Amenable Banach Algebras: A Panorama [1 ed.] 1071603493, 9781071603499

Table of contents :
Preface
Introduction
Contents
0 Paradoxical Decompositions
0.1 The Banach–Tarski Paradox
0.2 Tarski's Theorem
1 Amenable, Locally Compact Groups
1.1 Invariant Means and Asymptotic Invariance Properties
1.2 Hereditary Properties
1.3 Uniformly Bounded Representations
1.4 Leptin's Theorem
1.5 Fixed Point Theorems
2 Amenable Banach Algebras
2.1 Derivations from Group Algebras
2.2 Virtual and Approximate Diagonals
2.3 Hereditary and Splitting Properties
2.4 A First Look at Hochschild Cohomology
3 Examples
3.1 Measure Algebras
3.2 Fourier and Fourier–Stieltjes Algebras
3.3 Algebras of Approximable Operators
3.4 (Non-)Amenability of mathcalB(E)
3.5 An Amenable Radical Banach Algebra
4 Amenability-Like Properties
4.1 Contractibility
4.2 Weak Amenability
4.3 Character Amenability
4.4 Pseudo- and Approximate Amenability
4.5 Biflatness and Biprojectivity
5 Dual Banach Algebras
5.1 Connes-Amenability for Dual Banach Algebras
5.2 The Case of the Measure Algebra
5.3 Connes-Amenability without a Normal, Virtual Diagonal
5.4 Daws' Representation Theorem
5.5 Connes-Amenability and Connes-Injectivity
6 Banach Homological Algebra
6.1 Projectivity
6.2 Resolutions and Ext-Groups
6.3 Flatness and Injectivity
7 Operator Algebras on Hilbert Spaces
7.1 Amenable von Neumann Algebras
7.2 Injective von Neumann Algebras
7.3 Nuclear Cast-Algebras
7.4 Semidiscrete von Neumann Algebras
7.5 Normal, Virtual Diagonals
7.6 Commutative Operator Algebras
7.7 An Amenable Operator Algebra Not Similar to a Cast-Algebra
8 Operator Amenability
8.1 Operator Amenable, Completely Contractive Banach Algebras
8.2 Fourier Algebras
8.3 Fourier–Stieltjes Algebras
A Banach Spaces
A.1 Bases in Banach Spaces
A.2 Approximation Properties
A.3 The Radon–Nikodým Property
A.4 Local Theory
B Banach Algebras
B.1 Spectra and Gelfand Theory
B.2 Banach Modules and Bounded Approximate Identities
B.3 Multiplier Algebras
B.4 Prime and Primitive ideals
B.5 Structure of Semiprime and Semisimple Banach Algebras
C Cast- and von Neumann Algebras
C.1 ast-Algebras and -Homomorphisms
C.2 Cast-Algebras
C.3 Positivity in Cast-Algebras and Their Duals
C.4 ast-Representations of Cast-Algebras
C.5 von Neumann Algebras and Wast-Algebras
C.6 Multipliers of Cast-Algebras
C.7 Projections in von Neumann Algebras
C.8 Tensor Products of Cast- and von Neumann Algebras
C.9 Weights on von Neumann algebras
D Abstract Harmonic Analysis
D.1 Semitopological Semigroups and Locally Compact Groups
D.2 The Group Algebra
D.3 The Measure Algebra M(G)
D.4 Other Banach Algebras Associated with Locally Compact Groups
E Operator Spaces
E.1 Concrete and Abstract Operator Spaces
E.2 Completely Bounded Maps
E.3 Duality
E.4 The Projective Tensor Product of Operator Spaces
E.5 Completely Contractive Banach Algebras
F Fourier and Fourier–Stieltjes Algebras
F.1 Representations of Locally Compact Groups
F.2 The Fourier Algebra
F.3 The Fourier–Stieltjes Algebra
F.4 Cosets, Idempotents, and Piecewise Affine Maps
References
Index of Symbols
Index
Index of Names

Citation preview

Springer Monographs in Mathematics

Volker Runde

Amenable Banach Algebras A Panorama

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A. Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

More information about this series at http://www.springer.com/series/3733

Volker Runde

Amenable Banach Algebras A Panorama

123

Volker Runde Department of Mathematical and Statistical Science University of Alberta Edmonton, AB, Canada

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-1-0716-0349-9 ISBN 978-1-0716-0351-2 (eBook) https://doi.org/10.1007/978-1-0716-0351-2 Mathematics Subject Classification (2010): 46HXX, 22-02, 43-02, 46-02 © Springer Science+Business Media, LLC, part of Springer Nature 2020 This monograph is based on the author’s ‘Lectures on Amenability’ © Springer-Verlag Berlin Heidelberg 2002 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Science+Business Media, LLC part of Springer Nature. The registered company address is: 1 New York Plaza, New York, NY 10004, U.S.A.

Für Mieken (1939–1986) und Charlie (1939–2015), denen ich alles verdanke.

Preface

The notion of amenability has its origins in the beginnings of modern measure theory: Does a finitely additive set function exist which is invariant under a certain group action? Since the 1940s, amenability has become an important concept in abstract harmonic analysis (or rather, more generally, in the theory of semitopological semigroups). In 1972, B. E. Johnson showed that the amenability of a locally compact group G can be characterized in terms of the Hochschild cohomology of its group algebra L1 ðGÞ: this initiated the theory of amenable Banach algebras. Since then, amenability has penetrated other branches of mathematics, such as von Neumann algebras, operator spaces, and even differential geometry. In the summer term of 1999, I taught a course on amenability at the Universität des Saarlandes. My goals were lofty: I wanted to show my students how the concept of amenability originated from measure theoretic problems—of course, the Banach—Tarski Paradox would have to be covered—, how it moved from there to abstract harmonic analysis, how it then ventured into the theory of Banach algebras, and how it impacted areas as diverse as von Neumann algebras and differential geometry. I had also planned to include very recent developments such as C. J. Read’s construction of a commutative, radical, amenable Banach algebra or Z.-J. Ruan’s notion of operator amenability. On top of all this, I wanted my lectures to be accessible to students who had taken a one-year course in functional analysis (including the basics of Banach and C  -algebras), but who had not necessarily any background in operator algebras, homological algebra, or abstract harmonic analysis. Lofty as they were, these goals were unattainable, of course, in a one-semester course…. The present notes are an attempt to resurrect the original plan of my lectures at least in written form.

This quote is the beginning of the introduction to my book Lectures on Amenability ([297]). When [297] appeared, I didn’t waste a thought on a potential second, more permanent, and substantially revised edition. Still, with the years passing, I more and more felt the need for such a project. First of all, an embarrassing number of errors managed to get into [297], most of them minor, but some more serious. Secondly, [297] was my first attempt at writing a book, and it shows. Moreover, when writing [297], I was still learning some of the topics I was trying to explain to my audience—such as von Neumann algebras and operator spaces. Of course, the

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exposition suffered from this fact. With more than a decade and a half of additional experience behind me, I could now write a much better book on the same material. Thirdly—and most importantly—the area of amenable Banach algebras has made enormous progress since the publication of [297]: • The derivation problem for group algebras, which initiated the entire theory of amenable Banach algebras, was eventually solved by V. Losert ([231]), and only a few years ago, an astonishingly simple proof was discovered ([17]). • B. E. Forrest and myself were able to characterize those locally compact groups with an amenable Fourier algebras as those that have an abelian subgroup of finite index ([121]), as had been conjectured since the publication of [193]. • In [6], S. A. Argyros and R. G. Haydon solved the famous “Scalar-plusCompact Problem”. More specifically, they constructed a Banach space E that is a predual of ‘1 such that every bounded linear operator on E is a sum of a compact operator and a scalar multiple of the identity. The Banach algebra BðEÞ of all bounded linear operators on E is amenable as a by-product of this result: this settles an open question raised by B. E. Johnson in his groundbreaking memoir [188]. • On the other hand, Bð‘p Þ turned out to be non-amenable for all p 2 ½1; 1. This had long been known for p ¼ 2, but starting with C. J. Read’s paper [280], in which the case p ¼ 1 was settled, tremendous progress was made that eventually led to a solution for all p 2 ½1; 1 ([271], [257], [86], and [306]). • The area of dual Banach algebras, which seemed to be little more than a zoo of examples at the time [297] appeared, has evolved into a surprisingly rich theory. For instance, dual Banach algebras enjoy a representation theory that parallels very much that of von Neumann algebras, with reflexive Banach spaces taking on the rôle Hilbert spaces play for von Neumann algebras ([80] and [82]). • There is now an example—due to Y. Choi, I. Farah, and N. Ozawa—of an amenable operator algebra that is not isomorphic to a C -algebra ([49]). On the other hand, there is a proof—due to L. W. Marcoux and A. I. Popov—that amenable, commutative operator algebras are always similar to C -algebras ([240]). • Today, nobody in the field doubts anymore that operator amenability, as introduced by Z.-J. Ruan ([290]), is the “right” notion of amenability when it comes to dealing with Fourier, Fourier–Stieltjes, and related algebras. The present book is my reaction to these developments. It is much more than a mere second edition of [297] and consequently has a new title. Each chapter of [297] has been thoroughly reworked—with new material added throughout—in order to reflect what happened since the publication of [297]. Dual Banach algebras, for instance, now have a whole chapter of their own (as opposed to a mere section in [297]). On the other hand, I decided to drop [297, Chapter 8] altogether for two reasons: first, many readers of [297] perceived it as somewhat disconnected with the rest of the book, and secondly, with a lot of new material added, I had to keep the total number of pages within reasonable limits.

Preface

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It is a pleasure for me to thank those who discovered errors in [297] and let me know: M. Daws, N. Grønbæk, M. Mazowita, H. L. Pham, P. Rajendran, A. Skalski, R. Stokke, and A. Trichtchenko. Thanks to their diligence, I could eliminate some of the errors in [297]. Those that remain—and those that are new—are, of course, my fault alone. It is an equally pleasant task to thank those who read the present manuscript at various stages of its development and commented on it: Y. Choi, M. Daws, F. Ghahramani, N. Grønbæk, A. T.-M. Lau, R. J. Loy, Z. A. Lykova, M. S. Monfared, S. Öztop, K. Schlitt, M. Wiersma, G. A. Willis, and Y. Zhang. Finally, special thanks are due to D. P. Blecher: it was he who triggered this whole project by innocently asking if I was planning a second edition of [297]. Edmonton, Canada October 2019

Volker Runde

Introduction

The aim of this book is to give—as indicated in the title—a panorama of the theory of amenable Banach algebras and of its impact on various classes of Banach algebras. This places it in a limbo: it is neither a graduate student text nor a research monograph, but something in between. On the one hand, it contains exercises, and Chapters 1, 2, and 6 provide fairly self-contained introductions to amenable, locally compact groups and Banach algebras, respectively, and to Helemski’s Banach homological algebra. Starting with Chapter 3 (with the exception of Chapter 6), however, the book becomes less self-contained out of necessity. To treat the impact of amenability for particular classes of Banach algebras, substantial background is needed from other areas of mathematics: to grasp what amenability means for Banach algebras of operators on Banach spaces, for example, a certain understanding of Banach space theory is indispensable. Obviously, it would have been impossible to include all the background material necessary to bring the entire book up to the same level of self-containedness as Chapters 0–2, and 6. Chapter 2 is the backbone of the book. It provides an introduction to amenable Banach algebras as first introduced in [188]; in particular, B. E. Johnson’s classic theorem is proven: a locally compact group G is amenable if and only if its group algebra L1 ðGÞ is an amenable Banach algebra The characterization of amenable Banach algebras through virtual and bounded approximate diagonals is treated, and various hereditary properties are discussed. The chapter should be accessible to a student who has completed a one-year functional analysis course, including the basics of Banach and C  -algebras. For easier reference, background from Banach algebra theory is collected in Appendix B. Chapters 0 and 1 are the prelude to Chapter 2. Chapter 0 is a mathematical appetizer. It gives a brief discussion of paradoxical decomposition, leading up to a proof of the Banach–Tarski Paradox. Even though paradoxical decompositions play an important rôle in the historical development of the notion of amenability, the rest of the book is pretty much independent of Chapter 0. The mathematical background required for this chapter is modest: a

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sufficiently advanced undergraduate student should have no problems understanding it. As Johnson’s aforementioned theorem is the starting point for the theory of amenable Banach algebras, and since this theorem involves amenable, locally compact groups, Chapter 1 gives a concise introduction to amenable, locally compact groups, including Dixmier’s result that amenable groups are unitarizable, Leptin’s Theorem on bounded approximate identities in Fourier algebras, and—last, but not least—Day’s Fixed Point Theorem. Of course, some background in abstract harmonic analysis is indispensable for this chapter: see Appendices D and F. In Chapter 3, we add some more flesh to the theory of amenable Banach algebras by looking at various classes of Banach algebras. For the measure algebra MðGÞ and the Fourier algebra AðGÞ of a locally compact group G, we characterize those G for which MðGÞ and AðGÞ, respectively, are amenable. We exhibit amenable (and non-amenable) Banach algebras of compact operators on Banach spaces, present an example of an infinite-dimensional Banach space E for which BðEÞ, the Banach algebra of all bounded linear operators on E, is amenable, and prove that Bð‘p Þ is not amenable for p 2 ½1; 1. Finally, we present C. J. Read’s construction of a commutative, amenable, radical Banach algebra. It lies in the nature of things, that this chapter requires a much broader background from the reader than the previous ones. Some of it can be found in Appendices A, D, and F. There are various ways to tweak the definition of an amenable Banach algebra: such “amenability-like” properties are considered in Chapter 4. As the focus of this book is the theory of amenable Banach algebras, we deal only with a few of these properties and with how they relate to actual amenability: contractibiliy, weak amenability, character amenability, pseudo-amenability, approximate amenability, biprojectivity, and biflatness. (Biprojectivity and biflatness will be put in their proper context in Chapter 6). This chapter—the discussion of biprojectivity, to be precise—requires some background on Banach algebras beyond that for Chapter 2; it can also be found in Appendix B. Chapter 5 is devoted to the Connes-amenability of dual Banach algebras: dual Banach algebras are Banach algebras that are dual Banach spaces such that multiplication is separately weak* continuous. Connes-amenability is a notion weaker than Banach algebraic amenability for dual Banach algebras that takes the dual space structure of those algebras into account. In Sections 5.4 and 5.5, real interpolation of Banach spaces, as expounded in [20], plays a pivotal rôle; in the interest of readability, however, we have striven—wherever possible—to give ad hoc arguments avoiding the explicit use of interpolation theory. In Chapter 6, we present a concise introduction to Banach homological algebra according to Helemski. Beyond a basic familiarity with Banach algebras and modules, not much background is required from the reader. A background in homological algebra will, of course, help, but is not necessary. Of all the chapters of this book, Chapter 7 is certainly the least self contained: its first five section are about characterizing the Connes-amenable von Neumann algebras and, in the process, identifying the amenable C  -algebras as the nuclear ones. Not surprisingly, this requires deep results from von Neumann algebra theory.

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As the focus of this book is on Banach algebras—and C  - and von Neumann algebras are only viewed under the aspect of how they fit into the general theory— we feel justified to refer to the rich literature on those algebras instead of bloating the book with incorporating all the required background material. Some background on C  - and von Neumann algebras is given in Appendix C, but it’s far from exhaustive. When dealing with Banach algebras that arise as preduals of von Neumann algebras, it turns out that Banach algebraic amenability is true strong a notion. If one, however, modifies the definitions in way that takes the operator space structure into account one obtains attractive results. In Chapter 8, we sketch the theory of operator amenable completely contractive Banach algebras—its development very much parallels that of amenable Banach algebras expounded in Chapter 2—and then apply it to Fourier and Fourier–Stieltjes algebras. Background on operator space and Fourier and Fourier–Stieltjes algebras is contained in Appendices E and F. Each chapter ends with a section entitled “Notes and Comments”. These sections contain references to the original literature, outlines of results that were not included in the main text, as well as suggestions for further reading. The book concludes with six appendices: they collect background material that is necessary to understand certain parts of the book, but couldn’t have been incorporated into the main text, either because it is mathematically too far removed from the book’s main thrust or simply because including it would have meant to bloat the exposition to an unreasonable extent. As already indicated, these appendices are not meant to be exhaustive, but rather to serve as repositories for notation as well as for fundamental concepts and results, so that every reader is at least approximately on the same page. Unlike [297], this book does not have an appendix on Banach space tensor products: we only need very basic facts about the projective and the injective tensor product of Banach spaces, which are easily accessible through various sources, such as [91] or [311]. Unless explicitly stated otherwise, all linear spaces in this book are over the complex numbers C, and, of course, the universal disclaimer applies: just because I failed to explicitly attribute a result to someone else, this does by no means imply that it is due to me.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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0 Paradoxical Decompositions . . . 0.1 The Banach–Tarski Paradox 0.2 Tarski’s Theorem . . . . . . . . Notes and Comments . . . . . . . . .

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1 Amenable, Locally Compact Groups . . . . . . . . . . . . . . . 1.1 Invariant Means and Asymptotic Invariance Properties 1.2 Hereditary Properties . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Uniformly Bounded Representations . . . . . . . . . . . . . 1.4 Leptin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Amenable Banach Algebras . . . . . . . . . . . . . 2.1 Derivations from Group Algebras . . . . . . 2.2 Virtual and Approximate Diagonals . . . . . 2.3 Hereditary and Splitting Properties . . . . . 2.4 A First Look at Hochschild Cohomology . Notes and Comments . . . . . . . . . . . . . . . . . . .

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3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Measure Algebras . . . . . . . . . . . . . . . . 3.2 Fourier and Fourier–Stieltjes Algebras . 3.3 Algebras of Approximable Operators . . 3.4 (Non-)Amenability of BðEÞ . . . . . . . . . 3.5 An Amenable Radical Banach Algebra Notes and Comments . . . . . . . . . . . . . . . . .

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5 Dual Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Connes-Amenability for Dual Banach Algebras . . . . . . . 5.2 The Case of the Measure Algebra . . . . . . . . . . . . . . . . . 5.3 Connes-Amenability without a Normal, Virtual Diagonal 5.4 Daws’ Representation Theorem . . . . . . . . . . . . . . . . . . . 5.5 Connes-Amenability and Connes-Injectivity . . . . . . . . . . Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Banach Homological Algebra . . 6.1 Projectivity . . . . . . . . . . . . . 6.2 Resolutions and Ext-Groups 6.3 Flatness and Injectivity . . . . Notes and Comments . . . . . . . . .

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4 Amenability-Like Properties . . . . . . . . . . . 4.1 Contractibility . . . . . . . . . . . . . . . . . . 4.2 Weak Amenability . . . . . . . . . . . . . . . 4.3 Character Amenability . . . . . . . . . . . . 4.4 Pseudo- and Approximate Amenability 4.5 Biflatness and Biprojectivity . . . . . . . . Notes and Comments . . . . . . . . . . . . . . . . .

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7 Operator Algebras on Hilbert Spaces . . . . . . . . . . . 7.1 Amenable von Neumann Algebras . . . . . . . . . . . 7.2 Injective von Neumann Algebras . . . . . . . . . . . . 7.3 Nuclear C -Algebras . . . . . . . . . . . . . . . . . . . . . 7.4 Semidiscrete von Neumann Algebras . . . . . . . . . 7.5 Normal, Virtual Diagonals . . . . . . . . . . . . . . . . 7.6 Commutative Operator Algebras . . . . . . . . . . . . 7.7 An Amenable Operator Algebra Not Similar to a Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . .

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8 Operator Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Operator Amenable, Completely Contractive Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Fourier Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Fourier–Stieltjes Algebras . . . . . . . . . . . . . . . . . . . . Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Banach Spaces . . . . . . . . . . A.1 Bases in Banach Spaces. . . . . . . . A.2 Approximation Properties . . . . . . A.3 The Radon–Nikodým Property . . A.4 Local Theory . . . . . . . . . . . . . . . .

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Appendix B: Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Spectra and Gelfand Theory. . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Banach Modules and Bounded Approximate Identities . . . . . B.3 Multiplier Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Prime and Primitive ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Structure of Semiprime and Semisimple Banach Algebras . . .

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373 373 375 377 378 381

Appendix C: C - and von Neumann Algebras . . . . . . . . . . . . C.1  -Algebras and -Homomorphisms . . . . . . . . . . . . . . . . C.2 C -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Positivity in C  -Algebras and Their Duals . . . . . . . . . . C.4  -Representations of C  -Algebras. . . . . . . . . . . . . . . . . C.5 von Neumann Algebras and W  -Algebras . . . . . . . . . . C.6 Multipliers of C -Algebras . . . . . . . . . . . . . . . . . . . . . . C.7 Projections in von Neumann Algebras . . . . . . . . . . . . . C.8 Tensor Products of C  - and von Neumann Algebras . . C.9 Weights on von Neumann algebras . . . . . . . . . . . . . . .

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383 383 384 385 387 389 391 393 395 396

Appendix D: Abstract Harmonic Analysis . . . . . . . . . . . . . . . . . . . . D.1 Semitopological Semigroups and Locally Compact Groups . . D.2 The Group Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 The Measure Algebra MðGÞ . . . . . . . . . . . . . . . . . . . . . . . . . D.4 Other Banach Algebras Associated with Locally Compact Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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399 399 400 402

....

404

Appendix E: Operator Spaces . . . . . . . . . . . . . . . . . . . . . . . . . E.1 Concrete and Abstract Operator Spaces . . . . . . . . . . . . E.2 Completely Bounded Maps . . . . . . . . . . . . . . . . . . . . . E.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.4 The Projective Tensor Product of Operator Spaces . . . E.5 Completely Contractive Banach Algebras . . . . . . . . . .

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407 407 409 411 413 414

Appendix F: Fourier and Fourier–Stieltjes Algebras . . . F.1 Representations of Locally Compact Groups . . . . F.2 The Fourier Algebra . . . . . . . . . . . . . . . . . . . . . . . F.3 The Fourier–Stieltjes Algebra . . . . . . . . . . . . . . . . F.4 Cosets, Idempotents, and Piecewise Affine Maps .

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417 417 418 421 423

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Index of Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Chapter 0

Paradoxical Decompositions

There is a mathematical theorem implying: An orange can be cut into finitely many pieces, and these pieces can be reassembled to yield two oranges of the same size as the original one.

This sounds more like The Feeding of the Five Thousand than rigorous mathematics. Still, it is a perfectly sound consequence of the Banach–Tarski Paradox, the strongest form of which is: Let A and B be any two bounded sets in three-dimensional space with a nonempty interior. Then there is a partition of A into finitely many sets, which can be reassembled to yield B.

Consequences much more bizarre than just making two oranges out of one come to mind: A pea can be split into finitely many pieces which can be recombined to yield a life-sized statue of Stefan Banach (or a solid ball with a diameter larger than the distance of the Earth from the sun).

Can a theorem with consequences so obviously defying common sense be true? There is an element of faith to the Banach–Tarski Paradox (so that it is not all that far removed from the Feeding of the Five Thousand). Its proof rests on two pillars: • the Axiom of Choice (and you can believe in that or leave it), and • the fact that the free group on two generators lacks a property called amenability. The Banach–Tarski Paradox is just one instance of a paradoxical decomposition: a certain set can be split into finitely many pieces, which can be recombined to yield two copies of the original set. As we shall see in this chapter, paradoxical decompositions (or rather: their non-existence) are at the very heart of the notion of amenability.

© Springer Science+Business Media, LLC, part of Springer Nature 2020 V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-1-0716-0351-2 0

1

2

0 Paradoxical Decompositions

0.1 The Banach–Tarski Paradox We begin with the formal definition of a paradoxical set: Definition 0.1.1. Let G be a group acting on a set S. Then E ⊂ S is called G-paradoxical if there are pairwise disjoint subsets A1 , . . . , An , B1 , . . . , Bm of E, as well as elements x1 , . . . , xn , y1 , . . . , ym of G such that E=

n 

xj · Aj

j=1

and

E=

m 

yj · Bj .

j=1

Note that A1 ∪ · · · ∪ An ∪ B1 ∪ · · · Bm need not be all of E. We simply speak of paradoxical sets (without explicit reference to G) if no confusion can arise; in particular, we do so in the following two cases: • G acts on itself through multiplication from the left, or • G is the group of invertible isometries of a metric space X. We denote the free group in two generators by F2 . Theorem 0.1.2. F2 is paradoxical. Proof. Let a and b be two generators of F2 . For x ∈ {a, b, a−1 , b−1 }, set W (x) := {w ∈ F2 : w starts withx}. With  denoting the empty word, we have the disjoint union F2 = {} ∪ W (a) ∪ W (b) ∪ W (a−1 ) ∪ W (b−1 ). Let w ∈ F2 \ W (a). As w does not start with a, the word a−1 w has to be reduced, so that a−1 w ∈ W (a−1 ) and thus w ∈ aW (a−1 ). It follows that F2 = W (a) ∪ aW (a−1 ).  We obtain F2 = W (b) ∪ bW (b−1 ) in the same way. We shall see next that, given a paradoxical group G acting on a set S, a mild demand on the group action forces S to G-paradoxical. This is the first time that we encounter the Axiom of Choice; we indicate the dependence on the Axiom of Choice by (AC). We denote the identity element of a group G by eG or, if no confusion about G can arise, by e. We say that G acts on S without non-trivial fixed points if, given x ∈ G and s ∈ S such that x · s = s, we have x = e. Proposition 0.1.3 (AC). Let G be a paradoxical group acting on S without non-trivial fixed points. Then S is G-paradoxical. Proof. Let A1 , . . . , An , B1 , . . . , Bm ⊂ G and x1 , . . . , xn , y1 , . . . , ym be as in Definition 0.1.1. Choose a set T ⊂ S such that T contains exactly one element from every G-orbit.

0.1 The Banach–Tarski Paradox

3



Since T contains one point from every G-orbit, {g · T : g ∈ G} = S must hold. Let x, y ∈ G be such that x · T ∩ y · T = ∅, i.e., there are s, t ∈ T such that x · s = y · t. Then y −1 x · s = t, so that s, t ∈ T are in the same G-orbit and thus must be equal. Since G acts on S without non-trivial fixed points, this means y −1 x = e, i.e., x = y. Hence, {x · T : x ∈ G} is a disjoint partition of S. Set  A˜j := {x · T : x ∈ Aj } (j = 1, . . . , n) and ˜j := B



{x · T : x ∈ Bj }

(j = 1, . . . , m).

˜1 , . . . , B ˜m are pairwise disjoint subsets of S such that Then A˜1 , . . . , A˜n , B n 

xj · A˜j =

j=1

n  

{xj x · T : x ∈ Aj } =



{x · T : x ∈ G} = S

j=1

m ˜j = S. and—analogously— j=1 yj · B



Suppose that F2 acts without non-trivial fixed points on a set S. Then Theorem 0.1.2 and Proposition 0.1.3 combined yield that S is F2 -paradoxical. Since—in view of the Banach–Tarski Paradox—we want to show that certain subsets of R3 are paradoxical, we are thus faced with the problem of making F2 act on R3 as invertible isometries. For N ∈ N, we use the symbol O(N ) to denote the real N × N -matrices A such that At A = AAt = IN , where At is the transpose of A and IN is the identity matrix; O(N ) is called the N -dimensional orthogonal group. The N -dimensional special orthogonal group SO(N ) is the subgroup of O(N ) consisting of those matrices A ∈ O(N ) such that det A = 1. Theorem 0.1.4. There are rotations A and B about lines through the origin in R3 such that the subgroup of SO(3) generated by A and B is isomorphic to F2 . Proof. Set ⎡

⎤ √ −232 0 ⎢ ⎥ A = ⎣ 2 2 1 0⎦ 3 3 0 0 1 so that

1 3 √

⎡ A−1 =

⎤ 0 ⎥ 0⎦ 0 1

√ 2 2 1 3 √ ⎢ 2 2 13 ⎣− 3 3

0

and

⎤ ⎡ 1 0 0√ ⎢ 1 2 2⎥ B = ⎣0 √ 3 − 3 ⎦, 0 2 3 2 13

and

⎡ 1 0 ⎢ = ⎣0 13√ 0 −232

B −1

0 √

⎤

2 2⎥ 3 ⎦. 1 3

4

0 Paradoxical Decompositions

Obviously, A and B are rotations about lines through the origin. We claim that the subgroup of SO(3) generated by them is free. To prove this, it is enough to show that a non-empty reduced word in A, B, A−1 , and B −1 cannot act on R3 as the identity. Let w =  be a reduced word in A, B, A−1 , and B −1 . Obviously, w acts as the identity on R3 if and only if A±1 wA∓1 do. Hence, we may suppose without loss of generality that w ends in A or A−1 . We will show that there are a, b, c ∈ Z, with b not divisible by 3, such that ⎡ ⎤ ⎤ ⎡ a 1 √ 1 w · ⎣0⎦ = k ⎣b 2⎦ 3 0 c where k is the length of w. We proceed by induction on k. Suppose that k = 1. Then w = A±1 , so that ⎡ ⎤ ⎡ ⎤ 1√ 1 1 w · ⎣0⎦ = ⎣±2 2⎦ . 3 0 0 Suppose now that k > 1, so that w = A±1 w or w = B ±1 w with w = . By the induction hypothesis, there are a , b , c ∈ Z with b not divisible by 3 such that ⎡  ⎤ ⎡ ⎤ a√ 1 1 w · ⎣0⎦ = k−1 ⎣b 2⎦ 3 0 c It follows that

where

a = a ∓ 4b ,

⎡ ⎤ ⎡ ⎤ a 1 √ 1 w · ⎣0⎦ = k ⎣b 2⎦ , 3 0 c b = b ± 2a ,

and

c = 3c

if w = A±1 w and a = 3a ,

b = b ∓ 2c ,

and

c = c ± 4b

if w = B ±1 w . It is clear that a, b, c ∈ Z. What remains to be shown is that 3 does not divide b. We distinguish four cases. Case 1: w = A±1 B ±1 v. If v = , then b = ±2. If v = , we have ⎡  ⎤ ⎡ ⎤ a√ 1 1 v · ⎣0⎦ = k−2 ⎣b 2⎦ 3 0 c

0.1 The Banach–Tarski Paradox

5

with a , b , c ∈ Z, so that b = b ± 2a = b ± 6a . As 3 does not divide b by the induction hypothesis, b is not divisible by 3 either. Case 2: w = B ±1 A±1 v. This is very similar to Case 1. Case 3: w = A±1 A±1 v. The case v =  is straightforward. Otherwise, there are a , b , c ∈ Z such that ⎡  ⎤ ⎡ ⎤ a√ 1 1 v · ⎣0⎦ = k−2 ⎣b 2⎦ . 3 0 c It follows that b = b ± 2a = b ± 2(a ∓ 4b ) = b + (b ± 2a ) − 9b = 2b − 9b . By the induction hypothesis, 3 does not divide b ; hence, b is also not divisible by 3. Case 4: w = B ±1 B ±1 v. This is treated like Case 3.  Since SO(N ) contains SO(3) as a subgroup for each N ≥ 3, we conclude: Corollary 0.1.5. Let N ≥ 3. Then SO(N ) contains a subgroup isomorphic to F2 . With F2 as a subgroup of SO(3) and Proposition 0.1.3 at hand, we can now prove a first classical result on paradoxical decompositions. For N ≥ 2, we write SN −1 for the unit sphere in RN . Theorem 0.1.6 (Hausdorff Paradox; AC). There is a countable subset C of S2 such that S2 \ C is SO(3)-paradoxical. Proof. Let A and B be rotations about lines through the origin such that the subgroup G of SO(3) they generate is isomorphic to F2 ; such A and B exist according to Theorem 0.1.4. Since A and B are rotations about lines through the origin, each x ∈ G \ {e} has exactly two fixed points in S2 . As G is countable, so is F := {s ∈ S2 : sis a fixed point for somex ∈ G \ {e}}  and thus C := {x · F : x ∈ G}. It is obvious that G acts on S2 \ C without non-trivial fixed points. By Proposition 0.1.3, S2 \ C is, therefore, G-paradoxical and thus—trivially—also SO(3)-paradoxical.  We shall soon see that S2 itself is SO(3)-paradoxical. Definition 0.1.7. Let G be a group acting on a set S, and let A and B be subsets of S. Then A and B are called G-equidecomposable if there are disjoint partitions {A1 , . . . , An } of A and {B1 , . . . , Bn } of B along with x1 , . . . , xn ∈ G such that xj · Aj = Bj for j = 1, . . . , n.

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0 Paradoxical Decompositions

If no confusion can arise about G, we call two G-equidecomposable sets simply equidecomposable. If A and B are G-equidecomposable, we write A ∼G B or simply A ∼ B (if G is obvious). Proposition 0.1.8. Let C ⊂ S2 be countable. Then S2 and S2 \ C are SO(3)equidecomposable. Proof. Let  be a line in R3 through the origin that does not intersect C, and consider those θ ∈ [0, 2π) with the following property: there are x ∈ C and n ∈ N such that ρ · x ∈ C, where ρ is the rotation about  by the angle nθ. As C is countable, there are only countably many such θ; in particular, there is θ0 ∈ [0, 2π) lacking this property. Let ρ be the rotation about  by θ0 ; by the choice of θ0 , we have ρn ·C ∩C = ∅ for all n ∈ N. It follows that ρn · C ∩ ρm · C = ∅ (We write N0 for N ∪ {0}.) Set D =

(n, m ∈ N0 , n = m). ∞

n=0

ρn · C, and note that

S2 = D ∪ (S2 \ D) ∼ ρ · D ∪ (S2 \ D) = S2 \ C. This completes the proof.



Let G be a group acting on a set S, and let A and B be subsets of S. We write A G B (or simply: A  B) if A and a subset of B are Gequidecomposable. The following is an analogue of the Cantor–Bernstein Theorem for : Theorem 0.1.9. Let G be a group acting on a set S, and let A and B be subsets of S such that A G B and B G A. Then A ∼G B. Proof. Let B1 ⊂ B and A1 ⊂ A be such that A ∼ B1 and B ∼ A1 , and let φ : A → B1 and ψ : B → A1 be bijections as in Exercise 0.1.2(a). Set C0 :=A \ A1 , and define inductively Cn+1 := ψ(φ(Cn )) for n ∈ N0 ; set ∞ C := n=0 Cn . By Exercise 0.1.2(a), it is clear that C ∼ φ(C). By the definition of C, we have ψ −1 (A \ C) = B \ φ(C); again by Exercise 0.1.2(a), we obtain that that A \ C ∼ B \ φ(C). Exercise 0.1.2(b) now yields that A = (A \ C) ∪ C ∼ (B \ φ(C)) ∪ φ(C) = B, as claimed.



Corollary 0.1.10. Let G be a group acting on a set S. Then E ⊂ S is Gparadoxical if and only if there is a partition {A, B} of E such that A ∼G E ∼G B. Proof. Only the “only if” part needs proof. Let A1 , . . . , An , B1 , . . . , Bm ⊂ E and x1 , . . . , xn , y1 , . . . , ym ∈ G be as in in Definition 0.1.1. Set

0.1 The Banach–Tarski Paradox

A :=

n 

7

Aj

and

B :=

j=1

m 

Bj ,

j=1

We claim that E G A and E G B. Set A˜1 := x1 · A1 , and define inductively A˜j := xj · Aj \ (A˜1 ∪ · · · ∪ A˜j−1 )

(j = 2, . . . , n).

Then {A˜1 , . . . , A˜n } is a partition of E such that x−1 · A˜j ⊂ Aj , so that j n −1 ˜ E ∼G j=1 xj · Aj ⊂ A and thus E G A. Analogously, we obtain E G B. Since trivially A G E and B G E, Theorem 0.1.9 yields that A ∼G E and B ∼G E. Moreover, E ∼G B ⊂ E \ A ⊂ E holds, so that E \ A ∼G E.



The relevance of G-equidecomposability becomes apparent in the next corollary: Corollary 0.1.11. Let G be a group acting on a set S, and let E and E  be subsets of S with E ∼G E  . Then, if E is G-paradoxical, so is E  . Proof. Let {A, B} be a partition of E as in Corollary 0.1.10. It follows that A ∼G E  and B ∼G E  . This yields that E  is G-paradoxical as well.  Together, Theorem 0.1.6, Proposition 0.1.8, and Corollary 0.1.11 yield: Corollary 0.1.12 (AC). S2 is SO(3)-paradoxical. We can now prove a weak version of the Banach–Tarski Paradox (which is already sufficient if all you want is to make two oranges out of one): Corollary 0.1.13 (AC). Every closed ball in R3 is paradoxical. If X is a metric space, x0 ∈ X, and r > 0, we denote the open and the closed ball in X centered at x0 with radius r by ballr (x0 , X) and Ballr (x0 , X), respectively. If r = 1, we suppress the subscript, and if X is a Banach space E and x0 = 0, we write ballr (E) and Ballr (E); in particular, ball(E) and Ball(E) stand for the open and closed unit ball in E, respectively. Proof (of Corollary 0.1.13). It is sufficient to prove that Ball(R3 ) is paradoxical. We first show that Ball(R3 ) \ {0} is SO(3)-paradoxical. Since S2 is SO(3)paradoxical by Corollary 0.1.12, there are A1 , . . . , An , B1 , . . . , Bm ⊂ S2 and x1 , . . . , xn , y1 , . . . , ym ∈ SO(3) as in Definition 0.1.1. Set A˜j := {ta : t ∈ (0, 1], a ∈ Aj }

(j = 1, . . . , n)

8

0 Paradoxical Decompositions

and ˜j := {tb : t ∈ (0, 1], b ∈ Bj } B

(j = 1, . . . , m).

˜1 , . . . , B ˜m ⊂ Ball(R3 ) \ {0} are pairwise disIt is obvious that A˜1 , . . . , A˜n , B joint and that Ball(R3 ) \ {0} =

n  j=1

xj · A˜j =

m 

˜j . yj · B

j=1

Consequently, Ball(R3 ) \ {0} is indeed SO(3)-paradoxical. 3 3 Next, we show that  )\{0} and Ball(R ) are equidecomposable. Let

Ball(R 1  be a line through 0, 0, 2 that is parallel to the xy-plane, and let ρ be a rotation of infinite order about . Set C := {ρn · 0 : n ∈ N0 }, and note that ρ · C = C \ {0}. It follows that Ball(R3 ) = C ∪ (Ball(R3 ) \ C) ∼ ρ · C ∪ (Ball(R3 ) \ C) = Ball(R3 ) \ {0}. From Corollary 0.1.10, it follows that Ball(R3 ) is paradoxical.



Remark 0.1.14. A classical result by S. Mazur and S. Ulam ([243]) asserts that a surjective isometry between real Banach spaces is already linear if it maps the origin to the origin. It follows that any surjective isometry between Banach spaces is the composition of a surjective, R-linear isometry with a translation. For the invertible isometries on R3 , this means that every such isometry is the composition of an element of O(3) with a translation. It is not difficult to derive from Corollary 0.1.13 a much stronger form of the Banach–Tarski Paradox: Theorem 0.1.15 (Banach–Tarski Paradox; AC). Let A and B be bounded subsets of R3 , each with non-empty interior. Then A ∼ B. Proof. By symmetry and by Theorem 0.1.9, it is sufficient to prove that A  B. Since A is bounded, there is r > 0 such that A ⊂ Ballr (R3 ). Let x0 be an interior point of B. Then there is  > 0 such that Ball (x0 , R3 ) ⊂ B. Since Ballr (R3 ) is compact, there are invertible isometries x1 , . . . , xn : R3 → R3 — translations will do—such that Ballr (R3 ) ⊂

n 

xj · Ball (x0 , R3 ).

j=1

Choose isometries y1 , . . . , yn on R3 such that y1 · Ball (x0 , R3 ), . . . , yn · Ball n (x0 , R3 ) are pairwise disjoint (again, translations will do). Set C := j=1 yj · Ball (x0 , R3 ). It follows from the weak Banach–Tarski Paradox Corollary 0.1.13, that C  Ball (x0 , R3 ). Finally, note that

0.2 Tarski’s Theorem

A ⊂ Ballr (R3 ) ⊂

9 n 

xj · Ball (x0 , R3 )  C  Ball (x0 , R3 ) ⊂ B,

j=1

which completes the proof.



Exercises Exercise 0.1.1. Where exactly in the proof of Proposition 0.1.3 was the Axiom of Choice invoked? Exercise 0.1.2. Let G be a group acting on a set S. Show that: (a) if A, B ⊂ S are such that A ∼ B, then there is a bijection φ : A → B with C ∼ φ(C) for all C ⊂ A; (b) if A1 , A2 , B1 , B2 ⊂ S are such that A1 ∩ A2 = ∅ = B1 ∩ B2 , and and if suppose that Aj ∼ Bj for j = 1, 2, then A1 ∪ A2 ∼ B1 ∪ B2 . Exercise 0.1.3. Show that ∼G is an equivalence relation. Exercise 0.1.4. Verify the last sentence in the proof of Corollary 0.1.10. Exercise 0.1.5. Show that R3 is paradoxical. Exercise 0.1.6. Show that G is a reflexive and transitive relation on the equivalence classes of P(S), the power set of S, with respect to ∼G . Exercise 0.1.7. Where exactly in the proof of Theorem 0.1.15 was the axiom of choice used?

