (simply denoted by A ˛ ) and A .˛Cˇ / D A ˛A ˇ . For Borel functional calculus we have general formulas like .f C g/.A/ D f .A/ C g.A/;

.fg/.A/ D f .A/g.A/:

A point of the above proposition is that A˛Cˇ D A˛ B ˇ holds without closure for ˛; ˇ 2 R with ˛ˇ  0. The equality further extends to all ˛; ˇ 2 C with Re ˛
Re ˇ  0.

A.2 Positive quadratic forms An important aspect of positive self-adjoint operators on H is their correspondence to closed (densely defined) positive quadratic forms on H . We begin with their definition. Definition A.9. (1) A function qW D.q/ ! Œ0; 1/, where D.q/ is a dense subspace of H , is called a positive quadratic form on H if (a) q./ D jj2 q./,  2 D.q/,  2 C, (b) q. C / C q.

/ D 2q./ C 2q./, ;  2 D.q/.

(2) A positive quadratic form q is said to be closed if ¹n º  D.q/,  2 H , kn k ! 0 and q.n m / ! 0 as n; m ! 1, then  2 D.q/ and q.n / ! 0. Lemma A.10. Let qW D.q/ ! Œ0; 1/ be a positive quadratic form on H . Define q.; / WD

3 1X k i q. C i k /; 4 kD0

;  2 D.q/:

(A.6)

221

A.2 Positive quadratic forms

Then q.; / is a positive sesquilinear form on D.q/ such that q./ D q.; / for all  2 D.q/. Hence q./1=2 is a seminorm on D.q/, so that jq.1 /1=2

q.2 /1=2 j  q.1

Moreover, if ; n 2 D.q/ (n 2 N) and q.n

2 /1=2 ;

1 ; 2 2 D.q/:

/ ! 0, then q.n / ! q./.

Proof. The well-known Jordan–von Neumann theorem says that a (semi-) norm on a complex vector space arises from a (semi-) inner product if and only if the (semi-) norm satisfies the parallelogram law. Condition (b) of Definition A.9(1) is the parallelogram law for q./1=2 and (A.6) is the usual polarization formula to define a semi-inner product. Even when q./1=2 is not assumed to be a seminorm, the usual proof of the Jordan–von Neumann theorem can work to prove that q.; / is s sesquilinear form on D.q/. Although the details are left as an exercise to the reader, a point we have to check here is that the function  > 0 7! q.; / is continuous. For this, from conditions (a) and (b) one can see that for every ;  2 D.q/, the function  > 0 7! q. C / is midpoint convex and locally bounded. Hence it is continuous; see, e.g., [37, §3.18]. The remaining assertions of the lemma are now obvious.  The equivalence of (i) and (iii) in the next theorem is the most fundamental representation result for positive quadratic forms; see [54, Chap. 6, §2.6] for more details. The equivalence of (i) and (ii) was first given in [85, Thm. 2].1 Theorem A.11. Let q be a positive quadratic form on H . Then the following conditions are equivalent: (i) q is closed; (ii) qz is lower semicontinuous on H , where qz is the extension of q as ´ q./ if  2 D.q/; qz./ WD 1 if  2 H n D.q/I

(A.7)

(iii) there exists a positive self-adjoint operator A on H such that D.A1=2 / D D.q/ and q./ D kA1=2 k2 ;  2 D.A1=2 /: Moreover, A in condition (iii) is unique. Proof. (iii) H) (i). Assume that q is as given in (iii) by a positive self-adjoint operator R1 A with the spectral decomposition A D 0  dE . First let us confirm that q is a positive quadratic form. Property (a) of Definition A.9(1) is obvious. For ;  2 D.A1=2 / one has Z 1 Z 1 2 q. C / C q. / D  dkE . C /k C  dkE . /k2 0 0 Z 1 Z 1 2  dkE k C 2  dkE k2 D 2q./ C 2q./; D2 0

0

1In [85], Simon stated that the equivalence of (i) and (ii) in Theorem A.11 was Kato’s unpublished result.

222

Appendix Positive self-adjoint operators and positive quadratic forms

since kE . /k2 C kE . /k2 D 2kE k2 C 2kE k2 . So (b) holds as well. Moreover, the closedness of q immediately follows from that of A1=2 . (i) H) (iii). Assume that q is a closed positive quadratic form on H . It is easy to see (an exercise) that D.q/ becomes a Hilbert space with the inner product h; iq WD h; i C q.; /;

;  2 D.q/;

where q.; / is given in (A.6). For every  2 H , since jh; ij  kk kk  kk kkq , where kkq WD h; i1=2 q , the linear functional  2 D.q/ 7! h; i is bounded on D.q/. By the Riesz theorem there is a B 2 D.q/ such that h; i D hB; iq D hB; i C q.B; /;

 2 D.q/:

(A.8)

It follows from (A.8) that B is an injective linear operator from H into D.q/. If  2 D.q/ satisfies hB; iq D 0 for all  2 H , then h; i D 0 for all  2 H and hence  D 0. This means that R.B/ (the range of B) is dense in D.q/ and so dense in H . Hence we have a densely defined operator B 1 W D.B 1 / D R.B/ ! H . Define A WD B

1



;

 2 D.A/ WD R.B/:

It then follows from (A.8) that q.; / D hB

1

; i

h; i D hA; i;

 2 D.A/  D.q/;  2 D.q/:

(A.9)

Hence hA; i D q.; /  0 for all  2 D.A/, so A is positive. This implies that A and hence 1 C A D B 1 are symmetric. For every  2 H , letting  WD B, we have .1 C A/ D  and hB; i D h; .1 C A/i D h.1 C A/; i D h; Bi; which means that B is symmetric. The well-known Hellinger–Toeplitz theorem (see [77, p. 84], an easy corollary of the closed graph theorem) says that a symmetric operator defined on the whole H is a bounded self-adjoint operator. Hence B is a bounded selfadjoint operator, which implies that B 1 D 1 C A is self-adjoint and so A is self-adjoint (and positive). Moreover, by (A.9) we have q./ D kA1=2 k2 for all  2 D.A/  D.A1=2 /. For every  2 D.q/, since D.A/ is dense in D.q/ with h
;
iq , there is a sequence n 2 D.A/ such that kn k ! 0 and q.n / ! 0. By Lemma A.10 note that kA1=2 n

A1=2 m k D q.n

m /1=2  q.n

/1=2 C q.

as n; m ! 1. Since A1=2 is closed,  2 D.A1=2 / and kA1=2 n

m /1=2 ! 0

A1=2 k ! 0, so that

kA1=2 k2 D lim kA1=2 n k2 D lim q.n / D q./ n

n

by Lemma A.10 for the last equality above. On the other hand, for every  2 D.A1=2 /, since D.A/ is a core of A1=2 , there is a sequence n 2 D.A/  D.q/ such that kn k ! 0,

223

A.2 Positive quadratic forms

kA1=2 n A1=2 k ! 0 and q.n m / D kA1=2 n A1=2 m k2 ! 0. Since q is closed,  2 D.q/ and q.n / ! 0. Therefore, D.A1=2 / D D.q/ and (iii) has been shown. (ii) H) (i). Assume (ii). Let ¹n º  D.q/ and  2 H be such that kn and q.n m / ! 0 as n; m ! 1. For any " > 0 choose an n0 such that q.n for all n; m  n0 . From (ii) we find that for every n  n0 , qz.n

/  lim inf qz.n

k ! 0 m /  "

m /  ":

m

This in particular implies that n  2 D.q/ so that  2 D.q/ and q.n n  n0 . Therefore, q.n / ! 0 and (i) follows.

/  " for all

(iii) H) (ii). Assume (iii). To show (ii), it suffices to prove that qz./ D sup¹jhA; ij2 W  2 D.A/; hA; i D 1º

(A.10)

for all  2 H . Let  2 D.A/  D.A1=2 / with hA; i D 1. For every  2 D.q/ D D.A1=2 / one has jhA; ij2 D jhA1=2 ; A1=2 ij2  kA1=2 k2 kA1=2 k2 D hA; iq./ D qz./: If  2 H n D.q/, then jhA; ij2  1 D qz./. Therefore, qz./  RHS of (A.10) for all 2H. R1 On the other hand, taking the spectral decomposition A D 0  dE , for every  2 H one has Z hAEn ; i D

n 2

1

Z

 dkE k % 0

 dkE k2 D qz./:

0

If hAEn ; i D 0 for all n, then qz./ D 0. Otherwise, with n WD hAEn ; i D.A/ for n large, one has hAn ; n i D hAEn ; i jhAn ; ij2 D jhAEn ; i

1=2

1

1=2

En  in

hAEn ; i D 1;

hAEn ; ij2 D hAEn ; i % qz./:

Therefore, (A.10) holds. Finally, the uniqueness of A in condition (iii) is immediately seen from Lemma A.1.  Let p, q be positive quadratic forms on H . It is said that p is an extension of q if D.p/  D.q/ and p./ D q./ for all  2 D.q/. It is said that q is closable if q has a closed extension. The proof of the next theorem was given in [17].

224

Appendix Positive self-adjoint operators and positive quadratic forms

Theorem A.12. Let qW D.q/ ! Œ0; 1/ be a positive quadratic form on H . Then the following conditions are equivalent: (i) q is closable; (ii) if ¹n º  D.q/, kn k ! 0 and q.n

m / ! 0 as n; m ! 1, then q.n / ! 0;

(iii) q is lower semicontinuous on D.q/. In this case, there exists a closed extension q such that any closed extension of q is also an extension of q (and q is called the closure of q). Proof. (i) H) (ii). Assume that q is closable, and let p be a closed extension of q. Let ¹n º  D.q/, kn k ! 0 and q.n m / ! 0 as n; m ! 1. Then q.n / D p.n 0/ ! 0 by Definition A.9(2) for p. (ii) H) (i). Assume that (ii) holds, and define D.q/ WD ¹ 2 H W there exists a sequence n 2 D.q/ such that kn

k ! 0 and q.n

m / ! 0 as n; m ! 1º

(A.11)

and q./ WD lim q.n / n!1

for  2 D.q/:

(A.12)

Here, the limn q.n / above exists by Lemma A.10. Moreover, this limit is independent of the choice of ¹n º; indeed, if ¹n º  D.q/ is another sequence as above, then kn n k ! 0 and q..n n / .m m //  2q.n m / C 2q.n m / ! 0, so condition (ii) implies that q.n n / ! 0 and so limn q.n / D limn q.n /. It is clear that D.q/  D.q/ and q./ D q./ for all  2 D.q/. Let us prove that q is a closed positive quadratic form on H . For every ;  2 D.q/ choose ¹n º; ¹n º  D.q/ such that kn k ! 0, kn k ! 0, q.n m / ! 0 and q.n m / ! 0 as n; m ! 1. Then for every ;  2 C, k.n C n /

. C /k ! 0;

.m C m //1=2  jjq.n

q..n C n /

m /1=2 C jjq.n

m /1=2 ! 0;

so that  C  2 D.q/. Hence D.q/ is a dense subspace of H . Since q./ D lim q.n / D jj2 lim q.n / D jj2 q./; n

q. C / C q.

n

/ D lim¹q.n C n / C q.n n

n /º

D lim¹2q.n / C 2q.n /º D 2q./ C 2q./; n

it follows that q is a positive quadratic form on H . Furthermore, note that lim q.n n

/ D lim lim q.n

if n is a sequence as in (A.11).

n

m

m / D 0;

 2 D.q/;

(A.13)

225

A.2 Positive quadratic forms

To show the closedness of q, let ¹n º  D.q/,  2 H , kn k ! 0 and q.n m / ! 0 as n; m ! 1. For each n, apply (A.13) to  D n 2 D.q/ to find a n 2 D.q/ such that kn n k < 1=n and q.n n / < 1=n. Then kn k ! 0 and m /1=2 D q.n

q.n

 q.n

m /1=2 n /1=2 C q.n

m /1=2 C q.m

m /1=2 ! 0

as n; m ! 1. Hence we have  2 D.q/ and lim sup q.n n

/1=2  lim sup¹q.n n

n /1=2 C q.n

D lim sup lim q.n n

m

/1=2 º

m /1=2 D 0;

where q.n / D limm q.n m / holds since ¹n m ºm satisfies the condition in (A.11) for n . Therefore, q.n / ! 0, so that q is a closed extension of q. (i) H) (iii). Let p be a closed extension of q. Then pz is lower semicontinuous on H by Theorem A.11. Since q D pz on D.q/, it follows that q is lower semicontinuous on D.q/. (iii) H) (ii). Assume (iii). Let ¹n º  D.q/, kn k ! 0 and q.n m / ! 0 as n; m ! 1. For every " > 0 choose an n0 such that q.n m /  " for all n; m  n0 . From (iii) we find that q.n /  lim inf q.n m