0.2 Tarski’s Theorem We have mentioned in the introduction of this chapter that one of the key ingredients for the proof of the Banach–Tarski Paradox was that F2 lacks the property of amenability. We have not yet formally defined what amenability is, but for a (discrete) group it is equivalent to not being paradoxical. Nevertheless, the formal definition of amenability will look quite different from Definition 0.1.1. The reason why these two notions are equivalent is the following deep theorem due to A. Tarski: Theorem 0.2.1 (Tarski’s Theorem; AC). Let G be a group acting on a set S, and let E ⊂ S. Then the following are equivalent: (i) there is a finitely additive, G-invariant set function μ : P(S) → [0, ∞] with 0 < μ(E) < ∞;

10

0 Paradoxical Decompositions

(ii) E is not G-paradoxical. Here, we call μ G-invariant if μ(x · A) = μ(A) for all x ∈ G and A ⊂ S. The implication (i) =⇒ (ii) is easy to prove (Exercise 0.2.1 below). The proof of (ii) =⇒ (i) requires some preparation. We start with some notions from graph theory. A (undirected) graph is a triple (V, E, φ), where V and E are non-empty sets and φ is a map from E into the unordered pairs of elements of V . The elements of V are called vertices, and the elements of E are called edges. If e ∈ E and v, w ∈ V are such that φ(e) = (v, w), we say that e joins v and w or that v and w are the endpoints of e; we shall sometimes be sloppy and identify the edge e with its image under φ for the sake of simplicity. A path in (V, E, φ) is a finite sequence (e1 , . . . , en ) of edges together with a finite sequence (v0 , . . . , vn ) of vertices where v0 is an endpoint of e1 , vn is an endpoint of en , and vj+1 is an endpoint of ej for j = 1, . . . , n − 1. We say that v0 and vn are joined by the path. For formal reasons, we will also say that the empty path joins each vertex with itself. For notational simplicity, we will write (V, E) instead of (V, E, φ). Definition 0.2.2 Let (V, E) be a graph, and let k ∈ N. Then (V, E) is called: (a) k-regular if each vertex is the endpoint of exactly k edges; (b) bipartite if there is a partition {X, Y } of V such that each edge has one endpoint in X and one in Y . In what follows, we shall simply speak of a bipartite graph (X, Y, E) when we mean that (V, E) is a bipartite graph with V , X and Y as in Definition 0.2.2(b). Definition 0.2.3 Let (X, Y, E) be a bipartite graph, and let A ⊂ X and B ⊂ Y . A perfect matching of A and B is a subset F of E such that: (a) all endpoints of edges in F are in A ∪ B; (b) each element of A ∪ B is an endpoint of exactly one edge in F . Lemma 0.2.4 (Marriage Theorem). Let k ∈ N, and let (X, Y, E) be a k-regular, bipartite graph with |V | < ∞. Then there exists a perfect matching of X and Y . Proof. We begin with two preliminary observations. First note that, due to k-regularity, we have |E| = k|X| = k|Y |, so that |X| = |Y |. Secondly, for any S ⊂ V , set V (S) := {v ∈ V : vis joined by an edge with an element of S} For each v ∈ V (S), let Nv be the number of edges joining v with an element of S. As the number of edges joining elements of S with elements of V (S)

0.2 Tarski’s Theorem

11



is k|S| by k-regularity, we have v∈V (S) Nv = k|S|. Since Nv ≤ k for all v ∈ V (S), this means |V (S)| ≥ |S|. For the actual proof, let A ⊂ X and B ⊂ Y be such that there is a perfect matching F for A and B with |F | maximal. We claim that A = X (and, consequently, B = Y ). Assume that there is x ∈ X \ A. Call a path (e1 , . . . , en ) F -alternating if the ej s lie alternately in F and E \ F , and let Z denote the set of all vertices that are joined with x by an F alternating path. Let z ∈ Z, let (e1 , . . . , en ) be an F -alternating path joining x and z, and let (x, v1 , . . . , vn−1 , z) be finite sequence of vertices associated with this path. Then the vertices v2 , v4 , . . . in X must lie in A whereas the vertices v1 , v3 , . . . must lie in B (with the possible exception of z). Suppose that z ∈ Y ; note that this entails that n is odd. Assume that z ∈ / B. Set F˜ := (F \ {e2 , e4 , . . . , en−1 }) ∪ {e1 , e3 , . . . , en }. Then F˜ is a perfect matching of A ∪ {x} and B ∪ {z} with |F˜ | = |F | + 1, contradicting the maximality of |F |. We conclude that z ∈ B. Since Z ∩ Y ⊂ B, there is a bijection—via F —between Z ∩ Y and (Z ∩ X) \ {x}, so that |Z ∩ Y | = |Z ∩ X| − 1. On the other hand, it is clear that Z ∩Y ⊂ V (Z ∩X); we claim that even equality holds. Let z ∈ V (Z ∩X). Then there is a path (e1 , . . . , en−1 , en ) joining x and z such that (e1 , . . . , en−1 ) is F -alternating. It is easy to see that en−1 ∈ F (so that, in particular, n ≥ 3). / F , the path (e1 , . . . , en−1 , en ) is F -alternating, so that z ∈ Z ∩ Y . If en ∈ If en ∈ F , then en = en−1 by the definition of a perfect matching, and (e1 , . . . , en−2 ) is an F -alternating path joining x and z, i.e., z ∈ Z ∩ Y . It follows that |Z ∩ Y | ≥ |Z ∩ X|, contradicting |Z ∩ Y | = |Z ∩ X| − 1.  The reason why Lemma 0.2.4 carries the catchy name Marriage Theorem is the following “application”: if each girl in town likes exactly k boys and each boy in town likes exactly k girls, then it is possible to arrange marriages between them such that, in each marriage, the partners like one another. (How comforting to know. . . ) For the proof of Tarski’s Theorem, we need the conclusion of the Marriage Theorem without any finiteness requirement: Theorem 0.2.5 (K¨ onig’s Theorem; AC). Let k ∈ N, and let (X, Y, E) be a k-regular, bipartite graph. Then there exists a perfect matching of X and Y. Proof. Define an equivalence relation on V : two vertices are equivalent if they can be joined by a path. Each equivalence class with respect to this relation is again a k-regular, bipartite graph. If we can find a perfect matching for each such graph, we can put them together to form a perfect matching of X and Y . We can thus suppose without loss of generality that X and Y are infinite and that any two points in V can be joined by a path.

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Fix x ∈ X and y ∈ Y . Set Z1 := {x, y} ∪ V ({x, y}) and Zn+1 := V (Zn ) Since any two points in V can for n ∈ N; note that Zn ⊂ Zn+1 for n ∈ N. ∞ be joined by a path, it follows that V = n=1 Zn . For n ∈ N, set Xn := X ∩ Zn and Yn := Y ∩ Zn , and let En be the edges occurring in the inductive construction of Z1 , . . . , Zn . By Exercise 0.2.2 below, there is a k-regular, bipartite graph (Xn , Yn , En ) such that Xn ⊂ Xn , Yn ⊂ Yn , and En ⊂ En . By the Marriage Theorem, there is a perfect matching of Xn and Yn , which induces a perfect matching of Xm and a subset of Y for m = 1, . . . , n − 1. Let n ∈ N; we call F ⊂ E a n-matching if it is a perfect matching of Xn and a subset of Y . By the foregoing, there is an n-matching for each n ∈ N. Obviously, if F is an n-matching, it induces an j-matching for j = 1, . . . , n−1; abusing notation, we denote such a j-matching by F |Xj . We shall now inductively construct an increasing sequence (Fn )∞ n=1 of subsets of E. Due to k-regularity, there are only finitely many 1-matchings. Hence, there must be a 1-matching F1 such that, for each m ≥ 1, there is an   with Fm |X1 = F1 . Suppose that F1 ⊂ · · · ⊂ Fn have already m-matching Fm been constructed with the following properties: • Fn is an n-matching; • Fj = Fn |Xj for j = 1, . . . , n − 1;   such that Fm |Xn = Fn . • for each m ≥ 1, there is an m-matching Fm As there are only finitely many n + 1-matchings, there must be an n + 1matching Fn+1 with Fn = Fn+1 |Xn such that, for m ≥ ∞n + 1, there is an   with Fm |Xn+1 = Fn+1 . Finally, F := n=1 Fn is a perfect m-matching Fm matching of X and Y .  K¨ onig’s Theorem was only the first step on the way to a proof of Tarski’s Theorem. The next step is to associate, with each group action, an object called the type semigroup of that action. Let G be a group acting on a set S, and let SN0 denote the group of ˜ := G × SN acts canonically on permutations of N0 . Then the group G 0 ˜ ˜ S := S × N0 . Given A ⊂ S, we call those n ∈ N such that (s, n) ∈ A for some s ∈ S, the levels of A; if A has only finitely many levels, we call it bounded. For bounded A ⊂ S˜ the equivalence class of A with respect to ∼G˜ is called the type of A and is denoted by [A]. For E ⊂ X, set [E] := [E × {0}]. ˜ there is k ∈ N0 such that Given bounded A, B ⊂ S, B  := {(b, n + k) : (b, n) ∈ B} has empty intersection with A. Define [A] + [B] := [A ∪ B  ]. It is not difficult to see that + is well defined and turns S := {[A] : A ⊂ S˜ is bounded }. into a commutative semigroup (see Exercise 0.2.4). It is called the type semigroup of the action of G on S.

0.2 Tarski’s Theorem

13

If S is any commutative semigroup, set nα := α + · · · + α

 

(α ∈ S, n ∈ N).

n times

Also, we write α ≤ β for α, β ∈ S if there is γ ∈ S such that α + γ = β. We refer to Exercises 0.2.5 and 0.2.6 below for connections between type semigroups and paradoxical decompositions. For the type semigroup of a group action, the following cancellation law holds; for its proof, we require K¨ onig’s Theorem. Theorem 0.2.6 (AC). Let G be a group acting on a set S, and let S be the corresponding type semigroup. Then, if α, β ∈ S and n ∈ N are such that nα = nβ, we have α = β. ˜ Proof. If nα = nβ, then there are two disjoint, bounded G-equidecomposable  ˜ sets E, E ⊂ S with partitions {A1 , . . . , An } of E and {B1 , . . . , Bn } of E  such that (j = 1, . . . , n). [Aj ] = α and [Bj ] = β Let χ : E → E  and, for j = 1, . . . , n, let φj : A1 → Aj and ψj : B1 → Bj be bijective maps as in Exercise 0.1.2(i) (choose φ1 and ψ1 as the identity). For each a ∈ A1 and b ∈ B1 let a ¯ := {a, φ2 (a), . . . , φn (a)}

and

¯b := {b, ψ2 (b), . . . , ψn (b)}

Set X := {¯ a : a ∈ A1 } and Y := {¯b : b ∈ B1 }. Join vertices a ¯ ∈ X and ¯b ∈ Y by an edge whenever there is j ∈ {1, . . . , n} such that χ(φj (a)) ∈ ¯b. This produces an n-regular, bipartite graph. By K¨ onig’s Theorem, there is a perfect matching F of X and Y . Hence, for each a ¯ ∈ X, there are are unique ¯b ∈ Y and a unique edge in F joining a ¯ and ¯b, which means that χ(φj (a)) = ψk (b) for some j, k ∈ {1, . . . , n}. For any j, k ∈ {1, . . . , n}, set ¯ and ¯b are joined by an edge in F and χ(φj (a)) = ψk (b)}; Cj,k := {a ∈ A1 : a similarly, define Dj,k := {b ∈ B1 : a ¯ and ¯b are joined by an edge in F and χ(φj (a)) = ψk (b)}. Then ψk−1 ◦ χ ◦ φj maps Cj,k bijectively onto Dj,k as in Exercise 0.1.2(i), so that, in particular Cj,k ∼G˜ Dj,k . Since {Cj,k : j, k = 1, . . . , n} and {Dj,k : j, k = 1, . . . , n} are partitions of A1 and B1 , respectively, it follows from  Exercise 0.1.2(ii) that A1 ∼G˜ B1 , i.e., α = β. For the proof of Tarski’s Theorem, we will need the following corollary of the cancellation law: Corollary 0.2.7 (AC). Let S be the type semigroup of a group action, and let α ∈ S and n ∈ N be such that (n + 1)α ≤ nα. Then α = 2α.

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Proof. From the hypothesis, we obtain 2α + nα = (n + 1)α + α ≤ nα + α = (n + 1)α ≤ nα. Repeating this argument, we obtain nα ≥ nα + nα = 2nα. Since, trivially, nα ≤ 2nα, we have nα = 2nα = n(2α) by Exercise 0.2.6(b). From Theorem 0.2.6, α = 2α follows.  In order to complete the proof of Tarski’s Theorem, we need one more theorem, for which, in turn, we require a technical lemma: Lemma 0.2.8. Let S be a commutative semigroup, let S0 ⊂ S be finite, and let  ∈ S0 satisfy the following: (a) (n + 1) ≤ n for n ∈ N; (b) for each α ∈ S there is n ∈ N with α ≤ n. Then there is a function ν : S0 → [0, ∞] with the following properties: (i) ν() = 1; (ii) if α1 , . . . , αn , β1 , . . . , βm ∈ S0 are such that α1 +· · ·+αn ≤ β1 +· · ·+βm , then n m   ν(αj ) ≤ ν(βj ). j=1

j=1

Proof. We proceed by induction on the cardinality of |S0 |. If |S0 | = 1, then S0 = {}. In this case ν() := 1 satisfies (i). To verify (ii) let n, m ∈ N be such that n ≤ m, and assume that n ≥ m + 1. But then (m + 1) ≤ n ≤ m, which contradicts (a).s Suppose now that there is α0 ∈ S0 \ {}. By the induction hypothesis, there is a function ν : S0 \ {α0 } → [0, ∞] satisfying (i) and (ii). From (b), (i), and (ii), it follows that ν attains only finite values. Extend ν to S0 by letting ⎧ ⎛ ⎞⎫ p q ⎨1  ⎬  ⎝ ν(α0 ) := inf ν(γj ) − ν(δj )⎠ , ⎩r ⎭ j=1

j=1

where the infimum is taken over all r ∈ N and γ1 , . . . , γp , δ1 , . . . , δq ∈ S0 \{α0 } satisfying δ1 + · · · + δq + rα0 ≤ γ1 + · · · + γp . By (b) ν is well defined, i.e., the infimum is taken over a non-empty set; also, it is clear that ν(α0 ) ≥ 0. It remains to be shown that the extended ν satisfies (ii). Let α1 , . . . , αn , β1 , . . . , βm ∈ S0 \ {α0 } and s, t ∈ N0 be such that α1 + · · · αn + sα0 ≤ β1 + · · · + βm + tα0 . If s = t = 0, the claim is clear from the induction hypothesis.

(0.1)

0.2 Tarski’s Theorem

15

Case 1: s = 0 and t > 0. We have to show that n 

ν(αj ) ≤ tν(α0 ) +

m 

j=1

i.e.,

ν(βj ).

j=1

⎛ ⎞ n m  1 ⎝ ν(αp ) − ν(βj )⎠ =: w. ν(α0 ) ≥ t j=1 j=1

Let r ∈ N and γ1 , . . . , γp , δ1 , . . . , δq ∈ S0 \ {α0 } satisfy δ1 + · · · + δq + rα0 ≤ γ1 + · · · + γp .

(0.2)

By the definition of ν(α0 ), it is enough to check that ⎛ ⎞ p q  1 ⎝ ν(γj ) − ν(δj )⎠ ≥ w. r j=1 j=1

(0.3)

From (0.1)—note that s = 0—we obtain by multiplication with r and adding the same terms on both sides rα1 + · · · + rαn + tδ1 + · · · + tδq ≤ rβ1 + · · · + rβm + rtα0 + tδ1 + · · · + tδq . Combining this with (0.2) yields rα1 + · · · + rαn + tδ1 + · · · + tδq ≤ rβ1 + · · · + rβm + tγ1 + · · · + tγp . Applying the induction hypothesis, we obtain r

n 

ν(αj ) + t

j=1

q 

ν(δj ) ≤ r

j=1

m 

ν(βj ) + t

j=1

n 

ν(γj ),

j=1

which implies (0.3). Case 2: Suppose that s > 0. It suffices to show that sν(α0 ) +

n 

ν(αj ) ≤ z1 + · · · + zt +

j=1

m 

ν(βj ),

j=1

where z1 , . . . , zt are any of the numbers whose infimum defines ν(α0 ); we may suppose that z1 = · · · = zt =: z. We thus must prove that sν(α0 ) +

n  j=1

ν(αj ) ≤ tz +

m  j=1

ν(βj ).

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Multiplying (0.1) by r and adding the same terms on both sides yields: rα1 +· · ·+rαn +rsα0 +tδ1 +· · ·+tδq ≤ rβ1 +· · ·+rβm +rtα0 +tδ1 +· · ·+tδq . (0.4)

Let γ1 , . . . , γp , δ1 , . . . , δq ∈ S0 \ {α0 } satisfy (0.2) with ⎛ ⎞ p q   1 z= ⎝ ν(γj ) − ν(δj )⎠ . r j=1 j=1 Substituting (0.2) into (0.4) yields rα1 + · · · + rαn + tδ1 + · · · + tδq + rsα0 ≤ rβ1 + · · · + rβm + tγ1 + · · · + tγp . From this inequality and from the definition of ν(α0 ), it follows that sν(α0 ) +

n 

ν(αj )

j=1

⎛ ⎞ p q m n    s ⎝  ≤ r ν(αj ) + ν(βj ) + t ν(γj ) − r ν(αj ) − t ν(δj )⎠ sr j=1 j=1 j=1 j=1 j=1 n 

= tz +

m 

ν(βj ).

j=1

This completes the proof.



Theorem 0.2.9 (AC). Let (S, +) be a commutative semigroup with neutral element 0, and let  be an element of S. Then the following are equivalent: (i) (n + 1) ≤ n for all n ∈ N; (ii) there is a semigroup homomorphism ν : (S, +) → ([0, ∞], +) such that ν() = 1. Proof. The implication (ii) =⇒ (i) is easy (see Exercise 0.2.7 below). We shall thus focus on the proof of (i) =⇒ (ii). Without loss of generality suppose that, for each α ∈ S, there is n ∈ N such that α ≤ n (otherwise, first disregard those elements α lacking this property, and later define ν(α) := ∞ for all such α). For any finite subset S0 of S containing , let M (S0 ) be the set of all κ : S → [0, ∞] such that • κ() = 1, and • κ(α + β) = κ(α) + κ(β) for α, β, α + β ∈ S0 . Let S0 ⊂ S be finite, and let κ : S0 → [0, ∞] be a map as specified in Lemma 0.2.8. By Lemma 0.2.8(ii), it is clear that κ() = 1. Let α, β ∈ S0 be such that α + β ∈ S0 ; then Lemma 0.2.8(ii) yields

0.2 Tarski’s Theorem

17

ν(α + β) ≤ ν(α) + ν(β) ≤ ν(α + β), so that ν(α + β) = ν(α) + ν(β). We conclude that M (S0 ) = ∅. Let [0, ∞]S be equipped with the product topology; it is compact by Tychonoff’s theorem. It is easily seen that, for each finite subset S0 of S, the set M (S0 ) is closed in [0, ∞]S . It is equally easily seen that the family {M (S0 ) : S0 ⊂ S∫ {\} has the finite intersection property. Hence,  {M (S0 ) : S0 ⊂ S∫ {\} contains at least one map ν : S → [0, ∞]. It is clear that ν() = 1, and if α, β ∈ S, then ν(α + β) = ν(α) + ν(β) because ν ∈ M ({α, β, α + β}).  We can finally complete the proof of Tarski’s Theorem: Proof (of (ii) =⇒ (i) in Tarski’s Theorem). Let S be the type semigroup of the action of G on S. Suppose that E is not G-paradoxical. By Exercise 0.2.6(c), this means that [E] = 2[E], and from Corollary 0.2.7, it follows that (n + 1)[E] ≤ n[E] for all n ∈ N. Hence, Theorem 0.2.9 implies that there is an additive map ν : S → [0, ∞] such that ν([E]) = 1. Define a set function μ : P(S) → [0, ∞],

A → ν([A])

Obviously, μ(E) = 1, and from the definition of + in S, it is clear that μ is finitely additive. Since [x · A] = [A] for all x ∈ G and A ⊂ S, it is immediate that μ is G-invariant.  If S is any set, and μ : P(S) → [0, ∞] is a finitely additive set function with μ(X) < ∞, then we may define M ∈ ∞ (S)∗ through  φ(s) dμ(s) (φ ∈ ∞ (S)), (0.5) φ, M  := S

where ·, · stands for the pairing between a Banach space and its dual. (For a detailed exposition on integration with respect to not necessarily countably additive set functions, see [102, Chapter III].) Let G be a group acting on S. For any function φ on S, define its left translate Lx φ by x ∈ G on S through (Lx φ)(s) := φ(x · s) for s ∈ S. It is obvious that μ is G-invariant if and only if M defined as in (0.5) satisfies Lx φ, M  = φ, M 

(x ∈ G, φ ∈ ∞ (S)).

The following is thus an easy consequence of Tarski’s Theorem: Corollary 0.2.10 (AC). For a group G, the following are equivalent: (i) G is not paradoxical; (ii) there is a finitely additive, G-invariant set function μ : P(G) → [0, ∞) such that μ(G) = 1; (iii) there is M ∈ ∞ (G)∗ with 1, M  = M  = 1 such that Lx φ, M  = φ, M 

(x ∈ G, φ ∈ ∞ (S)).

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Groups satisfying the equivalent conditions of Corollary 0.2.10 are called amenable.

Exercises Exercise 0.2.1. Let G be a group acting on a set S, and let E ⊂ S be G-paradoxical, and let μ : P(S) → [0, ∞] be a finitely additive, G-invariant set function. Show that μ(E) = 0 or μ(E) = ∞. Do you need the axiom of choice? Exercise 0.2.2. Let k ∈ N, let (X, Y, E) be a bipartite graph with |X ∪ Y | < ∞ such that each vertex is an endpoint of at most k edges. Show that there is a k-regular, bipartite graph (X  , Y  , E  ) with |X  ∪ Y  | < ∞ such that X ⊂ X  , Y ⊂ Y  , and E ⊂ E  . Exercise 0.2.3. Where exactly in the proof of Theorem 0.2.5 was the axiom of choice used? Exercise 0.2.4. Let G be a group acting on a set S. Show that + is well defined on S, turning it into a commutative semigroup with identity element [∅]. Exercise 0.2.5. Let G be a group acting on a set S, and let E1 , E2 ⊂ X. Show that E1 ∼G E2 if and only if E1 × {n} ∼G˜ E2 × {m} for all n, m ∈ N0 . Exercise 0.2.6. Let G be a group acting on a set S, and let S be the corresponding type semigroup. Show that: (a) if A, B ⊂ S˜ are bounded then [A] ≤ [B] if and only if A G˜ B; (b) if α, β ∈ S are such that α ≤ β and β ≤ α, then α = β; (c) E ⊂ X is G-paradoxical if and only if [E] = 2[E]. Exercise 0.2.7. Verify (ii) =⇒ (i) of Theorem 0.2.9.

Notes and Comments Of course, the interest in paradoxical decompositions does not go as far back as to biblical times. Nevertheless, their origins can be traced back to the beginnings of modern measure theory. As is well known, N -dimensional Lebesgue measure on RN is (a) σadditive, (b) invariant under invertible isometries, and (c) normalized in the sense that a certain set, [0, 1]N say, is mapped to 1. It is a standard exercise in measure theory to show that N -dimensional Lebesgues measure cannot

Notes and Comments

19

be extended to all of P(RN ) in such a way that (a), (b), and (c) remain valid. In [165], F. Hausdorff raised the question of whether there was a (a)’ finitely additive set function on P(RN ) that still satisfied (b) and (c). With the help of his paradox, Hausdorff was able to answer this question in the negative for N ≥ 3. Interestingly, for N = 1, 2, there is a set function on all of P(RN ) for which (a)’, (b), and (c) still hold ([18]). Further investigations in this direction led S. Banach and A. Tarski to the paradox that now carries their name ([19]). Tarski’s Theorem first saw the light of day in [341]. Our exposition is based on [350], which is a monograph devoted entirely to paradoxical decompositions; another modern account of the Banach–Tarski paradox is [331]. A remarkable result is that paradoxical decompositions of the unit ball in R3 already exist using just five pieces whereas there is no paradoxical decomposition involving four or fewer pieces ([350, Theorem 4.7]). Popular expositions of the Banach–Tarski Paradox are the articles [126] and [294] and the book [351]. The article [126] contains three pictures showing its author with one orange, then cutting up the orange with a knife into finitely many pieces, and eventually with two oranges. Why is any attempt to implement the Banach–Tarski Paradox using a knife bound to fail?

Chapter 1

Amenable, Locally Compact Groups

In view of Corollary 0.2.10, there are three possible ways of extending the notion of amenability from discrete groups to general locally compact groups: • require that the group G be not paradoxical with some Borel overtones added to Definition 0.1.1, i.e., A1 , . . . , An and B1 , . . . , Bm in Definition 0.1.1 be Borel sets; • require the existence of a finitely additive set function μ on the Borel subsets of G with μ(G) = 1 such that μ(xB) = μ(B) for all x ∈ G and all Borel subsets B of G; • require the existence of a linear functional M as in Corollary 0.2.10(iii) on some space that can replace ∞ (G) in the general locally compact situation, such as L∞ (G) or C(G). It is this third definition of amenability which turns out to be the “right” one: the class of amenable, locally compact groups is large enough to encompass many interesting examples but, on the other hand, is small enough to allow for the development of a strong theory. As we define amenable, locally compact groups in terms of bounded, linear functionals, we are now entering the realm of functional analysis, where the Axiom of Choice is indispensable; thus, in this chapter—and in all subsequent ones—we suppose that the Axiom of Choice holds. For notation and background material from abstract harmonic analysis and Banach algebras, we refer to Appendices D and B.

1.1 Invariant Means and Asymptotic Invariance Properties We start with the definition of a mean:

© Springer Science+Business Media, LLC, part of Springer Nature 2020 V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-1-0716-0351-2 1

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22

1 Amenable, Locally Compact Groups

Definition 1.1.1. Let G be a locally compact group, and let E be a subspace of L∞ (G) containing the constant functions. A mean on E is a functional M ∈ E ∗ such that 1, M  = M  = 1. For the subspaces of L∞ (G) we are interested in—L∞ (G) itself, C(G), LUC(G), RUC(G), and UC(S) (see Section D.4 for the definitions of these spaces)—the following characterization of means holds: Proposition 1.1.2. Let G be a locally compact group, and let E be a subspace of L∞ (G) containing the constant functions and closed under complex conjugation. Then, for a linear functional M : E → C with 1, M  = 1, the following are equivalent: (i) M is a mean on E; (ii) M is positive, i.e., φ, M  ≥ 0 for all nonnegative φ ∈ E. Proof. (i) =⇒ (ii): We first prove that φ, M  ∈ R for all R-valued φ ∈ E. Let φ ∈ E be R-valued, and suppose without loss of generality that φ∞ ≤ 1. Let α, β ∈ R be such that φ, M  = α + βi. For any t ∈ R, we then have (β + t)2 ≤ |α + i(β + t)|2 = |φ + it1, M |2 ≤ φ + it12∞ ≤ 1 + t2 and thus 2βt ≤ 1 − β 2 ; this is possible only if β = 0. Now, let φ ∈ E be nonnegative. Without loss of generality, suppose that 0 ≤ φ ≤ 1. Set ψ := 2φ − 1, so that ψ is R-valued with ψ∞ ≤ 1. Since M  = 1, we have |ψ, M | ≤ ψ∞ ≤ 1, so that φ, M  =

1 1 1 + ψ, M  = (1 + ψ, M ) ≥ 0. 2 2

(ii) =⇒ (i): Let φ ∈ E be R-valued. Then ψ := φ∞ 1 − φ ≥ 0, so that ψ, M  ≥ 0. It follows that φ, M  ∈ R and φ, M  ≤ φ∞ . Replacing φ with −φ, we obtain −φ, M  ≤ φ∞ , so that, as a whole, |φ, M | ≤ φ∞ . For arbitrary φ ∈ E, choose λ ∈ C such that λφ, M  = |φ, M |. Since E is closed under complex conjugation, there are R-valued φ1 , φ2 ∈ E such that λφ = φ1 + iφ2 . Since φ1 , M  + iφ2 , M  = λφ, M  ≥ 0 and since φ1 , M , φ2 , M  ∈ R, it follows that φ2 , M  = 0. We thus have |φ, M | = φ1 , M  ≤ φ1 ∞ ≤ λφ∞ = φ∞ , which proves (i).



We call a subspace E of L∞ (G) left invariant if Lx φ ∈ E for all φ ∈ E and x ∈ G. Similarly, we call right invariant if Rx φ ∈ E for all φ ∈ E and x ∈ G. Finally, if E is both left and right invariant, we simply speak of an invariant subspace. Definition 1.1.3. Let G be a locally compact group, and let E be a subspace of L∞ (G) containing the constant functions. A mean M on E is called left invariant if

1.1 Invariant Means and Asymptotic Invariance Properties

Lx φ, M  = φ, M 

23

(φ ∈ E, x ∈ G).

It is clear, how right invariant means on right invariant subspaces of L∞ (G) and invariant means on invariant subspaces of L∞ (G) can be defined in an analogous way. Definition 1.1.4. A locally compact group G is amenable if there is a left invariant mean on L∞ (G). By Corollary 0.2.10, for a discrete group, being amenable is the same as not being paradoxical. Example 1.1.5. By Theorem 0.1.2 and Corollary 0.2.10, F2 is not amenable (since this conclusion only depends on the easy direction of Tarski’s theorem, it is independent of the axiom of choice). Example 1.1.6. Let G be a compact group, so that L∞ (G) ⊂ L1 (G). Then integration with respect to normalized (left) Haar measure defines a left invariant mean on L∞ (G), so that G is amenable. Example 1.1.7. Let G be a locally compact, abelian group. To see that G is amenable, let K be the set of all means on L∞ (G); by Exercise 1.1.1 below, it is convex and weak∗ compact. For x ∈ G, define Tx : L∞ (G)∗ → L∞ (G)∗ as the adjoint of Lx : L∞ (G) → L∞ (G); then Tx is weak∗ continuous and leaves K invariant. Furthermore, it is clear that Txy = Tx Ty for x, y ∈ G. By the Markov–Kakutani Fixed Point Theorem ([102, Theorem V.10.6]), there is thus M ∈ K such that Tx M = M for all x ∈ G, i.e., M is a left invariant mean on L∞ (G). An apparent drawback of Definition 1.1.4 is its lack of symmetry: we have defined amenability through left invariant means, so that groups satisfying that definition ought to be rather called “left amenable”. We could as well have defined amenability via right invariant means or invariant means (it is also clear what that is supposed to mean). Fortunately, all these variants of Definition 1.1.4 characterize the same class of locally compact groups. We first characterize amenability through an asymptotic invariance property. Lemma 1.1.8. The following are equivalent for a locally compact group G: (i) G is amenable; (ii) there is a net (mα )α of nonnegative functions of norm one in L1 (G) such that (x ∈ G). (1.1) δx ∗ mα − mα 1 → 0 Proof. If (mα )α is a net of nonnegative norm one functions in L1 (G) satisfying (1.1), then each of its weak∗ accumulation points in L∞ (G)∗ is a left invariant mean on L∞ (G). So, only (i) =⇒ (ii) needs proof.

24

1 Amenable, Locally Compact Groups

Suppose that G is amenable, and let M be a left invariant mean on L∞ (G). By Exercise 1.1.4 below, there is a net (mβ )β of nonnegative norm one functions in L1 (G) such that mβ

σ(L∞ (G)∗ ,L∞ (G))

δ x ∗ mβ − m β

−→

M and, consequently,

σ(L∞ (G)∗ ,L∞ (G))

−→

0

(x ∈ G).

(1.2)

(By σ(L∞ (G)∗ , L∞ (G)), we mean the weak∗ topology on L∞ (G)∗ induced by L∞ (G).) As σ(L∞ (G)∗ , L∞ (G)) restricted to L1 (G) is the weak topology is, in fact, in the weak topology of L1 (G). on L1 (G), the  limit (1.2) 1 Let X := x∈G L (G) be equipped with the product topology arising from the norm topology on L1 (G). From the definition of the product topology, it is immediate that a linear functional φ : X → C is continuous if and only if there are x1 , . . . , xn ∈ G and φ1 , . . . , φn ∈ L∞ (G) such that (fx )x∈G , φ =

n 

fxj , φj 

((fx )x∈G ∈ X ).

j=1

In view of (1.2), this means that the net ((δx ∗ mβ − mβ )x∈G )β in X converges to 0 with respect to the weak topology on X . In particular, 0 lies in the weak closure of the convex set K := {(δx ∗ m − m)x∈G : m ∈ L1 (G) is positive and of norm one} in X . As is well known ([292, 3.12 Theorem]), the closures of a convex subset of a locally convex space coincide in the given topology and in the weak topology. Hence, 0 lies also in the closure of K with respect to the given topology of X . In view of the definition of the product topology, this guarantees the existence of a net as in (ii).  If we had defined amenability in terms of right invariant means, the net (mα )α in Lemma 1.1.8(ii) would have had to satisfy mα ∗ δx − mα 1 → 0

(x ∈ G)

(1.3)

instead of (1.1). Proposition 1.1.9. For a locally compact group G, the following are equivalent: (i) G is amenable; (ii) there is a right invariant mean on L∞ (G); (iii) there is an invariant mean on L∞ (G). Proof. (i) =⇒ (ii): Let M be a left invariant mean on L∞ (G). For φ ∈ L∞ (G) ˇ := φ(x−1 ) for x ∈ G. Define define φˇ ∈ L∞ (G) by letting φ(x) ˇ : L∞ (G) → C, M

ˇ M . φ → φ,

1.1 Invariant Means and Asymptotic Invariance Properties

25

ˇ is a right invariant mean on L∞ (G). Then M (ii) =⇒ (i) is proven in exactly the same fashion. Suppose that (i)—and, equivalently, (ii)—holds. Let (mα )α be a net as in Lemma 1.1.8(ii). By the analog of Lemma 1.1.8 for right invariant means, we obtain another net (mβ )β of non-negative norm one functions in L1 (G) for which the “right version” (1.3) of (1.1) holds. Then any weak∗ accumulation point of (mα ∗ mβ )α,β in L∞ (G)∗ is an invariant mean on L∞ (G). Finally, (iii) =⇒ (i) is trivial.  We have defined amenable, locally compact groups G in terms of left invariant means on L∞ (G). This definition is not always convenient to work with. For example, if we have a closed subgroup H of G, then Haar measure on H need not be the restriction to H of Haar measure on G, so that there is, in general, no relation between L∞ (H) and L∞ (G). This is particularly awkward if we want to investigate the hereditary properties of amenability. We shall next characterize amenable, locally compact groups in terms of other (left invariant) subspaces of L∞ (G). In what follows let, for a locally compact group G, the Banach spaces L∞ (G) and L∞ (G)∗ ∼ = L1 (G)∗∗ be equipped with their canonical Banach 1 L (G)-bimodule structures as dual and second dual of the Banach algebra L1 (G) (see Section B.2). Definition 1.1.10. Let G be a locally compact group, and let E be a closed right L1 (G)-submodule of L∞ (G) containing the constants. Then a mean M ∈ E ∗ is called topologically left invariant if f · M = f, 1M

(f ∈ L1 (G)).

(Here, · denotes the canonical module action of L1 (G) on its second dual L∞ (G)∗ .) Of course, one can also define topologically right invariant means and simply topologically invariant means on suitable submodules of L∞ (G), such as C(G), LUC(G), RUC(G), and UC(G) (Proposition D.4.2(i)). We first clarify the relationship between left invariant means and topologically invariant means: Lemma 1.1.11. Let G be a locally compact group, and let E be L∞ (G), C(G), LUC(G), RUC(G), or UC(G). Then every topologically left invariant mean on E is left invariant. Proof. Let M ∈ E ∗ be a topologically left invariant mean. Let φ ∈ E, and let g ∈ G. Fix f ∈ L1 (G) with 1, f  = 1, and note that Lx φ, M  = (Lx φ) · f, M  = φ · (δx ∗ f ), M  = φ, M  so that M is left invariant, as claimed.

(φ ∈ E, x ∈ G), 

Lemma 1.1.12. Let G be a locally compact group, and let M ∈ UC(G)∗ be a left invariant mean. Then M is topologically left invariant.

26

1 Amenable, Locally Compact Groups

Proof. Let φ ∈ UC(G), and note that Lx (φ · f ) = δ(x−1 )(φ · Rx−1 f )

(x ∈ G, f ∈ L1 (G)).

(1.4)

As L1 (G)  f → φ, f · M  is bounded, there is ψ ∈ L∞ (G) such that φ, f · M  = f, ψ for f ∈ L1 (G). Note that f, Rx ψ = δ(x−1 )Rx−1 f, ψ = δ(x−1 )(φ · Rx−1 f ), M  = Lx (φ · f ), M , = φ · f, M  = f, ψ

by (1.4),

(x ∈ G, f ∈ L1 (G))

and thus Rx ψ = ψ for all x ∈ G, i.e., ψ is constant. For any φ ∈ UC(G), there is, therefore, a unique cφ ∈ C such that φ, f · M  = f, 1cφ ,

(f ∈ L1 (G)).

We claim that cφ = φ, M  for each φ ∈ UC(G): this establishes that M is indeed a topologically invariant mean on UC(G). To see this, let (eα )α be a bounded approximate identity for L1 (G) consisiting of nonnegative norm one functions (which exists according to Theorem D.2.5). Then (eα )α is also bounded right approximate identity for the L1 (G)-module UC(G). Finally, note that cφ = limφ, eα · M  = limφ · eα , M  = φ, M  α

α

(φ ∈ UC(G)), 

which completes the proof.

Theorem 1.1.13. For a locally compact group G, the following are equivalent: (i) G is amenable; (ii) there is a topologically left invariant mean on (iii) there is a left invariant mean on C(G); (iv) there is a topologically left invariant mean on (v) there is a left invariant mean on LUC(G); (vi) there is a topologically left invariant mean on (vii) there is a left invariant mean on RUC(G); (viii) there is a topologically left invariant mean on (ix) there is a left invariant mean on UC(G). (x) there is a topologically left invariant mean on

L∞ (G); C(G); LUC(G); RUC(G); UC(G).

Proof. In view of Lemma 1.1.11, (ii) =⇒ (i), (iv) =⇒ (iii), (vi) =⇒ (v), (viii) =⇒ (vii), and (x) =⇒ (ix) are clear, and both (i) =⇒ (iii) =⇒ (v) =⇒ (vii) =⇒ (ix) and (ii) =⇒ (iv) =⇒ (vi) =⇒ (viii) =⇒ (x) are trivial. Hence, only (ix) =⇒ (ii) still needs proof.

1.1 Invariant Means and Asymptotic Invariance Properties

27

Let M be a left invariant mean on UC(G). By Lemma 1.1.12, M is also topologically left invariant. Let (eα )α∈A be a bounded approximate identity for L1 (G) consisting of nonnegative norm one functions. By Proposition D.4.2(ii), (eα · φ · eα )α∈A is a (necessarily bounded) net in UC(G) for each φ ∈ L∞ (G). Choose an ultrafilter U on A that dominates the order filter, and define ˜ : L∞ (G) → C, φ → lim eα · φ · eα , M . M α→U

∞

˜ is a mean on L (G). For f ∈ L1 (G) and φ ∈ L∞ (G), It is easily seen that M we have ˜  = lim (eα · φ · f ) · eα , M  φ · f, M α→U

= lim (eα · φ · eα ) · f, M  α→U

= f, 1 lim eα · φ · eα , M  α→U

˜ , = f, 1φ, M ˜ is topologically left invariant. so that M



Of course, there are analogues of Theorem 1.1.13 for right invariance and (two-sided) invariance. Corollary 1.1.14. For a locally compact group G, let Gd be the same group equipped with the discrete topology. Then, if Gd is amenable, so is G. Proof. If Gd is amenable, then there is a left invariant mean M on ∞ (G). Since C(G) ⊂ ∞ (G), M |C(G) is a left invariant mean on C(G), so that G is amenable.  As we shall see in the next section, the converse of Corollary 1.1.14 is false. The following is a an analog of Lemma 1.1.8 for topological left invariance: Lemma 1.1.15. The following are equivalent for a locally compact group G: (i) G is amenable; (ii) there is a net (mα )α of nonnegative functions of norm one in L1 (G) such that (f ∈ L1 (G)). (1.5) f ∗ mα − f, 1mα → 0 Proof. Exercise 1.1.6 below.