m /  ";

n  n0 ;

so that q.n / ! 0 and (ii) follows. Finally, we show that q given in the above proof of (ii) H) (i) is the smallest closed extension of q. For this, let p be any closed extension of q. If  2 D.q/, then by the definition of q in (A.11) and (A.12) there is a sequence n 2 D.q/  D.p/ such that kn k ! 0 and p.n m / D q.n m / ! 0, so  2 D.p/ and p.n / ! 0. Hence q./ D limn q.n / D limn p.n / D p./. Hence the result follows.  Remark A.13. Let q be a positive quadratic form on H and assume that q is lower semicontinuous on D.q/. Then by Theorems A.12 and A.11 there exists a unique positive self-adjoint operator A1 on H such that D.A1=2 1 / D D.q/ and 2 q./ D kA1=2 1 k ;

 2 D.q/:

In particular, D.A1=2 1 /  D.q/

2 and q./ D kA1=2 1 k ;

 2 D.q/:

(A.14)

Furthermore, note that D.q/ is a core of A1=2 and A1 is the largest (in the sense of 1 Definition A.2) positive self-adjoint operator on H satisfying (A.14). Indeed, that D.q/

226

Appendix Positive self-adjoint operators and positive quadratic forms

is immediately seen from (A.11) and (A.13). Let A be any positive selfis a core of A1=2 1 adjoint operator satisfying (A.14). For every  2 D.A1=2 1 / D D.q/ there is a sequence n 2 D.q/ such that kn k ! 0 and q.n / ! q./. Hence one has 2 kA1=2 k2  lim inf kA1=2 n k2 D lim q.n / D q./ D kA1=2 1 k ; n

n

implying that A  A1 in the sense of Definition A.2. In view of condition (ii) of Theorem A.11, it is convenient to reformulate positive quadratic forms as those defined on the whole H but allowed to have the value 1 in the following way. Definition A.14. We call a function qW H ! Œ0; 1 a positive form on H if properties (a) and (b) of Definition A.9(1) hold for all ;  2 H , with the convention 01 D 0. Here we use the term “positive form” instead of “positive quadratic form” to distinguish the present definition from Definition A.9(1). Lemma A.15. Let qW H ! Œ0; 1 be a positive form on H , and set D.q/ WD ¹ 2 H W q./ < 1º. Then K WD D.q/ is a closed subspace of H and qjD.q/ is a positive quadratic form on K . Conversely, let qW D.q/ ! Œ0; 1/ be a positive quadratic form on a closed subspace of H , and set qzW H ! Œ0; 1 by (A.7). Then qz is a positive form. Proof. For the first assertion, it suffices to show that D.q/ is a subspace of H . But this is clear from (a) and (b) for q. Next let q and qz be as stated in the second assertion. We show that qz satisfies (a) and (b) for all ;  2 H . Obviously, when  D 0, both sides of (a) for qz are 0 for all  2 H . When  ¤ 0, both sides of (a) for qz are 1 for all  2 H n D.q/. Hence qz satisfies (a) for all  2 H . When  62 D.q/ or  62 D.q/, it is clear that  C  62 D.q/ or   62 D.q/, so that both sides of (b) are 1. Hence qz satisfies (b) for all ;  2 H .  The next theorem is a reformulation of Theorem A.11. Theorem A.16. Let q be a positive form on H in the sense of Definition A.14. Set D.q/ WD ¹ 2 H W q./ < 1º and K WD D.q/. Then the following conditions are equivalent: (i) qjD.q/ is a closed positive quadratic form on K in the sense of Definition A.9; (ii) q is lower semicontinuous on H ; (iii) there exists a positive self-adjoint operator A on K such that D.A1=2 / D D.q/ and ´ kA1=2 k2 if  2 D.A1=2 /; q./ D (A.15) 1 if  2 H n D.A1=2 /: Moreover, A in condition (iii) is unique.