The relevance of the following definition—another asymptotic invariance property—will become apparent in Section 1.4. Definition 1.1.16. Let G be a locally compact group, and let p ∈ [1, ∞). We say that G has Reiter’s Property (Pp ) if there is a net (ξα )α of nonnegative norm one functions in Lp (G) such that

28

1 Amenable, Locally Compact Groups

sup Lx ξα − ξα p → 0

x∈K

for all compact subsets K of G. By Lemma 1.1.8, G is amenable if it has Reiter’s Property (P1 ), but more is true: Theorem 1.1.17 (Reiter’s Theorem). The following are equivalent for a locally compact group G: (i) (ii) (iii) (iv)

G is amenable; G has Reiter’s Property (P1 ); there is p ∈ [1, ∞) such that G has Reiter’s Property (Pp ); G has Reiter’s Property (Pp ) for all p ∈ [1, ∞).

Proof. (ii) =⇒ (iii) and (iv) =⇒ (ii) are trivial, and—as noted before—(ii) =⇒ (i) follows from (the easy direction of) Lemma 1.1.8. From Exercise 1.1.7 below, we conclude that (iii) =⇒ (ii) and (ii) =⇒ (iv). (i) =⇒ (ii): We need to show that, given  > 0 and K ⊂ G compact, there is a nonnegative norm one function m ∈ L1 (G) such that supx∈K Lx m − m < . By Lemma 1.1.15, there is a net (mα )α of non-negative norm one functions in L1 (G) such that (1.5) holds. Let  > 0, and let K ⊂ G be compact; suppose without loss of generality that eG ∈ K. Fix an arbitrary nonnegative f ∈ L1 (G) with f 1 = 1. As a consequence of Proposition D.2.1, the subset L := {Lx f : x ∈ K} of L1 (G) is compact, so that sup g ∗ mα − mα 1 → 0. g∈L

Consequently, there is an index α such that sup Lx (f ∗ mα ) − mα 1 = sup g ∗ mα − mα 1
0}. Then U is an open, symmetric, relatively

34

1 Amenable, Locally Compact Groups

compact neighborhood of eG . Choose S be as in Lemma 1.2.4, and define f : G → [0, ∞) through  g(xs−1 ) (x ∈ G). (1.10) f (x) := s∈S

By Lemma 1.2.4(ii), the sum in (1.10) is finite whenever x ranges through an arbitrary compact subset of G. This establishes at once that f is well-defined and continuous. (i) is now an immediate consequence of Lemma 1.2.4(i) and the definition of U . For (ii), let K ⊂ G be compact. By Lemma 1.2.4(ii), there are s1 , . . . , sn such that  {x ∈ KH : f (x) > 0} = U s ∩ KH ⊂ U s1 ∪ · · · ∪ U sn . s∈S

Since U s1 ∪ · · · ∪ U sn is relatively compact, it follows that supp(f |KH ) is compact.  Proposition 1.2.6. Let G be a locally compact group, and let H be a closed subgroup of G. Then there is a Bruhat function for H. Proof. Let f be as in Lemma 1.2.5, and define  α : G → C, x → f (xy) dmH (y). H

It follows from Lemma 1.2.5(ii) that α is well defined, and Exercise 1.2.1 below establishes its continuity. Moreover, it follows from Lemma 1.2.5(i) and standard properties of Haar measure (see, e.g., [56, Lemma 9.2.5]) that α(x) > 0 for all x ∈ G. Define f (x) . β : G → C, x → α(x) Then β is continuous and satisfies Definition 1.2.3(a) by Lemma 1.2.5(ii). Since mH is left invariant, α is constant on the left cosets of H. Hence, β satisfies Definition 1.2.3(b) as well.  Theorem 1.2.7. Let G be an amenable, locally compact group, and let H be a closed subgroup of G. Then H is amenable. Proof. Let β : G → C be a Bruhat function for H, and define T : C(H) → ∞ (G) through  (T φ)(x) := φ(y)β(x−1 y) dmH (y) (φ ∈ C(H), x ∈ G). H

1.2 Hereditary Properties

35

It is immediate that T is a contraction such that T 1 = 1, and by Exercise 1.2.1 below, T maps into C(G). Let M be a left invariant mean on C(G), and ˜ is a mean on C(H). ˜ := T ∗ M , so that M define M For φ ∈ C(H) and z ∈ H, we have:  T (Lz φ)(x) = φ(zy)β(x−1 y) dmH (z) H  = φ(y)β(x−1 z −1 y) dmH (k) H  φ(y)β((zx)−1 y) dmH (k) = H

= (T φ)(zx) = Lz (T φ)(x)

(x ∈ G).

˜ is left invariant. This shows that M



Together, Theorem 1.2.7 and Example 1.1.5 yield: Corollary 1.2.8. Let G be a locally compact group that contains a closed subgroup isomorphic to F2 . Then G is not amenable. As a consequence, the converse of Corollary 1.1.14 does not hold: Example 1.2.9. For N ≥ 3, the group SO(N ) is compact and thus amenable. However, by Corollary 0.1.5, SO(N ) has a subgroup isomorphic to F2 . Since in the discrete topology every subgroup is closed, SO(N )d is cannot be amenable. For N ∈ N, let GL(N, R) denote the group of all invertible N ×N matrices over R, and let SL(N, R) stand for all the subgroup of GL(N, R) consisting of all matrices of determinant 1 with determinant 1; analogously, GL(N, C) and SL(N, C) are defined. In the next example, we use Theorem 1.2.7 to see that, if N ≥ 2, the all those matrix groups—equipped with their Euclidean topology—are not amenable: Example 1.2.10. Recall, from complex variables, the notion of a M¨ obius transform ([60, Definition III.3.5]). The collection of all M¨obius transforms forms a group under composition, which we denote by G. For each matrix

A = ac db ∈ GL(2, C), there is an element gA of G canonically associated with it through az + b . gA (z) = cz + d It is routinely seen that the assignment GL(2, C) → G,

A → gA

(1.11)

36

1 Amenable, Locally Compact Groups

is a group homomorphism. Let PSL(2, R) be the image of SL(2, R) under (1.11), equipped with the quotient topology. It is elementary to check that that PSL(2, R) ∼ = SL(2, R)/{±I2 }. We shall first show that PSL(2, R) is not amenable. Define 2z + 3 5z + 24 and g2 (z) := , g1 (z) := z+2 z+5 so that g1 , g2 ∈ PSL(2, R). We claim that the subgroup H of PSL(2, R) generated algebraically by g1 and g2 is closed and isomorphic to F2 , which establishes the non-amenability of PSL(2, R) by Corollary 1.2.8. Consider the circles K1 := {z ∈ C : |z + 2| = 1}, ˜ 1 := {z ∈ C : |z − 2| = 1}, K K2 := {z ∈ C : |z + 5| = 1}, and ˜ 2 := {z ∈ C : |z − 5| = 1}, K and note that each of them lies in the exterior of any other. Also, note that ˜ j , the interior of Kj onto the exterior of K ˜ j , and the that gj maps Kj onto K ˜ j for j = 1, 2. exterior of Kj onto the interior of K For a non-empty reduced word w over {g1 , g2 , g1−1 , g2−1 }, let hw ∈ H be the corresponding M¨ obius transformation. We claim the following for j = 1, 2 ˜ 1 , K2 , and K ˜ 2 : if w starts and any point z exterior to all four circles K1 , K ˜ j , and if w starts with g −1 , then with gj , then hw (z) lies in the interior of K j hw (z) lies in the interior of Kj . We proceed by induction on the length of w. The length one case is clear because gj maps the exterior of Kj onto ˜ j onto the ˜ j , and, consequently, g −1 maps the exterior of K the interior of K j interior of Kj . For the induction step, first consider the case w = gj v with v = . As w is reduced, v can only start with gj , gk , or gk−1 where k = j. By the ˜j, K ˜ k , or Kk and thus induction hypothesis, hv (z) lies in the interior of K in the exterior of Kj . Consequently, hw (z) = gj (hv (z)) is an interior point ˜ j . If w = g −1 v, then v has to start with g −1 , gk , or g −1 , so that hv (z) of K j j k ˜ k , or Kk and thus in the exterior of K ˜ j . It follows that is interior to Kj , K hw (z) = gj−1 (hv (z)) lies in the interior of Kj . Fix a point z exterior to all four circles, and let d > 0 be its distance from ˜ 1 ∪K2 ∪ K ˜ 2 . Then, by the foregoing, hw (z) lies in the interior of at least K1 ∪ K ˜ ˜ 2 for each non-empty word w over {g1 , g2 , g −1 , g −1 }, one of K1 , K1 , K2 , and K 1 2 so that (1.12) |hw (z) − z| > d.

1.2 Hereditary Properties

37

It is immediate from (1.12) that H is the free group generated by g1 and g2 . Moreover, as PSL(2, R)  A → gA (z) is continuous, it also follows from (1.12) that H is discrete in its relative topology (and thus closed). Consequently, PSL(2, R) is not amenable, and neither is SL(2, R) by Corollary 1.2.2. As SL(2, R) embeds canonically as a closed subgroup into SL(N, R), GL(N, R), SL(N, C), and GL(N, C), these groups cannot be amenable either. Next, we consider short, exact sequences: Theorem 1.2.11. Let G be a locally compact group, and let N be a closed, normal subgroup such that both N and G/N are amenable. Then G is amenable. Proof. Let MN be a left invariant mean on C(N ). For φ ∈ LUC(G), define T φ : G → C through (T φ)(x) := (Lx φ)|N , MN 

(x ∈ G)

As φ ∈ LUC(G), it follows T φ ∈ C(G). Obviously, T : LUC(G) → C(G) is a linear contraction. Let x1 , x2 ∈ G belong to the same coset of N , i.e.,there is y ∈ N such that x1 = x2 y. For φ ∈ LUC(G), this means that (T φ)(x1 ) = (Lx1 φ)|N , MN  = Ly ((Lx2 φ)|N ), MN  = (Lx2 φ)|N , MN  = (T φ)(x2 ). Hence, for any φ ∈ LUC(G), the value of T φ at x ∈ G depends only on the coset of xN , so that T φ drops to a function in C(G/N ). Consequently, T induces a linear contraction T˜ : LUC(G) → C(G/N ); it is immediate that T˜1 = 1. Let MG/N be a left invariant mean on C(G/N ). Define M ∈ LUC(G)∗ by letting (φ ∈ LUC(G)). φ, M  := T˜φ, MG/N  Then it is clear that M is a mean on LUC(G). To see that M is left invariant, first observe that T˜(Lx φ)(yN ) = LxN (T˜φ)(yN )

(φ ∈ LUC(G), x, y ∈ G).

(1.13)

Hence, we have for φ ∈ LUC(G) and x ∈ G: Lx φ, M  = T˜(Lx φ), MG/N  = LxN (T˜φ), MN/G ,

by (1.13),

= T˜φ, MN/G  = φ, M . Thus, M is indeed left invariant.



38

1 Amenable, Locally Compact Groups

Example 1.2.12. A group G is called solvable if there are normal subgroups N0 , . . . , Nn of G such that: • {eG } = N0 ⊂ · · · ⊂ Nn = G; • Nj /Nj−1 is abelian for j = 1, . . . , n. Since abelian groups are amenable by Example 1.1.7, it follows from successively applying Theorem 1.2.11 that all solvable groups are amenable when equipped with the discrete topology. If a locally compact group G is solvable, Gd is amenable—as solvability is a purely algebraic property—and by Corollary 1.1.14, G is amenable. Finally, we consider directed unions: Proposition 1.2.13. Let G be a locally compact group, and let H be a directed family of closed subgroups of G such that: (a) each  H ∈ H is amenable; (b) {H : H ∈ H} is dense in G. Then G is amenable. Proof. For each H ∈ H, let MH be a left translation invariant mean on ˜ H ∈ LUC(G)∗ through LUC(H). Define M ˜ H  := φ|H , MH  φ, M

(φ ∈ LUC(G)),

˜ H is a mean on LUC(G). so that M ˜ H )H∈H in LUC(G)∗ . Let M be a weak∗ accumulation point of the net (M Clearly, M is a mean on LUC(G).  To see that M is left invariant, let φ ∈ LUC(G), and let x ∈ {H : H ∈ H}. Choose H0 ∈ H such that x ∈ H0 . It follows for all H ∈ H with H ⊃ H0 that ˜ H  = Lx (φ|H ), MH  = φ|H , MH  = φ, M ˜H Lx φ, M and thus Lx φ, M  = φ, M . (1.14)  Since φ ∈ LUC(G) and {H : H ∈ H} is dense in G, we see—just as in the proof of Proposition 1.2.1—that (1.14) holds for any x ∈ G.  Example 1.2.14. A group G is called locally finite if every finite set of elements of G generates a finite subgroup of G. Let G be a locally finite group, and let F be the collection of all finite subsets of G. For each F ∈ F, let F  denote the group generated  by F , so that (F )F ∈F is a directed family of subgroups of G such that F ∈F F  = G. Since finite groups are amenable, G is amenable.

1.2 Hereditary Properties

39

Exercises Exercise 1.2.1. Let G be a locally compact group, let H be a closed subgroup of G, let φ ∈ L∞ (H), and let f : G → C be continuous such that supp(f |KH ) is compact for every compact subset K of G. Then  G → C, x → φ(y)f (xy) dmH (y) H

is continuous. (Hint: Use the fact that continuous functions with compact support on locally compact groups are uniformly continuous, e.g., [118, (2.6) Proposition].) Exercise 1.2.2. What’s wrong with the following simple “proof” of Theorem 1.2.7? Define a map T : C(H) → L∞ (G) as follows. Let (xα )α be a family of representatives of the right cosets of H. Let x ∈ G. Since G is the disjoint union of (Hxα )α , there is a unique xα such that x ∈ Hxα , so that xx−1 α ∈ H. Define (T φ)(x) := φ(xx−1 α )

(φ ∈ C(H)).

It follows immediately from that definition that T φ|Hxα ∈ C(Hxα ) for all φ ∈ C(H) and any xα , so that T φ ∈ L∞ (G) for all φ ∈ C(H). The following properties of T are straightforward: • T is a linear isometry; • T 1 = 1; • T (Ly φ) = Ly (T φ)

(φ ∈ C(H)).

Consequently, if M is a left invariant mean on L∞ (G), then T ∗ M is a left invariant mean on C(H).

Show that this argument does indeed work if H is open. Exercise 1.2.3. Show that the Heisenberg group ⎧⎡ ⎫ ⎤ ⎨ 1xy ⎬ ⎣0 1 z ⎦ : x, y, z ∈ R ⎩ ⎭ 001 is solvable and thus amenable. Exercise 1.2.4. Let SN denote the group of all permutations of N, i.e., the set of all bijective maps from N to itself with the composition of maps as product, equipped with the discrete topology. Show that {π ∈ SN : there is N ∈ N such that π(n) = n for n ≥ N } is an amenable subgroup of SN whereas SN itself is not amenable.

40

1 Amenable, Locally Compact Groups

1.3 Uniformly Bounded Representations To further illustrate the many nice properties of amenable, locally compact groups, we present a classical application of invariant means to representation theory. We denote by B(E) the Banach algebra of all bounded linear operators on a Banach space E. Recall that the the weak operator topology on B(E) is defined by the family of seminorms (px,φ )x∈E,φ∈E ∗ , where px,φ (T ) := |T x, φ|

(x ∈ E, φ ∈ E ∗ , T ∈ B(E)).

Similarly, the strong operator topology on B(E) is given through the family of seminorms (px )x∈E , where px (T ) := T x

(x ∈ E, T ∈ B(E)).

It is obvious, that the weak operator topology on B(E) is coarser than the strong operator topology; for more on how these two topologies relate, see Exercises 1.3.1 and 1.3.2 below. Definition 1.3.1. Let G be a locally compact group, and let E be a Banach space. A representation of G on E is a group homomorphism π from G into the invertible bounded operators on E, which is continuous with respect to the given topology on G and the weak operator topology on B(E). Despite Exercise 1.3.2 we have: Lemma 1.3.2. Let G be a locally compact group, let E be a Banach space, and let π : G → B(E) be a representation of G on E. Then π is continuous with respect to the given topology on G and the strong operator topology on B(E). Proof. Of course, it is enough to show that π is continuous at eG with respect to the strong operator topology on B(E). Choose a compact neighborhood K of eG in G. Then {π(x) : x ∈ K} is compact in the weak operator topology, so that {π(x)ξ : x ∈ K} is compact in the weak topology of E for each ξ ∈ E. The Uniform Boundedness Principle yields C > 0 with supx∈K π(x) ≤ C. Let U be a symmetric neighborhood of eG such that U U ⊂ K. Let V be the family of all neighborhoods of eG contained in U ordered by reversed set inclusion. Let (eV )V ∈V be a bounded approximate identity for L1 (G), consisting of continuous functions, as specified in Theorem D.2.5. Fix ξ ∈ E. As {π(x)ξ : x ∈ K} is weakly compact, so is its closed convex hull by [102, Theorem V.6.4]. Standard results on vector valued integration, such as [292, Theorem I.3.27], thus guarantee that the integrals in (1.15) below all converge in the weak topology of E. The identity

1.3 Uniformly Bounded Representations

41

 π(y)

  eV (x)π(x)ξ dx = eV (x)π(yx)ξ dx G G  = eV (y −1 x)π(x)ξ dx

(1.15) (y ∈ U )

G

is immediate.  Define a net (ξV )V ∈V in E through ξV := G eV (x)π(x)ξ dx for V ∈ V. We claim that ξV → ξ in the weak topology of E. To see this, let φ ∈ E ∗ and  > 0. As G  x → π(x)ξ, φ is continuous at eG , there is V0 ∈ V such that |π(x)ξ, φ − ξ, φ| <  for all x ∈ V0 . For V ∈ V with V ⊂ V0 , this means that     |ξV − ξ, φ| =  eV (x)π(x)ξ, φ dx − ξ, φ G    =  eV (x)(π(x)ξ, φ − ξ, φ) dx G ≤ eV (x)|π(x)ξ, φ − ξ, φ| dx G

< . As φ ∈ E ∗ and  > 0 were arbitrary, this yields the claim. Set F := {ζ ∈ E : G  x → π(x)ζis norm continuous at eG }. Obviously, F is a linear subspace of E. As     −1  π(y)ξV − ξV  =  (eV (y x) − eV (x))π(x)ξ dx ,

by (1.15),

G

≤ CξLy−1 eV − eV 1

(y ∈ U, V ∈ V).

the maps G → E,

x → π(x)ξV

are continuous, for all V ∈ V, at eG by Proposition D.2.1, i.e., the net (ξV )V ∈V lies in F . We claim that F is closed. To see this, let (ζn )∞ n=1 be a sequence in F converging to ζ ∈ E, and let (xα )α∈A be a net in K converging to eG . Note that π(xα )ζ − ζ = π(xα )ζ − π(xα )ζn  + π(xα )ζn − ζn  + ζn − ζ (1.16) ≤ (C + 1)ζn − ζ + π(xα )ζn − ζn   for all n ∈ N and α ∈ A. Let  > 0. Choose n0 ∈ N such that ζn −ζ < 2(C+1)  for n ≥ n0 and α0 such that π(xα )ζn0 − ζn0  < 2 for α  α0 . From (1.16), it follows that π(xα )ζ − ζ <  for α  α0 . As  > 0 was arbitrary, this

42

1 Amenable, Locally Compact Groups

means that ζ ∈ F as well. Hence, F is indeed a closed—and thus weakly closed—subspace of E. As ξ is the weak limit of (ξV )V , it follows that ξ must also lie in F . Since ξ ∈ E was arbitrary, we conclude that F = E.  Definition 1.3.3. Let G be a locally compact group, and let E be a Banach space. Then: (a) two representations π1 and π2 of G on E are similar if there is an invertible operator T ∈ B(E) such that π1 (x) = T π2 (x)T −1

(x ∈ G);

(b) a representation π of G on E is uniformly bounded if supx∈G π(x) < ∞; (c) a representation π of G on a Hilbert space is unitary if π(G) consists of unitary operators. In the representation theory of locally compact groups, the unitary representations play the predominant rˆ ole because they establish a link between locally compact groups and C ∗ -algebras (see, e.g., [97, Chapter 13]). It is natural to ask when an arbitrary representation of a locally compact group is similar to a unitary representation. A necessary requirement is, obviously, that the representation be uniformly bounded. But is this necessary condition also sufficient? This is indeed the case for amenable groups: Theorem 1.3.4. Let G be an amenable locally compact group, let H be a Hilbert space, and let π : G → B(H) be a uniformly bounded representation. Then π is similar to a unitary representation. Proof. For ξ, η ∈ H, define a bounded function φξ,η : G → C,

x → π(x−1 )ξ|π(x−1 )η.

(To tell a Hilbert space inner product—linear in the first variable and conjugate linear in the second one—apart from a Banach space duality, we use the symbol ·|· for an inner product as opposed to ·, ·.) Note that |φξ,η (x) − φξ,η (y)| ≤ |π(x−1 )ξ|π(x−1 )η − π(y −1 )ξ|π(y −1 )η| ≤ |π(x−1 )ξ|π(x−1 )η − π(x−1 )ξ|π(y −1 )η| + |π(x−1 )ξ|π(y −1 )η − π(y −1 )ξ|π(y −1 )η| ≤ Cξπ(x−1 )η − π(y −1 )η + Cηπ(x−1 )ξ − π(y −1 )ξ

(x, y ∈ G),

where C := supx∈G π(x). Since π is continuous with respect to the strong operator topology on B(H) by Lemma 1.3.2, we conclude that φξ,η ∈ C(G). Let M be a left invariant mean on C(G), and define

1.3 Uniformly Bounded Representations

[ξ|η] := φξ,η , M 

43

(ξ, η ∈ H).

It is easy to see that [·, ·] is a bounded, positive semidefinite, sesquilinear form on H. By standard Hilbert space theory (e.g., [292, Theorem III.12.8]), there is a unique S ∈ B(H) such that [ξ|η] = Sξ|η

(ξ, η ∈ H).

(1.17)

As [·, ·] is positive semidefinite, it is immediate that S is positive; we claim that it is invertible as well. Set 1 |||ξ||| := [ξ, ξ] 2 (ξ ∈ H). Then it is immediate that ||| · ||| ≤ C ·  on H. Also, since ξ2 = π(x)∗ π(x)π(x−1 )ξ|π(x−1 )ξ ≤ π(x)∗ π(x)φξ,ξ (x) ≤ C 2 φξ,ξ (x)

(ξ ∈ H, x ∈ G),

we have  ·  ≤ C||| · ||| as well. Consequently,  ·  and ||| · ||| are equivalent norms on H. From (1.17), we conclude that S is invertible. 1 Set T := S 2 . Noting that φπ(x)ξ,π(x)η (y) = π(y −1 )π(x−1 )ξ|π(y −1 )π(x−1 )η = π((xy)−1 )ξ|π((xy)−1 )η = (Lx−1 φξ,η )(y)

(ξ, η ∈ H, x, y ∈ G), (1.18)

we obtain for ξ, η ∈ H and x ∈ G: T π(x)T −1 ξ|T π(x)T −1 η = [π(x)T −1 ξ|π(x)T −1 η] = φπ(x)T −1 ξ,π(x)T −1 η , M  = Lx−1 φT −1 ξ,T −1 η , M ,

by (1.18),

= φT −1 ξ,T −1 η , M  = [T −1 ξ|T −1 η] = ξ|η. Consequently, the representation G  x → T π(x)T −1 of G is unitary.



We conclude this section with a somewhat unusual application of Theorem 1.3.4. (For the notions of an unconditional basis and the equivalence of bases, we refer to Section A.1 below; for more background, see [225], [360], or [112].) Corollary 1.3.5 (K¨ othe–Lorch Theorem). Every normalized unconditional basis of a Hilbert space is equivalent to an orthonormal basis. Proof. Let (xn )∞ n=1 be a normalized unconditional basis for a Hilbert space H.

44

1 Amenable, Locally Compact Groups

Let the group G := {−1, 1}N be equipped with the discrete topology; it ∞ is abelian and thus amenable. For ∞ := (∗n )n=1 ∈ G, define a linear map π() : H → H by letting π()x := n=1 n xn (x)xn ; by Theorem A.1.7, this is well defined, π() is bounded for each  ∈ G, and we have sup∈G π() < ∞. Consequently, π : G → B(H) is equivalent to a unitary representation of G on H, i.e., there is an invertible operator T ∈ B(H) such that G   → T −1 π()T is a unitary representation. It is clear that (T −1 xn )∞ n=1 is an unconditional basis of H equivalent to . (xn )∞ n=1 Let n = m. Choose  = (n )∞ n=1 ∈ G such that n = 1 and m = −1. Then both T −1 xn and T −1 xm are eigenvectors of T −1 π()T , the first with corresponding eigenvalue 1 and the second with corresponding eigenvalue −1. As T −1 π()T is unitary and thus normal, standard spectral theory of normal operators ([292, 12.12 Theorem (e)]) yields that T −1 xn and T −1 xm are orthogonal. Since 1 ≤ T −1 xn  ≤ T −1  (n ∈ N), T  ∞  −1 we conclude from [225, Proposition 1.c.7] that TT −1 xxnn  is an unconn=1

∞ ditional basis of H equivalent to (T −1 xn )∞ n=1 —and thus to (xn )n=1 —that is obviously orthonormal. 

Exercises Exercise 1.3.1. Let E be a Banach space. Show that the following are equivalent for a linear functional φ : B(E) → C: (i) φ is continuous with respect to the strong operator topology; (ii) φ is continuous with respect to the weak operator topology; (iii) there are x1 , . . . , xn ∈ E and ψ1 , . . . , ψn ∈ E ∗ such that T, φ =

n 

T xj , ψj 

(T ∈ B(E)).

j=1

Exercise 1.3.2. Let E be a Banach space. Show that the strong and the weak operator topology on B(E) coincide if and only if dim E < ∞. Exercise 1.3.3. Let H be a Hilbert space, and let U(B(H)) denote the group of unitary operators on H. Show that: (a) the weak and the strong operator topology on B(H) coincide on U(B(H)); (b) equipped with the weak/strong operator topology, U(B(H)) is a topological group;

1.3

Uniformly Bounded Representations

45

(c) equipped with the weak/strong operator topology, U(B(H)) is locally compact if and only if dim H < ∞. Exercise 1.3.4. What is an example of a representation of R on a Hilbert space which is not uniformly bounded? Exercise 1.3.5. Show that the operator T obtained in the proof of Theorem 1.3.4 satisfies T T −1  ≤ supx∈G π(x)2 . Exercise 1.3.6. Let G be a locally compact group, let E be a Banach space, and let π : G → B(E) be a uniformly bounded representation. Show that there is an equivalent norm on E such that, with respect to this new norm, π(G) consists of isometries.

1.4 Leptin’s Theorem In this section, we characterize the amenable, locally compact groups in terms of their Fourier algebras. For the necessary background on Fourier and Fourier–Stieltjes algebras, we refer to Appendix F. Our main result is: Theorem 1.4.1 (Leptin’s Theorem). The following are equivalent for a locally compact group G: (i) (ii) (iii) (iv) (v)

G is amenable; A(G) has a bounded approximate identity consisting of states of VN(G); A(G) has a bounded approximate identity; Br (G) contains the constant function 1; Br (G) = B(G).

In the literature, “Leptin’s Theorem” is usually used to refer to the equivalence between (i) and (iii). To prove Theorem 1.4.1, we require some preparations. ˆ ∈ B(L2 (G× Proposition 1.4.2. Let G be a locally compact group, and let W G)) be defined as in (F.1). Furthermore, let (ξα )α∈A be a net of unit vectors in L2 (G), and let eα (x) := λ(x)ξα |ξα 

(x ∈ G, α ∈ A).

Then the following are equivalent: (i) (ii) (iii) (iv)

(eα )α∈A is a bounded approximate identity for A(G); eα → 1 uniformly on compact subsets of G; eα → 1 with respect to σ(B(G), C ∗ (G)); ˆ (ξα ⊗ η) − ξα ⊗ η2 → 0 for all η ∈ L2 (G). W

(1.19)

46

1 Amenable, Locally Compact Groups

Proof. (i) =⇒ (ii): Let K ⊂ G be compact. As A(G) is regular by Theorem F.2.2(iv), there is f ∈ A(G) such that f |K = 1 by Proposition B.1.7. With  · ∞ denoting the supremum norm, we have sup |eα (x) − 1| ≤ eα f − f ∞ ≤ eα f − f A(G) → 0.

x∈K

(ii) =⇒ (iii): This is clear because C00 (G) is dense in L1 (G) and thus in C (G). (iii) =⇒ (iv): Let η ∈ L2 (G) be, and set f := η η¯; it follows that ˜ λ(f ), eα  → 1. We have ∗

˜ ), eα  = eα η|η λ(f   = ξα (y −1 x)ξα (x)η(y)η(y) dx dy G G   ξα (x)ξα (yx)η(y)η(y) dx dy = G G   ˆ (ξα ⊗ η)(x, y) dx dy (ξα ⊗ η)(x, y)W = G

G

ˆ (ξα ⊗ η) = ξα ⊗ η|W

(α ∈ A).

˜ ), eα  → 1, we conclude that As λ(f ˆ (ξα ⊗ η) − ξα ⊗ η22 W ˆ (ξα ⊗ η)22 − 2 Re ξα ⊗ η|W ˆ (ξα ⊗ η) + ξα ⊗ η22 = W ˆ (ξα ⊗ η) = 2 − 2 Reξα ⊗ η|W → 0. (iv) =⇒ (i): Let f ∈ A(G). Without loss of generality suppose that there is a unit vector η ∈ L2 (G) such that f (x) = λ(x)η|η for x ∈ G. For x ∈ VN(G), we have by (E.5) and (F.2): |eα f − f, x| ˆ (ξα ⊗ η)|ξα ⊗ η − (1 ⊗ x)(ξα ⊗ η)|ξα ⊗ η| ˆ ∗ (1 ⊗ x)W = |W ˆ (ξα ⊗ η) − (1 ⊗ x)(ξα ⊗ η)|ξα ⊗ η| ˆ (ξα ⊗ η)|W = |(1 ⊗ x)W ˆ (ξα ⊗ η) − (1 ⊗ x)W ˆ (ξα ⊗ η)|ξα ⊗ η| ˆ (ξα ⊗ η)|W ≤ |(1 ⊗ x)W ˆ (ξα ⊗ η)|ξα ⊗ η − (1 ⊗ x)(ξα ⊗ η)|ξα ⊗ η| + |(1 ⊗ x)W ˆ (ξα ⊗ η) − ξα ⊗ η| ˆ (ξα ⊗ η)|W = |(1 ⊗ x)W ˆ (ξα ⊗ η) − ξα ⊗ η))|ξα ⊗ η| + |(1 ⊗ x)(W ˆ (ξα ⊗ η) − ξα ⊗ η2 . ≤ 2xW

1.4

Leptin’s Theorem

47

It follows that ˆ (ξα ⊗ η − ξα ⊗ η2 → 0, eα f − f A(G) ≤ 2W 

which proves (i). We need one more lemma.

Lemma 1.4.3. Let G be a locally compact group such that there is a bounded net in A(G) consisting of states of VN(G) that converges to the constant function 1 pointwise on G. Then G is amenable. Proof. Let (eα )α∈A be a bounded net in A(G) consisting of states of VN(G) such that eα (x) → 1 for x ∈ G. By Proposition F.2.6, there is a net of unit vectors (ξα )α∈A such that eα (e) = λ(x)ξα , ξα 

(x ∈ G, α ∈ A).

For x ∈ G, we then have: λ(x)ξα − ξα 22 = λ(x)ξα − ξα |λ(x)ξα − ξα  = λ(x)ξα 22 − 2 Re λ(x)ξα |ξα  + ξα 22 = 2(1 − Re eα (x)) → 0. For α ∈ A, set mα := ξα ξ¯α . It is obvious that (mα )α is a net of nonnegative norm one functions in L1 (G). Moreover, we have for x ∈ G: δx ∗ mα − mα 1 = (λ(x)ξα )(λ(x)ξα ) − ξα ξ¯α 1 ≤ (λ(x)ξα )(λ(x)ξα ) − (λ(x)ξα )ξ¯α 1 + (λ(x)ξα )ξ¯α − ξα ξ¯α 1 ≤ λ(x)ξα 2 λ(x)ξα − ξα 2 + λ(x)ξα − ξα 2 ξ¯α 2 = 2λ(x)ξα − ξα  → 0. The amenability of G then follows from (the easy direction of) Lemma 1.1.8.  Proof (of Leptin’s Theorem). (i) =⇒ (ii): By Theorem 1.1.17, G has Reiter’s Property (P2 ). Let (ξα )α∈A be a net in L2 (G) as required in the definition of Reiter’s Property (P2 ), and define eα (x) := λ(x)ξα |ξα 

(x ∈ G, α ∈ A).

Clearly, eα → 1 uniformly on compact subsets of G. By Proposition 1.4.2, (eα )α∈A is thus a bounded approximate identity for A(G), which obviously consists of states.

48

1 Amenable, Locally Compact Groups

(ii) =⇒ (iii) is trivial. (iii) =⇒ (iv): Let (eα )α be a bounded approximate identity for A(G). As in the proof of Proposition 1.4.2, we see that 1 is the σ(B(G), C ∗ (G)) limit of (eα )α and thus, obviously, lies in Br (G). (iv) =⇒ (v) is clear because Br (G) is an ideal of B(G), and (v) =⇒ (iv) is trivial. (iv) =⇒ (i): The constant function 1 is a multiplicative functional of Cr∗ (G) and thus a state. By Exercise 1.4.1 below, there is thus a net (eα )α of positive σ(Br (G),C ∗ (G))

definite functions in A(G) such that eα → r 1. By Proposition 1.4.2, this means that eα → 1 uniformly on compact subsets of G and thus, in particular, pointwise. Lemma 1.4.3 then yields the amenability of G. 

Exercises Exercise 1.4.1. Let M be a von Neumann algebra with predual M∗ , and let A be a σ(M, M∗ ) dense C ∗ -subalgebra of M. Show that, for each state φ of A, there is a net (φα )α of states in M∗ such that φα |A

σ(A∗ ,A)

→

φ.

Exercise 1.4.2. Let G be an amenable, locally compact group, and let f ∈ C0 (G). Show that there are g ∈ A(G) and h ∈ C0 (G) such that f = gh.

1.5 Fixed Point Theorems In Example 1.1.7, we used the Markov–Kakutani Fixed Point Theorem to show that abelian groups are amenable. In this section, we give a characterization of amenable, locally compact groups—in principle due to M. M. Day—in terms of a fixed point property. We say that a semigroup S acts: • affinely on a convex subset C of a vector space if, for each s ∈ S, the map C  x → s · x is affine, i.e., satisfies s · (tx + (1 − t)y) = t s · x + (1 − t) s · y for x, y ∈ C and t ∈ [0, 1]; • isometrically on a Banach space E if, for each s ∈ S, the map E  x → s·x is a—not necessarily linear—isometry, i.e., s · x − s · y = x − y holds for all x, y ∈ E. Theorem 1.5.1 (Day’s Fixed Point Theorem). For a locally compact group G, the following are equivalent: (i) G is amenable; (ii) if G acts affinely on a compact, convex subset C of a locally convex vector space E such that

1.5

Fixed Point Theorems

49

G × C → C,

(x, c) → x · c

(1.20)

is separately continuous, then there is c ∈ C such that x · c = c for all x ∈ G. Proof. (i) =⇒ (ii): Fix c0 ∈ C once and for all. Let Aff(C) denote the set of all continuous, affine functions on C; note that that the restrictions of all continuous linear functionals on E to C lie in Aff(C). For each f ∈ Aff(C), define φf : G → C, x → f (x · c0 ). Since (1.20) is continuous in the first variable, it is clear that φf lies in C(G). Let M be a left invariant mean on C(G). We claim: • there is c˜ ∈ C such that φf , M  = φ(˜ c)

(f ∈ Aff(C));

• this c˜ is the desired fixed point. Let (mα )α∈A be a net of means of the form (1.22) on C(G) converging to M in the weak∗ topology of C(G)∗ . For each α ∈ A, there are nα ∈ N, x1,α , . . . , xnα ,α ∈ G, and t1,α , . . . , tnα ,α ≥ 0 with t1,α + · · · + tnα ,α = 1 such that nα  tj,α δxj,α . mα = j=1

Set cα :=

nα 

tj,α xj,α · c0 ,

j=1

so that cα ∈ C and φf , mα  =

nα 

tj,α f (xj,α · c0 ) = f (cα )

(f ∈ Aff(C)).

j=1

Passing to a subnet, we can suppose that (cα )α converges to some c˜ ∈ C. It follows that f (˜ c) = lim f (cα ) = limφf , mα  = φf , M  α

α

(f ∈ Aff(C)),

as claimed. For x ∈ G and ψ ∈ E ∗ , the function fx,ψ : C → C,

c → x · c, ψ

belongs to Aff(C): here, the continuity of (1.20) in the second variable is used. Note that

50

1 Amenable, Locally Compact Groups

φfx,ψ (y) = fx,ψ (y · c0 ) = xy · c0 , ψ = (Lx φfeG ,ψ )(y) and thus

(x, y ∈ G, ψ ∈ E ∗ )

(x ∈ G, ψ ∈ E ∗ ).

φfx,ψ = Lx φfeG ,ψ

(1.21)

It follows that x · c˜, ψ = fx,ψ (˜ c) = φfx,ψ , M  = Lx φψeG ,ψ , M ,

by (1.21),

= φfeG ,ψ , M  = ψeG ,ξ (˜ c) = ˜ c, ψ

(x ∈ G, ψ ∈ E ∗ ).

Since E ∗ separates the points of E, it follows that x · c˜ = c˜ for all x ∈ G. (ii) =⇒ (i): Exercise 1.5.3 below.  We shall conclude this section with the proof of another fixed point theorem that we shall require in the sequel. Definition 1.5.2. Let E be a Banach space, and let ∅ = B ⊂ E be bounded. Then: (a) the circumradius of S in E is defined as (B, E) := inf{r ≥ 0 : there is x ∈ E such that B ⊂ Ballr (x, E)}; (b) the Chebyshev center of B in E is defined to be C(B, E) := {x ∈ E : B ⊂ Ball(B,E) (x, E)} Lemma 1.5.3. Let E be a Banach space, and let ∅ = B ⊂ E be bounded. Then  Cr (B, E) C(B, E) = r>(B,E)

holds where Cr (B, E) :=



Ballr (x, E)

(r > (B, E))

x∈B

Proof. Let x ∈ C(B, E), and let r > (B, E). By the definition of C(B, E), we have B ⊂Ball(B,E) (x, E), i.e., for each y ∈ B, we have y−x ≤ (B, E), so that x ∈ y∈B Ball(B,E)  (y, E) ⊂ Cr (B, E). As r > (B, E) was arbitrary, this proves C(B, E) ⊂ r>(B,E) Cr (B, E). For the reverse inclusion, let x ∈ Cr (B, E) for all r > (B, E), i.e., for all r > (B, E) and all y ∈ B, we have x − y ≤ r. This, of course, means

1.5

Fixed Point Theorems

51

that x − y ≤ (B, E) for all y ∈ B and thus B ⊂ Ball(B,E) (x, E), i.e., x ∈ C(B, E).  Corollary 1.5.4. Let E be a Banach space, and let ∅ = B ⊂ E be bounded. Then C(B, E) is a closed, convex subset of E. Moreover, if E is a dual Banach space, C(B, E) is weak∗ compact and non-empty. Proof. By Lemma 1.5.3, C(B, E) is an intersection of closed, convex sets and thus closed and convex itself. Suppose that E is a dual Banach space. Then C(B, E) is an intersection of weak∗ compact sets and thus weak∗ compact itself. Assume that C(B, E) = ∅. Note that Cr1 (B, E) ⊂ Cr2 (B, E) for (B, E) < r1 ≤ r2 . From the finite intersection property and Lemma 1.5.3, we thus conclude that there is r > (B, E), it is clear that (B, E) such that Cr (B, E) = ∅. By the definition of  there is y ∈ E such that B ⊂ Ballr (y, E), i.e., y ∈ x∈B Ballr (x, E). This  contradicts Cr (B, E) = ∅. Let E and F be Banach spaces, and let p ∈ [1, ∞]. We denote by E ⊕p F the p - direct sum of E and F , i.e., the direct sum E ⊕ F equipped with the norm 1 (x ∈ E, y ∈ F ) (x, y) := (xp + xp ) p if p ∈ [1, ∞) and (x, y) := max{x, y}

(x ∈ E, y ∈ F )

if p = ∞. We call a Banach space E L-embedded if there is a closed subspace Y of E ∗∗ such that E ∗∗ = E ⊕1 Y , where we indentify E with its canonical image in E ∗∗ . For instance, the preduals of von Neumann algebras are Lembedded (Theorem C.5.15). Lemma 1.5.5. Let E be an L-embedded Banach space, and let ∅ = B ⊂ E be bounded. Then C(B, E) is weakly compact and non-empty. Proof. By Corollary 1.5.4, C(B, E ∗∗ ) is non-empty and weak∗ compact. We claim that C(B, E ∗∗ ) = C(B, E). Let Y be a closed subspace of E ∗∗ such that E ∗∗ = E ⊕1 Y . Let x ∈ C(B, E ∗∗ ), and let xE ∈ E and xY ∈ Y be such that x = xE + xY . As B ⊂ E, we have y − x = y − xE − xY  = y − xE  + xY 

(y ∈ B).

It follows that (B, E ∗∗ ) = sup y − x = sup y − xE  + xY  ≥ (B, E) + xY . y∈B

y∈B

Since trivially (B, E ∗∗ ) ≤ (B, E), we conclude that xY = 0 and (B, E ∗∗ ) = (B, E). This proves the claim. 

52

1 Amenable, Locally Compact Groups

Theorem 1.5.6. Let E be an L-embedded Banach space, let ∅ = B ⊂ E be bounded, and let S be a semigroup acting affinely and isometrically on E such that s · B ⊂ B for each s ∈ S. Then there is x ∈ E such that: (i) s · x = x for all s ∈ S; (ii) supy∈B x − y is minimal. Proof. ByLemma1.5.3,C(B, E)isconvex,andbyLemma1.5.5,itisweaklycompact. As S acts isometrically on E, it is immediate from the definition of C(B, E) that s · C(B, E) ⊂ C(B, E) for all s ∈ S. Hence, the Ryll-Narzewksi Fixed Point Theorem ([145, (9.6) Theorem]) applies and yields x ∈ C(B, E) such that s · x = x for all s ∈ S. From the definition of C(B, E), it is clear that (ii) holds. 

Exercises Exercise 1.5.1. Let E and F be be linear spaces. Show that a map T : E → F is affine if and only if there are an R-linear map S : E → F and y ∈ F such that T (x) = Sx + y for x ∈ E. Exercise 1.5.2. Let G be a locally compact group. Show that the set of means of the form n 

tj δxj

(n ∈ N, x1 , . . . , xn ∈ G, t1 , . . . , tn ≥ 0, t1 + · · · + tn = 1) (1.22)

j=1

is weak∗ dense in the set of all means on C(G). Exercise 1.5.3. Prove implication (ii) =⇒ (i) of Theorem 1.5.1. (Hint: Let C be the set of all means on LUC(G) equipped with the weak∗ topology, and define a group action of G on C by letting f, x · M  = Lx f, M  for x ∈ G, M ∈ C, and f ∈ LUC(G).)

Notes and Comments Amenable (discrete) groups were first considered by J. von Neumann in [253]. He called them “messbare Gruppen”, which is German for “measurable groups”, and defined them in terms of finitely additive set functions as in Corollary 0.2.10(ii). The first to use the adjective “amenable” appears to be M. M. Day in [88]. Apparently, he had a pun in mind: these groups G are called amenable because they have an invariant mean on L∞ (G), but also since they are particularly pleasant to deal with and thus are truly amenable—just in the sense of that adjective in colloquial English. The equivalence of amenability and Reiter’s property (P1 ) is due to H. Reiter ([281]),

Notes and Comments

53

in whose honor the properties (Pp ) for p ∈ [1, ∞) are named. Theorem 1.1.19 is essentially [188, Proposition 2.6], which, in turn, builds on earlier work of Reiter ([282]). The introductions of [267] and [262] give sketches of the historical development of the subject (including more references to early papers). Our exposition borrows heavily from the monographs [146], [267], and [262]. An alternative (but not easier) proof of Theorem 1.2.7, which avoids Bruhat functions, is given in [146]; the underlying idea is essentially the same as in the false “proof” in Exercise 1.2.2. Corollary 1.2.8 begets the natural conjecture that every (discrete) nonamenable group contains a subgroup isomorphic to F2 : this conjecture is known as von Neumann’s Conjecture—even though it seems to be actually due to Day. The conjecture was eventually refuted by A. Yu. Ol’shanski˘ı ([254]), relying on an amenability criterion due to R. I. Grigorchuk ([147]). Further counterexamples to von Neumann’s Conjecture can be found in [1], [255], [256], [247], [234], and [246]; the counterexample from [246] is comparatively simple. Let G be a locally compact group which has a paradoxical decomposition such that the sets A1 , . . . , An and B1 , . . . , Bm in Definition 0.1.1 are Borel sets. Then G is not amenable: this can either be seen directly, as in Exercise 0.2.1, or be deduced from Corollaries 0.2.10 and 1.1.14. The question of whether the non-existence of such Borel paradoxical decompositions characterizes the amenable, locally compact groups seems to be open ([350]). Theorem 1.3.4 is due to J. Dixmier ([96]); the special case G = R had been established earlier by B. Sz.-Nagy ([333]). Lemma 1.3.2 is [141, Theorem 2.8]. The idea to use Theorem 1.3.4 to prove the K¨ othe–Lorch Theorem is due to G. A. Willis ([358]). Groups with the property that every uniformly bounded representation on a Hilbert space is similar to a unitary one are often referred to as unitarizable. In this terminology, Theorem 1.3.4 asserts that all amenable groups are unitarizable. Free groups in finitely or countably many generators are not unitarizable: the probably easiest example of this kind is due M. Bo˙zejko and G. Fendler ([37]; see also [270, Chapter 2], especially its Notes and Remarks section). The counterexamples to von Neumann’s Conjecture from [256] and [247] are also not unitarizable. The natural question if unitarizability characterizes the amenable, locally compact groups were already raised by Dixmier. It appears to be open to this day: it is one instance of so-called similarity problems.For an account of the progress towards the converse of Theorem 1.3.4 in the discrete case— up to the time of its publication—see [271]. In [273], G. Pisier proved the following partial converse to Theorem 1.3.4 for discrete groups G: if, for every uniformly bounded representation π of G on a Hilbert space, there is T in the von Neumann algebra generated by π such that T π(·)T −1 is unitary, then G is amenable.

54

1 Amenable, Locally Compact Groups

Theorem 1.4.1—or rather the equivalence of Theorem 1.4.1(i), (ii), and (iii)—is due to H. Leptin ([223]); an alternative proof was later given by A. Derighetti ([92]). Both in [223] and in [92], Følner type conditions (see below) are used to prove the implication (i) =⇒ (ii) of Theorem 1.4.1. Theorem 1.4.1 extends naturally to the Fig` a-Talamanca–Herz algebras Ap (G) (see [267] for the definition): G is amenable if and only if Ap (G) has a bounded approximate identity for one—and, equivalently, all—p ∈ (1, ∞). This generalization of Leptin’s Theorem was proved by C. Herz in [176] through an obvious modification of Leptin’s original proof. (Herz himself claims the result to be folklore.) Day’s Fixed Point Theorem was first stated in [90], but the proof given in [90] of (ii) =⇒ (i) contained an error (see [89]). A correct proof was finally presented in [285]. Theorem 1.5.6 is from [17]. Our treatment of amenable, locally compact groups in this chapter is far from exhaustive: as this book focuses on the Banach algebraic aspects of amenability, we have dealt with them mainly to set the stage for Johnson’s Theorem in the next chapter. For more information, on amenable, locally compact groups, we refer to the aforementioned monographs [146], [267], and and [262]. Still, we would like to at least sketch two more characterizations of amenability for locally compact groups.

Følner type conditions A locally compact group G with Haar measure mG is said to satisfy the Følner condition if, for each compact subset K of G and for each  > 0, there is a Borel subset B of G with 0 < mG (B) < ∞ such that mG ((xB \ B) ∪ (B \ xB)) < mG (B)

(x ∈ K).

This condition is named in the honor of E. Følner, who studied it for the first time in [119] (in the discrete case only). There are variants of the Følner condition—the weak Følner condition, the strong Følner condition, the Leptin condition—some of which are formally weaker, some formally stronger (see [146], [267], or [262]). They all turn out to be equivalent to G being amenable.

The multiplier norm of the Fourier algebra Given a locally compact group G, the multiplier norm on A(G) is defined as f M := sup{f gA(G) : g ∈ Ball(A(G))}

(f ∈ A(G)).

Notes and Comments

55

From Leptin’s Theorem, it is immediate that  · M coincides with the given norm for amenable G. A remarkable result due to V. Losert asserts that, if ·M and the given norm on A(G) are equivalent, then G has to be amenable ([230]).

Chapter 2

Amenable Banach Algebras

We now move from groups to Banach algebras. There are various Banach spaces and algebras associated with a locally compact group G: see Appendices D and F. Perhaps the most important one is the group algebra L1 (G). It is a complete invariant for G: if G1 and G2 are locally compact groups such that L1 (G1 ) and L1 (G2 ) are isometrically isomorphic, then G1 and G2 are topologically isomorphic, i.e., all information about G is already encoded in L1 (G). In 1972, B. E. Johnson discovered that a locally compact group G is amenable precisely if every derivation from L1 (G) into a dual Banach L1 (G)bimodule is inner. In the language of Hochschild cohomology, this means that the amenable locally compact groups G can be characterized by the vanishing of certain cohomology groups of L1 (G). This is interesting for two reasons: • Homological algebra is a powerful mathematical toolkit. So, knowing that a certain property can be characterized in terms of cohomology is thus good in itself. • The cohomological triviality condition used by Johnson to characterize the amenable, locally compact groups can be studied for arbitrary Banach algebras, which may have nothing to do with locally compact groups. We can thus meaningfully speak of amenable Banach algebras. As in the case of amenable, locally compact groups, the notion of an amenable Banach algebra has turned out to be a very fruitful one. We have a variety of interesting examples—L1 (G) for amenable, locally compact groups G, nuclear C ∗ -algebras, algebras of compact operators on Banach spaces with certain properties, and even radical Banach algebras—whose only common feature seems to be their amenability. On the other hand, there is still a substantial general theory for amenable Banach algebras. In this chapter, our main focus is to develop the general theory of amenable Banach algebras with relatively little focus on examples: they will be dealt with in subsequent chapters. © Springer Science+Business Media, LLC, part of Springer Nature 2020 V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-1-0716-0351-2 2

57

58

2 Amenable Banach Algebras

We have collected the necessary background from Banach algebra theory in Appendix B.

2.1 Derivations from Group Algebras Let A be a Banach algebra, and let E be a Banach A-bimodule. A bounded linear map D : A → E is called a derivation if the product rule D(ab) = a · Db + (Da) · b

(a, b ∈ A)

holds. Let x ∈ E. We define adx : A → E,

a → a · x − x · a

It is easily checked that adx is a derivation. Derivations of this form are called inner derivations. In general, it can be difficult to come up with examples of derivations that are not inner: in many situations, there simply aren’t any. The following is the first result of this kind. Proposition 2.1.1. Let A be a Banach algebra with a bounded right approximate identity, and let E be a Banach A-bimodule such that a · x = 0 for all a ∈ A and x ∈ E. Then every derivation from A into E ∗ is inner. Proof. From the definition of the module actions of A on E ∗ , it is clear that φ · a = 0 for a ∈ A and φ ∈ E ∗ . Let D : A → E ∗ be a derivation; it follows that D(ab) = a · Db (a, b ∈ A). Let (eα )α be a bounded right approximate identity for A, and let φ ∈ E ∗ be a weak∗ accumulation point of (Deα )α . Without loss of generality, we may suppose that φ = weak∗ - limα Deα . It follows that Da = lim D(aeα ) = lim a · Deα = a · φ α

so that D = adφ .

α

(a ∈ A), 

Of course, there is an analog of Proposition 2.1.1 for Banach algebras with a bounded left approximate identity when the module action from the right is trivial. We will now use fixed point theorems in order to show that, under certain conditions, derivations from the measure algebra of a locally compact group into dual Banach bimodules are always inner. Let A be a Banach algebra, and let I be a closed ideal of A. Then the I-strict topology on A is defined through the family of seminorms (px )x∈I where

2.1 Derivations from Group Algebras

px (a) := ax + xa

59

(a ∈ A, x ∈ I).

Note that the I-strict topology is Hausdorff only if {a ∈ A : a·I = I ·a = {0}} = {0}. If G is a locally compact group, we can view the group algebra L1 (G) as a closed ideal of M (G) (Theorem D.3.7), so that we may speak of the L1 (G)-strict topology on M (G). Proposition 2.1.2. Let G be a locally compact group, let E be a Banach M (G)-bimodule such that the module actions of M (G) on E are continuous with respect to the L1 (G)-strict topology on M (G) and the norm topology on E, let D : M (G) → E ∗ be a derivation which is continuous with respect to the L1 (G)-strict topology on M (G) and the weak∗ topology on E ∗ , and suppose that one of the following holds: (a) G is amenable; (b) E ∗ is L-embedded, and δx · φ · δx−1  = φ

(φ ∈ E ∗ , x ∈ G).

Then D is inner. For the proof, we require the following lemma: Lemma 2.1.3. Let G be a locally compact group. Then the linear span of {δx : x ∈ G} is L1 (G)-strictly dense in M (G). Proof. Assume towards a contradiction that there is μ0 ∈ M (G) that does not belong to the L1 (G)-strictly closed linear span of {δx : x ∈ G}. As the L1 (G)strict topology on M (G) is locally convex, the Hahn–Banach Theorem yields a L1 (G)-strictly continuous functional F on M (G) such that μ0 , F  = 0 whereas δx , F  = 0 for all x ∈ G. An argument closely akin to the one used for Exercise 1.3.1 shows that F must be of the following form: there are f1 , . . . , fn ∈ L1 (G), as well as φ1 , ψ1 , . . . , φn , ψn ∈ L∞ (G) such that μ, F  =

n 

μ ∗ fj , φj  + fj ∗ μ, ψj 

(μ ∈ M (G)).

j=1 1 ∞ Let n · denote the module action of L (G) on its dual L (G), and set g := j=1 fj · φj + ψj · fj . By Proposition D.4.2(ii), we have g ∈ RUC(G) + LUC(G) ⊂ C(G), and it is routinely checked that  μ, F  = g(x) dμ(x) (μ ∈ M (G)). G

In particular, g must be non-zero while, on the other hand, δx , F  = g(x) = 0 for all x ∈ G. This is impossible. 

60

2 Amenable Banach Algebras

Proof (of Proposition 2.1.2). The continuity hypothesis for the module actions of M (G) on E yields that the module actions of M (G) on E ∗ are continuous with respect to the L1 (G)-strict topology on M (G) and the weak∗ topology on E ∗ . As the linear span of {δx : x ∈ G} is L1 (G)-strictly dense in M (G) by Lemma 2.1.3, it is thus sufficient to show that there is φ0 ∈ E ∗ such that (x ∈ G). (2.1) D(δx ) = δx · φ0 − φ0 · δx Define x ◦ φ := δx · φ · δx−1 − D(δx ) · δx−1

(x ∈ G, φ ∈ E ∗ ).

For x, y ∈ G and φ ∈ E ∗ , we have: xy ◦ φ = δxy · φ · δ(xy)−1 − D(δxy ) · δ(xy)−1 = δx · (δy · φ · δy−1 ) · δx−1 − (δx · D(δy ) + D(δx ) · δy ) · (δy−1 ∗ δx−1 ) = δx · (δy · φ · δy−1 ) · δx−1 − δx · (D(δy ) · δy−1 ) · δx−1 − D(δx ) · δx−1 = δx · (δy · φ · δy−1 − D(δy ) · δy−1 ) · δx−1 − D(δx ) · δx−1 = x ◦ (y ◦ φ). (2.2) Hence,

G × E∗ → E∗,

(x, φ) → x ◦ φ

(2.3)

is a group action of G on E ∗ , which is obviously affine. It is routinely checked that this action is separately continuous with respect to the given topology on G and the weak∗ topology on E ∗ (see Exercise 2.1.3 below). Letting φ = 0 in (2.2), we obtain that x ◦ (−D(δy ) · δy−1 ) = −D(δxy ) · δ(xy)−1

(x, y ∈ G),

i.e., by restriction, (2.3) yields an action of G on the bounded set B := {−D(δy ) · y : y ∈ G}. Suppose that (a) holds. Let C be the weak∗ closed convex hull of B in ∗ E . As the group action (2.3) is affine and weak∗ -weak∗ continuous in the second variable, we obtain through restriction a separately continuous affine action of G on C. Day’s Fixed Point Theorem then yields φ0 ∈ C such that x ◦ φ0 = φ0 for all x ∈ G, i.e., (2.1) holds. Suppose that (b) holds. Clearly, (2.3) is then an isometric action of G on  E ∗ . We obtain the desired φ0 ∈ E ∗ from Theorem 1.5.6. We shall use Proposition 2.1.2 to show that, under certain conditions, derivations from group algebras are inner. We therefore need to extend derivations from a group algebra to the measure algebra in such a way that Proposition 2.1.2 becomes applicable. If A is a unital Banach algebra with identity eA , a left Banach A-bimodule E is called unital if eA · x = x holds for x ∈ E; similarly, unital Banach right

2.1 Derivations from Group Algebras

61

and bimodules are defined. This motivates the choice of terminology in the following definition: Definition 2.1.4. Let A be a Banach algebra. We call a Banach A-module neo-unital if: (a) E is a left Banach A-module and E = {a · x : a ∈ A, x ∈ E}; (b) E is a right Banach A-module and E = {x · a : a ∈ A, x ∈ E}; (c) E is a Banach A-bimodule and E = {a · x · b : a, b ∈ A, x ∈ E}. A related definition is: Definition 2.1.5. Let A be a Banach algebra. We call a Banach A-module essential if: (a) E is a left Banach A-module, and the closed linear span of {a · x : a ∈ A, x ∈ E} is dense in E; (b) E is a right Banach A-module, and the closed linear span of {x · a : a ∈ A, x ∈ E} is dense in E; (c) E is a Banach A-bimodule, and the closed linear span of {a · x · b : a, b ∈ A, x ∈ E} is dense in E. From Corollary B.2.5 and its right module version, it is immediate that any essential Banach module over a Banach algebra with a bounded approximate identity is neo-unital. Proposition 2.1.6. Let A be a Banach algebra, let I be a closed ideal of A with a bounded approximate identity, let E be a neo-unital Banach Ibimodule, and let D : I → E ∗ be a derivation. Then: (i) letting a · (x · y) := ax · y

and

(y · x) · b := y · xb

(a, b ∈ A, x ∈ I, y ∈ E)

turns E into a Banach A-bimodule; ˜ : A → E ∗ , extending D and continuous (ii) there is a unique derivation D with respect to the I-strict topology on A and the weak∗ topology on E ∗ . Proof. For (i), all that needs to be shown is that the module actions are well defined. Let a ∈ A, and let x, x ∈ I and y, y  ∈ E be such that x · y = x · y  . We claim that ax · y = ax · y  . Note that abx · y = (ab) · (x · y) = (ab) · (x · y  ) = abx · y 

(b ∈ I).

Let (eα )α∈A be a bounded approximate identity for I. Then ax · y = lim aeα x · y = lim aeα x · y  = ax · y  α

α

holds, i.e., the left module action of A on E is well defined. The well definedness of the right module action is obtained analogously.

62

2 Amenable Banach Algebras

For (ii), define a → weak∗ - lim(D(aeα ) − a · Deα ).

˜ : A → E∗, D

α

(2.4)

˜ is well defined, i.e., the limit in (2.4) exists. Let a ∈ A, We claim that D x ∈ I, and y ∈ E. Then we have for all α ∈ A: y · x, D(aeα ) − a · Deα  = y, x · D(aeα ) − xa · Deα  = y, D(xaeα ) − (Dx) · aeα − D(xaeα ) + D(xa) · eα  = y, −(Dx) · aeα + D(xa) · eα  = eα · y, −(Dx) · a + D(xa) As (eα )α∈A is a bounded approximate identity for E, it follows that the limit in (2.4) does indeed exist. Furthermore, ˜ = weak∗ - lim(D(xeα ) − x · Deα ) = weak∗ - lim(Dx) · eα = Dx Dx α

α

(x ∈ I)

˜ extends D. holds, so that D Let a ∈ A and x ∈ I, and note that ˜ · x = weak∗ - lim(D(aeα ) · x − a · (Deα ) · x) (Da) α

= weak∗ - lim(D(aeα x) − aeα · Dx − a · D(eα x) + aeα · Dx) (2.5) α

= D(ax) − a · Dx. Let (aγ )γ be a net in A converging to 0 in the I-strict topology. Let x ∈ I and y ∈ E; from (2.5), we conclude that ˜ γ ) = y, D(a ˜ γ ) · x x · y, D(a = y, D(aγ x) − aγ · Dx = y, D(aγ x) − y · aγ , Dx → 0. ˜ is continuous with respect to the I-strict topology on A and Consequently, D the weak∗ topology on E ∗ . ˜ is a derivation. From the definition of the It remains to be shown that D I-strict topology, it is clear that aeα → a in this topology for all a ∈ A. Since ˜ is continuous with respect to the I-strict topology on A and the weak∗ D topology on E ∗ , we obtain for a, b ∈ A that ˜ D(ab) = weak∗ - lim weak∗ - lim D((aeα )(beβ )) α

β

∗

= weak - lim weak∗ - lim(aeα · D(beβ ) + D(aeα ) · beβ ) α

˜ + (Da) ˜ · b; = a · Db

β

2.1 Derivations from Group Algebras

63

this completes the proof.



Combining Propositions 2.1.2 and 2.1.6, we obtain: Corollary 2.1.7. Let G be a locally compact group, let E be neo-unital Banach L1 (G)-bimodule, let D : L1 (G) → E ∗ be a derivation, and suppose that one of the following holds: (a) G is amenable; (b) E ∗ is L-embedded, and δx · φ · δx−1  = φ

(φ ∈ E ∗ , x ∈ G).

(2.6)

Then D is inner. Proof. Use Proposition 2.1.6 to extend the module actions, as well as D to M (G), and note that the hypotheses of Proposition 2.1.2 are satisfied.  The module actions of M (G) on E ∗ in (2.6) are, of course, those given by Proposition 2.1.6. Let G be any locally compact group. Then C0 (G) is a closed submodule of L∞ (G) contained in UC(G) (Proposition D.4.2(i) and (ii)). Since L1 (G) has a bounded approximate identity, C0 (G) has to be essential as a consequence of Proposition D.4.2(ii) and thus is neo-unital. Furthermore, M (G) = C0 (G)∗ is the dual space of a C ∗ -algebra, thus the predual of a von Neumann algebra, and, consequently, L-embedded. We, therefore, obtain from Corollary 2.1.7: Corollary 2.1.8. Let G be a locally compact group, and let D : L1 (G) → L1 (G) be a derivation. Then there is μ ∈ M (G) such that D = adμ . Corollary 2.1.7 asserts that, if G is an amenable, locally compact group, then every derivation from L1 (G) into the dual of a neo-unital Banach L1 (G)bimodule is inner. This conclusion remains true even if we consider derivations with values in arbitrary dual Banach L1 (G)-bimodules, as follows from the next proposition: Proposition 2.1.9. Let A be a Banach algebra with a bounded approximate identity. Then the following are equivalent: (i) for every Banach E-bimodule, every derivation D : A → E ∗ is inner; (ii) for every neo-unital Banach E-bimodule, every derivation D : A → E ∗ is inner. Proof. Obviously, only (ii) =⇒ (i) needs proof. Let E be a Banach A-bimodule, and let E0 and E1 denote the closed linear spans of {a · x · b : a, b ∈ A, x ∈ E} and {x · b : b ∈ A, x ∈ E}, respectively. Then E0 and E1 are closed submodules of E with E0 ⊂ E1 , and E0 is an essential Banach A-bimodule and, therefore, neo-unital. We shall see that every derivation from A to E ∗ is inner. Before we do this, we show that every derivation from A into E1∗ is inner.

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2 Amenable Banach Algebras

Let D1 : A → E1∗ be a derivation, and let π0 : E1∗ → E0∗ be the restriction map; then π0 ◦ D1 : A → E0∗ is a derivation. As E0 is neo-unital, π0 ◦ D1 is inner, i.e., there is there is φ0 ∈ E0∗ such that π0 ◦ D1 = adφ0 . Use the Hahn– Banach Theorem to extend φ0 to E1 , and let that extension be denoted by φ1 . It follows that D1 − adφ1 is a derivation from A into E1∗ that vanishes on E0 , i.e., attains its values in the Banach A-bimodule E1∗ ∩ E0⊥ . (Here, E0⊥ denotes the annihilator of E0 in E1∗ , i.e., those functionals in E1∗ that vanish on E0 .) As E1∗ ∩ E0⊥ ∼ = (E1 /E0 )∗ as Banach A-bimodules, and since A·(E1 /E0 ) = {0}, it follows from Proposition 2.1.1 that there is ψ1 ∈ E1∗ ∩E0⊥ such that D1 − adφ1 = adψ1 and thus D1 = adφ1 +ψ1 . Now, let D : A → E ∗ be a derivation, and let π1 : E ∗ → E1∗ be the restriction map; then π1 ◦ D : A → E1∗ is a derivation. By the foregoing, there is ψ1 ∈ E1∗ such that π1 ◦ D = adψ1 . Via Hahn–Banach, we obtain an extension ψ0 ∈ E ∗ of ψ1 , so that D − adψ0 is a derivation from A into E0⊥ . Since E1⊥ ∼ = (E/E1 )∗ , and since (E/E1 ) · A = {0}, the right module variant  of Proposition 2.1.1 yields φ0 ∈ E1⊥ such that D = adφ0 +ψ0 . We can now prove the main result of this section: Theorem 2.1.10 (Johnson’s Theorem). For a locally compact group G, the following are equivalent: (i) G is amenable. (ii) for every Banach L1 (G)-bimodule E, every derivation D : L1 (G) → E ∗ is inner. Proof. (i) =⇒ (ii) follows from Proposition 2.1.9 and Corollary 2.1.7. (ii) =⇒ (i): As the second dual of L1 (G), the space L∞ (G)∗ is a Banach 1 L (G)-bimodule in a canonical manner; let the corresponding module actions be denoted by ·. We define: f ◦ F := f · F

and F ◦ f := f, 1F

(f ∈ L1 (G), F ∈ L∞ (G)).

Then ◦ turns L∞ (G)∗ into a Banach L1 (G)-bimodule. Fix F0 ∈ L1 (G)∞ such that 1, F0  = 1, and set D := adF0 . Set E = {F ∈ L∞ (G)∗ : 1, F  = 0}. Then E is a weak∗ closed submodule of L∞ (G)∗ and thus a dual Banach L1 (G)-bimodule. Since 1, Df  = 1 · f, F0  − f, 1 = f, 1 − f, 1 = 0

(f ∈ L1 (G)),

the derivation D attains its values in E. Consequently, there is F1 ∈ E such that D = adF1 , i.e., f · (F0 − F1 ) = f, 1(F0 − F1 )

(f ∈ L1 (G)).

(2.7)

2.1 Derivations from Group Algebras

65

Set M := (F0 − F1 ); since 1, M  = 1, F0  = 1, it is clear that M = 0. Fix f ∈ L1 (G) such that f, 1 = 1; from (2.7), it follows that f ·M = M . Finally, note that Lx φ, M  = Lx φ, f · M  = φ, (δx ∗ f ) · M  = δx ∗ f, 1φ, M  = φ, M 

(φ ∈ L∞ (G), x ∈ G).

The amenability of G then follows from Lemma 1.1.18.



In light of Theorem 2.1.10, the use of the word amenable in the following definition makes sense: Definition 2.1.11. A Banach algebra A is called amenable if, for every Banach A-bimodule E, every derivation D : A → E ∗ is inner.

Exercises Exercise 2.1.1. Let A be a Banach algebra, let E be a Banach A-module— left, right, or bi-—, and let F be a weak∗ closed submodule of E ∗ . Show that F is a dual Banach A-module. Exercise 2.1.2. Wendel’s Theorem asserts: Let G be a locally compact group, and let T ∈ B(L1 (G)) be such that T (f ∗g) = f ∗T g for all f, g ∈ L1 (G). Then there is µ ∈ M (G) such that T f = f ∗ µ for all f ∈ L1 (G).

Derive Wendel’s Theorem from Proposition 2.1.1. Exercise 2.1.3. Show that, under the hypotheses of Proposition 2.1.2, the group action (2.3) is separately continuous with respect to the given topology on G and the weak∗ topology on E ∗ . Exercise 2.1.4. Let G be a locally compact group, let E be a Banach L1 (G)bimodule such that the underlying Banach space is reflexive, and let D : L1 (G) → E ∗ be a derivation. Show that D is inner. (Hint: Reflexive Banach spaces are trivially L-embedded, and reflexivity is a property not affected by renorming.) Exercise 2.1.5. Let A be a Banach algebra, let I be a closed ideal with a left or right approximate identity, and let D : I → A be a derivation. Show that DI ⊂ I. Exercise 2.1.6. Let A be a Banach algebra. Show that A is amenable if and only if, for every Banach A-bimodule E, every derivation D : A → E ∗∗ is inner. Exercise 2.1.7. Let A be a Banach algebra. Show that A is amenable if and only if, for every Banach A-bimodule E and every derivation D : A → E, there is a bounded net (xα )α in E such that Da = limα adxα a for all a ∈ A.

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2 Amenable Banach Algebras

2.2 Virtual and Approximate Diagonals In this section, we give an intrinsic characterization of amenable Banach algebras. We begin with a preliminary result: Proposition 2.2.1. Let A be an amenable Banach algebra. Then A has a bounded approximate identity. Proof. Let A be the Banach A-bimodule whose underlying space is A, but on which A acts via a · x := ax and x · a := 0

(a ∈ A, x ∈ A).

Let D : A → A∗∗ be the canonical embedding of A into its second dual; it is routinely seen to be a derivation. As A is amenable, D is inner, i.e., there is E ∈ A∗∗ such that a = a · E for all a ∈ A. Let (eα )α be a bounded net in A such that E = σ(A∗∗ , A∗ )- limα eα . As σ(A∗∗ , A∗ ) restricted to A is the weak topology of A, it follows that a = weak- limα aeα for all a ∈ A. Passing to convex combinations, we can suppose that a = limα aeα in the norm topology of A for all a ∈ A, i.e., (eα ) is a bounded right approximate identity for A. Analogously, we obtain a bounded left approximate identity for A. By Proposition B.2.3, A thus has a bounded approximate identity.  Following the notation of [106], we use ⊗γ to denote the projective tensor product of Banach spaces. If A is a Banach algebra, E is a left and F a right Banach A-module, the space E ⊗γ F becomes a Banach A-bimodule via a · (x ⊗ y) := a · x ⊗ y

and

(x ⊗ y) · a := x ⊗ y · a

(a ∈ A, x ∈ E, y ∈ F ).

In particular, A⊗γ A is a Banach A-bimodule in a canonical way. The diagonal operator is defined as ΔA : A ⊗γ A → A,

a ⊗ b → ab;

if it is clear to which algebra A we refer, we simply write Δ. Obviously, ΔA is a bimodule homomorphism from A ⊗γ A to A. Definition 2.2.2. Let A be a Banach algebra. Then: (a) an approximate diagonal for A is a net (dα )α in A ⊗γ A such that a · dα − dα · a → 0 and aΔA dα → a

(a ∈ A);

(b) a virtual diagonal for A is an element D ∈ (A ⊗γ A)∗∗ such that a · D = D · a and a · Δ∗∗ A D =a

(a ∈ A).

Remark 2.2.3. In most of the literature, approximate diagonals are also required to be bounded. We do not make this requirement because in Section

2.2 Virtual and Approximate Diagonals

67

4.4, we will study Banach algebras with possibly unbounded approximate diagonals (Definition 4.4.1). We begin with a very basic example: Example 2.2.4. Let A be the Banach algebra of complex N × N matrices, and let {ej,k : j, k = 1, . . . , N } be a set of matrix units for A. Set N 1  d := ej,k ⊗ ek,j . N j,k=1

Since Δd =

N 1  N ej,j = IN , N j=1

it is clear that aΔd = a

(a ∈ A).

(2.8)

Fix , m ∈ {1, . . . , N }, and note that e,m · d =

N 1  e,m ej,k ⊗ ek,j N j,k=1

=

N N 1  1  e,k ⊗ ek,m = ej,k ⊗ ek,j e,m = d · e,m . N N k=1

j,k=1

By linearity, we conclude that a·d=d·a

(a ∈ A).

(2.9)

Thus, d is a virtual diagonal for A. If A is any Banach algebra, we call an element d of A ⊗γ A satisfying (2.8) and (2.9) a diagonal for A. Our next theorem shows that the existence of virtual and approximate diagonals, respectively, characterizes the amenable Banach algebras Theorem 2.2.5. For a Banach algebra A, the following are equivalent: (i) A is amenable; (ii) there is a bounded approximate diagonal for A; (iii) there is a virtual diagonal for A. Proof. (i) =⇒ (iii): By Proposition 2.2.1, A has a bounded approximate identity, say (eα )α . Let E ∈ (A ⊗γ A)∗∗ be a weak∗ accumulation point of (eα ⊗ eα )α . By Exercise 2.2.2 and its right hand version, (e2α )α is also a bounded approximate identity for A, so that, in particular, limα ae2α −e2α a = 0 for each a ∈ A. Now note that

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2 Amenable Banach Algebras

Δ∗∗ (a · E − E · a) = weak∗ - lim Δ(aeα ⊗ eα − eα ⊗ eα a) = lim(ae2α − e2α a) = 0 α

α

(a ∈ A).

This means that the inner derivation adE attains its values in ker Δ∗∗ . Since Δ is a bimodule homomorphism, so is Δ∗∗ , and consequently, ker Δ∗∗ is a Banach A-bimodule. As Δ∗∗ is weak∗ -weak∗ continuous, its kernel is closed in (A ⊗γ A)∗∗ and, therefore, in fact, a dual Banach A-bimodule. By the definition of an amenable Banach algebra, there is thus F ∈ ker Δ∗∗ such that adE = adF . Set D := E − F . Then we obtain a · Δ∗∗ D = a · Δ∗∗ E = lim ae2α = a α

(a ∈ A)

and a · D − D · a = adD a = adE −F a = adE a − adF a = 0

(a ∈ A),

i.e., D is a virtual diagonal for A. (iii) =⇒ (ii): Let D be a virtual diagonal for A. By the Bipolar Theorem, there is a net (dα )α in A⊗γ A bounded by D such that D = weak∗ - limα dα . As σ(A∗∗ , A∗ ) and σ((A ⊗γ A)∗∗ , (A ⊗γ A)∗ ) restricted to A and A ⊗γ A, respectively, are just the respective weak topologies, we have weak- lim(a·dα −dα ·a) = 0 and α

weak- lim aΔdα = a α

(a ∈ A). (2.10)

Passing to convex combinations, we can replace the weak limits in (2.10) by norm limits, thus obtaining a bounded approximate diagonal for A. (ii) =⇒ (i): Let E be a Banach A-bimodule, and let D : A → E ∗ be a derivation. Let (dα )α∈A be a bounded approximate diagonal for A. Then (Δdα )α∈A is a bounded approximate identity for A. Hence, by Proposition 2.1.9, we can suppose without loss of generality that E is neo-unital. Suppose that ∞  (α) dα = a(α) (α ∈ A) n ⊗ bn n=1

  (α) (α) (α) (α) ∞ with supα∈A n=1 an bn  < ∞. Then a · Db is a bounded n n n=1 α ∗ ∗ ∗ net in E with a weak -accumulation point, say φ ∈ E ; without loss of gen(α) (α) ∞ erality, suppose that φ is the weak∗ limit of . Then we n=1 an · Dbn α have for a ∈ A and x ∈ E: ∞

2.2 Virtual and Approximate Diagonals

 x, a · φ = lim x, α



= lim x, α



= lim x, α



∞  n=1 ∞ 

(α) aa(α) n · Dbn



 (α) a(α) n · D bn a

n=1 ∞   n=1 ∞ 

= lim x · α

69



(α) a(α) n bn



· Da +

a(α) n

·

Db(α) n

 ·a



 (α) a(α) n bn , Da

+ x, φ · a

n=1

= x, Da + x, φ · a. 

It follows that D = adφ .

An advantage of the characterization in Theorem 2.2.5 over Definition 2.1.11 is, that it allows to easily introduce a quantitative aspect: Definition 2.2.6. Let A be a Banach algebra, and let C ≥ 0. We say that A is C-amenable if it has an approximate diagonal bounded by C. It is immediate that a Banach algebra A is C-amenable if and only if it has a virtual diagonal of norm less than or equal to C. By Theorem 2.2.5, a Banach algebra is amenable if and only if it is C-amenable for some C ≥ 0. Definition 2.2.7. Let A be a Banach algebra. The amenability constant of A is defined as AM(A) := inf{C ≥ 0 : A is C-amenable}. It is easy to see that the infimum in Definition 2.2.7 is, in fact, a minimum. If we adopt the usual convention that the infimum over the empty set is ∞, we can rephrase Theorem 2.2.5 as: a Banach algebra A is amenable if and only if AM(A) < ∞. It is clear that AM(A) ≥ 1 for any non-zero Banach algebra, and it is not difficult to see that the amenability constant of an amenable Banach algebra can be arbitrary large (see Exercise 2.2.5 below). Here is an example that will become relevant later on: Example 2.2.8. For N ∈ N and p ∈ [1, ∞], let pN be CN equipped with the corresponding p -norm, i.e., ⎛ (λ1 , . . . , λN )p := ⎝

N 

⎞ p1 |λj |p ⎠

((λ1 , . . . , λN ) ∈ CN )

j=1

in the case where p < ∞ and (λ1 , . . . , λN )∞ := max{|λ1 |, . . . , |λN |}

((λ1 , . . . , λN ) ∈ CN )

70

2 Amenable Banach Algebras

if p = ∞. We can algebraically identify B( pN ) with the algebra of all complex N × N matrices. By Example 2.2.4, B( pN ) is, therefore, amenable. We claim that AM(B( pN )) = 1. To see this, suppose that G is a finite group of invertible isometries on pN whose linear span is all of B( pN ). Set d :=

1  x ⊗ x−1 . |G| x∈G

It is immediate that Δd is the identity on pN and that y·d=

1  1  −1 yx ⊗ x−1 = yy x ⊗ (y −1 x)−1 = d · y |G| |G| x∈G

(y ∈ G).

x∈G

Since G spans B( p ), we conclude that d is a diagonal for B( p ). (As d is also a diagonal for B( pN )op , it follows from Exercise 2.2.4 below that d and the diagonal from Example 2.2.4 actually coincide; in particular, d is independent of the choice of G.) Furthermore, since G consists of isometries, we have d ≤

1  xx−1  = 1, |G| x∈G

so that B( pN ) is 1-amenable, as claimed. To come up with a suitable group G, let, for instance, SN be the group of permutations of {1, . . . , N }, and let, for each σ ∈ SN , the corresponding permutation matrix be denoted by Aσ . Moreover, for every  ∈ {−1, 1}N , let D be the corresponding diagonal matrix. Set G := {D Aσ :  ∈ {−1, 1}N , σ ∈ SN }. It is routinely checked that G is a finite group, spanning B( pN ) and consisting of invertible isometries on pN . We conclude this section with an improvement of Theorem 2.1.10. We require the following result, which is of independent interest. Proposition 2.2.9. Let G be an amenable, locally compact group. Then L1 (G) has a bounded approximate identity (eα )α consisting of non-negative functions of norm one such that δx ∗ eα − eα ∗ δx → 0 uniformly in x on compact subsets of G. Proof. Let (uα )α∈A be a bounded approximate identity for L1 (G) consisting of non-negative norm one functions, and let (mβ )β∈B be a net in L1 (G) as specified in the definition of Reiter’s Property (P1 ). Define  eα,β := mβ (x)(δx ∗ uα ∗ δx−1 ) dx (α ∈ A, β ∈ B), G

where the integral is a Bochner integral (see [56, Appendix E]); it is immediate that eα,β ≥ 0 and eα,β 1 = 1 for all α ∈ A and β ∈ B.

2.2 Virtual and Approximate Diagonals

71

For x ∈ G, note that  δx ∗ eα,β ∗ δx−1 = mβ (y)(δxy ∗ uα ∗ δ(xy)−1 ) dy G (δx ∗ mβ )(y)(δy ∗ uα ∗ δy−1 ) dy = G

(α ∈ A, β ∈ B),

so that δx ∗ eα,β ∗ δx−1 − eα,β 1 ≤ δx ∗ mβ − mβ 1 . From the definition of Reiter’s property (P1 ), we conclude that δx ∗ eα,β ∗ δx−1 − eα,β → 0 uniformly in x on compact subsets of G. To see that (eα,β )α,β is a bounded approximate identity for L1 (G), fix f ∈ L1 (G). If K ⊂ G is compact, then {δx−1 ∗ f : x ∈ K} is a compact subset of L1 (G), so that lim sup uα ∗ δx−1 ∗ f − δx−1 ∗ f 1 = 0. α x∈K

It follows that (δx ∗ uα ∗ δx−1 ) ∗ f → f uniformly in x on compact subsets of G. Since  mβ (x)(δx ∗ uα ∗ δx−1 ) ∗ f dx (α ∈ A, β ∈ B), eα,β ∗ f = G

we conclude that limα,β eα,β ∗ f = f , i.e., (eα,β )α,β is a bounded left approximate identity for L1 (G). In the same way, one sees that (eα,β )α,β is also a bounded right approximate identity.  Theorem 2.2.10. The following are equivalent for a locally compact group G: (i) G is amenable; (ii) AM(L1 (G)) < ∞; (iii) AM(L1 (G)) = 1. Proof. In view of Theorem 2.1.10, only (i) =⇒ (iii) needs proof. Let (mα )α∈A be a net as in the definition of Reiter’s property (P1 ), and let (eβ )β∈B be a bounded approximate identity for L1 (G) as specified in Proposition 2.2.9. For α ∈ A and β ∈ B, define dα,β : G × G → C,

(x, y) → mα (x)eβ (xy).