A.2 Positive quadratic forms

227

Proof. It is clear that (ii) is equivalent to the lower semicontinuity of qjK . Hence the equivalence between (i)–(iii) immediately follows from Lemma A.15 and Theorem A.11 applied to a positive quadratic form qjD.q/ on K . Moreover, formula (A.15) implies that D.A1=2 / D D.q/, so the uniqueness of A follows from that in Theorem A.11.  We write q D qA for the positive form determined by A as in (iii) above. In view of Remark 7.4 for M D B.H /, such an A is regarded as a generalized positive operator in B .H /C , so Theorem A.16 shows a one-to-one correspondence between the set of lower semicontinuous positive forms on H and B .H /C .

2

2

Example A.17. Here we recall the notion of form sums of two positive self-adjoint operators (on closed subspaces of H ), which was introduced in [54]. Let A, B be positive self-adjoint operators on some respective closed subspaces of H (or A; B 2 B .H /C ; see Remark 7.4). Define D.q/ WD D.A1=2 / \ D.B 1=2 / and ´ kA1=2 k2 C kB 1=2 k2 if  2 D.q/; q./ WD 1 if  2 H n D.q/:

2

Then it is immediate to see that q is a positive form on H satisfying condition (ii) of Theorem A.16. Hence there exists a unique positive self-adjoint operator C on D.q/ such that D.C 1=2 / D D.q/ and kC 1=2 k2 D kA1=2 k2 C kB 1=2 k2 ;

 2 D.A1=2 / \ D.B 1=2 /:

P B and called the That is, C is determined by the equality qC D qA C qB , denoted by A C form sum of A and B (to distinguish it from the operator sum A C B with D.A C B/ D D.A/ \ D.B/). In the rest of the section we further discuss relations between lower semicontinuous positive forms on H and their representing generalized positive operators. Let A 2 B .H /C and take the spectral decomposition Z 1 AD  dE C 1P;

2

0

where .E /0 is a spectral resolution on a closed subspace K of H and P is the projection onto K ? ; see Section 7.1. We define Z 1 1 .1 C A/ 1 WD 0P C dE ; 1 C  0 Z 1  1 A.1 C A/ WD dE C 1P D 1 .1 C A/ 1 : (A.16) 1 C  0 The next proposition extends Lemma A.1 to generalized positive operators.

228

Appendix Positive self-adjoint operators and positive quadratic forms

2

Proposition A.18. For A; B 2 B .H /C the following conditions are equivalent: (i) qA ./  qB ./ for all  2 H , that is, D.B 1=2 /  D.A1=2 / and kA1=2 k2  kB 1=2 k2 for all  2 D.B 1=2 /; (ii) .1 C B/

1

(iii) A.1 C A/

 .1 C A/ 1

1

;

 B.1 C B/

1

.

Proof. (ii) ” (iii) is obvious from (A.16). Next let K be a closed subspace of H given above for A, and L be similar for B. Note that D.A1=2 / D D..1 C A/1=2 / D R..1 C A/ 1=2 / is dense in K and D.B 1=2 / D D..1 C B/1=2 / D R..1 C B/ 1=2 / is dense in L . Then the proof of (i) ” (ii) is similar to that of (i) ” (iii) of Lemma A.1 with modifications as follows: A contraction C in the proof of (i) H) (iii) of Lemma A.1 is given as C W L ! K and extended to H as C D 0 on L ? . Also, C in the proof of (iii) H) (i) of Lemma A.1 is given as C W K ! L and extended to H as C D 0 on K ?. 

2

For every A 2 B .H /C the following variational formula was observed in [60, p. 400]: h; A.1 C A/

1

i D inf¹kk2 C qA ./ W  D  C º;

2H;

which explicitly shows (i) H) (iii) of Proposition A.18. We write A  B if the conditions of Proposition A.18 hold. In fact, condition (i) means that A.! /  B.! / for all  2 H , so that A.'/  B.'/ for all ' 2 B.H /C  , where A, B are considered in the original sense of Section 7.1. Furthermore, for A; An 2 B .H /C (n 2 N), extending Definition A.6 we say that An converges to A in the strong resolvent sense if .1 C An / 1 ! .1 C A/ 1 strongly.