It is immediate that (dα,β )α,β is a net in L1 (G × G) ∼ = L1 (G) ⊗γ L1 (G), consisting of non-negative norm one functions. We claim that (dα,β )α,β is an approximate diagonal for L1 (G). Let φ ∈ L∞ (G), and note that

72

2 Amenable Banach Algebras

ΔL1 (G) dα,β , φ = dα,β , Δ∗L1 (G) φ   = mα (x)eβ (xy)φ(xy) dy dx G G   mα (x)eβ (y)φ(y) dy dx = G G = eβ (y)φ(y) dy G

= eβ , φ

(α ∈ A, β ∈ B).

It follows that ΔL1 (G) dα,β = eβ for α ∈ A and β ∈ B. Hence, (dα,β )α,β satisfies (2.9). Next, note that the L1 (G)-module actions on L1 (G) ⊗γ L1 (G) extend naturally to M (G). For x, y, z ∈ G, we have (δx · dα,β )(y, z) = mα (x−1 y)eβ (x−1 yz) = (δx ∗ mα )(y)(δx ∗ eβ )(yz) and (dα,β · δx )(y, z) = mα (y)δ(x−1 )eβ (yzx−1 ) = mβ (y)(eβ ∗ δx )(yz) for all α ∈ A and β ∈ B. It follows that δx · dα,β − dα,β · δx 1   = |(δx ∗ mα )(y)(δx ∗ eβ )(yz) − mβ (y)(eβ ∗ δx )(yz)| dz dy G G |(δx ∗ mα )(y)(δx ∗ eβ )(z) − mβ (y)(eβ ∗ δx )(z)| dz dy = G G |(δx ∗ mα )(y) − mα (y)|(δx ∗ eβ )(z) dz dy ≤ G G   mα (y)|(δx ∗ eβ )(z) − (eβ ∗ δx )(z)| dy dz + G

G

= δx ∗ mα − mα 1 + δx ∗ eβ − eβ ∗ δx 1 →0 uniformly in x on all compact subsets of G and thus—by the inner regularity of the elements of M (G)— μ · dα,β − dα,β · μ → 0 In particular, (dα,β )α,β satisfies (2.8).

(μ ∈ M (G)). 

2.2 Virtual and Approximate Diagonals

73

Exercises Exercise 2.2.1. For p ∈ [1, ∞), let p be equipped with coordinatewise multiplication. Show that p lacks a bounded approximate identity (and, consequently, is not amenable). Exercise 2.2.2. Let A be a Banach algebra with a bounded left approximate identity (eα )α . Show that (enα )α is a bounded left approximate identity for A for each n ∈ N. Exercise 2.2.3. Let A be a unital, amenable Banach algebra. Show that A has a bounded approximate diagonal (dα )α∈A such that Δdα = eA for all α ∈ A. Exercise 2.2.4. For an algebra A, let Aop denote its opposite algebra, i.e., the algebra whose underlying linear space is the same as for A, but whose multiplication is the multiplication in A reversed. For N ∈ N, let A be the Banach algebra of complex N × N matrices. Show that the diagonal d for A given in Example 2.2.4 is the only element of A ⊗γ A that is a diagonal for both A and Aop . Exercise 2.2.5. For N ∈ N, let 1N be equipped with coordinatewise multiplication. Show that AM( 1N ) = N .

2.3 Hereditary and Splitting Properties As in our discussion of amenable, locally compact groups, we now study the hereditary properties of amenability for Banach algebras. Instead of merely giving qualitative hereditary properties, we often provide relations between amenability constants. For this purpose, we adopt the usual rules of arithmetic in [0, ∞], including that 0 · ∞ = ∞ · 0 = 0. Proposition 2.3.1. Let A and B be Banach algebras, and let θ : A → B be a (continuous) algebra homomorphism with dense range. Then AM(B) ≤ θ2 AM(A) holds. In particular, if A is amenable, then so is B. Proof. If A is not amenable, the claimed inequality is trivial. Therefore, suppose that A is amenable with an approximate diagonal (dα )α such that supα dα  ≤ AM(A). Let a ∈ A, set b := θ(a), and note that b · ((θ ⊗ θ)dα ) − ((θ ⊗ θ)dα ) · b = (θ ⊗ θ)(a · dα − dα · a) → 0

(2.11)

and bΔB ((θ ⊗ θ)dα ) = θ(aΔA dα ) → θ(a) = b.

(2.12)

74

2 Amenable Banach Algebras

As θ(A) is dense in B, (2.11) and (2.12) do not only hold for arbitrary b ∈ θ(A), but for all b ∈ B. Hence, ((θ ⊗ θ)dα )α is an approximate diagonal for B with sup (θ ⊗ θ)dα  ≤ θ2 sup dα  ≤ θ2 AM(A). α

α

The claimed inequality is then an immediate consequence of the definition of AM(B).  Corollary 2.3.2. Let A be a Banach algebra, and let I be a closed ideal in A. Then AM(A/I) ≤ AM(A) holds. In particular, if A is amenable, then so is A/I. What about passing to substructures such as closed subalgebras and closed ideals? There is virtually nothing that can be said about closed subalgebras (see Exercise 2.3.2 below). By Proposition 2.2.1, a necessary condition for a closed ideal of an amenable Banach algebra to be amenable is that it has a bounded approximate identity. We shall see that this condition is also necessary: Proposition 2.3.3. Let A be an amenable Banach algebra, and let I be a closed ideal of A. Then the following are equivalent: (i) I is amenable; (ii) I has a bounded approximate identity. Moreover, if I has an approximate identity of bound C ≥ 1, then AM(I) ≤ C 2 AM(A) holds. Proof. In view of Proposition 2.2.1, only (ii) =⇒ (i) needs proof. Let (dα )α∈A be an approximate diagonal with supα dα  ≤ AM(A), and let (eβ )β∈B be an approximate identity for I of bound C ≥ 1. For β ∈ B, define Lβ : A → I as left multiplication by eβ , i.e., Lβ x = eβ x. Then ((Lβ ⊗ Lβ )dα )α,β is a net in I ⊗γ I bounded by C 2 AM(A). We claim that a subnet of ((Lβ ⊗ Lβ )dα )α,β is an approximate diagonal for I. Let a ∈ I, α ∈ A, and β ∈ B. Then we have a · ((Lβ ⊗ Lβ )dα ) − ((Lβ ⊗ Lβ )dα ) · a ≤ a · ((Lβ ⊗ Lβ )dα ) − (Lβ ⊗ Lβ )(a · dα ) + (Lβ ⊗ Lβ )(a · dα ) − ((Lβ ⊗ Lβ )dα ) · a = (id ⊗ Lβ )((aeβ − eβ a) · dα ) + (Lβ ⊗ Lβ )(a · dα − dα · a) ≤ Caeβ − eβ adα  + C 2 a · dα − dα · a. as well as,

(2.13)

2.3

Hereditary and Splitting Properties

75

aΔI ((Lβ ⊗ Lβ )dα ) − aΔA (dα ) ≤ aΔA ((Lβ ⊗ Lβ )dα ) − aΔA ((id ⊗ Lβ )dα ) + aΔA ((id ⊗ Lβ )dα ) − aΔA (dα )

(2.14)

= (aeβ − a)ΔA ((id ⊗ Lβ )dα ) + aΔA ((id ⊗ Lβ )dα ) − aΔA (dα ). (By idE —or simply id—we denote the identity map on a Banach space E.) For fixed α ∈ A, the first summand in the last line of (2.13), as well as both summands of the last line of (2.14) tend to zero in β, so that lim sup a · ((Lβ ⊗ Lβ )dα ) − ((Lβ ⊗ Lβ )dα ) · a ≤ C 2 a · dα − dα · a β

(a ∈ I)

and lim aΔI ((Lβ ⊗ Lβ )dα ) = aΔA (dα ) β

(a ∈ I)

for each α ∈ A. As (dα )α is an approximate diagonal for A, picking a suitable subnet of ((Lβ ⊗ Lβ )dα )α,β , therefore, yields an approximate diagonal for I.  Example 2.3.4. Let A be a commutative C ∗ -algebra. First, consider the case where A is unital. With A being commutative, U(A) is abelian and, therefore, amenable as a discrete group. Consider  f (u)u; θ : 1 (U(A)) → A, f → u∈U(A)

it is routinely checked that θ is a contractive algebra homomorphism. It is well known that every element in a unital C ∗ -algebra is a linear combination of no more than four unitaries ([338, Proposition I.4.9]): if x ∈ A is self-adjoint 1 with x ≤ 1, then u := x + i(1 − x2 ) 2 is a unitary such that x = 12 (u + u∗ ). As a consequence, π is onto. From Proposition 2.3.1, we conclude that A is amenable; as 1 (U(A)) is not only amenable, but already 1-amenable by Theorem 2.2.10, we even obtain that AM(A) = 1. Suppose now that A is not unital. The unitization A# of A is a unital ∗ C -algebra (Corollary C.4.7) and thus 1-amenable by the foregoing. Since A is a closed ideal in A# and, like any C ∗ -algebra, has an approximate identity of bound 1 (Theorem C.3.4), A itself is also 1-amenable by Proposition 2.3.3. There is another, deeper characterization of the closed ideals of amenable Banach algebras that are amenable themselves, which is entirely Banach space theoretic. We require the following definition: Definition 2.3.5. Let E be a Banach space. Then a closed subspace F of E is called weakly complemented in E if its annihilator F ⊥ in E ∗ is complemented in E ∗ . Obviously, every complemented subspace is weakly complemented. The converse, however, is not true, as shown in Exercise 2.3.3 below.

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2 Amenable Banach Algebras

Lemma 2.3.6. Let A be an amenable Banach algebra, and let L be a closed left ideal of A. Then the following are equivalent: (i) L has a bounded right approximate identity; (ii) L is weakly complemented in A; (iii) L∗∗ is complemented in A∗∗ . Proof. (i) =⇒ (ii): Let (eα )α∈A be a bounded right approximate identity for L. Let U be an ultrafilter over A that dominates the order filter, and define P : A∗ → A∗ ,

φ → weak∗ - lim (φ − eα · φ). α→U

(2.15)

(The weak∗ limit in (2.15) exists due to the weak∗ compactness of Ball(A∗ ).) As a, P φ = lim a, φ − eα · φ = lim a − aeα , φ = 0 α→U

α→U

(a ∈ L, φ ∈ A∗ ),

it is clear that P A∗ ⊂ L⊥ . Also, as aeα ∈ L for each α ∈ A, we have a, P φ = lim (a, φ − aeα , φ) = a, φ α→U

(a ∈ A, φ ∈ L⊥ ),

i.e., the restriction of P to L⊥ is the identity. (Note that we didn’t require the amenability of A at all for this part of the proof.) (ii) =⇒ (iii): Let P : A∗ → A∗ be a projection onto L⊥ , and set Q := idA∗∗ − P ∗∗ ; then Q : A∗∗ → A∗∗ is a projection onto L⊥⊥ ∼ = L∗∗ . (iii) =⇒ (i). Let (dα )α∈A be a bounded approximate diagonal for A, and suppose that ∞  (α) a(α) (α ∈ A) dα = n ⊗ bn ∞

n=1 (α)

(α)

with supα n=1 an bn  < ∞. Let Q : A∗∗ → A∗∗ be a projection onto L∗∗ , and define ∞  (α) a(α) (α ∈ A). Eα := n · Qbn n=1

Then (Eα )α∈A is a bounded net in L∗∗ and, consequently, has a weak∗ accumulation point in that space, say E. Passing to a subnet, we can suppose without loss of generality that E = weak∗ - limα Eα . We obtain:

2.3

Hereditary and Splitting Properties

77

a · E = weak∗ - lim a · Eα α

= weak∗ - lim α

= weak∗ - lim α

= weak∗ - lim α

=a

∞  n=1 ∞  n=1 ∞ 

(α) aa(α) n · Qbn (α) a(α) n · Q(bn a) (α) a(α) n bn a,

as b(α) n a ∈ L,

n=1

(a ∈ L).

As in the proof of Proposition 2.2.1, we obtain a bounded right approximate identity for L.  Remark 2.3.7. An closer inspection of the proof of Lemma 2.3.6 shows that, if Q : A∗∗ → A∗∗ is a projection onto L∗∗ , then L has right approximate identity of bound QAM(A). For closed ideals (two-sided), we now obtain: Theorem 2.3.8. Let A be an amenable Banach algebra, and let I be a closed ideal of A. Then the following are equivalent: (i) I is amenable; (ii) I has a bounded approximate identity; (iii) I is weakly complemented; (iv) I ∗∗ is complemented in A∗∗ . Proof. Most of the theorem has already been proven (Proposition 2.3.3 and Lemma 2.3.6). Only (iv) =⇒ (ii) still needs some consideration. By Lemma 2.3.6, I has a bounded right approximate identity. Passing to Aop and applying Lemma 2.3.6 once again, we obtain a bounded left approximate identity for I. Proposition B.2.3 then yields a bounded approximate identity for I.  Remark 2.3.9. Let Q : A∗∗ → A∗∗ be a projection onto I ∗∗ . By Remark 2.3.7, I then has both a left and a right approximate identity of bound QAM(A) and thus—as a consequence of Proposition B.2.3—an approximate identity of bound QAM(A)(2 + QAM(A)). From Proposition 2.3.3, we obtain that AM(I) ≤ Q2 (2 + QAM(A))2 AM(A)3 . As ideals with finite dimension or codimension are trivially complemented, we have: Corollary 2.3.10. Let A be an amenable Banach algebra, and let I be a closed ideal of A with finite dimension or codimension. Then I is amenable.

78

2 Amenable Banach Algebras

This corollary applies, in particular, to the maximal modular ideals of amenable, commutative Banach algebras. As an application of Theorem 2.3.8, we can improve Theorem 1.1.19: Corollary 2.3.11. For a locally compact group G, the following are equivalent: (i) G is amenable; (ii) L10 (G) has an approximate identity of bound 2; (iii) every closed ideal of L1 (G) with finite codimension has a bounded approximate identity; (iv) every weakly complemented closed ideal of L1 (G) has a bounded approximate identity; (v) every weakly complemented closed ideal of L1 (G) has a left or right bounded approximate identity. Next, we show that amenability is preserved by short, exact sequences: Theorem 2.3.12. Let A be a Banach algebra, and let I be a closed ideal of A such that both I and A/I are amenable. Then A is amenable. Proof. Let E be a Banach A-bimodule, and let D : A → E ∗ be a derivation. Of course, the restriction of D to I is also a derivation, and the amenability of I yields φ1 ∈ E ∗ such that Da = a · φ1 − φ1 · a

(a ∈ I).

˜ = D − adφ . Then D ˜ is a derivation from A into E ∗ that vanishes on Set D 1 I and thus drops to a map from A/I into E ∗ , which we denote likewise by ˜ Set D. F := {ψ ∈ E ∗ : a · ψ = ψ · a = 0 for all a ∈ I}. It is routinely checked that F is a weak∗ closed submodule of E ∗ and thus a dual Banach A-bimodule itself. ˜ vanishes on I, we have As D ˜ = D(ab) ˜ ˜ ·b=0 a · Db − (Da)

(a ∈ I, b ∈ A)

and, analogously, ˜ ·a=0 (Db)

(a ∈ I, b ∈ A).

˜ It follows that D(A/I) ⊂ F . As A/I is amenable, there is φ2 ∈ F such that ˜ D = adφ2 . Consequently, D = adφ1 +φ2 holds.  Corollary 2.3.13. A Banach algebra A is amenable if and only if its unitization A# is amenable. If A and B are algebras, their algebraic tensor product A ⊗ B becomes an algebra in a canonical fashion via

2.3

Hereditary and Splitting Properties

(a1 ⊗ b1 ) • (a2 ⊗ b2 ) := a1 a2 ⊗ b1 b2

79

(a1 , a2 ∈ A, b1 , b2 ∈ B).

(2.16)

In case both A and B are Banach algebras, the product • extends canonically to A ⊗γ B, turning it into a Banach algebra. Proposition 2.3.14. Let A and B be Banach algebras. Then AM(A⊗γ B) ≤ AM(A)AM(B) holds. In particular, if both A and B are amenable, then so is A ⊗γ B. Proof. Only the case where both A and B are amenable needs consideration. Let (dA,α )α and (dB,β )β be approximate diagonals for A and B, bounded by AM(A) and AM(B), respectively. Consider the isometric “shuffle” map θ : (A ⊗γ A) ⊗γ (B ⊗γ B) → (A ⊗γ B) ⊗γ (A ⊗γ B). It is immediate that (θ(dA,α ⊗ dB,β ))α,β is an approximate diagonal for A ⊗γ A, bounded by AM(A)AM(B).  The next proposition can be viewed as the Banach algebra analog of Proposition 1.2.13 Proposition 2.3.15. Let A be a Banach algebra, and let B be a directed

family of closed subalgebras of A such that {B : B ∈ B} is dense in A. Then AM(A) ≤ supB∈B AM(B) holds. In particular, if supB∈B AM(B) is finite, then A is amenable. Proof. Suppose that supB∈B AM(B) < ∞ because the claim is trivial otherwise.

Let F be the family of all finite subsets of {B : B ∈ B}. Let F ∈ F, and let > 0. Choose B ∈ B with F ⊂ B. Then there is dF, ∈ B ⊗γ B with dF,  ≤ AM(B) such that a · dF, − dF, · a < and

aΔdF, − a <

(a ∈ F ).

Order (0, ∞) × F as follows: ( 1 , F1 )  ( 2 , F2 )

:⇐⇒

2 ≤ 1 and F1 ⊂ F2

(( 1 , F1 ), ( 2 , F2 ) ∈ (0, ∞) × F)

Then, as one routinely shows, the net (d ,F ) ,F is a bounded approximate  diagonal for A, bounded by supB∈B AM(B). Remark 2.3.16. In order to obtain the amenability of A in Proposition 2.3.15, it is not enough only to require that all the algebras in B are amenable: see Exercise 2.3.6 below. Example 2.3.17. Let E = c0 or E = p with p ∈ (1, ∞). For N ∈ N, let PN : E → E be the projection onto the first N coordinates, and set

80

2 Amenable Banach Algebras

AN := PN K(E)PN . (For a Banach space E, we denote by K(E) the compact

∞ p operators on E.) It follows that N =1 AN = K(E). Since AN ∼ = B( N )—if p ∞ ∼ E = with p ∈ (1, ∞)—and AN = B( N )—if E = c0 —for all N ∈ N, we can combine Example 2.2.8 with Proposition 2.3.15, to conclude that AM(K(E)) = 1. We will prove one more hereditary property of amenability. Given two Banach spaces E and F , we denote their ∞ -sum, i.e., their direct sum E ⊕ F with the norm (x, y) := max{x, y} for x ∈ E and y ∈ F by E ⊕∞ F . Of course, if A and B are Banach algebras, then A ⊕∞ B, equipped with coordinatewise multiplication, is again a Banach algebra. Lemma 2.3.18. Let A and B be Banach algebras. Then AM(A ⊕∞ B) = max{AM(A), AM(B)} holds. In particular, A ⊕∞ B is amenable if and only if A and B are amenable. Proof. If A ⊕∞ B is amenable, then both A and B are quotients of A ⊕∞ B and thus amenable with max{AM(A), AM(B)} ≤ AM(A ⊕∞ B). Conversely, suppose that both A and B are amenable, and let (dA,α )α and (dB,β )β be approximate diagonals for A and B bounded by AM(A) and AM(B), respectively. The map (A × A) ⊕∞ (B × B) → (A ⊕∞ B) ⊗γ (A ⊕∞ B), ((a1 , a2 ), (b1 , b2 )) → (a1 , b1 ) ⊗ (a2 , b2 ) induces a contraction θ : (A ⊗γ A) ⊕∞ (B ⊗γ B) → (A ⊕∞ B) ⊗γ (A ⊕∞ B). It is routinely checked that (θ((dA,α , dB,β )))α,β is an approximate diagonal  for A ⊕∞ B bounded by max{AM(A), AM(B)}. If (E ι )ι∈I is any family of Banach spaces, we denote its c0 -direct sum by c 0 ι∈I Eι , i.e., the space of all those (xι )ι∈I in the Cartesian product  ι∈I Eι with the property that, for each > 0, thereis a finite subset F of I such that supι∈I\F xι  < . It is clear that c0 - ι∈I Eι is again a Banach space via (xι )ι∈I  := supι∈I xι  for (xι )ι∈I ∈ c0 -  ι∈I Eι . It is also clear that, if (Aι )ι∈I is a family of Banach algebras, then c0 - ι∈I Aι —under coordinatewise multiplication—is also a Banach algebra. We have: Proposition  2.3.19. Let (Aι )ι∈I be a family of Banach algebras, and let A := c0 - ι∈I Aι . Then AM(A) = supι∈I AM(Aι ) holds. In particular, A is amenable if and only if supι∈I AM(Aι ) < ∞. Proof. For finite I, the claim follows from Lemma 2.3.18 by induction. The general case, let F(I) denote the family of all finite subsets of I, ordered by set inclusion. For F ∈ F(I), let AF denote the subalgebra of A consisting of those families (aι )ι∈I for which aι = 0 for ι ∈ I \ F . Clearly, AF and

2.3

Hereditary and Splitting Properties

81



c0 - ι∈F Aι are isometrically isomorphic. Also, {AF : F ∈ F(I)} is dense in A. From Proposition 2.3.15 and the finite case, we thus conclude that AM(A) ≤ sup AM(AF ) = sup max{AM(Aι ) : ι ∈ F } = sup AM(Aι ). F ∈F(I)

ι∈I

F ∈F(I)

On the other hand, Aι is a quotient of A for each ι ∈ I, so that AM(A) ≥ supι∈I AM(Aι ).  Next, we take a first look at splitting properties of exact sequences of Banach modules over amenable Banach algebras. Let A be a Banach algebra, let E be a left Banach A-module, and let F be a closed submodule of A. We denote by ι : F → E the inclusion map, and by π : E → E/F the quotient map. Then we call ι

π

{0} −→ F −→ E −→ E/F −→ {0}

(2.17)

a short exact sequence of left Banach A-modules. In the same vein, once considers short exact sequences of right Banach A-modules and of Banach A-bimodules. Definition 2.3.20. Let A be a Banach algebra. Then we say that a short exact sequence (2.17) of left Banach A-modules: (a) is admissible if π : E → E/F has a bounded linear right inverse; (b) splits if π has a bounded right inverse that is also a left module homomorphism. It is obvious how these definitions are adapted to the right modules and to bimodules. It is clear that (2.17) is admissible if and only if F is complemented in E, and (2.17) splits if and only there is a projection onto F that is also a module homomorphism. In particular, (2.17) is admissible whenever: • F or E/F is finite-dimensional, or • the underlying Banach space of E is a Hilbert space. Trivially, every short exact sequence of Banach modules that splits is admissible. For modules over amenable Banach algebras we have a partial converse: Theorem 2.3.21. Let A be an amenable Banach algebra. Then every admissible, short, exact sequence (2.17) of left Banach A-modules splits, provided that F is a dual module. Proof. Without loss of generality, we may suppose that A is unital, and that E is a unital left Banach A-module, i.e., eA · x = x for all x ∈ E. Otherwise, replace A by its unitization, which is also amenable by Corollary 2.3.13, and extend the module action by letting (a + λe) · x := a · x + λx

(a ∈ A, λ ∈ C, x ∈ E).

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2 Amenable Banach Algebras

Let (dα )α∈A be a bounded approximate diagonal for A, where dα =

∞ 

(α) a(α) n ⊗ bn

(α ∈ A)

n=1

∞ (α) (α) with supα n=1 an bn  < ∞. By Exercise 2.2.3, we may suppose that ∞ (α) (α) = eA for all α ∈ A. Let Q : E → E be a projection onto F . n=1 an bn Let U be an ultrafilter on A that dominates the order filter, and define P : E → F,

x → weak∗ - lim

α→U

∞ 

  (α) a(α) n · Q bn · x ;

n=1

here the weak∗ topology referred to is induced by a predual F∗ of F that turns F into a dual left Banach A-module. For a ∈ A and x ∈ E, we have: P (a · x) = weak∗ - lim

α→U

= weak∗ - lim

α→U

∞  n=1 ∞ 

  (α) a(α) · Q b a · x n n   (α) aa(α) n · Q bn · x

n=1

= a · P x, i.e., P is a left module homomorphism. Furthermore, note for x ∈ F : P x = weak∗ - lim

α→U

= weak∗ - lim

α→U

∞  n=1 ∞ 

  (α) a(α) n · Q bn · x (α) a(α) n bn · x,

because b(α) n · x ∈ F,

n=1

= x. Hence, P is the identity on F . Since P E ⊂ F by definition, it follows that P is a projection onto F .  Of course, there are a right module and a bimodule version of Theorem 2.3.21. They can be proven by simply modifying the proof of Theorem 2.3.21 accordingly. We prefer, however, to deduce them from Theorem 2.3.21 through two tricks that allow us to deduce statements on right modules or bimodules from their counterparts for left modules; they will turn out to be useful in the sequel. Corollary 2.3.22. Let A be an amenable Banach algebra. Then every admissible, short, exact sequence (2.17) of right Banach A-modules splits, provided that F is a dual module.

2.3

Hereditary and Splitting Properties

83

Proof. Turn E into a left Banach Aop -module by letting a ◦ x := x · a

(a ∈ A, x ∈ E).

By Exercise 2.3.8 below, Aop is amenable as well. Apply Theorem 2.3.21.  Corollary 2.3.23. Let A be an amenable Banach algebra. Then every admissible, short, exact sequence (2.17) of Banach A-bimodules splits, provided that F is a dual module. Proof. Turn E into a left Banach A ⊗γ Aop -module by letting (a ⊗ b) ◦ x := a · x · b

(a, b ∈ A, x ∈ E).

By Exercise 2.3.8 below and Proposition 2.3.14, A⊗γ Aop is amenable as well. Apply Theorem 2.3.21.  In fact, the amenable Banach algebras can be characterized through the splitting of particular exact sequences of Banach bimodules (see Exercise 2.3.9 below). We conclude this section with two applications of Theorem 2.3.21, which both show that the seemingly weak condition of amenability can impose surprisingly strong structural constraints on a Banach algebra. A commutative Banach algebra A is called uniform if the Gelfand transform GA : A → C0 (ΦA ) (see Theorem B.1.5) is an isometry. The next theorem asserts that, in this setting, we can have amenability only in the trivial situation: Theorem 2.3.24. Let A be an amenable, uniform Banach algebra, i.e., a closed subalgebra of a commutative C ∗ -algebra. Then G : A → C0 (ΦA ) is an isometric isomorphism. Proof. Without loss of generality, suppose that A be unital, so that K := ΦA is compact. For the sake of simplicity, we identify A with GA ⊂ C(K), and assume towards a contradiction that A  C(K). By the Hahn–Banach Theorem and the Riesz Representation Theorem ([56, Theorem 7.3.6]), there is a non-zero, regular complex Borel measure μ on K such that  f dμ = 0 (f ∈ A). K

2

Set H := L (|μ|), where |μ| is the variation of μ, and let K be the closure of A in H. Then both H and K are (left) Banach A-modules via left multiplication. Trivially, the short, exact sequence {0} −→ K −→ H −→ H/K −→ {0} is admissible and thus splits by Theorem 2.3.21. Let P : H → H be a projection onto K which is also a left A-module homomorphism. For f ∈ C(K), let

84

2 Amenable Banach Algebras

Lf : H → H be the corresponding multiplication operator; clearly, Lf is normal with L∗f = Lf¯. As P is an A-module homomorphism, Lf P = P Lf

(f ∈ A)

(2.18)

holds. The Fuglede–Putnam Theorem ([61, IX.6.7]), implies that also Lf¯P = P Lf¯

(f ∈ A).

(2.19)

Clearly, {f ∈ C(K) : Lf P = P Lf } is a closed subalgebra of C(K) which, by (2.18), (2.19), and the Stone–Weierstraß Theorem must be all of C(K). It follows that f = Lf 1 = Lf P 1 = P (Lf 1) = P f ∈ K

(f ∈ C(K)),

so that H = K. Let f ∈ C(K). Then there is a sequence (fn )∞ n=1 in A with f − fn 2 → 0 in H. As fn , μ = 0 for all n ∈ N, we obtain: |f, μ|2 = |f − fn , μ|2  2    =  (f − fn ) dμ K



2 |f − fn | d|μ| K  ≤ |μ|(K) |f − fn |2 d|μ| ≤

K

= |μ|(K)f − fn 22

(n ∈ N).

As f − fn 2 → 0, this contradicts μ = 0.



For n ∈ N, we denote the n × n matrices with complex entries by Mn . Theorem 2.3.25. Let A be a Banach algebra such that the underlying Banach space is reflexive and that each maximal modular ideal of A is complemented in A. Then the following are equivalent: (i) A is amenable; (ii) A is semisimple and finite-dimensional; (iii) there are N1 , . . . , Nn ∈ N such that A∼ = MN1 ⊕ · · · ⊕ MNn . Proof. First, note that A must be unital: as an amenable Banach algebra, it has a bounded approximate identity, which—due to the reflexivity of A—has a weak accumulation point, say eA , in A; clearly, eA is an identity for A. (i) =⇒ (ii): We first suppose that A is semisimple.

2.3

Hereditary and Splitting Properties

85

Note that A must be unital: as an amenable Banach algebra, it has a bounded approximate identity, which—due to the reflexivity of A—has a weak accumulation point in A. This element of A must be its identity. Let L be a maximal (modular) left ideal of A. Then the short exact sequence π

{0} −→ L −→ A −→ A/L −→ {0} of left Banach A-bimodules is admissible splits and, therefore splits, i.e., there is a bounded right inverse ρ : A/L → A of π that is also a left module homomorphism. As π(ρ(eA + L)) = eA + L = L, it is clear that ρ(eA + L) = 0 whereas (x ∈ L). xρ(eA + L) = ρ(x + L) = ρ(L) = 0 Hence, A satisfies Definition B.5.6, i.e., is a modular annihilator algebra and thus finite-dimensional by Corollary B.5.8. We now consider the general case; in fact, we will rule out that A is not semisimple. Let rad(A) denote the Jacobson radical of A (see Definition B.4.9). By Corollary 2.3.2 and Proposition B.4.11(ii), A/rad(A) is an amenable, semisimple Banach algebra whose underlying Banach space is reflexive. Moreover, by the definition of rad(A), the maximal left ideals of A/rad(A) are in canonical one-to-one correspondence with those of A; in particular, all maximal left ideals of A/rad(A) are complemented in A/rad(A). By our discussion of the semisimple case, we therefore, know that A/rad(A) is finite-dimensional. Consequently, rad(A) is complemented in A and thus has a bounded approximate identity by Theorem 2.3.8. As rad(A) is also reflexive, this means that rad(A) has an identity, say erad(A) . By Proposition B.4.11(i), we have σA (erad(A) ) = {0}; in particular, eA − erad(A) is an invertible idempotent, which is possible only if erad(A) = 0 and thus rad(A) = {0}. Hence, A must be semisimple and thus is finite-dimensional. (ii) =⇒ (iii): This is a direct consequence of Wedderburn’s Theorem (Theorem B.5.9). (iii) =⇒ (i): This is clear by Example 2.2.4, Lemma 2.3.18, and induction on n.  Corollary 2.3.26. Let A be an amenable, commutative Banach algebra such that the underlying Banach space is reflexive. Then there is N ∈ N such that A∼ = CN . Corollary 2.3.27. Let A be a Banach algebra such that the underlying Banach space is a Hilbert space. Then the following are equivalent: (i) A is amenable; (ii) there are N1 , . . . , Nn ∈ N such that A∼ = MN1 ⊕ · · · ⊕ MNn .

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2 Amenable Banach Algebras

Exercises Exercise 2.3.1. Let A and B be Banach algebras, let θ : A → B be an algebra homomorphism with dense range, and suppose that A is amenable. Show that B is amenable, using only Definition 2.1.11. Exercise 2.3.2. Let B be the subalgebra of 1 (Z) consisting of all those f : Z → C such that f (n) = 0 for n < 0. Show that B is a closed, complemented subalgebra of 1 (Z) that fails to be amenable. (Hint: Identify the character space of B with the closed unit disc, and show that the Gelfand transforms of all elements of B are holomorphic on the open unit disc.) Exercise 2.3.3. Show that c0 is weakly complemented in ∞ , but not complemented. Proceed as follows: (a) Show that there is S ⊂ P(N) such that |S| = |R|, i.e., S has the same cardinality as R, consisting of infinite sets such that S1 ∩ S1 is finite for all S1 , S2 ∈ S with S1 = S2 . (Hint: Replace N by the set Q of rational numbers—using the fact that both sets are countably infinite—and, for each r ∈ R, define Sr ⊂ Q to be an infinite set of which r is the only cluster point.) (b) Show that here is no countable subset of ( ∞ /c0 )∗ that separates the points of ∞ /c0 . (Hint: Choose S as in (a), and show that, for fixed F ∈ ( ∞ /c0 )∗ , the set {S ∈ S : χS + c0 , F  = 0} is at most countable, where χS stands for the indicator function of S.) (c) Conclude from (b) that c0 is not complemented in ∞ . (d) Show that c0 is weakly complemented in ∞ . Exercise 2.3.4. Let A be an amenable Banach algebra, and let I be a closed ideal of A with a one-sided, i.e., left or right, approximate identity. Show that I has a two-sided approximate identity. Exercise 2.3.5. For a locally compact Hausdorff space X and a Banach space E, we write C0 (X, E) for the E-valued continuous functions on X that vanish at infinity; it can be identified with the injective Banach space tensor product C0 (X) ⊗λ E (notation as in [106]). Show that, if X is a locally compact Hausdorff space and A is a Banach algebra, then C0 (X, A) equipped with the pointwise product is a Banach algebra such that AM(C0 (X, A)) = AM(A). (Hint: Propositions 2.3.1 and 2.3.14.) Exercise 2.3.6. Give an example of a Banach algebra

A such that there is a directed family B of closed subalgebras of A with {B : B ∈ B} dense in A and AM(B) < ∞ for each B ∈ B, but which fails to be amenable itself. Exercise 2.3.7. Does the argument in Example 2.3.17 also establish the amenability of K( p ) for p = 1, ∞? Exercise 2.3.8. Let A be a Banach algebra. Show that AM(A) = AM(Aop ).

2.3

Hereditary and Splitting Properties

87

Exercise 2.3.9. Let A be a Banach algebra with a left or right bounded approximate identity. Show that: (a) the short, exact sequence Δ∗

{0} −→ A∗ −→ (A ⊗γ A)∗ −→ (ker Δ)∗ −→ {0}

(2.20)

of Banach A-bimodules is admissible; (b) the following are equivalent: (i) A is amenable; (ii) A has a bounded approximate identity, and (2.20) splits; (iii) A has a bounded approximate identity, and the short, exact sequence Δ∗∗

{0} −→ (ker Δ)∗∗ −→ (A ⊗γ A)∗∗ −→ A∗∗ −→ {0} of Banach A-bimodules splits.

2.4 A First Look at Hochschild Cohomology So far, we haven’t formally introduced Hochschild cohomology yet. We conclude this chapter with a brief look at its very basics. (The topic of cohomology of Banach algebras will be treated in much greater depth in Chapter 6.) Definition 2.4.1. Let A be a Banach algebra, let E be a Banach A-bimodule, and let n ∈ N0 . Then: (a) the space C n (A, E) of n-cochains is defined as E if n = 0 and as the space of all bounded n-linear maps from An to E otherwise; (b) the n-coboundary operator δ n : C n (A, E) → C n+1 (A, E) is defined through (δ n T )(a1 , . . . , an+1 ) := a1 · T (a2 , . . . , an+1 ) n  + (−1)k T (a1 , . . . , ak ak+1 , . . . , an+1 ) k=1

+(−1)n+1 T (a1 , . . . , an ) · an+1 (T ∈ C n (A, E), a1 , . . . , an+1 ∈ A); (c) the space of n-cocycles Z n (A, E) is defined as ker δ n ; (d) the space of n-coboundaries B n (A, E) is defined as {0} if n = 0 and as the range of δ n−1 otherwise. The sequence

88

2 Amenable Banach Algebras δ0

δ1

δ2

{0} −→ E −→ C 1 (A, E) −→ C 2 (A, E) −→ δ n−1

δn

δ n+1

· · · −→ C n (A, E) −→ C n+1 (A, E) −→ · · ·

(2.21)

is called the Hochschild cochain complex. Example 2.4.2. Let A be a Banach algebra, and let E be a Banach Abimodule. Then (δ 0 x)(a) = a · x − x · a

(x ∈ E, a ∈ A),

so that Z 0 (A, E) consists precisely of those x ∈ E for which a · x = x · a for all a ∈ A whereas B 1 (A, E) is the linear space of all inner derivations from A to E. For T ∈ C 1 (A, E) note that (δ 1 T )(a, b) = a · T b − T (ab) + (T a) · b

(a, b ∈ A).

Hence, Z 1 (A, E) is the Banach space of all derivations from A to E. The next lemma is crucial for the definition of Hochschild cohomology: Lemma 2.4.3. Let A be a Banach algebra, and let E be a Banach Abimodule. Then B n (A, E) is a linear subspace of Z n (A, E) for all n ∈ N0 . For n = 0, 1, we have already verified the claim in Example 2.4.2. The proof of the general case is equally elementary but considerably lengthier and more tedious: we leave it to the reader to check it (Exercise 2.4.1 below). In view of Lemma 2.4.3, the following definition makes sense. Definition 2.4.4. Let A be a Banach algebra, and let E be a Banach Abimodule. For n ∈ N0 , the n-th Hochschild cohomology group of A with coefficients in E is defined as the quotient space Hn (A, E) := Z n (A, E)/B n (A, E). Remark 2.4.5. As B n (A, E) is a linear subspace of Z n (A, E)—and not just a subgroup of its additive group—the term Hochschild cohomology space would be more appropriate. However, the term “Hochschild cohomology group” seems to be too entrenched be replaced by something else. Note that B n (A, E) need not be a closed subspace of Z n (A, E) (see Exercise 2.4.3 below), so that Hn (A, E) is generally not a Banach space. With Example 2.4.2 in mind, we can thus rephrase the definition of an amenable Banach algebra: Proposition 2.4.6. Let A be a Banach algebra. Then A is amenable if and only if H1 (A, E ∗ ) = {0} for each Banach A-bimodule E. With Definition 2.4.4 in place, two questions arise naturally:

2.4

A First Look at Hochschild Cohomology

89

• What relevant information about A is encoded in Hn (A, E)? Thanks to Proposition 2.4.6, the first Hochschild cohomology group determines when A is amenable. But what about Hn (A, E) if n ≥ 2? • Are there effective methods to compute Hn (A, E)? We don’t have the space to deal with these questions in any depth and will only indicate possible answers. We deal with the second question first by stating two elementary lemmas, the proofs of which we leave to the reader (Exercise 2.4.2 below). Lemma 2.4.7. Let A be a Banach algebra, let E be a Banach A-bimodule, and let k ∈ N. Then C k (A, E) becomes a Banach A-bimodule via (a · T )(a1 , . . . , ak ) := a · T (a1 , . . . , ak ) (a ∈ A, T ∈ C k (A, E), a1 , . . . , ak ∈ A) and (T · a)(a1 , . . . , ak ) := T (aa1 , . . . , ak ) +

k−1 

(−1)j T (a, a1 , . . . , aj aj+1 , . . . , ak )

j=1

+ (−1)k T (a, a1 , . . . , ak−1 ) · ak (a ∈ A, T ∈ C k (A, E), a1 , . . . , ak ∈ A). Let A be a Banach algebra, let E be a Banach A-bimodule, let k ∈ N, and suppose that C k (A, E) is equipped with the module actions described in Lemma 2.4.7. Then, in order to avoid confusion, we let δkn : C n (A, C k (A, E)) → C n+1 (A, C k (A, E))

(n ∈ N0 )

denote the corresponding n-coboundary operator whereas the coboundary operators of (2.21) are still denoted by δ n for n ∈ N0 . Lemma 2.4.8. Let A be a Banach algebra, and let E be a Banach Abimodule. Then, for each k ∈ N and n ∈ N0 , the linear map τ n : C n+k (A, E) → C n (A, C k (A, E)) defined by ((τ n T )(a1 , . . . , an ))(an+1 , . . . , an+k ) := T (a1 , . . . , an , an+1 , . . . , an+k ) (T ∈ C n+k (A, E), a1 , . . . , an , an+1 , . . . , an+k ∈A) is an isometric isomorphism such that the diagram

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2 Amenable Banach Algebras δ n+k

C n+k (A, E) τn

/ C (n+1)+k (A, E)

(2.22)

τn+1



C n (A, C k (A, E))

n δk

 / C n+1 (A, C k (A, E))

commutes. With Lemmas 2.4.7 and 2.4.8 in place, we can prove: Theorem 2.4.9 (Reduction of Dimension). Let A be a Banach algebra, let E be a Banach A-bimodule, and let k ∈ N0 . Then we have linear isomorphisms Hn+k (A, E) ∼ = Hn (A, C k (A, E))

(n ∈ N0 ).

Proof. As the claim is trivially true for k = 0, suppose that k ∈ N. For n ∈ N0 , let τ n : C n+k (A, E) → C n (A, C k (A, E)) be as in Lemma 2.4.8. We claim that τ n induces an isomorphism Hn+k (A, E) ∼ = Hn (A, C k (A, E)). Chasing the diagram (2.22) yields immediately for T ∈ C n+k (A, E) that T ∈ Z n+k (A, E) ⇐⇒ τ n T ∈ Z n (A, C k (A, E)), i.e., τ n induces an isomorphism between Z n+k (A, E) and Z n (A, C k (A, E)). If n = 0, this already proves the claim. Suppose thus that n ∈ N. Chasing (2.22) again—with n − 1 en lieu of n—we see that τ n also induces an isomorphism between B n+k (A, E) and B n (A, C k (A, E)). This completes the proof.  Roughly speaking, Theorem 2.4.9 asserts that every Hochschild cohomology group is already a first Hochschild cohomology group, albeit with coefficients in a different module. This does not means that higher Hochschild cohomology groups are uninteresting: in order to reduce the order of the cohomology group as in Theorem 2.4.9, we have to trade the (perhaps quite simple) coefficient module E for the much more complicated module C k (A, E). An application of Theorem 2.4.9 is the following: Theorem 2.4.10. For a Banach algebra A, the following are equivalent: (i) A is amenable; (ii) Hn (A, E ∗ ) = {0} for each Banach A-bimodule E and for all n ∈ N. Proof. Of course, only (i) =⇒ (ii) needs proof. Let E be a Banach A-bimodule, and let n ∈ N. To avoid triviality, suppose that n ≥ 2. Set F := A ⊗γ · · · ⊗γ A ⊗γ E,    (n−1)-times

and turn F into a Banach A-bimodule by letting

2.4

A First Look at Hochschild Cohomology

91

a · (a1 ⊗ · · · ⊗ an−1 ⊗ x) := aa1 ⊗ · · · ⊗ an−1 +

n−2 

(−1)j a ⊗ a1 ⊗ · · · ⊗ aj aj+1 ⊗ · · · ⊗ an−1 ⊗ x

j=1

+(−1)n−1 a ⊗ a1 ⊗ · · · ⊗ an−2 ⊗ an−1 · x (a,a1 , . . . , an−1 ∈ A, x ∈ E) and (a1 ⊗ · · · ⊗ an−1 ⊗ x) · a := a1 ⊗ · · · ⊗ an−1 ⊗ x · a (a, a1 , . . . , an−1 ∈ A, x ∈ E). We can canonically identify the dual space F ∗ with C n−1 (A, E ∗ ) via a1 ⊗ · · · ⊗ an−1 ⊗ x, T  =x, T (a1 , . . . , an−1 ) (a1 , . . . , an−1 ∈ A, x ∈ E, T ∈ C n−1 (A, E)). It is routinely checked that the dual module actions of A on F ∗ are the module actions of A on C n−1 (A, E ∗ ) described in Lemma 2.4.7. Theorem 2.4.9 thus yields Hn (A, E ∗ ) ∼ = H1 (A, C n−1 (A, E ∗ )) ∼ = H1 (A, F ∗ ) = {0} as claimed.



We want to conclude this section with an application of second Hochschild cohomology groups. Given an algebra A, a subalgebra B of A, and an ideal I of A, we write A = B  I if A = B + I and B ∩ I = {0}. If A is finite-dimensional with Jacobson radical rad(A), then the classical Wedderburn Decomposition Theorem ([71, Corollary 1.5.19]) asserts that there is a subalgebra B of A such that A = B  rad(A). If an arbitrary algebra has such a decomposition, we say that A admits a Wedderburn decomposition. For Banach algebras, the following variant is often more appropriate: Definition 2.4.11. Let A be a Banach algebra. Then we say that A admits a strong Wedderburn decomposition if there is a closed subalgebra B of A such that A = B  rad(A). We will give a necessary condition in terms of second Hochschild cohomology groups which guarantee that a Banach algebra has a strong Wedderburn decomposition. Recall that an ideal I of a Banach algebra A is called n-nilpotent for some n ∈ N if x1 · · · xn = 0 for all x1 , . . . , xn ∈ I; if there is n ∈ N such that I is n-nilpotent, we call I simply nilpotent. Lemma 2.4.12. Let A be a Banach algebra such that rad(A) is 2-nilpotent, and let ρ : A/rad(A) → A be a bounded right inverse of the quotient map

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2 Amenable Banach Algebras

π : A → A/rad(A). Then rad(A) is turned into a Banach A/rad(A)-bimodule via (a + rad(A)) · r := ρ(a + rad(A))r

and

r · (a + rad(A)) := rρ(a + rad(A))

(a ∈ A, r ∈ rad(A)).

Furthermore, this module action is independent of the choice of ρ. Proof. Let b1 , b2 ∈ A/rad(A). Then π(ρ(b1 )ρ(b2 ) − ρ(b1 b2 )) = π(ρ(b1 ))π(ρ(b2 )) − π(ρ(b1 b2 )) = 0 holds, so that ρ(b1 )ρ(b2 ) − ρ(b1 b2 ) ∈ rad(A). As rad(A) is 2-nilpotent, this means that b1 · (b2 · r) − (b1 b2 ) · r = (ρ(b1 )ρ(b2 ) − ρ(b1 b2 ))r = 0

(r ∈ rad(A))

so that rad(A) is a left Banach A/rad(A)-module. Analogously, one sees that rad(A) is also a right Banach A/rad(A)-module. It is routinely checked that the two module actions are compatible in such a way that rad(A) becomes a Banach A/rad(A)-bimodule. Let ρ˜ be another continuous right inverse of π, so that ρ(b) − ρ˜(b) ∈ rad(A) for all b ∈ A/rad(A). Since rad(A) is 2-nilpotent, we see that ρ(b) · r = ρ˜(b) · r

(b ∈ A/rad(A), r ∈ rad(A)).

Therefore, the left module action of A/rad(A) on rad(A) is independent of the choice of ρ; the same is true for the right module action (as one sees in exactly the same way).  In the next theorem, we refer to the module action defined in Lemma 2.4.12. Theorem 2.4.13. Let A be a Banach algebra such that rad(A) is complemented and 2-nilpotent, and suppose that H2 (A/rad(A), rad(A)) = {0}. Then A has a strong Wedderburn decomposition. Proof. Let ρ : A/rad(A) → A be a bounded right inverse of the quotient map π : A → A/rad(A), and define T ∈ C 2 (A/rad(A), rad(A)) through T (b1 , b2 ) = ρ(b1 )ρ(b2 ) − ρ(b1 b2 ) For b1 , b2 , b3 ∈ A/rad(A), we have

(b1 , b2 ∈ A/rad(A)).

2.4

A First Look at Hochschild Cohomology

93

(δ 2 T )(b1 , b2 , b3 ) = b1 · T (b2 , b3 ) − T (b1 b2 , b3 ) + T (b1 , b2 b3 ) − T (b1 , b2 ) · b3 = ρ(b1 )(ρ(b2 )ρ(b3 ) − ρ(b2 b3 )) − ρ(b1 b2 )ρ(b3 ) + ρ(b1 b2 b3 ) + ρ(b1 )ρ(b2 b3 ) − ρ(b1 b2 b3 ) − (ρ(b1 )ρ(b2 ) − ρ(b1 b2 ))ρ(b3 ) = ρ(b1 )ρ(b2 )ρ(b3 ) − ρ(b1 )ρ(b2 b3 ) − ρ(b1 b2 )ρ(b3 ) + ρ(b1 b2 b3 ) + ρ(b1 )ρ(b2 b3 ) − ρ(b1 b2 b3 ) − ρ(b1 )ρ(b2 )ρ(b3 ) + ρ(b1 b2 )ρ(b3 ) = 0, so that T ∈ Z 2 (A/rad(A), rad(A)). As H2 (A/rad(A), rad(A)) = {0}, this means there is S ∈ C 1 (A/rad(A), rad(A)) with T = δ 1 S. Set θ := ρ − S. Then θ is a right inverse of π, and for b1 , b2 ∈ A/rad(A), we obtain: θ(b1 )θ(b2 ) − θ(b1 b2 ) = (ρ(b1 ) − Sb1 )(ρ(b2 ) − Sb2 ) − ρ(b1 b2 ) + S(b1 b2 ) = ρ(b1 )ρ(b2 ) − ρ(b1 b2 ) − ρ(b1 )Sb2 + S(b1 b2 ) − (Sb1 )ρ(b2 ) = T (b1 , b2 ) − b1 · Sb2 + S(b1 b2 ) − (Sb1 ) · b2 = (T − δ 1 S)(b1 , b2 ) = 0. Hence, θ : A/rad(A) → A is a bounded homomorphism of Banach algebras. Set B := θ(A/rad(A)). It is straightforward that B is a subalgebra of A with B ∩ rad(A) = {0} and B + rad(A) = A. What remains to be shown that B is a closed subalgebra of A. Let (bn )∞ n=1 be a sequence in B converging in A to some element a. It is immediate that π(a) = limn→∞ π(bn ), so that a = lim bn = lim (θ ◦ π)(bn ) = θ(π(a)) ∈ B. n→∞

n→∞

Consequently, B is closed.



Corollary 2.4.14. Let A be a Banach algebra such that rad(A) is finitedimensional and 2-nilpotent, and suppose that A/rad(A) is amenable. Then A has a strong Wedderburn decomposition. Proof. Since rad(A) is finite-dimensional, it is complemented in A and reflexive. As A/rad(A) is amenable, Theorem 2.4.10 yields H2 (A/rad(A), rad(A)) ∼ = H2 (A/rad(A), rad(A)∗∗ ) = {0}. By Theorem 2.4.13, this yields the claim.

Exercises Exercise 2.4.1. Verify Lemma 2.4.3.



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2 Amenable Banach Algebras

Exercise 2.4.2. Verify Lemmas 2.4.7 and 2.4.8 Exercise 2.4.3. Let A be a Banach algebra such that the left multiplier norm  · LM defined on A by aLM := sup{ax : x ∈ Ball(A)}

(a ∈ A)

is not equivalent to the given norm, and let E be the Banach A-bimodule whose underlying Banach space is A with the module actions a · x := ax and

x · a := 0

(a ∈ A, x ∈ E).

Show that B 1 (A, E) is not closed in Z 1 (A, E).

Notes and Comments B. E. Johnson’s memoir [188] is the big bang of amenable Banach algebras: it contains Theorem 2.1.10, as well as Definition 2.1.11. The result in Exercise 2.1.7 is from [143]. Corollary 2.1.8 answers the so-called derivation problem for group algebras: Given a locally compact group G and a derivation D : L1 (G) → L1 (G), is there always μ ∈ M (G) such that D = adμ ? In a certain sense, this problem can be considered the starting point of the entire theory of amenable Banach algebras. Apparently, it was given to Johnson by his thesis supervisor J. Williamson, and his attempts to solve it ultimately prompted him to develop the theory of amenable Banach algebras. He was able to answer the question affirmatively first for discrete G ([197]), and then, in [188], for amenable G— as a consequence of Theorem 2.1.10—as well as for [SIN]-groups, and certain matrix groups. For over a quarter of a century, there was virtually no progress towards a solution of the derivation problem. The first paper to pick it up again was [135]. Subsequently, Johnson gave an affirmative answer for connected G ([195]). Eventually, the problem was solved in full generality by V. Losert ([231]). The short and elegant approach via Theorem 1.5.6 we present is due to U. Bader, T. Gelander, and N. Monod ([17]). Theorem 2.2.5 is from [187]. The amenability constant of a Banach algebra seems to have been considered for the first time in [193]. Our proof of the 1-amenability of L1 (G) for amenable G is due to R. Stokke ([329]) with Proposition 2.2.9 being [233, Theorem 3]. If A is a Banach algebra, then an element of A ⊗γ A is called symmetric if it is fixed by the flip map A ⊗γ A  a ⊗ b → b ⊗ a. If A has a bounded approximate diagonal consisting of symmetric tensors, it is called symmetrically amenable. The symmetrically amenable Banach algebras were introduced and studied in [194]; they have hereditary properties similar to those of amenable Banach algebras. If G is

Notes and Comments

95

an amenable, locally compact group, then L1 (G) is already symmetrically amenable. The hereditary properties of amenable Banach algebras, as we present them, go back to [188]. Corollary 2.3.11 originates from [227]. Theorem 2.3.21 was first obtained by the Russian school of Banach algebraists, led by A. Ya. Helemski˘ı ([174]); the proof we give is is from [68]. Theorem 2.3.24 is due to M. V. She˘ınberg ([321]) whereas Theorem 2.3.25 was proven by Johnson ([192]), relying results from [127]. Corollary 2.3.27 was discovered independently of [192] and is from [130]. Only recently, a converse of Proposition 2.3.14 was proven ([129]; compare also [194]): if A and B are Banach algebras such that A ⊗γ B = {0} is amenable, then both A and B are amenable. Is every amenable Banach algebra with an underlying reflexive Banach space—in short: a reflexive, amenable Banach algebra—finite-dimensional (and thus, automatically, classically semisimple)? This question is open, and the consensus seems to be that the answer is “yes”. The papers [192] and [127] contain first partial results, and further results in this direction can be found in [293], [296], or [364]. A related question also studied in [192] and [127], is as follows: If A is an amenable Banach algebra, and θ is a weakly compact algebra homomorphism, does θ(A) have to be finite dimensional? An affirmative answer to this question would imply that all reflexive amenable Banach algebras are finite-dimensional: just take θ as the identity on A. It is a by now classical result that every weakly compact operator between Banach spaces factors through a reflexive Banach space ([78]). In [32], A. Blanco, S. Kaijser, and T. J. Ransford proved an analog of this result in the Banach algebra context: every weakly compact algebra homomorphism between Banach algebras factors through a reflexive Banach algebra as an algebra homomorphism. Hence, both questions are, in fact, equivalent: every reflexive, amenable Banach algebra is finite-dimensional if and only if every weakly compact algebra homomorphism from an amenable Banach algebra has a finite rank. Hochschild cohomology is named in the honor of G. Hochschild who introduced it in [180] and [181]; one of his motivations was to generalize Wedderburn’s Decomposition Theorem. Hochschild’s original definitions were for arbitrary associative algebras, i.e., without any topology. It was H. Kamowitz ([201]) who first adapted Hochschild’s ideas to the Banach algebra context by restricting himself to bounded cochains (compare also [157]). For this reason, the Banach algebraic Hochschild cohomology we are discussing is sometimes called Kamowitz–Hochschild cohomology. Theorem 2.4.10 motivates the following definition: For n ∈ N, we call a Banach algebra A n-amenable if Hm (A, E ∗ ) = {0} for all m ≥ n ([263]). There is a characterization of n-amenable Banach algebras similar to Theorem 2.2.5; see [263] for details. Already Kamowitz used Hochschild cohomology to show that certain Banach algebras admit strong Wedderburn decompositions. Theorem 2.4.13 is essentially due to him (Kamowitz only considers commutative Banach alge-

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2 Amenable Banach Algebras

bras, but all the ideas are already given in [201]). In fact, Theorem 2.4.13 is rather feeble: the hypotheses concerning the radical can be substantially weakened. Strong Wedderburn decompositions are just one instance of what are called strong splittings of extensions of Banach algebras: see [16], which also has an extensive list of references. One of the driving forces behind the research on (strong) splittings of extensions of Banach algebras was the following question: Does a commutative Banach algebra A with nilpotent radical such that A/rad(A) ∼ = C(Ω) for some compact Hausdorff space Ω admit a strong Wedderburn decomposition? This question was first studied by W. G. Bade and P. C. Curtis, Jr., in [13]. In the same paper, they were able to settle the question affirmatively for totally disconnected Ω. It was almost four decades later, when E. Albrecht and O. Ermert—adapting ideas due to M. Solovej ([323])—finally answered the question affirmatively in full generality ([3]). Subsequently, a more elementary proof of a slightly more general result was given in [15]. Third Hochschild cohomology groups occur naturally in the study of perturbations of Banach algebras (see [189] and [277]). I am unaware of any Banach algebraic interpretation of n-th Hochschild cohomology groups for n ≥ 4. Of course, for a Banach algebra A and a Banach A-bimodule E, one can also consider the purely algebraic cohomology groups H n (A, E) as originally defined by Hochschild. For each n ∈ N0 , there is a canonical group homomorphism cn : Hn (A, E) → H n (A, E), the cohomology comparison map. Obviously, c0 is an isomorphism, and c1 is injective. Already for c2 , the situation becomes more complicated (see [324] and [77]). In [359], M. Wodzicki showed that, if A is amenable and has the cardinality ℵn , then cm ≡ 0 for all m ≥ n + 3. If we assume the continuum hypothesis, this means that for amenable Banach algebras with the cardinality of the continuum, cm = 0 for all m ≥ 4, i.e., algebraic and Banach algebraic Hochschild cohomology can be surprisingly unrelated. Nothing seems to be known about c3 .

Chapter 3

Examples

So far, the collection of amenable Banach algebras we have assembled is not very impressive: we have the group algebras of amenable groups, the commutative C ∗ -algebras, the algebras of compact operators on certain classical sequence spaces, and we can use standard constructions like quotients, tensor products, c0 -direct sums, etc., to get further examples from the old ones. The purpose of this chapter, is to obtain more examples of amenable— and non-amenable—Banach algebras. The grand theme is to determine, for a given class of Banach algebras, what it means for algebras in that class to be amenable. The general thrust of the results presented is that amenable Banach algebras tend to be “small”—whatever that may mean precisely. Therefore, “large” Banach algebras—like the measure algebras M (G) for nondiscrete, locally compact groups G and the algebras B(E) for most infinitedimensional Banach spaces E—ought not to be amenable. But still, there are surprising exceptions, such as the fairly recent example of an infinitedimensional Banach space E, for which B(E) is indeed amenable.

3.1 Measure Algebras If G is a discrete, amenable group, then M (G) = 1 (G) = L1 (G) is an amenable Banach algebra according to Theorem 2.1.10. This begets the natural question, for which—not necessarily discrete— locally compact groups G precisely, the measure algebra M (G) is amenable. The following theorem gives the answer: Theorem 3.1.1. The following are equivalent for a locally compact group G: (i) G is discrete and amenable; (ii) M (G) is amenable. Proof. Of course, only (ii) =⇒ (i) needs proof.

© Springer Science+Business Media, LLC, part of Springer Nature 2020 V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-1-0716-0351-2 3

97

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3 Examples

Suppose that M (G) is amenable. If G is discrete, the identifications M (G) = 1 (G) = L1 (G) immediately yield the amenability of G. Hence, assume toward a contradiction that G is not discrete. Then, by Theorem D.3.6, Theorem D.3.7(i), and Theorem D.3.9(ii), Mc (G) is a nonzero complemented ideal of M (G). By Theorem 2.3.8, this ideal has a bounded approximate identity. In particular, the closed linear span of {μ ∗ ν : μ, ν ∈ Mc (G)}  is dense in Mc (G). But this contradicts Proposition 3.1.2 below. The proof of Theorem 3.1.1 thus rests entirely on: Proposition 3.1.2. Let G be a non-discrete, locally compact group. Then there are a probability measure μ0 ∈ Mc (G) and a positive element F0 ∈ M (G)∗ with μ0 , F  = F0  = 1 such that F0 |Md (G) ≡ 0 and μ ∗ ν, F0  = 0 for all μ, ν ∈ Mc (G). Here, we mean by a positive element of M (G)∗ a functional that maps positive elements of M (G) into [0, ∞). We establish Proposition 3.1.2 by proceeding through a series of auxiliary assertions, the first three of which focus on the metrizable case. If S is a nonempty subset of a metric space (X, d), we write diam S for its diameter, i.e., diam S := sup{d(x, y) : x, y ∈ S}. Lemma 3.1.3. Let G be a non-discrete, metrizable, locally compact group. Then there is a decreasing sequence (Kn )∞ n=1 of compact subsets of G with the following properties: (i) for each n ∈ N, there is a family (Kn (j))j ∈([1,4]∩N)n of pairwise disjoint, compact sets, each with nonempty interior—the sets of level n—such that  Kn = {Kn (j) : j ∈ ([1, 4] ∩ N)n } and

1 (j ∈ ([1, 4] ∩ N)n ); 2n (ii) for each n ∈ N, j ∈ ([1, 4] ∩ N)n , and k ∈ [1, 4] ∩ N, we have diam Kn (j) ≤

Kn+1 (j, k) ⊂ Kn (j); (iii) for n ∈ N and x1 , x2 , x3 , and x4 belonging to four distinct sets of level −1 n, we have x1 x−1 2 x3 x4 = eG . Proof. We construct the sequence (Kn )∞ n=1 inductively, along with—for each n ∈ N—the corresponding sets of level n. Fix a nonempty, open, relatively compact subset U of G, and set V := U × U × U × U . For σ ∈ S4 , define   Sσ := (x1 , x2 , x3 , x4 ) ∈ V : xσ(1) = xσ(2) and

3.1 Measure Algebras

99

  −1 . Tσ := (x1 , x2 , x3 , x4 ) ∈ V : xσ(1) x−1 x x = e G σ(3) σ(2) σ(4) (Recall that S4 denotes the group of permutations of {1, 2, 3, 4}.) Obviously, Sσ and  Tσ are closed, nowhere dense subsets of V for each σ ∈ S4 . Therefore, {Sσ ∪ Tσ : σ∈ S4 } is also a closed, nowhere dense subset of V , so that W := V \ {Sσ ∪ Tσ : σ ∈ S4 } is a dense, open subset of V . Fix (a1 , a2 , a3 , a4 ) ∈ W . By the definition of V , we have aj = ak for all j, k ∈ [1, 4] ∩ N with j = k. Consequently, we can find r1 ∈ 0, 12 such that the open balls ballr1 (a1 , G), . . . , ballr1 (a4 , G) are pairwise disjoint; making r1 smaller, if necessary, we can also achieve that j

ballr1 (aj , G) ⊂ W.

j=1

For j = 1, 2, 3, 4, now set K1 (j) := Ball r21 (aj , G); then define K1 := 4 j=1 K1 (j). It is clear by construction that the family (K1 (j))j∈[1,4]∩N consists of pairwise disjoint, compact sets, each with nonempty interior, such 4 that K1 = j=1 K1 (j) and diam K1 (j) ≤ 12 for j = 1, 2, 3, 4. Let x1 , x2 , x3 , x4 ∈ K1 be such that there is exactly one point in each of the four sets K1 (1), K1 (2), K1 (3), and K1 (4), and let σ ∈ S4 be such that / Tσ−1 , it follows xσ(j) ∈ K1 (j) for j = 1, 2, 3, 4. As (xσ(1) , xσ(2) , xσ(3) , xσ(4) ) ∈ −1 x x =

e . that x1 x−1 3 G 2 4 Let n ∈ N, and suppose that Kn and (Kn (j))j ∈([1,4]∩N)n have been defined with the required properties. For each j ∈ ([1, 4] ∩ N)n , let U (j) denote the interior of Kn (j), and set V (j) := U (j) × U (j) × U (j) × U (j); then define V :=

{V (j) : j ∈ ([1, 4] ∩ N)n }.

Each element of V is thus a finite family (xj ,k )j ∈([1,4]∩N)n , k∈[1,4]∩N of elements of U (j). For each j ∈ ([1, 4] ∩ N)n and σ ∈ S4 , set   Sj ,σ := (xl,k )l∈([1,4]∩N)n , k∈[1,4]∩N ∈ V : xj ,σ(1) = xj ,σ(2) ; furthermore, for each injective τ : [1, 4] ∩ N → ([1, 4] ∩ N)n+1 , set   −1 . Tτ := (xl,k )l∈([1,4]∩N)n , k∈[1,4]∩N ∈ V : xτ (1) x−1 x x = e G τ (2) τ (3) τ (4) These sets are closed and nowhere dense in V , as thus is their union S, so that W := V \ S is a dense open subset of V . Let (aj ,k )j ∈([1,4]∩N)n , k∈[1,4]∩N ∈ W . For each j ∈ ([1, 4]∩N)n , the points aj ,1 , aj ,2 , aj ,3 , and aj ,4 are distinct points in U (j). Consequently, {aj ,k : j ∈ ([1, 4] ∩ N)n , k ∈ [1, 4] ∩ N} is a collection 1 such that of 4n+1 distinct points in Kn . We can thus choose rn+1 ∈ 0, 2n+1

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the open balls ballrn+1 (aj ,k , G) with j ∈ ([1, 4] ∩ N)n and k ∈ [1, 4] ∩ N are pairwise disjoint and such that moreover ballrn+1 (aj ,k , G) ⊂ U (j)

(j ∈ ([1, 4] ∩ N)n , k ∈ [1, 4] ∩ N)

and

{ballrn+1 (aj ,k , G) : j ∈ ([1, 4] ∩ N)n , k ∈ [1, 4] ∩ N} ⊂ W.

(3.1)

Now, define Kn+1 (j, k) := Ball rn+1 (aj ,k , G) 2

(j ∈ ([1, 4] ∩ N)n , k ∈ [1, 4] ∩ N);

each of these sets is compact, has a nonempty interior, and its diameter does 1 . Setting not exceed 2n+1 Kn+1 :=



{Kn+1 (j, k) : j ∈ ([1, 4] ∩ N)n , k ∈ [1, 4] ∩ N}

completes the construction. To verify that (iii) holds for n + 1, let x1 , x2 , x3 , x4 ∈ Kn+1 belong to four distinct sets of level n + 1. Let τ : [1, 4] ∩ N → ([1, 4] ∩ N)n+1 be injective such that xl ∈ Kn+1 (τ (l)) for l = 1, 2, 3, 4. Let (yj ,k )j ∈([1,4]∩N)n , k∈[1,4]∩N be such that yj ,k = xk if (j, k) = τ (l) and yj ,k = aj ,k otherwise. It follows that yj ,k ∈ ballrn+1 (aj ,k , G) for j ∈ ([1, 4] ∩ N)n and k ∈ [1, 4] ∩ N; by (3.1), this means that (yj ,k )j ∈([1,4]∩N)n , k∈[1,4]∩N ∈ W . In particular, (yj ,k )j ∈([1,4]∩N)n , k∈[1,4]∩N does −1  not lie in Tτ , so that x1 x−1 2 x3 x4 = eG . Lemma 3.1.4. Let G be a non-discrete, metrizable, locally compact group, ∞ be as specified in Lemma 3.1.3. Then K(G) := K let (Kn )∞ n is a n=1 n=1 −1 x x =

e for any four distinct points compact subset of G such that x1 x−1 3 4 G 2 x1 , x2 , x3 , x4 ∈ K. Proof. Obviously, K is compact. Let x1 , x2 , x3 , x4 be four distinct points in K(G), and let δ be the minimum distance between any two of them. Choose n ∈ N so large that 21n < δ. By Lemma 3.1.3(i), this means that those points must lie in four distinct sets of −1  level n. From Lemma 3.1.3(iii), we conclude that x1 x−1 2 x3 x4 = eG . Lemma 3.1.5. Let the setting of Lemma 3.1.4 be given, and let L(G) denote the collection of all sets Kn (j) with n ∈ N and j ∈ ([1, 4] ∩ N)n . Then, for each L ∈ L(G), there is a probability measure μL ∈ Mc (G) such that μL (K(G) ∩ L) = 1

and

μL (G \ (K(G) ∩ L)) = 0.

(3.2)

Proof. Fix L ∈ L(G), say L = Kn0 (j) with n0 ∈ N and j ∈ ([1, 4] ∩ N)n0 . Let μ denote left Haar measure on G. For n ≥ n0 and j ∈ ([1, 4] ∩ N)n such that Kn (j) ⊂ L, let μn,j be the restriction of μ to Kn (j) divided by

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4n−n0 μ(Kn (j))—as Kn (j) has nonempty interior, this is no problem—so that μn,j (Kn (j)) = 4n0 −n holds. Define a sequence (μn )∞ n=n0 of probability measures in M (G) by letting {μn,j : j ∈ ([1, 4] ∩ N)n , Kn (j) ⊂ L} (n ≥ n0 ). μn := This sequence has a weak∗ accumulation point in M (G), say μL . Of course, μL must be positive and have norm at most one. To see that μL is a continuous measure, let x ∈ G; we need to show that / K(G) ∩ L, we can suppose μL ({x}) = 0. As this is clearly the case if x ∈ that x ∈ K(G) ∩ L. For each n ≥ n0 , there is a unique j ∈ ([1, 4] ∩ N)n such that x ∈ Kn (j). Choose fn ∈ C0 (G) such that fn (x) = 1, fn (G) ⊂ [0, 1], and f |Kn (k) ≡ 0 for all k ∈ ([1, 4] ∩ N)n with k = j. For any m ≥ n, we have

fn dμm ≤ μm (Kn (j)) ≤ 4n0 −n G

and thus μL ({x}) ≤

fn dμL ≤ 4n0 −n .

G

As n ≥ n0 was arbitrary, this proves that indeed μL ({x}) = 0. To see that (3.2) holds, let n ≥ n0 , and let U be a relatively compact neighborhood of Kn ∩L. Choose f ∈ C0 (G) with f (G) ⊂ [0, 1] such that f |U ≡ 1. It follows that f, μm  = 1 for m ≥ n, thus f, μL  = 1, and consequently μL (U ) ≥ 1. From the regularity of μL , it follows ∞ that μL (Kn ∩ L) ≥ 1. Since is a decreasing sequence with (Kn ∩ L)∞ n=n0 n=n0 Kn ∩ L = K(G) ∩ L, we obtain (3.2).  Next, we move from metrizable, to general locally compact groups with the help of the following proposition: Proposition 3.1.6. Let G be a non-discrete, locally compact group. Then there are a closed, separable, σ-compact, non-discrete subgroup G0 of G, and a compact, normal subgroup N of G0 such that G0 /N is a non-discrete, metrizable, locally compact group. Proof. Let U be a symmetric, relatively compact neighborhood of eG , and set ∞  G1 := {x1 · · · xn : x1 , . . . , xn ∈ U }. n=1

It is routine to see that G1 is an open—and thus closed—σ-compact subgroup of G (see Exercise 3.1.1 below). As G is not discrete, U must contain infinitely many points. By Exercise 3.1.2 below, there is thus a countably infinite subset D of U that is discrete in the relative topology. Set

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3 Examples

G2 :=

∞ 

{x1 · · · xn : x1 , . . . , xn ∈ D ∪ D−1 },

n=1

which is obviously a subgroup of G. Let G0 be the closure of G2 in G. As G2 is countable, G0 is separable, and since G0 ⊂ G1 , it is also σ-compact. As U is relatively compact, D has at least one cluster point in U \ D; therefore G0 cannot be discrete. Let μ be left Haar measure on G0 . As G0 is not discrete, μ({e}) = 0 must hold. Invoking regularity, there is thus a sequence (Un )∞ n=1 of open (8.7)], neighborhoods of e in G0 such that μ(Un ) → 0. From [178, Theorem ∞ we conclude that there is a compact normal subgroup N ⊂ n=1 Un of G0 such that G0 /N is metrizable. Clearly, μ(N ) = 0, so that N is not open.  Hence, G0 /N is not discrete. Lemma 3.1.7. Let G be a locally compact group, let G0 and N be as in Proposition 3.1.6, let π : G0 → G0 /N denote the quotient map, and let K(G0 /N ) be as in Lemma 3.1.5. Then there is a Borel subset B of G with the following properties: (i) π|B is injective, and π(B) = K(G0 /N ); −1 (ii) x1 x−1 2 x3 x4 = eG for any four distinct points x1 , x2 , x3 , x4 ∈ B; (iii) xB ∩ yB contains at most three elements for any x, y ∈ G with x = y. Proof. To obtain B, first use Exercise 3.1.3 below, to get a compact subset ˜ of G0 with π(K) ˜ = K(G0 /N ). By [262, Appendix C, Theorem], there is K ˜ = K(G0 /N ) such that π|B is a Borel subset B of G0 with π(B) = π(K) injective. Then B satisfies (i). Let x1 , x2 , x3 , and x4 be four distinct points in B. Then π(x1 ), π(x2 ), π(x3 ), and π(x4 ) are distinct points in K(G0 /N ). By Lemma 3.1.4(ii), this means that π(x1 )π(x2 )−1 π(x3 )π(x4 )−1 = eG0 /N and, consequently, −1 x1 x−1 2 x3 x4 = eG , as claimed in (ii). For (iii), let x, y ∈ G with x = y. Without loss of generality suppose that y = eG and that xB ∩ B = ∅. Let x1 ∈ xB ∩ B, so that x1 = xx2 for some x2 ∈ B. As x = eG , we have x1 = x2 . We claim that xB ∩ B ⊂ {x1 , x2 , xx1 }. To see this, we assume toward a contradiction that there is x3 ∈ xB ∩ B / {x1 , x2 , xx1 }. Let x4 ∈ B be such that x3 = xx4 . As x = eG , such that x3 ∈ we have x4 = x3 ; as x3 = xx1 , we have x4 = x1 ; and as x3 = x1 , we also have x4 = x2 . All in all, x1 , x2 , x3 , and x4 are four distinct points of B. By −1 −1 −1 hold, so that x1 x−1 construction, x1 x−1 2 = x and x3 x4 = x 2 x3 x4 = eG , which contradicts (ii).  We can finally put it all together and prove Proposition 3.1.2: Proof. (of Proposition 3.1.2). Let G0 and N be as in Proposition 3.1.6, let π : G0 → G0 /N be the quotient map, and let B be as specified in Lemma 3.1.7.

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103

Let L(G0 /N ) be as in Lemma 3.1.5, and fix L ∈ L(G0 /N ). Define FL ∈ Mc (G)∗ by letting μ, FL  = μ(B ∩ π −1 (L))

(μ ∈ Mc (G));

obviously, FL is non-negative and of norm at most one. We claim that μ ∗ ν, FL  = 0

(μ, ν ∈ Mc (G)).

For arbitrary ν ∈ Mc (G), set   1 Gn (ν) := x ∈ G : |ν|(xB) > n

(3.3)

(n ∈ N).

Assume toward a contradiction that there is n0 ∈ N such that Gn0 is infinite. If so, there is a subset {yn : n ∈ N} of Gn0 (ν) with yn = ym for n, m ∈ N with n = m. For n ∈ N, set Cn := yn B; it follows from Lemma 3.1.7(iii) that Cn ∩ Cm contains at most three elements whenever n, m ∈ N are such that n = m. Set  C := {Cn ∩ Cm : n, m ∈ N, n = m}. Then C is countable, so that |ν|(C) = 0 because ν lies in Mc (G). It follows that 1 (n ∈ N). |ν|(Cn \ C) = |ν|(Cn ) > n0 As (Cn \ C) ∩ (Cm ∩ C) = ∅ for all n, m ∈ N with n = m, we conclude that  n  n  n |ν|(G) ≥ |ν| Cj \ C = |ν|(Ck \ C) ≥ (n ∈ N). n0 k=1

k=1

As |ν|(G) < ∞, this is a contradiction, so that Gn (ν) is finite for each n ∈ N. Set ∞  G(ν) := {x ∈ G : |ν|(xB) > 0} = Gn (ν). n=1

By the foregoing, G(ν) is countable, and by definition, |ν|(xB) = 0 holds for each x ∈ G \ G(ν). To prove (3.3), let μ, ν ∈ Mc (G), and note that

μ ∗ ν, FL  = χB∩π−1 (L) (xy) dμ(x) dν(y)

G G ν(x−1 (B ∩ π −1 (L))) dμ(x). = G

As |ν|(x−1 B) = 0 for all x ∈ G \ G(ν)−1 , we obtain

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3 Examples

      −1 −1 |μ∗ν, FL | =  ν(x (B ∩ π (L))) dμ(x) ≤ ν |μ|(G(ν)−1 ) (3.4)  G(ν)−1  Since G(ν) is countable, so is G(ν)−1 ; by the continuity of μ, this means that |μ|(G(ν)−1 ) = 0. Hence, the right hand side of—and, consequently, each term in—(3.4) is zero. Let μL ∈ Mc (G0 /N ) be as described in Lemma 3.1.5. Slightly abusing notation, denote the inverse of π|B to K(G0 /N ) by π −1 . As π is open, π −1 is continuous. Consequently, π −1 : K(G0 /N ) → B is a homeomorphism by standard topology ([301, Theorem 3.3.11]). For any Borel subset A of G, the set B ∩ A is a Borel subset of G0 , and thus, π(B ∩ A) is a Borel subset of G0 /N by the foregoing. For every Borel subset A of G, we can thus define μ0 (A) := μL (π(B ∩ A)). Clearly, μ0 is a σ-additive set function on the Borel σ-algebra over G— because π|B is injective—which is regular by standard measure theory ([56, Proposition 7.2.3]). It is also obvious that μ0 ∈ Mc (G). Finally, note that μ0 , FL  = μ0 (B ∩ π −1 (L)) = μL (π(B ∩ π −1 (L))) = μL (K(G0 /N ) ∩ L) = 1. Using Theorem D.3.9(ii), we obtain F0 ∈ M (G)∗ of norm one such that  F0 |Mc (G) = FL and F0 |Md (G) ≡ 0. This completes the proof.

Exercises Exercise 3.1.1. Verify that G1 in the proof of Proposition 3.1.6 is indeed an open, σ-compact subgroup of G, as claimed. Exercise 3.1.2. Let X be a completely regular topological space (see [301, Definition 3.5.6], for instance) with infinitely many points. Show that there is a sequence (Un )∞ n=1 of open, nonempty subsets of X such that Un ∩ Um = ∅ for n = m and conclude that X contains a countably infinite subset that is discrete in the relative topology. (Hint: First, show that there is a nonempty, open subset U of X such that X \ U is infinite.) Exercise 3.1.3. Let X and Y be Hausdorff spaces with X locally compact, let π : X → Y be an open and continuous mapping, and let K ⊂ Y be ˜ ⊂ X such that π(K) ˜ = K. compact. Show that there is a compact set K Exercise 3.1.4. Give a direct, i.e., without invoking Theorem 3.1.1, proof of the following claim: Let G be a locally compact group such that M (G) is amenable. Then Gd is amenable.

(Hint: Consider M (G)/Mc (G).)