2

2

2

Proposition A.19. Let An 2 B .H /C , n 2 N, be such that A1  A2 


. Then there exists an A 2 B .H /C such that An ! A in the strong resolvent sense and qA ./ D limn qAn ./ (D supn qAn ./) for all  2 H . Proof. Since .1 C An / 1 is decreasing, we have the strong limit T WD limn .1 C An / 1 R1 so that 0  T  1. With the spectral decomposition T D 0  dF , one can define an A 2 B .H /C by Z

2

A WD

.

1

1/ dF C 1F0 :

.0;1

Then it is easy to verify that T D .1 C A/ 1 , so that An  A for all n and An ! A in the strong resolvent sense. On the other hand, since qAn ./ is increasing by Proposition A.18, one can define q0 ./ WD lim qAn ./ D sup qAn ./; n

n

2H:

By definition it is clear that q is a positive form on H and lower semicontinuous. Hence by Theorem A.16 there exists an A0 2 B .H /C such that q D qA0 . Since

2

A.2 Positive quadratic forms

229

qAn  qA0 , Proposition A.18 gives An  A0 for all n, which implies that A  A0 . Since qAn  qA  qA0 and qAn ! qA0 , we have qA D qA0 and hence A D A0 by the uniqueness assertion of Theorem A.16. Thus the result has been shown.  Remark A.20. Proposition A.19 says that the limit of an increasing sequence of lower semicontinuous positive forms on H is compatible with the limit in the strong resolvent sense of the corresponding generalized positive operators. However, this is not true for a decreasing sequence. Let An 2 B .H /C be such that A1  A2 


. As in the increasing case, there is an A 2 B .H /C such that .1 C An / 1 % .1 C A/ 1 strongly. But this does not imply that qA ./ D inf n qAn ./, because q0 ./ WD inf n qAn ./ is not necessarily lower semicontinuous on H . In the decreasing case, it is known (see [86, §7.5], [60]) that qA is the maximum of all the lower semicontinuous positive forms q satisfying q./  q0 ./ for all  2 H .

2

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Index

C  -algebra, 2

-homogeneous, 203 K-functional, 54 Lp –Lq -duality, 69, 153, 209 Lp -space Haagerup’s, 145 Kosaki’s, 158 spatial, 203 with respect to a trace, 57 M 0 -support, 163 M -support, 163 n-positive map, 71 !-conditional expectation, 82 !-dual map, 81 ‰-bounded, 181 ‰-integral, 203, 208 -exponential type, 105 -finite von Neumann algebra, 13 -weak topology, 1 S -set, 6 -compact, 55 -dense, 41 -measurable, 42 T -set, 6 W  -algebra, 2 W  -dynamical system, 123 W  -probability space, 11 AFD von Neumann algebra, 8 affiliated with, 39 analytic element, 21 analytic generator, 101 approximate identity, 88 Araki–Woods factor, 6 atomic von Neumann algebra, 4 balanced functional, 173 balanced weight, 90 Borel functional calculus, 211

canonical trace, 136 Carlson’s theorem, 107, 203 center, 2 centralizer, 23, 111 chain rule, 117, 178 Ciorˇanescu and Zsidó’s theorem, 100, 216 Clarkson’s inequality, 68 closable positive quadratic form, 223 closed positive quadratic form, 220 cocycle identity, 92, 176 commutant, 2 commutant theorem, 3 commutativity for normal positive functionals, 180 compatible couple, 157 completely positive map, 71 complex interpolation method, 157 complex interpolation space, 157 conditional expectation, 74, 76 Connes’ cocycle derivative, 92, 174 Connes’ inverse theorem, 92 continuous crossed product decomposition, 137 covariance property, 123 crossed product, 7, 124 second, 126 cyclic vector, 13

decreasing rearrangement, 47 discrete crossed product decomposition, 137 discrete von Neumann algebra, 4 domain, 211 double commutation theorem, 2 dual action, 7, 125 dual weight, 131