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105

3.2 Fourier and Fourier–Stieltjes Algebras Let G be a locally compact group such that its Fourier algebra A(G) is amenable. Then A(G) must have a bounded approximate identity by Proposition 2.2.1, so that G is amenable by Leptin’s Theorem. It is thus impossible for a non-amenable group G to have an amenable Fourier algebra. If G is ˆ then the Fourier transform yields an isometabelian with dual group G, ˆ In view of ric isomorphism between the Banach algebras A(G) and L1 (G). Theorem 2.1.10, one is therefore tempted to conjecture that an analog of Theorem 2.1.10 holds for Fourier algebras: A(G) is amenable if and only if G is amenable. As we shall see in this section, the amenability of A(G) imposes far stronger constraints on G than mere amenability. As is customary, we write [G : H] for the (left) index of a subgroup H of a group G, i.e., the number of left cosets of H in G. Definition 3.2.1. Let G be a group. Then we call G almost abelian if it has an abelian subgroup H such that [G : H] < ∞. We will see that the almost abelian locally compact groups are precisely those with an amenable Fourier algebra. One direction is relatively easy: Proposition 3.2.2. Let G be an almost abelian, locally compact group. Then A(G) is amenable. Proof. Let H be an abelian subgroup of G with N := [G : H] < ∞. Replacing H by its closure if necessary, we can suppose without loss of generality that H is closed and thus locally compact itself; note that H has to be open because ˆ it is clear that A(H) is amenable. [G : H] < ∞. As A(H) ∼ = L1 (H), Let x1 , . . . , xN be representatives of the left cosets of H, and suppose without loss of generality that x1 = eG . For j = 1, . . . , N , let Aj (H) := {f ∈ A(G) : supp f ⊂ xj H}; it is straightforward that A1 (H), . . . , AN (H) are closed subalgebras of A(G). Moreover, A1 (H) ∼ = A(H) by Theorem F.2.8(iii), so that A1 (H) is amenable. In view of Remark F.2.7, Aj (H) → A1 (H),

f → Lxj f

(j = 1, . . . , N )

are (isometric) isomorphisms of Banach algebras. Consequently, Aj (H) is amenable for j = 1, . . . , N as is thus A1 (H) ⊕ · · · ⊕ AN (H). We claim that A(G) ∼ = A1 (H) ⊕ · · · ⊕ AN (H). To see this, first note that χH , the indicator function of H, is positive definite ([177, (32.43) Some positive definite functions (a)]) and thus lies in B(G) (see Remark F.3.7). In view of Remark F.3.6, this implies that χxj H = Lx−1 χH ∈ B(G) as well for j j = 2, . . . , N . Consequently,

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3 Examples

A(G) → A1 (H) ⊕ · · · ⊕ AN (H),

f → (χx1 H f, . . . , χxN H f )

(3.5)

is well defined and easily seen to be an isomorphism of Banach algebras. This completes the proof.  For the converse, we need some deeper insight into the structure of almost abelian groups ˆ denote the set of (equivalence classes Let A be a C ∗ -algebra, and let A up to unitary equivalence) of irreducible ∗ -representations of A. There is a ˆ (see [97, Chapter canonical, albeit not necessarily Hausdorff topology on A 3, §1]), the so-called Fell topology. As is customary, if G is a locally compact ∗ (G) (for abelian G, this notation is consistent ˆ instead of C group, we write G ˆ denoting the dual group of G). with G If N ∈ N, we call A N -subhomogeneous if supπ∈Aˆ deg π ≤ N . If A is N -subhomogeneous for some N , we often call A simply subhomogeneous The following is [248, Theorem 1], which we quote without proof: Theorem 3.2.3. The following are equivalent for a locally compact group G: (i) G is almost abelian; (ii) C ∗ (G) is subhomogeneous. Let A be an algebra, and let N ∈ N. We say that A satisfies the standard polynomial identity SN = 0 if (sgn σ)aσ(1) · · · aσ(N ) (a1 , . . . , aN ∈ A); SN (a1 , . . . , aN ) := σ∈SN

here, sgn σ denotes the sign of the permutation σ. The following lemma is well known, but for lack of a suitable reference, we sketch a proof: Lemma 3.2.4. The following are equivalent for a C ∗ -algebra A and N ∈ N: (i) A is N -subhomogeneous; (ii) A satisfies S2N = 0. ˆ As A is N -subhomogeneous, we have π(A) ∼ Proof. (i) =⇒ (ii): Let π ∈ A. = Mn for some n ≤ N . From [286, Theorem 1.4.1], we conclude that 0 = S2N (π(a1 ), . . . , π(a2N )) = π(S2N (a1 , . . . , a2N ))

(a1 , . . . , a2N ∈ A).

By [338, Theorem I.9.23], this means that S2N (a1 , . . . , a2N ) = 0

(a1 , . . . , a2N ∈ A),

i.e., A satisfies S2N = 0. ˆ and suppose that deg π > N , i.e., the corresponding (ii) =⇒ (i): Let π ∈ A, Hilbert space H is of dimension at least N +1. Let P ∈ B(H) be an orthogonal

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107

projection onto an N + 1-dimensional subspace of H. Since π is irreducible, Jacobson’s Density Theorem ([36, Corollary III.25.5]) , yields that P π(A)P = P B(H)P ∼ = MN +1 . As A satisfies S2N = 0, so does P π(A)P and thus MN +1 . This contradicts [286, Theorem 1.4.5].  With the help of Lemma 3.2.4, we can mildly extend Theorem 3.2.3: Proposition 3.2.5. The following are equivalent for a locally compact group: (i) G is almost abelian; (ii) C ∗ (G) is subhomogeneous; (iii) Cr∗ (G) is subhomogeneous; (iv) VN(G) is subhomogeneous. Proof. (i) ⇐⇒ (ii) is Theorem 3.2.3. (ii) =⇒ (iii): If C ∗ (G) is subhomogeneous, then so is each of its quotients, such as Cr∗ (G). (iii) =⇒ (iv): Let N ∈ N be such that Cr∗ (G) is N -subhomogeneous. Then ∗ Cr (G) satisfies S2N = 0 by Lemma 3.2.4, as does its closure in B(L2 (G)) with respect to the weak (or strong) operator topology, i.e., VN(G). Again by Lemma 3.2.4, this means that VN(G) is N -subhomogeneous. (iv) =⇒ (ii): Let N be such that VN(G) is N -subhomogeneous, so that VN(G) satisfies S2N = 0. Consequently, L1 (G)—a subalgebra of Cr∗ (G) via λ—satisfies S2N = 0, too. As C ∗ (G) is just the completion of L1 (G) with respect to another norm, we conclude that C ∗ (G) satisfies S2N = 0 as well and thus is N -subhomogeneous.  We will establish the converse of Proposition 3.2.2 by showing that if a locally compact group G is such that A(G) is amenable, then VN(G) must be subhomogeneous. To this end, we require the theory of operator spaces, as sketched in Appendix E. Given a linear space E, we denote by E its conjugate linear space, i.e., the vector space over C whose additive structure is that of E and whose scalar ¯ for λ ∈ C and x ∈ E. If E and multiplication is defined through λ  x := λx F are vector spaces, then a map T : E → F is linear if and only if T : E → F is conjugate linear. It is easy to verify that, if E is an operator space, then so is E. If A is a C ∗ -algebra, then A → A,

a → a∗

(3.6)

is a linear isometry. In contrast, we have in the operator space category: Lemma 3.2.6. Let N ∈ N, and let A = B(2N ). Then (3.6) has cb-norm N .

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3 Examples

Proof. Let J : 2N → 2N stand for coordinatewise complex conjugation; obviously, J unitary. As B(2N ) can be canonically identified with B(2N ), we obtain a complete isometry B(2N ) → B(2N ),

T → JT J.

(3.7)

For T ∈ B(2N ), let T t ∈ B(2N ) denote its Banach space adjoint. Then B(2N ) → B(2N ),

T → T t

(3.8)

has cb-norm N (Example E.2.5). Since (3.6) is the composition of (3.8) and (3.7), the claim follows.  Proposition 3.2.7. Let A be a C ∗ -algebra such that (3.6) is completely bounded. Then A is subhomogeneous. Proof. Let C denote the cb-norm of (3.6), and assume that A is not subhomogeneous. ˆ such that the corresponding Hilbert space H is of Then there is π ∈ A dimension at least C + 1. Let P ∈ B(H) be an orthogonal projection onto an N -dimensional subspace of H with N > C. By Kadison’s Transitivity Theorem ([251, Theorem 5.2.2]), we have P π(A)P = P B(H)P ∼ = B(P H).

(3.9)

Also, note that A → P π(A)P,

a → P π(a)P

(3.10)

is a complete quotient map. Consider the commutative diagram A

(3.10)

(3.6)

 A

(3.10)

/ P π(A)P ,  / P π(A)P

where the right vertical arrow is P π(A)P → P π(A)P ,

x → x∗ .

(3.11)

Since the left vertical arrow has cb-norm C, and since the horizontal arrows are complete quotient maps, it follows that the right vertical arrow, i.e., (3.11) also has cb-norm C < N . In view of (3.9), this contradicts Lemma 3.2.6.  We now put Proposition 3.2.7 to use in the context of Fourier algebras: Lemma 3.2.8. Let G be a locally compact group such that

3.2 Fourier and Fourier–Stieltjes Algebras

A(G) → A(G),

109

f → fˇ

(3.12)

is completely bounded. Then G is almost abelian. Proof. First note that the adjoint map of (3.12) is simply taking the Banach space adjoint VN(G) → VN(G), x → xt , (3.13) which is also completely bounded by Proposition E.3.6. Letting J : L2 (G) → L2 (G) denote pointwise conjugation, we obtain a complete isometry VN(G) → VN(G),

x → JxJ.

Since Jxt J = x∗ for x ∈ VN(G), we conclude that (3.6)—with A = VN(G)— is completely bounded. By Proposition 3.2.7 this means that VN(G) is subhomogeneous, so that G is almost abelian by Proposition 3.2.5.  We are one more lemma away from a proof of a converse of Proposition 3.2.2. Given any group G, we call the set {(x, x−1 ) : x ∈ G} its anti-diagonal and denote it by ∇. Lemma 3.2.9. Let G be a locally compact group such that A(G) is amenable. Then χ∇ lies in B(Gd × Gd ) with χ∇ B(Gd ×Gd ) ≤ AMA(G) . Proof. Let (dα )α be an approximate diagonal for A(G) bounded by AMA(G) . Being the preadjoint of the invertible isometry (3.13), ∨ : A(G) → A(G) is itself an isometry. Consequently, ((id ⊗ ∨ )dα )α is a net in A(G) ⊗γ A(G) that is also bounded by AMA(G) . We have a canonical contraction from A(G) ⊗γ A(G) into A(G ⊗ G) ⊂ B(G × G) and from there—see Remark F.3.10—into B(Gd × Gd ). We may thus view ((id ⊗ ∨ )dα )α as a net in B(Gd × Gd ) bounded by AMA(G) . From Definition 2.2.2(b), it follows that ((id ⊗ ∨ )dα )α converges to χ∇ pointwise on G × G. By Proposition F.3.8, this means that  χ∇ ∈ B(Gd × Gd ) with χ∇ B(Gd ×Gd ) ≤ AMA(G) . We can finally prove the main result of this section: Theorem 3.2.10. The following are equivalent for a locally compact group G: (i) A(G) is amenable; (ii) G is almost abelian. Proof. In view of Proposition 3.2.2, only (i) =⇒ (ii) needs proof. Suppose that A(G) is amenable. By Lemma 3.2.9, this means that χ∇ ∈ B(Gd ×Gd ), so that ∇ ∈ Ω(Gd ×Gd ) by Theorem F.4.1. Since ∇ is the graph of α : Gd → Gd , x → x−1 ,

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3 Examples

it thus follows from Theorem F.4.5 that α is piecewise affine. Consequently, α∗ : A(Gd ) → B(Gd ) is completely bounded by Proposition F.4.3. But α∗ is just (3.12); hence, Gd is almost abelian. As being almost abelian is a purely algebraic property, this completes the proof.  Corollary 3.2.11. The following are equivalent for a locally compact group G: (i) B(G) is amenable; (ii) G has a compact abelian subgroup of finite index. Proof. (i) =⇒ (ii): It will follow from Proposition 8.3.1 below that A(G) is a closed, complemented ideal of B(G) and thus is amenable by Theorem 2.3.8. By Theorem 3.2.10, G thus has an abelian subgroup H of finite index. Replacing H by its closure, we can suppose that H is closed and thus open. We claim that the restriction map (see Remark F.3.11) B(G) → B(H),

f → f |H

(3.14)

is surjective. Let f ∈ B(H); we need to find f˜ ∈ B(G) such that f˜|H = f . In view of Remark F.3.7, there is no loss of generality if we suppose that f is positive definite. Define an extension f˜ : G → C by setting f˜(x) := 0 for x ∈ G \ H. By [177, (32.43) Some positive definite functions (a)], f˜ is also positive definite and thus lies in B(G). Hence, (3.14) is indeed surjective, and ˆ be the from Proposition 2.3.1, we conclude that B(H) is amenable. Let H ˆ via the Fourier–Stieltjes transform; dual group of H, so that B(H) ∼ = M (H) ˆ is amenable. By Theorem 3.1.1, this means that H ˆ has in particular, M (H) to be discrete, i.e., H is compact. (ii) =⇒ (i): By Proposition 3.2.2, A(G) is amenable. As G has a compact abelian subgroup of finite index, G itself must be compact. In view of Theorem  F.3.4(iv), this means that B(G) = A(G). Theorem 3.2.10 characterizes those locally compact groups G for which AM(A(G)) < ∞. Theorem 2.2.10 states that AM(L1 (G)) < ∞ already implies that AM(L1 (G)) = 1. As we shall now see, the corresponding statement for A(G) fails. We first require a variant of Theorem F.4.1. Proposition 3.2.12. Let G be a locally compact group. Then the following are equivalent for a nonzero idempotent χ ∈ C(G): (i) χ ∈ B(G) with χ = 1; (ii) χ is the indicator function of an open coset in G. Proof. (i) =⇒ (ii): Let π be a unitary representation of G on a Hilbert space H, and let ξ, η ∈ H with ξ = η = 1 such that χ(x) = π(x)ξ|η

(x ∈ G).

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111

Let H ⊂ G be such that χ = χH ; due to the continuity of χ, it is clear that H is open. Replacing χ by a translate, if necessary, we can suppose without loss of generality that eG ∈ H. We shall see that H is a subgroup of G. Since 1 = χ(eG ) = ξ, η, and since ξ = η, the Cauchy–Schwarz inequality immediately yields that ξ = η. Let x ∈ H. Then 1 = π(x)ξ, ξ, and since π(x)ξ = ξ = 1, the Cauchy–Schwarz inequality again yields that π(x)ξ = ξ. On the other hand, if x ∈ G is such that π(x)ξ = ξ, then x ∈ H holds trivially. All in all, we see that H = {x ∈ G : π(x)ξ = ξ}. Clearly, H is a subsemigroup of G. Further, if x ∈ H, then ξ = π(eG )ξ = π(x−1 x)ξ = π(x−1 )π(x)ξ = π(x−1 )ξ, so that x−1 ∈ H as well. All in all, H is a subgroup of G. (ii) =⇒ (i): Let H be an open subgroup of G. Then χH lies in B(G) with χH  = χH (eG ) = 1. Now, let C be an open coset in G, and let x ∈ G and an open subgroup H of G be such that C = xH. By the foregoing χH ∈ B(G) with χH  = 1. As χC = Lx−1 χH , we conclude that χC ∈ B(G)  with χC B(G) = 1 as well. We can now prove: Theorem 3.2.13. The following are equivalent for a locally compact group G: (i) AM(A(G)) = 1; (ii) G is abelian. Proof. (i) =⇒ (ii): By Lemma 3.2.9, χ∇ is a norm one idempotent in B(Gd × Bd ), which means, by Proposition 3.2.12, that ∇ is a coset in Gd × Gd . As ∇ contains the identity of G × G, it follows that ∇ is already a subgroup of G × G. This is possible only if G is abelian. ˆ holds isometrically via the (ii) =⇒ (i): If G is abelian, A(G) ∼ = L1 (G) ˆ = 1 by Theorem 2.2.10, (i) holds.  Fourier transform. As AM(L1 (G))

Exercises Exercise 3.2.1. Show that a group G is almost abelian if and only if it has a normal abelian subgroup N such that [G : N ] < ∞. Exercise 3.2.2. Let G be a locally compact group, and let H be an abelian subgroup of G. Show that AM(A(G)) ≤ [G : H]2 , where ∞2 := ∞.

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3 Examples

3.3 Algebras of Approximable Operators In Example 2.3.17, we saw that K(E) is amenable whenever E = c0 or E = p for p ∈ (1, ∞). A natural question is thus to ask for which Banach spaces E precisely the Banach algebra K(E) is amenable. For every E, perhaps? Given two Banach spaces E and F , we write F(E, F ) for the bounded finite rank operators from E to F and A(E, F ) for their closure in B(E, F ), the space of all bounded linear operators from E to F , to which we refer to as the approximable operators from E to F ; if F = E, we simply write F(E) and A(E), respectively. In this section, we focus on the question for which Banach spaces E the Banach algebra A(E) is amenable (or not). Since amenable Banach algebra have bounded approximate identities by Proposition 2.2.1, we first look at the question of when A(E) has a (bounded) approximate identity. For background on the approximation property and related notions, we refer to Section A.2. Proposition 3.3.1. Let E be a Banach space with the approximation property. Then K(E) has a left approximate identity belonging to F(E). In particular, K(E) = A(E) holds. Proof. Let I := {(K, ) : K ⊂ E is compact and  > 0} be ordered via (K1 , 1 ) ≺ (K2 , 2 )

:⇐⇒

K1 ⊂ K2 and 1 ≥2 ((K1 , 1 ), (K2 , 2 ) ∈ I);

this turns it into a directed set. For each compact subset (K, ) ∈ I, there is SK, ∈ F(E) such that sup{x − SK, x : x ∈ K} < . Since, for any T ∈ K(E), the image of Ball(E) under T is relatively compact, it follows that lim(K,) SK, T − T  = 0, so that (SK, )(K,)∈I is a left approximate identity for K(E).  We don’t know if the converse of Proposition 3.3.1 holds, i.e., if E necessarily has the approximation property when K(E) has a left approximate identity belonging to F(E). For the bounded approximation property, the situation is better understood: Theorem 3.3.2. Let E be a Banach space, and let C ≥ 1. Then the following are equivalent: (i) E has the C-approximation property; (ii) K(E) has a left approximate identity bounded by C and belonging to F(E);

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113

(iii) A(E) has a left approximate identity bounded by C. Proof. (i) =⇒ (ii) is proven like Proposition 3.3.1, and (ii) =⇒ (iii) is trivial. (iii) =⇒ (i): Let E be such that A(E) has a left approximate identity (Sα )α bounded by C. We may suppose that (Sα )α lies in F(E). It is clear that Sα → idE in the strong operator topology, and thus on all finite subsets of E. Since (Sα )α is bounded, it follows that Sα → idE uniformly on all compact subsets of E.  Corollary 3.3.3. The following are equivalent for a Banach space E: (i) E has the bounded approximation property; (ii) A(E) has a bounded left approximate identity. Hence, if we are looking for Banach spaces E such that A(E) is amenable, we can restrict ourselves to spaces with the bounded approximation property. Next, we consider the implications for a Banach space E of the existence of a (right) approximate identity for A(E). For the proof of the next theorem, we introduce the following notation: given two Banach spaces E and F and φ ∈ E ∗ and y ∈ F , we denote the rank one operator E  x → x, φy by y  φ. Also, recall that, following [106], we denote the injective tensor product of Banach space by ⊗λ . Theorem 3.3.4. Let E be a Banach space, and let C ≥ 1. Then the following are equivalent: (i) E ∗ has the C-approximation property; (ii) there is a net (Sα )α in F(E) bounded by C such that limα Sα∗ = idE ∗ uniformly on the compact subsets of E ∗ ; (iii) A(E ∗ ) has a left approximate identity bounded by C; (iv) A(E) has a right approximate identity bounded by C. Proof. (i) =⇒ (ii): Let φ1 , . . . , φn ∈ E ∗ , and let  > 0. It is enough to show that there is S∗ ∈ F(E) with S∗  < C(1 + ) such that φj − S∗∗ φj  <  for j = 1, . . . , n. By (i), there is S ∈ F(E ∗ ) with S < C such that φj −Sφj  <  for  j = 1, . . . , n. Let ψ1 , . . . , ψm ∈ E ∗ and X1 , . . . , Xm ∈ E ∗∗ be such that m S = j=1 ψj  Xj , and let F be the linear span of X1 , . . . , Xm in E ∗∗ . By the Local Reflexivity Principle (Theorem A.4.2), there is τ : F → E such that τ  < 1 +  and τ (X), φj  = φj , X

(X ∈ F, j = 1, . . . , n).

(3.15)

From (3.15), it is clear that φj − S∗∗ φj  = φj − Sφj  < 

(j = 1, . . . , n)

holds. Consider S ∗ an element of F(E ∗∗ , F ), and note that S∗ = τ S ∗ |E . This yields S∗∗  = S∗  ≤ τ S ∗  ≤ τ S < C(1 + ).

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3 Examples

(ii) =⇒ (iv): As in the proof of Proposition 3.3.1, we see that limα Sα∗ T = T for all T ∈ A(E ∗ ). In particular, we have limα Sα∗ T ∗ = T ∗ for all T ∈ A(E). Consequently, limα T Sα = T holds for all T ∈ A(E). (iv) =⇒ (ii): Let (Sα )α be a right approximate identity for A(E) bounded by C. We may suppose that (Sα )α is contained in F(E). Then limα Sα∗ T ∗ = T ∗ holds for all T ∈ A(E). As in the proof of Theorem 3.3.2, one sees that limα Sα∗ = idE ∗ uniformly on the compact subsets of E ∗ . (ii) =⇒ (i) is trivial, and (i) ⇐⇒ (iii) is immediate by Theorem 3.3.2.  Since a Banach space E such that E ∗ has the C-approximation property for some C ≥ 1 has itself the C-approximation property (see Exercise 3.3.1 below), we obtain: Corollary 3.3.5. The following are equivalent for a Banach space E: (i) E ∗ has the bounded approximation property; (ii) A(E) has a bounded right approximate identity; (iii) A(E) has bounded approximate identity. Proof. Suppose that E ∗ has the bounded approximation property. By Theorem 3.3.4, A(E) has a bounded right approximate identity. Since E has the bounded approximation property as well, A(E) also has a bounded left approximate identity by Theorem 3.3.2. From Proposition B.2.3, we obtain a bounded approximate identity.  Hence, for A(E) to be amenable we have to demand at least E ∗ have the bounded approximation property. Example 3.3.6. The Banach space B(2 ) ∼ = (2 ⊗γ 2 )∗ lacks the approximation property ([334]). Consequently, A(2 ⊗γ 2 ) does not have a bounded approximate identity and thus is not amenable. However, it is straightforward to verify that 2 ⊗γ 2 has the 1-approximation property (see Exercise 3.3.2 below), so that, as a consequence of Theorem 3.3.2, A(2 ⊗γ 2 ) has a left approximate identity bounded by 1. We now introduce a rather strong approximation property for a Banach space E which forces A(E) to be amenable. The idea is to construct a bounded approximate diagonal for A(E) by “gluing together” diagonals of a form as in Example 2.2.8. Definition 3.3.7. Let E be a Banach space. A finite, biorthogonal system for E is a set (3.16) {(xj , φk ) : j, k = 1, . . . , N } with x1 , . . . , xN ∈ E and φ1 , . . . , φN ∈ E ∗ such that xj , φk  = δj,k

(j, k = 1, . . . , N ).

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115

Given a Banach space E and a finite, biorthogonal system (3.16), define θ : MN → F(E),

[aj,k ]j,k=1,...,N →

N

aj,k xj  φk ;

j,k=1

it is routinely checked that θ is an algebra homomorphism. Definition 3.3.8. A Banach space E is said to have property (A) if there is a net of finite, biorthogonal systems    (α) (α) xj , φk : j, k = 1, . . . , Nα for E with corresponding algebra homomorphisms θα : MNα → F(E) such that: (a) limα θα (INα ) = idE uniformly on the compact subsets of E; (b) limα θα (INα )∗ = idE ∗ uniformly on the compact subsets of E ∗ ; (c) for each index α, there is a finite subgroup Gα of GL(Nα , C) whose linear span is all of MNα , such that sup sup θα (x) < ∞. α x∈Gα

It is immediate from Definition 3.3.8(b) that the dual of each Banach space with property (A) must have the bounded approximation property (as does, consequently, the space itself). Theorem 3.3.9. Let E be a Banach space with property (A). Then A(E) is amenable. More specifically, if C denotes the supremum in Definition 3.3.8(c), we have AM(A(E)) ≤ C 2 . Proof. With notation as in Definition 3.3.8, set dα :=

1 θα (x) ⊗ θα (x−1 ) |Gα | x∈Gα

for each index α, so that dα  ≤ C 2 . We claim that (dα )α is an approximate diagonal for A(E). For each index α, set Pα := θα (INα ) = ΔA(E) dα . Then (Pα )α is a bounded approximate identity for A(E), and, in particular, we have T ΔA(E) dα → T

(T ∈ A(E)).

Now, let T ∈ A(E) be arbitrary, and note that Pα T Pα · dα = dα · Pα T Pα .

(3.17)

for each index α due to the construction of (dα )α (compare Example 2.2.8). Further note that

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3 Examples

(T − Pα T Pα ) · dα − dα · (T − Pα T Pα ) → 0 because limα Pα T Pα = T and (dα )α is bounded. We thus obtain: T · dα − dα · T = (T − Pα T Pα ) · dα − dα · (T − Pα T Pα ) + Pα T Pα · dα − dα · Pα T Pα = (T − Pα T Pα ) · dα − dα · (T − Pα T Pα ), → 0.

by (3.17)

Therefore, (dα )α is indeed an approximate diagonal for A(E).



It is easy to see that what we did in Example 2.3.17 in order to show that K(E) = A(E) is amenable for E = c0 or E = p with p ∈ (1, ∞) was, in fact, a demonstration that these spaces enjoy property (A). Before we exhibit further examples of spaces with property (A), we first take a look at the hereditary properties of (A): Theorem 3.3.10. Let E be a Banach space such that E ∗ has property (A). Then E has property (A) as well. Proof. Let



(α)

(α)

φj , Xk



 : j, k = 1, . . . , Nα

α∈A

(3.18)

be a net of finite, biorthogonal systems for E ∗ , as required in the definition of property (A). Let F(E) and F(E ∗ ) be the collections of all finite subsets of E and E ∗ , respectively. For each index α, for each F ∈ F(E), and for each Φ ∈ F(E ∗ ), local reflexivity yields a linear map τα,F,Φ from the closed linear span of  (α)

Xj : j = 1, . . . , Nα ∪ F into E, which has norm at most 2, whose restriction to F is the identity on F , and which satisfies       (α) (α) τα,F,Φ Xj , φ = φ, Xj     (α) j = 1, . . . , nα , φ ∈ φj : j = 1, . . . , nα ∪ Φ . Obviously,

   (α) τα,F,Φ (Xj ), φj : j, k = 1, . . . , Nα

(3.19)

is a finite, biorthogonal system for E. Ordering A × F(E) × F(E ∗ ) in the natural fashion, we thus obtain a net of finite, biorthogonal systems for E. Setting Gα,F,Φ := Gα for (α, F, Φ) ∈ A × F(E) × F(E ∗ ), it is clear that this net satisfies Definition 3.3.8(c): the supremum increases at most by a factor 2. For (α, F, Φ) ∈ A × F(E) × F(E ∗ ), let θα : MNα → F(E ∗ ) and θα,F,Φ : MNα → F(E) be the algebra homomorphisms corresponding to (3.18) and (3.19), respectively; set Pα := θα (INα ) and Pα,F,Φ := θα,F,Φ (INα ). The net

3.3 Algebras of Approximable Operators

117

(Pα,F,Φ )α,F,Φ is clearly bounded, and for any F ∈ F(E) and Φ ∈ F(E ∗ ), we have Pα,F,Φ x − x =τα,F,Φ (Pα∗ x) − x

= τα,F,Φ (Pα∗ x − x) ≤ 2Pα∗ x − x

(x ∈ F );

it follows that Definition 3.3.8(a) holds. As ∗ Pα,F,Φ φ=

Nα     (α) (α) τα,F,Φ Xj , φ Xj j=1

=

Nα 

(α)

φ, Xj



(α)

Xj

= Pα∗ φ

(φ ∈ Φ),

j=1



Definition 3.3.8(b) is equally satisfied.

We shall soon encounter a Banach space enjoying (A) whose dual lacks that property. For a discussion of the Radon–Nikod´ ym Property, see Section A.3. Theorem 3.3.11. Let E and F be Banach spaces with property (A) such that: (a) E ∗ or F ∗ has the approximation property; (b) E ∗ or F ∗ has the Radon–Nikod´ym Property. Then E ⊗λ F has property (A) as well. Proof. Let



(α)

(α)



and



(β)

(β)

y j , ψk

 : j, k = 1, . . . , Nα

xj , φk



α∈A

 : j, k = 1, . . . , Nβ

β∈B

be nets of finite, biorthogonal systems for E and F , respectively, as required by Definition 3.3.8. For α ∈ A and β ∈ B, define a finite, biorthogonal system E ⊗λ F :    (α) (α) xj ⊗ yν(β) , φk ⊗ ψμ(β) : j, k = 1, . . . , Nα , ν, μ = 1, . . . , Nβ . Ordering A × B in the canonical way, we thus obtain a net of finite, biorthogonal systems for E ⊗λ F . For each (α, β) ∈ A × B, we canonically identify MNα ⊗MNβ with MNα Nβ . It is then immediate that Definition 3.3.8(a) is satisfied. By Theorem A.3.6 (E ⊗λ F )∗ ∼ = E ∗ ⊗γ F ∗ holds. In particular, E ∗ ⊗ F ∗ λ ∗ is dense in (E ⊗ F ) , so that Definition 3.3.8(b) is satisfied as well. Finally, if Gα and Hβ are finite subgroups of GL(Nα , C) and GL(Nβ , C), respectively, spanning MNα and MNβ , respectively, then Gα × Hβ (with the appropriate

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3 Examples

identifications) is a finite subgroup of GL(Nα Nβ , C) that spans MNα Nβ . It is then straightforward to verify that Definition 3.3.8(c) is also satisfied.  Theorem 3.3.11 is no longer true if we replace the injective tensor product by the projective tensor product, as we will soon see (Remark 3.3.17 below). Example 3.3.12. Let (Ω, S, μ) be a measure space, and let p ∈ (1, ∞). Consider the collection of all families T consisting of finitely many, pairwise disjoint sets in S such that 0 < μ(A) < ∞ for each A ∈ T . Turn this collection into a directed set as follows: for two such families T1 and T2 , set T1 ≺ T2 if each member of T1 is a union of elements of T2 . For each such T , we have a corresponding finite, biorthogonal system for Lp (Ω, S, μ):    1 1 : A, B ∈ T , 1 χA , 1 χB μ(A) p μ(B) q where q ∈ (1, ∞) is dual to p, i.e., p1 + 1q = 1. Let θT : MNT → F(Lp (Ω, S, μ)) be the corresponding algebra homomorphism. It is immediate that θT (NNT )χA = χA

and θT (INT )∗ χA = χA

(A ∈ S, {A} ≺ T ).

This implies that Definition 3.3.8(a) and (b) are satisfied. Let GT be the finite subgroup of GL(NT , C) described in Example 2.2.8. For σ ∈ SNT ,  = (1 , . . . , NT ) ∈ {−1, 1}NT and T = {A1 , . . . , ANT }, we have:  p    

NT   1  j p   θT (D Aσ )f p =  f dμ 1 1 χAσ(j)  μ(Aσ(j) ) q  j=1 μ(Aj ) p Aj  p   p

NT   1   = f dμ p    q Aj j=1 μ(Aj )

NT p 1 q ≤ |f |p dμ, by H¨ older’s Inequality, p μ(Aj ) q μ(A ) A j j j=1 NT |f |p dμ = j=1

= f pp

Aj

(f ∈ Lp (Ω, S, μ)).

It follows that Lp (Ω, S, μ) has property (A), which, by Theorem 3.3.9, entails that A(Lp (Ω, S, μ)) is amenable (in fact, 1-amenable). Example 3.3.13. For any measure space (Ω, S, μ), the space L1 (Ω, S, μ) has property (A): see Exercise 3.3.4 below.

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119

Example 3.3.14. Let X be a locally compact Hausdorff space. By Riesz’ Representation Theorem ([56, Theorem 7.3.6]), the dual space C0 (X)∗ is isometrically isomorphic to the space of all regular, complex Borel measures on X. That space, however, is an L1 -space for a suitable measure space: this follows, for instance, from the Kakutani–Bohnenblust–Nakano Theorem ([4, Theorem 12.26]). Thus, C0 (X)∗ has property (A) by Example 3.3.13 as does C0 (X) by Theorem 3.3.10. Example 3.3.15. Let (Ω, S, μ) be a measure space. Then the Banach space L∞ (Ω, S, μ) equipped with pointwise multiplication is a commutative C ∗ algebra with identity. By the Gelfand–Naimark Theorem, there is thus a compact Hausdorff space K such that L∞ (Ω, S, μ) ∼ = C(K). From Example 3.3.14, it follows that L∞ (Ω, S, μ) has property (A). Example 3.3.16. Let Ω be a locally compact Hausdorff space, and let E be ym a Banach space with property (A) such that E ∗ has the Radon–Nikod´ Property. As C0 (X, E) ∼ = C0 (X) ⊗λ E, it follows from Theorem 3.3.11 that C0 (X, E) has property (A). Remark 3.3.17. By Example 3.3.12, 2 has property (A) as does 2 ⊗λ 2 by Theorem 3.3.11. However, as we have seen in Example 3.3.6, A(2 ⊗γ 2 ) is not amenable, so that (2 ⊗λ 2 )∗ ∼ = 2 ⊗γ 2 cannot have property (A). This shows at the same time that, in Theorem 3.3.10, the rˆ oles of E and E ∗ cannot be interchanged and that, in Theorem 3.3.11, the injective tensor product cannot be replaced by the projective one. Concluding this section, we give an example of a—otherwise very nice— Banach space E such that A(E) is not amenable; in fact, E = p ⊕ q will do if p ∈ (1, ∞) \ {2} and p = q (see Example 3.3.21 below). We first need some purely Banach algebraic prerequisites: Definition 3.3.18. A Banach algebra A is said to have trivial virtual center if, for each X ∈ A∗∗ such that a · X = X · a for all a ∈ A, there is λ ∈ C such that a · X = λa = X · a (a ∈ A). For the definition of the multiplier algebra M(A) of a Banach algebra A—often also referred to as the double centralizer algebra of A—we refer to Section B.3. Proposition 3.3.19. Let A be an amenable Banach algebra, let P1 ∈ M(A) be a projection, let P2 := idA − P1 , and suppose that: (a) both P1 AP1 and P2 AP2 have trivial virtual center; (b) one of P1 AP1 and P2 AP2 is nonzero. Then one of the following holds: (i) ΔA (P1 AP2 ⊗γ P2 AP1 ) = P1 AP1 ;

120

3 Examples

(ii) ΔA (P2 AP1 ⊗γ P1 AP2 ) = P2 AP2 . Proof. For notational simplicity, let Aj,k := Pj APk for j, k = 1, 2. For j, k = 1, 2 with k = j, let A◦j := ΔA (Aj,k ⊗γ Ak,j ) be equipped with the quotient norm, which we denote by  · ◦j ; note that A◦j is an ideal in Aj,j for j = 1, 2. We need to show that there is j ∈ {1, 2} such that Aj,j = A◦j,j . Set E := {a ∈ A : Pj aPj ∈ A◦j , j = 1, 2}. For x ∈ E, define |||x||| := max{P1 xP1 ◦1 , P1 xP2 , P2 xP1 , P2 xP2 ◦2 }. It is routinely checked ||| · ||| is a norm on E turning it into a Banach Abimodule. Define D : A → E, a → P1 aP2 − P2 aP1 , and note that Da =P1 aP2 − P2 aP1 = P1 aP2 + P1 aP1 − (P1 aP1 + P2 aP1 ) = P1 a − aP1

(a ∈ A),

so that D is a derivation. As A is amenable, there is thus X ∈ E ∗∗ such that D = adX . For j, k = 1, 2, set Xj,k := Pj XPk ∈ Pj E ∗∗ Pk , and note that Xj,j ∈ ◦ ∗∗ (Aj ) . If a ∈ Aj,j for some j ∈ {1, 2}, we have Da = 0, i.e., a · X − X · a = 0 and thus a · Xj,j − Xj,j · a = 0. Let ι : E → A be the inclusion map, and let, for j = 1, 2, the restriction of ι to A◦j,j be denoted by ιj ; then ιj : A◦j,j → Aj,j is an Aj,j -bimodule homomorphism for j = 1, 2. As ι restricted to P1 EP2 and P2 EP1 is an isometry on A1,2 and A2,1 , respectively, and since both A1,1 and A2,2 have trivial virtual center, we conclude that there are λ1 , λ2 ∈ C such that ∗∗ a · ι∗∗ j (Xj,j ) = λj a = ιj (Xj,j ) · a

(j = 1, 2, a ∈ Aj,j ).

(3.20)

Without loss of generality suppose that A1,2 = {0}, and choose a ∈ A1,2 \ {0}. As ι restricted to P1 EP2 and P2 EP1 is an isometry onto A1,2 and A2,1 , respectively, ι∗∗ restricted to P1 E ∗∗ P2 and P2 E ∗∗ P1 is an isometry onto A∗∗ 1,2 ∗∗ ∗∗ and A∗∗ 2,1 , respectively. As a · X2,2 ∈ P1 E P2 and X1,1 · a ∈ P2 E P1 , we therefore obtain that a = P1 aP2 = P1 aP2 − P2 aP1 = a · X − X · a ∗∗ = a · X2,2 − X1,1 · a = a · ι∗∗ 2 (X2,2 ) − ι1 (X1,1 ) · a, (3.21) holds in A∗∗ . It is not difficult to see (Exercise 3.3.5 below) that A1,2 is a neo-unital left Banach A1,1 -module as well as a neo-unital right Banach A2,2 -

3.3 Algebras of Approximable Operators

121

module. Hence, there are a ∈ A1,2 , b ∈ A1,1 and c ∈ A2,2 such that a = ba c. By (3.20), we have c · ι∗∗ 2 (X2,2 ) = λ2 c

and

ι∗∗ 1 (X1,1 ) · b = λ1 b.

Hence, (3.21) becomes ∗∗ a = ba c · ι∗∗ 2 (X2,2 ) − ι1 (X1,1 ) · ba c = ba λ2 c − λ1 ba c = (λ2 − λ1 )a.

As a = 0, this mean that λ2 − λ1 = 1; in particular, not both of λ1 and λ2 can be zero. Without loss of generality suppose that λ1 = 0, and set Y := λ−1 1 X1,1 ∈ ◦ (Y ) = a for all a ∈ A . Let (y ) be a bounded—with (A1,1 )∗∗ , so that a·ι∗∗ 1,1 α α 1 respect to  · ◦1 —net in A◦1,1 that converges to Y in the weak∗ topology of (A◦1,1 )∗∗ ; it follows that ayα → a weakly in A◦1,1 for each a ∈ A1,1 . Passing to convex combinations, we obtain a net (eβ )β in A◦1 which is a right approximate identity for A1,1 such that C := supβ eβ ◦1 < ∞. Thus, for each a ∈ A1,1 and for each  > 0, there is b ∈ A◦1 with a − b <  and b◦1 ≤ Ca (just arbitrary, and choose b := aeβ for sufficiently large β). Let a ∈ A1,1 \ {0} be ∞ be a sequence of strictly positive reals such that let (n )∞ n=1 n=1 n < ∞; ◦ for convenience, set 0 := a. Choose b1 ∈ A1,1 with a − b1  < 1 and b1 ◦1 ≤ Ca = C0 . Suppose that b1 , . . . , bn ∈ A◦1,1 have already been chosen such that a − (b1 + · · · + bk ) < k

and

bk ◦1 ≤ Ck−1

(k = 1, . . . , n).

Choose bn+1 ∈ A◦1,1 such that a − (b1 + · · · + bn + bn+1 ) 0 such that, ∞ p q ∞ for each T ∈ A(p ), thereare sequences (R n )n=1 in A( ,  ) and (Sn )n=1 in ∞ ∞ A(q , p ) such that T = n=1 Sn Rn and n=1 Sn Rn  ≤ CT ; without

3.3 Algebras of Approximable Operators

loss of generality, we can suppose that C p T p . Define R : p → q (q ),

123

∞ n=1

Rn q ≤ 1 and

n=1

Sn p ≤

x → (R1 x, R2 x, R3 x, . . .)

and S : q (q ) → p ,

∞

(x1 , x2 , x3 , . . .) →

∞

Sn xn .

n=1

It is routinely checked that R and S are bounded linear operators with R ≤ 1 and S ≤ CT  such that SR = T . As we have isometric isomorphisms q (q ) ∼ = q (N × N) ∼ = q , we conclude that, for each T ∈ A(p ), there are R ∈ A(p , q ) and S ∈ A(q , p ) with SR ≤ CT  and T = SR. Let F be a finite-dimensional subspace of p . For n ∈ N, let Pn ∈ F(p ) be the projection onto the first n coordinates. As (Pn )∞ n=1 converges to id p in the converges to idF in the norm topology strong operator topology, (Pn |F )∞ n=1 of B(F, p ). For sufficiently large N ∈ N, PN |F is thus an isomorphism onto its image such that, say, (PN |F )−1  ≤ 2. Choose R ∈ A(p , q ) and S ∈ A(p , q ) with SR ≤ C and PN = SR, and set τ := R|F . Then τ is an isomorphism onto its image with τ −1 = (PN |F )−1 S, so that τ τ −1  ≤ 2C. This means that p is 2C-finitely representable in q in the sense of Definition A.4.3(a). As p, q ∈ (1, ∞) \ {2} and p = q, this is impossible by Example A.4.10. (Note that, despite failing to be amenable, A(p ⊕ q ) still has a bounded approximate identity.)