240 ergodic action, 124 exact interpolation functor of exponent , 157 extended positive part, 95 factor, 4 Fatou’s lemma, 52 finite von Neumann algebra, 4 fixed-point, 82 flow of weights, 8, 137 form sense, 45, 213 form sum, 227 free entropy, 11 free group factor, 11 free probability theory, 11 free product, 11 Fuglede–Kadison determinant, 56 generalized conditional expectation, 82 generalized positive operator, 95 generalized s-number, 47 GNS construction, 3, 5, 13, 88 GNS cyclic representation, 13, 32 group measure space construction, 8 group von Neumann algebra, 124 Hölder’s inequality, 64, 149, 209 Haagerup’s L1 -space, 145 Haagerup’s Lp -space, 145 Hilbert–Schmidt class, 58 hyperfinite von Neumann algebra, 8 injective factor, 137 injective von Neumann algebra, 8 ITPFI factor, 6 Jones’ index, 10, 202 Kaplansky density theorem, 2 KMS condition, 5, 21 Kosaki’s Lp -space, 158 Krieger factor, 8

Index

left Hilbert algebra, 89 localizable measure space, 46 McCarthy’s inequality, 68 measure topology, 45 Minkowski’s inequality, 61, 148 modular automorphism group, 5, 14, 89 modular conjugation, 5, 14 modular operator, 5, 14, 164 modular theory, 5 module, 9 monotone convergence theorem, 52 multiplicative domain, 82 Murray–von Neumann equivalence, 4 natural positive cone, 9, 25 non-commutative Lp -space, 57 non-commutative Stein–Weiss interpolation theorem, 160 norm-one projection, 76 normal, 2 normal state, 3 obstruction, 9 operator algebra, 1 operator-valued weight, 98 faithful, 99 normal, 99 semifinite, 99 outer period, 9 Pedersen–Takesaki construction, 111 Petz recovery map, 81 positive form, 226 positive map, 71 positive quadratic form, 220 closable, 223 closed, 220 positive self-adjoint operator, 211 Powers factor, 6 predual, 3 properly infinite von Neumann algebra, 4 purely infinite von Neumann algebra, 4

Index

Radon–Nikodym derivative, 70, 119 reduction theory, 5 relative modular operator, 163 representation, 2 Riesz–Thorin theorem, 157 right Hilbert algebra, 89 rigidity theory, 11 Schatten–von Neumann p-class, 58 Schwarz map, 71 second crossed product, 126 semifinite von Neumann algebra, 4 separating vector, 13 singular value, 47 smooth flow of weights, 137 spatial derivative, 183, 189 spatial Lp -space, 203 standard form, 9, 28 *-representation, 2 *-subalgebra, 1 Stinespring representation theorem, 74 Stone’s representation theorem, 101, 119, 195, 216 strictly semifinite, 120 strong resolvent sense, 46, 216, 228 structure theorem, 137 subalgebra, 1 subfactor theory, 10 support M -, 163 M 0 -, 163

Takesaki’s duality theorem, 7, 126 tensor product Hilbert space, 3 von Neumann algebra, 3 Tomita algebra, 89 Tomita’s fundamental theorem, 5, 14 Tomita–Takesaki theory, 5 trace, 3 trace class, 58 trace scaling property, 136 triangle inequality, 46 uniformly convex, 68 vector representative, 33 von Neumann algebra, 2 -finite, 13 AFD, 8 atomic, 4 discrete, 4 finite, 4 hyperfinite, 8 injective, 8 of type III, 4 properly infinite, 4 purely infinite, 4 semifinite, 4 von Neumann density theorem, 2 weight, 3, 87 faithful, 87 normal, 87 semifinite, 87

241

EMS SERIES OF LECTURES IN MATHEMATICS

Fumio Hiai

Lectures on Selected Topics in von Neumann Algebras The theory of von Neumann algebras, originating with the work of F. J. Murray and J. von Neumann in the late 1930s, has grown into a rich discipline with connections to different branches of mathematics and physics. Following the breakthrough of Tomita–Takesaki theory, many great advances were made throughout the 1970s by H. Araki, A. Connes, U. Haagerup, M. Takesaki and others. These lecture notes aim to present a fast-track study of some important topics in classical parts of von Neumann algebra theory that were developed in the 1970s. Starting with Tomita–Takesaki theory, this book covers topics such as the standard form, Connes’ cocycle derivatives, operator-valued weights, type III structure theory and non-commutative integration theory. The self-contained presentation of the material makes this book useful not only to graduate students and researchers who want to know the fundamentals of von Neumann algebras, but also to interested undergraduates who have a basic knowledge of functional analysis and measure theory.

https://ems.press ISBN 978-3-98547-004-4