Exercises Exercise 3.3.1. Let E be a Banach space, and let C ≥ 1 be such that E ∗ has the C-approximation property. Show that E also has the C-approximation property. Exercise 3.3.2. For j = 1, 2, let Cj ≥ 1, and let Ej be a Banach space with the Cj -approximation property. Show that both E1 ⊗γ E2 and E1 ⊗λ E2 have the C1 C2 -approximation property. Exercise 3.3.3. Let A be a Banach algebra. An element of A ⊗γ A is called symmetric if it is invariant under the flip map A ⊗γ A  a ⊗ b → b ⊗ a. If A has a bounded approximate diagonal consisting of symmetric tensors, it is called symmetrically amenable (see the Notes and Comments at the end of the previous chapter). Show that, if E is a Banach space with property (A), then A(E) is symmetrically amenable.

124

3 Examples

Exercise 3.3.4. Verify Example 3.3.13 by proceeding as follows: (a) Prove the case where μ(Ω) < ∞ along the lines of Example 3.3.12. (b) For each A ∈ S, set A ∩ S := {A ∩ B : B ∈ S}. Deduce, for general (Ω, S, μ), that L1 (Ω, S, μ) has property (A) from the fact that, by (a), L1 (A, A ∩ S, μ|A∩S ) has property (A) for each A ∈ S with μ(A) < ∞. Exercise 3.3.5. Let the hypotheses of Proposition 3.3.19 be given, and let notation be as in its proof. Verify that A1,2 is a neo-unital left Banach A1,1 module and a neo-unital right Banach A2,2 -module. (Hint: Given a bounded approximate identity (eα )α for A, show that (P1 eα P1 )α is a bounded left approximate identity for A1,2 , and apply Cohen’s Factorization Theorem; proceed similarly with (P2 eα P2 )α .) Exercise 3.3.6. Let E be a Banach space. Show that A(E) has trivial virtual center by proceeding as follows: (a) Show that F(E) is simple, i.e., has no (two-sided) ideals other than {0} and itself. (b) Conclude from (a) that A(E) is topologically simple, i.e., has no closed (two-sided) ideals other than {0} and itself. (c) Let X ∈ A(E)∗∗ be such that T · X = X · T for all T ∈ A(E). Pick a rank one projection P on E and show that there is λ ∈ C such that P · X = λP . Then use (b) to conclude that T · X = λT = X · T for all T ∈ A(E).

3.4 (Non-)Amenability of B(E) The following is the main result of [6] and solves the long open “Scalar-plusCompact Problem”: Theorem 3.4.1. There is a Banach space E with E ∗ = 1 such that every bounded linear operator on E is the sum of a compact operator and a scalar multiple of idE . Given this, it is surprisingly easy to come up with an example of an infinitedimensional Banach space E for which B(E) is amenable: Example 3.4.2. Let E be a Banach space as in Theorem 3.4.1. As 1 has property (A), so does E by Theorem 3.3.10. By Theorem 3.3.9, this means that A(E) = K(E) is amenable. Since B(E)/K(E) ∼ = C is trivially amenable, the amenability of B(E) follows from Theorem 2.3.12. What makes Example 3.4.2 work is the fact that the space E in question has very few bounded linear operators beyond the compact ones, so that B(E) is “small” relative to A(E).

3.4 (Non-)Amenability of B(E)

125

The remainder of this section will be devoted to showing that B(E) is not amenable for certain Banach spaces E. We begin with a result that has a surprisingly easy proof. (For any two Banach spaces E and F , we write K(E, F ) for the compact operators from E to F .) Theorem 3.4.3. Let E and F be Banach spaces such that B(E, F ) = K(E, F ) and K(F, E)  B(E, F ). Then B(E ⊕ F ) is not amenable. Proof. The Banach algebra B(E ⊕ F ) has a block matrix structure     B(E) B(F, E) B(E) B(F, E) , B(E ⊕ F ) = = B(E, F ) B(F ) K(E, F ) B(F ) where the second equality is due to the fact that B(E, F ) = K(E, F ); its closed ideal K(E ⊕ F ) has a similar block matrix structure:   K(E) K(F, E) K(E ⊕ F ) = . K(E, F ) K(F ) The quotient algebra B(E ⊕ F )/K(E ⊕ F ) consequently has the block matrix structure   B(E)/K(E) B(F, E)/K(F, E) ; B(E ⊕ F )/K(E ⊕ F ) = 0 B(F )/K(F ) it is straightforward to verify that   0 B(F, E)/K(F, E) I := . 0 0 is a nonzero, closed, complemented ideal of B(E ⊕ F )/K(E ⊕ F ). Assume that B(E ⊕ F ) is amenable, then so is B(E ⊕ F )/K(E ⊕ F ). By Theorem 2.3.8, this means that I has a bounded approximate identity. As I is 2-nilpotent, however, this is impossible.  Corollary 3.4.4. The Banach algebra B(E) is not amenable for the following spaces E: (a) E = p ⊕ q for p, q ∈ [1, ∞) with p = q; (b) E = c0 ⊕ p or E = p ⊕ c0 for p ∈ [1, ∞). Proof. Of course it is enough to consider the cases where E = p ⊕ q with p > q or E = c0 ⊕ p , and in those cases Pitt’s Theorem ([112, Proposition 6.25]) guarantees that the hypotheses of Theorem 3.4.3 are satisfied.  The remainder of this section is devoted to showing that B(p ) is not amenable for any p ∈ [1, ∞]. At the heart of the argument is Lemma 3.4.5 below. We begin with outlining the setup for it.

126

3 Examples

Let Zprime denote the set of prime numbers; fix p ∈ Zprime , so that Z/pZ is a finite field. We define two nonzero points x and y of (Z/pZ)3 to be equivalent if there is λ ∈ Z/pZ such that λx = y. The resulting set of equivalence classes is the projective plane over Z/pZ, which we denote by Pp . The group SL(3, Z) consists of all 3 × 3 matrices with integer entries and with determinant one. It acts on (Z/pZ)3 through matrix multiplication, which induces an action of SL(3, Z) on Pp ; this action, in turn, induces a unitary representation πp of SL(3, Z) on 2 (Pp ). |P |−1 Choose a subset Sp of Pp such that |Sp | = p2 , and define an invertible 2 isometry vp ∈ B( (Pp )) through  ex , x ∈ Sp , vp ex = , / Sp . −ex , x ∈ where (ex )x∈Pp is the canonical basis of 2 (Pp ). The group SL(3, Z) is finitely generated: this follows, e.g., from the fact that it has Kazhdan’s property (T ) (see Proposition 3.4.7 below), but can also be deduced by entirely elementary means ([21, Theorem 4.1.3]). With generators x1 , . . . , xm of SL(3, Z) fixed, we shall write πp (xm+1 ) instead of vp for the sake of notational convenience. (We would like to stress that πp (xm+1 ) does not lie in πp (SL(3, Z)).) We can now formulate: Lemma 3.4.5. It is impossible to find, for each  > 0, a number r ∈ N with the following property: r for each p ∈ Zprime , there are ξ1,p , η1,p , . . . , ξr,p , ηr,p ∈ 2 (Pp ) such that k=1 ξk,p ⊗ ηk,p = 0 and  r      ξk,p ⊗ ηk,p − (πp (xj ) ⊗ πp (xj ))(ξk,p ⊗ ηk,p )    k=1

2 (Pp )⊗γ 2 (Pp )   r     ≤  ξk,p ⊗ ηk,p  (j = 1, . . . , m + 1). (3.23)  

2 (Pp )⊗γ 2 (Pp )

k=1

The proof has two main ingredients. The first is that SL(3, Z) has Kazhdan’s property (T ) ([21, Example 1.7.4]). We state the definition of this property (for discrete groups only): Definition 3.4.6. A discrete group G is said to have Kazhdan’s property (T ) if there are a finite subset F of G and  > 0 such that, for every unitary representation π of G on a Hilbert space H with the property that there is ξ ∈ H \ {0} such that max π(x)ξ − ξ < ξ, x∈F

there is η ∈ H \ {0} with π(x)η = η for x ∈ G.

3.4 (Non-)Amenability of B(E)

127

For a exhaustive treatment of Kazhdan’s property (T ), we refer to [21]. We use it a “black box” for the proof of Lemma 3.4.5 through the following proposition: Proposition 3.4.7. Let G be a group with Kazhdan’s property (T ). Then G is finitely generated, by x1 , . . . , xm , say, and there is a constant C > 0 such that, for each unitary representation π of G on a Hilbert space H and each ξ ∈ H, there is η ∈ H satisfying π(x)η = η for all x ∈ G and ξ − η ≤ C

max π(xj )ξ − ξ.

j=1,...,m

Proof. By [21, Proposition 1.3.2], G is finitely generated; moreover, the set F in Definition 3.4.6 can be chosen to be a set of generators {x1 , . . . , xm }. Pick  > 0 accordingly as in Definition 3.4.6. Set K := {η ∈ H : π(x)η = η for x ∈ G}. It is clear that K is a closed subspace of H and invariant under π(G) as is its orthogonal complement K⊥ . Since G has Kazhdan’s property (T ) and since K ∩ K⊥ = {0}, this means that there must be j ∈ {1, . . . , m} such that π(xj )ξ − ξ ≥ ξ for all ξ ∈ K⊥ \ {0}, i.e., max π(xj )ξ − ξ ≥ ξ

j=1,...,n

(ξ ∈ K⊥ ).

(3.24)

Let P be the orthogonal projection onto K, let ξ ∈ H be arbitrary, and set η := P ξ. In view of (3.24), we obtain ξ − η = (idH − P )ξ ≤ max π(xj )(idH − P )ξ − (id − P )ξ = max π(xj )ξ − ξ. j=1,...,m

Letting C :=

1 

completes the proof.

j=1,...,m



The second ingredient for the proof of Lemma 3.4.5 is the uniform continuity of the noncommutative Mazur map. Given a Hilbert space H, we use T (H) and HS(H) to denote the trace class and the Hilbert—Schmidt operators on H, respectively; the symbol Tr stands for the canonical trace on T (H). The noncommutative Mazur map Φ : T (H) → HS(H) is defined as follows. Given T ∈ T (H) with polar decomposition T = U |T |, the positive operator |T | lies in T (H) as well, so that 1 |T | 2 —obtain through continuous functional calculus—belongs to HS(H). We 1 set Φ(T ) := U |T | 2 . It is clear from this definition that Φ(V T V ∗ ) = V Φ(T )V ∗ for all T ∈ T (H) and all unitary V ∈ B(H). The following is known as the Powers–Størmer Inequality: Lemma 3.4.8 (Powers–Størmer Inequality). Let H be a Hilbert space, and let S, T ∈ T (H) be positive. Then

128

3 Examples

 1  1 2  2 S − T 2 

HS(H)

≤ S − T T (H)

holds. 

Proof. See [276, Lemma 4.1]. We also require: Lemma 3.4.9. Let H be a Hilbert space, and let S, T ∈ T (H). Then |S| − |T |2T (H) ≤ 2S + T T (H) S − T T (H) holds. Proof. This is a special case of the main result of [208].



Proposition 3.4.10. Let H be a Hilbert space. Then Φ : T (H) → HS(H) restricted to the unit sphere of T (H) is uniformly continuous. In particular, if we define ωΦ () := sup{Φ(S) − Φ(T ) : S, T ∈ T (H), ST (H) = 1 = T T (H) , S − T T (H) ≤ }

( > 0),

then lim→0 ωΦ () = 0. Proof. Let S, T ∈ T (H) with ST (H) = 1 = T T (H) with corresponding polar decompositions S = U |S| and T = V |T |, and note that

3.4 (Non-)Amenability of B(E)

129

Φ(S) − Φ(T )2HS(H) = 2(1 − Re Tr(Φ(T )∗ Φ(S)))   1 1 = 2 Tr |S| − V ∗ U |S| 2 |T | 2   1 1  ≤ 2 |S| − V ∗ U |S| 2 |T | 2  T (H)    1 1  ≤ 2 |S| − V ∗ ST (H) + V ∗ S − V ∗ U |S| 2 |T | 2  T (H)     1 1 1   ≤ 2 |S| − V ∗ ST (H) + V ∗ U |S| 2 |S| 2 − |T | 2  T (H)      1 1  12  ≤ 2 |S| − V ∗ ST (H) + V ∗ U |S| 2  |S| − |T | 2  HS(H) HS(H)   1  1   ≤ 2 |S| − V ∗ ST (H) + |S| 2 − |T | 2  HS(H)   1  1  ≤ 2 |S| − |T |T (H) + S − T T (H) + |S| 2 − |T | 2  HS(H)   1 ≤ 2 |S| − |T |T (H) + S − T T (H) + |S| − |T |T2 (H) , 

by Lemma 3.4.8, 1

≤ 2 2S − T T2 (H) + S − T T (H) +

 √ 1 2S − T T4 (H)

by Lemma 3.4.9. This proves the claim about the uniform continuity of Φ. The statement about ωΦ is just a reformulation of it.



Proof. (of Lemma 3.4.5). Assume toward a contradiction that, for each  > 0, there is r ∈  N and, for each p ∈ Zprime , there are ξ1,p , η1,p , . . . , ξr,p , ηr,p ∈ r 2 (Pp ) with k=1 ξk,p ⊗ ηrk,p = 0 such that (3.23) holds. Without loss of generality suppose that k=1 ξk,p ⊗ ηk,p has norm one in 2 (Pp ) ⊗γ 2 (Pp ) Let P denote the disjoint union of {Pp : p ∈ Zprime }. We can identify the projective tensor product 2 (P) ⊗γ 2 (P) with T (2 (P)). With ⊗2 denoting the Hilbert space tensor product, we obtain a similar identification of 2 (P)⊗2 2 (P) with HS(2 (P)). With these identifications, the noncommutative Mazur 2 γ 2 2 2 2 map rbecomes a map Φ :  (P) ⊗  (P) →  (P) ⊗  (P). Setting ξ := Φ ( k=1 ξk,p ⊗ ηk,p ), we obtain a unit vector in 2 (Pp ) ⊗2 2 (Pp ) satisfying ξ − (πp (xj ) ⊗ πp (xj ))ξ 2 (Pp )⊗2 2 (Pp ) ≤ ωΦ ()

(j = 1, . . . , m + 1).

With C > 0 as in Proposition 3.4.7, there is η ∈ 2 (Pp ) ⊗2 2 (Pp ) such that (πp (x) ⊗ πp (x))η = η for all x ∈ SL(3, Z) which satisfies ξ − η 2 (Pp )⊗2 2 (Pp ) ≤ C ωΦ ().

130

3 Examples

The product action of SL(3, Z) on Pp × Pp has two orbits: the diagonal and its complement. Consequently, there are α, β ∈ C such that η = αζ 1 + βζ 2 with 1 ζ 1 = |Pp |− 2 ex ⊗ ex and ζ 2 = |Pp |−1 ex ⊗ ey . x∈Pp

x,y∈Pp

From the definition of πp (xm+1 ), it is clear that (πp (xm+1 )⊗πp (xm+1 ))ζ 1 = 0, so that  1 |β| ≤ |β| 2 − 2|Pp |−2 2 = |β|ζ 2 − (πp (xm+1 ) ⊗ πp (xm+1 ))ζ 2  2 (Pp )⊗2 2 (Pp ) = η − (πp (xm+1 ) ⊗ πp (xm+1 ))η 2 (Pp )⊗2 2 (Pp ) ≤ 2C ωΦ () + ξ − (πp (xm+1 ) ⊗ πp (xm+1 ))ξ 2 (Pp )⊗2 2 (Pp ) ≤ (2C + 1) ωΦ () and thus ξ − αζ 1  2 (Pp )⊗2 2 (Pp ) ≤ ξ − η 2 (Pp )⊗2 2 (Pp ) + |β| ≤ (3C + 1) ωΦ (); (3.25) in particular, |α| ≥ 1 − (3C + 1) ωΦ ()

(3.26)

holds. On the other hand, we may view ξ as a Hilbert–Schmidt operator on 2 (Pp ) 1 of rank at most r, so that |Tr(ξ)| ≤ r 2 . Cauchy–Schwarz then yields 1

1

1

|Pp | 2 ξ − αζ 1  2 (Pp )⊗2 2 (Pp ) ≥ |Tr(αζ 1 − ξ)| ≥ |α||Pp | 2 − r 2 . 1

Dividing by |Pp | 2 and taking (3.25) and (3.26) into account, we obtain  (3C + 1) ωΦ () ≥ ξ − αζ 1  2 (Pp )⊗2 2 (Pp ) ≥ 1 − (3C + 1) ωΦ () − and thus

 2(3C + 1) ωΦ () ≥ 1 −

r |Pp |

r |Pp |

1 2

1 2

.

Since lim→0 ωΦ () = 0 by Proposition 3.4.10, we can choose  > 0 so small that (3C + 1) ωΦ () ≤ 14 , so that 1 ≥1− 2 which is impossible.



r |Pp |

1 2

(p ∈ Zprime ), 

3.4 (Non-)Amenability of B(E)

131

For our next non-amenability result, we require two more lemmas. Lemma 3.4.11. Let p, q ∈ {1, 2, ∞} be such that p1 + 1q = 1, and let (en )∞ n=1 p q and (e∗n )∞ n=1 be the canonical unit vectors in  and  , respectively. Then ∞

Sen  2N T e∗n  2N ≤ N ST 

n=1

(N ∈ N, S ∈ B(p , pN ), T ∈ B(q , qN )) holds. Proof. Fix N ∈ N, S ∈ B(p , pN ), and T ∈ B(q , qN ). If p = 2, we obtain ∞

 Sen  2N T e∗n  2N ≤

n=1

∞

 12  Sen 2 2

N

n=1

 ≤

N ∞

∞

≤

∞ N

=

N

N

 12 

|Sen , e∗k |2

N ∞

 12 |T e∗n , ek |2

n=1 k=1

 12  |en , S ∗ e∗k |2

n=1 k=1



T e∗n 2 2

n=1

n=1 k=1



 12

 12 

S ∗ e∗k 2 2

n=1

∞ N

 12 |e∗n , T ∗ ek |2

n=1 k=1 N

 12

T ∗ ek 2 2

k=1

≤ N ST , which proves the claim in this case. If p = 1, note that ∞ n=1

T e∗n  1 =

N

T ∗ ek  1 ≤ N T 

k=1

so that ∞

Sen  ∞ T e∗n  1N ≤ S N

n=1

and

∞ n=1

It follows that

Sen  1N T e∗n  ∞ ≤ S N

∞

T e∗n  1 ≤ N ST 

n=1 ∞ n=1

T e∗n  1N ≤ N ST .

132

3 Examples ∞

Sen  2N T e∗n  2N

n=1

≤

∞

1

n=1

 ≤

1

(Sen  ∞ T e∗n  1N ) 2 (Sen  1N T e∗n  ∞ )2 N N

∞

 12  Sen  ∞ T e∗n  1N N

n=1

∞

 12 Sen  1N T e∗n  ∞ N

n=1

≤ N T S as claimed. The p = ∞ case follows by symmetry.



We are now ready to prove: Theorem 3.4.12. Let p ∈ {1, 2, ∞}. Then B(p ) is not amenable. Proof. Assume toward a contradiction that B(p ) is amenable. Let P beas in the proof of Lemma 3.4.5; we can identify p with p  (P) = p - p∈Zprime p (Pp ). Let x1 , . . . , xm be generators of SL(3, Z); for each p ∈ Zprime , let the operators πp (x1 ), . . . , πp (xm ), πp (xm+1 ) on p (Pp ) be defined as for Lemma 3.4.5. (In Lemma 3.4.5, these operators were considered only for p = 2, but it is obvious that they can be defined for every  p ∈ [1, ∞].). We can identify ∞ - p∈Zprime B(p (Pp )) with the subalgebra of block diagonal matrices in B(p (P)) corresponding to the decomposition  p p  (P) =  - p∈Zprime p (Pp ); in particular, we can view (πp (xj ))p∈Zprime as an element of B(p (P)) for j = 1, . . . , m + 1. Let  >0, and use Exercise 2.2.3 to obtain  a1 , b1 , . . . , ar , br ∈ B(p (P)) r r such that k=1 ak bk = id p (P) and—with T := k=1 ak ⊗ bk ∈ B(p (P)) ⊗γ p B( (P))— (πp (xj ))p∈Zprime · T − T · (πp (xj ))p∈Zprime 
0. Then there is a Banach FDNC-extension of A that contains a metric, δ-approximate unit for A. We postpone the proof. Definition 3.5.3. Let A be a commutative Banach algebra, let a, u ∈ A be such that u ≤ 1, and let ζ, η > 0. We call d ∈ Ball(A ⊗γ A): (a) a metric, ζ-approximate commutant for a with image u if ΔA d = u and a · d − d · a ≤ ζa; (b) a weak metric, (η, ζ)-approximate commutant for a with image u if ΔA d = u and if there is x ∈ A such that x − a ≤ ηa

and

x · d − d · x ≤ ζa.

There is an analogue of Lemma 3.5.2 for weak metric, approximate commutants: Lemma 3.5.4. Let ! 9 A" be a Banach FDNC-algebra, let u ∈ Ball(A), let ζ > , 1 . Then, for each a ∈ A, there is a Banach FDNC0, and let η ∈ 10 extension B of A such that B⊗γ B contains a weak metric, (η, ζ)-approximate commutant for a with image u. Again, we postpone the proof. Once Lemmas 3.5.2 and 3.5.4 have been established, the remainder of the construction is relatively easy.

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3 Examples

Lemma 3.5.5. " A be a Banach FDNC-algebra, let u ∈ Ball(A), let ζ > 0, ! 9 Let , 1 . Then, for each a ∈ Ball(A) and for each n ∈ N, there and let η ∈ 10 is a Banach FDNC-extension Bn of A such that Bn ⊗γ Bn contains a weak metric, (η n , nζ)-approximate commutant for a with image un . Proof. We proceed by induction on n. For n = 1, the claim is immediate by Lemma 3.5.4. For the induction step, we suppose without loss of generality that a = 1. Let n ∈ N be such that Bn as required has been found. By Definition 3.5.3(b), there are thus x ∈ Bn and dn ∈ Bn ⊗γ Bn with ΔBn dn = un such that and x · dn − dn · x ≤ nζ. x − a ≤ η n Let b := η1n (x − a), so that b ∈ Ball(Bn ). Again by Lemma 3.5.4, there is a Banach FDNC-extension Bn+1 of Bn containing a weak metric, (η, ζ)approximate commutant d ∈ Bn+1 ⊗γ Bn+1 for b with image u; in particular, there is y ∈ Bn+1 such that y − b ≤ η

and

y · d − d · y ≤ ζ.

Set dn+1 := d • dn , so that dn+1 ∈ Ball(Bn+1 ⊗γ Bn+1 ) and ΔBn+1 dn+1 = un+1 . Set z := x − η n y. We obtain z − a = x − a − η n y = η n b − y ≤ η n+1 and z · dn+1 − dn+1 · z ≤ x · dn+1 − dn+1 · x + η n y · dn+1 − dn+1 · η n y ≤ dx · dn − dn · x + η n dn y · d − d · y ≤ x · dn − dn · x + η n y · d − d · y ≤ nζ + η n ζ ≤ (n + 1)ζ, so that dn+1 is a weak metric, (η n+1 , (n + 1)ζ)-approximate commutant for  a with image un+1 . Lemma 3.5.6. Let A be a Banach FDNC-algebra, and let δ > 0. Then there is a Banach FDNC-extension B of A such that: (i) B contains a metric, δ-approximate unit u for A; (ii) B ⊗γ B contains an element that is a metric, δ-approximate commutant with image u for each a ∈ A. and choose Proof. Choose a basis N a1 , . . . , aN of A consisting of unit vectors,  N C > 0 such that j=1 |λj | ≤ Ca for each a ∈ A with a = j=1 λj aj . !9 Further, choose η ∈ 10 , 1 and n ∈ N such that 2η n C ≤ 2δ ; then choose ζ > 0 such that nζC < 2δ .

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137

By Lemma 3.5.2, there is a Banach FDNC-extension B0 of A containing a metric, Nδ2 -approximate unit u0 for A. Then use Lemma 3.5.5, to find a Banach FDNC-extension B1 of B0 such that B1 ⊗γ B1 contains a weak metric, (η n , nζ)-approximate commutant d1 for a1 with image uN 0 . Successively invoking Lemma 3.5.5, we thus obtain Banach FDNC-algebras B1 , . . . , BN such that, for j = 1, . . . , N : • Bj is a Banach FDNC-extension of Bj−1 ; • Bj ⊗γ Bj contains a metric, (η n , nζ)-approximate commutant dj for aj with image uN 0 . Set B := BN , and define d := d1 • · · · • dN ∈ Ball(B ⊗γ B). With 2 u := uN 0 = ΔB d, we have ua − a ≤

2 N −1

j 2 uj+1 0 a − u0 a ≤ N

j=0

δ a = δa N2

(a ∈ A),

so that u is a metric, δ-approximate unit for A. This proves (i). N Let a = j=1 λj aj ∈ A, and note that a·d−d·a=

N

λj (aj · dj − dj · aj ) •



dk .

k =j

j=1

By the choice of C, this implies that a·d−d·a ≤

N

|λj |aj ·dj −dj ·aj  ≤ Ca max aj ·dj −dj ·aj . (3.30) j=1,...,N

j=1

For j = 1, . . . , N , there is xj ∈ Bj ⊂ B such that xj − aj  ≤ η n

and

xj · dj − dj · xj  ≤ nζ.

As (xj − aj ) · dj − dj · (xj − aj ) ≤ 2xj − aj dj  ≤ 2η n

(j = 1, . . . , N ),

it follows that aj · dj − dj · aj  ≤ 2η n + nζ

(j = 1, . . . , N ).

Together with (3.30), this yields a · d − d · a ≤ Ca(2η n + nζ) ≤ δa due to the choices for n, η, and ζ. Since a ∈ A was arbitrary, this proves (ii). 

138

3 Examples

We are now in a position to prove the main theorem of this section: Theorem 3.5.7. Let A be a a Banach FDNC-algebra. Then there is a commutative, radical Banach algebra R with AM(R) = 1 containing A as a closed subalgebra. Proof. Choose a sequence (δn )∞ n=1 of positive numbers such that limn→∞ δn = 0. Applying Lemma 3.5.6, we inductively obtain a sequence (An )∞ n=1 of Banach FDNC-algebras with the following properties: • A1 = A; • for each n ∈ N, An+1 is a Banach FDNC-extension of An ; • for each n ∈ N, there is dn ∈ An+1 ⊗γ An+1 which is a metric, δn approximate commutant for each a ∈ An and such that ΔAn+1 dn is a metric, δn -approximate unit for An . ∞ Let R be the completion of n=1 An . We claim that the sequence (dn )∞ n=1 — or ratherits image in R ⊗γ R—is an approximate diagonal for R. To see this, ∞ let a ∈ n=1 An be arbitrary, and let N ∈ N be such that a ∈ AN . For n ≥ N , we thus have (ΔR dn )a − a ≤ δn a

and

a · dn − dn · a ≤ δn a.

∞ As n=1 An is dense in R, and since (dn )∞ n=1 is bounded (by 1), this this proves the claim. Consequently, R is 1-amenable. We claim that R is radical. As R is commutative, it is sufficient to show that there is no nonzero multiplicative linear functional on R (see Proposition B.4.13). Indeed, by construction, the set of nilpotent elements is dense in R, and as any multiplicative linear functional vanishes on this set, it must vanish on all of R.  Corollary 3.5.8. There is a nonzero, commutative, radical, amenable Banach algebra. What remains to be done, of course, is to prove Lemmas 3.5.2 and 3.5.4; in fact, this is where most of the work for the proof of Theorem 3.5.7 is done. We begin with the proof of Lemma 3.5.2 because it is easier and will thus help to clarify the idea common to the proofs of both Lemma 3.5.2 and Lemma 3.5.4. Proof. (Proof of Lemma 3.5.2). Let the unitization A# of A be equipped with the 1 -norm, i.e., λe + a := |λ| + a

(λ ∈ C, a ∈ A).

(3.31)

Consider the algebra A# [X] of all polynomials in one variable with coefficients in A# . We use A# [X]0 to denote the subalgebra of A# [X] consisting of those polynomials with constant term in A. For N ∈ N, let IN be the principal ideal in A# [X] generated by X N . Obviously, IN is contained in A# [X]0 so

3.5 An Amenable Radical Banach Algebra

139

that B := A# [X]0 /IN is well defined. It is clear from this definition that B is an FDNC-algebra. In fact, if we fix m ∈ N such that am = 0 for all a ∈ A, then bN +m−1 = 0 for all b ∈ B. For a0 , . . . , aN −1 ∈ A# , set    N −1 N −1   j  := a X aj ; (3.32) j     j=0 j=0 this defines an algebra norm on A# [X]/IN ∼ = B# that extends the norm on A# . Set ⎫ ⎧ k ⎬ ⎨1

# (a X − a ) : k ∈ N , a . . . , a ∈ A , a  = · · · = a  = 1 , S := j j 0 1 k 1 k ⎭ ⎩ δk j=1 so that S is a subsemigroup of the multiplicative semigroup of B# containing the identity of B# . with aα ∈ A# for |α| < N 2 m. This norm, in turn, induces a quotient norm on B# , which we will likewise denote by  · . (For the sake of notational convenience, we have suppressed and will continue to suppress for the remainder of this proof the term “+I” when dealing with elements of B# .) With n := dim A, note that a1 · · · anm = 0 for all a1 , . . . , anm ∈ A.  nm−1 It follows that sups∈S s ≤ 2δ . Let  · S be the algebra norm on B# defined as in Exercise 3.5.1 below. Then, for each a ∈ A with a = 1, we have 1δ (aX − a) ∈ S, so that aX − aS ≤ δaS

(a ∈ A).

As XS ≤ X = 1, it follows that X (or rather its coset in B) is a metric, δ-approximate unit for A—if the algebra is equipped with the restriction of  · S to A. Next, we show that, for sufficiently large N , the norm  · S coincides on A with the given norm, so that B becomes a Banach FDNC-extension of A. We do so by dualizing: we show that, for N sufficiently large, each φ ∈ A∗ of norm one has an extension to (B# ,  · S ) of norm at most one. First, extend φ to A# by letting e, φ := 0 and then define φ¯ : B# → C by letting  ¯ := 1 − l a, φ aX l , φ (a ∈ A# , l = 0, . . . , N − 1) N ¯ ≤ 1 for all s ∈ S By Exercise 3.5.1(ii), it is sufficient to show that |bs, φ| and b ∈ B# with b = 1. In view of (3.32), we can limit ourselves to b of the form aX l with a ∈ A# such that a = 1 and l ∈ {0, . . . , N − 1}. The element bs is thus of the form k

1 l aX (aj X − aj ), δk j=1

(3.33)

140

3 Examples

where a1 , . . . , ak ∈ A are such that a1  = · · · = ak  = 1. If k = 0, it is ¯ ≤ 1. We can thus suppose that k∈{1, . . . , nm−1}. straightforward that |bs, φ| Case 1: k + l < N . In this )k case, the highest power of X occurring in bs is N − 1. Setting a ˜ := a j=1 aj , we thus obtain   k 1 ν+l ν k ¯ bs, φ = k ˜ a, φ (−1) 1− ν δ N ν=0

.

(3.34)

k  For k ≥ 1, clearly ν=0 νk (−1)ν = (1 − 1)k = 0 holds, and for k > 1, we  k have ν=0 ν νk (−1)ν = 0 as well (see Exercise 3.5.2 below). Hence, (3.34) becomes  1 a, φ, k = 1, ¯ = δN ˜ bs, φ 0, k > 1. ¯ ≤ 1. For N > 1δ , we therefore obtain |bs, φ| Case 2: k + l ≥ N . Let j ∈ {0, . . . , k}. If j + l ≥ N , we have a ˜X j+l ∈ IN ; if j + l < N , on the other hand, we obtain  j+l nm j+l ¯ |˜ aX , φ| = 1 − |˜ a, φ| ≤ N N ¯ we thus get because k < nm. Using (3.33) to compute bs, φ,  k  2 1 k nm ¯ = |bs, φ| ≤ k δ ν=0 ν N δ

k

nm ≤ N



2 δ

nm

nm , N

where the last inequality holds  ifnmδ ≤ 2, which we can suppose without loss ¯ ≤ 1. of generality. Choosing N > 2δ nm, we see again that |bs, φ|  Proof (of Lemma 3.5.4). Without loss of generality, suppose that a = 1. The idea of the proof is similar to that of the proof of Lemma 3.5.2: equip A# with the norm defined in (3.31), and consider the algebra A# [X1 , . . . , XN ] of polynomials in N variables with coefficients in A# . We write A# [X1 , . . . , XN ]0 for the subalgebra of A# [X1 , . . . , XN ] consisting of those polynomials in A# [X1 , . . . , XN ] with constant term in A. The desired Banach FDNC-extension B of A will be defined as a quotient of A# [X1 , . . . , XN ]0 . We use the customary multiindex notation, i.e., for α = (α1 , . . . , αN ) ∈ αN α := X1α1 · · · XN , write |α| for α1 + · · · + αN , and, for k ∈ NN 0 , we set X  k k! N, denote the multinomial coefficient α1 !···α ; one particular by α1 ,...,α N N multiindex is 1 := (1, . . . , 1). Let I0 be the ideal of A# [X1 , . . . , XN ] generated by the set {X α : |α| ≥ 2 ˜m = 0 N m}, where—as in the proof of Lemma 3.5.2—m ∈ N is such that a

3.5 An Amenable Radical Banach Algebra

141

for each a ˜ ∈ A. Furthermore, let I1 be the principal ideal of A# [X1 , . . . , XN ] generated by X N 1 −u. Then I := I0 +I1 is contained in A# [X1 , . . . , XN ]0 , so that B := A# [X1 , . . . , XN ]0 /I is well defined with B# ∼ = A# [X1 , . . . , XN ]/I. It is straightforward to check that B is an FDNC-algebra. We first extend the norm on A# to A# [X1 , . . . , XN ]/I0 by setting       α  := a X aα  α   |α| m 1 + log 6 5 we obtain from (3.39) that

/ ⎛

⎞k 0 N 1 α ¯ ⎝ ⎠ a ˜ Xj − a X ,φ 2N j=1  k 1 = ˜, φ (−1)k−|β| uν ak+|γ| a (2N )|β| β1 , . . . , βN |β|≤k



= so that

1 −1 2

k

uν ak+|γ| a ˜, φ,

  ⎞k / ⎛ 0  N   1 1   ˜⎝ Xj − a⎠ X α , φ¯  ≤  a   2N j=1 2  

k

≤ ηk .

This, again, establishes (3.37). log 4 Case 3: k ≤ dlog 6 , and α assumes all values not covered under Case 2. 5 Unless every summand in the multinomial expansion ⎞k 0 N 1 α ¯ ⎠ ⎝ Xj − a X ,φ a ˜ 2N j=1  k 1 = ˜X α+β , φ (−1)k−|β| ak−|β| a (2N )|β| β1 , . . . , βN

/ ⎛

|β|≤k

is zero—in which case nothing has to be shown—it follows from the definition of φ¯ that at least for some β ∈ NN 0 with |β| ≤ k, we have α+β = νN   1+γ with 2 0 ≤ ν < m and γ ∈ NN 0 such that |γ| < m. If we choose N > m 1 + 2

log 4 log 65

,

the any two possible values of |α| + |β| must be less than N − m apart from each other. Hence, there is only one possible value 0 ≤ ν < m—independent of β—for which α + β = νN 1 + γ with γ ∈ NN 0 such that |γ| < m. Let β0 be the positive part of νN 1 − α, and note that |β0 | ≥ 1. Then all other multiindices β for which α + β can be expressed in the aforementioned way must satisfy β ≥ β0 . Hence, we obtain

3.5 An Amenable Radical Banach Algebra

145

  ⎞k / ⎛ 0 N   1   ˜⎝ Xj − a⎠ X α , φ¯   a   2N j=1   ≤

β≥β0 , |β|≤k



k−|β0 | k 1 k! (k + 1)! . ≤ Nμ ≤ |β| μ+|β0 | β1 , . . . , βN (2N ) 2N (2N ) μ=0

(3.41) log 4 1 (M +1)! Let M be the integer part of mlog . Then (3.37) 6 , and choose N > η M 2 5 follows from (3.41), which completes the proof of Lemma 3.5.4. 

Exercises Exercise 3.5.1. Let A be a commutative, normed algebra with identity, and let S be a subsemigroup of the multiplicative semigroup of A containing eA such that C := sups∈S s < ∞. For a ∈ A, define aS as ⎧ ⎫ n n ⎨ ⎬ inf aj  : n ∈ N, a1 , . . . , an ∈ A, s1 , . . . , sn ∈ S, a = aj sj . ⎩ ⎭ j=1

j=1

Show that: (i)  · S is a submultiplicative norm on A with sups∈S sS ≤ 1 and satisfying 1 a ≤ aS ≤ a (a ∈ A); C (ii) φ ∈ A∗ has norm at most one with respect to  · S if and only if |as, φ| ≤ 1 for all s ∈ S and a ∈ A with a = 1. Exercise 3.5.2. Show that

k ν=0

ν

k ν

(−1)ν = 0 if k > 1.

Notes and Comments It seems that Theorem 3.1.1 had already been conjectured by B. E. Johnson at the time he wrote [188]. Still, it appears that it was stated in writing for the first time in [215], where also the connected case was proven. The abelian case had been settled much earlier in [38] (even though the term “amenable Banach algebra” does not show up in [38]). The general case was ultimately established by H. G. Dales, F. Ghahramani, and A. Ya. Helemski˘ı ([73]).

146

3 Examples

As stated earlier, it is a tempting conjecture that the Fourier algebra A(G) of a locally compact group G is amenable if and only if G is amenable: indeed, for a long time, this seems to have been a widely held belief among mathematicians working in the field. It was B. E. Johnson who, in [193], showed that there are compact groups G—among them SO(3)—for which A(G) is not amenable. In view of the results from [193], the conjecture came up that A(G) is amenable if and only if G is almost abelian. The “only if” part of Proposition 3.2.2 was proven in [217], but was almost simultaneously discovered by B. E. Forrest even though his findings never made it into publication. Eventually, the conjecture was confirmed by Forrest and the author in [121]; the proof of Theorem 3.2.10 we present is based on [302]. The paper [193] is also the one in which the amenability constant of an amenable Banach algebra is formally defined for the first time. For finite groups G, Johnson derives a remarkable formula:  3 ˆ (deg π) . AM(A(G)) = π∈G 2 ˆ (deg π) π∈G From this formula, Johnson obtains the estimate AM(A(G)) ≥ 32 for any non-abelian, finite group G. For general locally compact groups G, a fairly crude estimate is given in Exercise 3.2.2, but, in fact, a better estimate holds: AM(A(G)) ≤ supπ∈Gˆ deg π ([302]). Theorem 3.2.13 is proven in [302]; Proposition 3.2.12, on which the proof of Theorem 3.2.13 relies, is from [184]. In [122], it is shown that AM(A(G)) ≥ √23 whenever G is a non-abelian, locally compact group. This estimate is not as good as Johnson’s for finite G; its proof makes use of the theory of Schur multipliers. Let p ∈ (1, ∞), and suppose that the Fig` a-Talamanca–Herz algebra Ap (G) is amenable. Then Ap (G) has a bounded approximate identity, so that G must be amenable. In view of Theorem 3.2.10, it is natural to conjecture that Ap (G) is amenable if and only if G is almost abelian, but so far, a proof is lacking. The main obstacle seems to be the lack of an analog of Theorem F.4.1 in a-Talamanca–Herz the Lp -context. Still, Theorem 3.2.13 does extend to Fig` algebras: AM(Ap (G)) = 1 for one—and, equivalently, for all—p ∈ (1, ∞) if and only if G is abelian ([304]). The results we present on bounded approximate identities in Banach algebras of compact or, rather, approximable operators can be found in [23], [54], [99], [154], and [314]; there appear to have been a lot of re- and parallel discoveries on this particular topic. Already in [188], Johnson raised the question if the Banach algebra K(E) was amenable for every Banach space E; he was able to establish the amenability for K(E) for certain spaces E, among them p for p ∈ (1, ∞). About twenty years later—together with N. Grønbæk and G. A. Willis— Johnson revisited the problem in [153]. Our exposition is based on (parts of) [153]. In particular, [153] introduces property (A) as presented in Definition 3.3.8. With an eye on Theorem 3.3.9, it is, of course, tempting to ask if

Notes and Comments

147

A(E) for a Banach space E is amenable if and only if E has property (A). In view of Exercise 3.3.3, however, a positive answer seemed doubtful for several years: it would entail that, if A(E) is amenable, it is already symmetrically amenable. However, A. Blanco was able to prove that, for a Banach space E, the Banach algebra A(E) is amenable if and only if it is symmetrically amenable ([29, Theorem 3.1]). Still, to this day, there seems to be no example of a Banach space E without property (A) for which A(E) is amenable (see [31]). In [31], Blanco and Grønbæk resume the investigation begun in [153] and push it further. In particular, they formulate an intrinsic condition for a Banach space E—formally weaker than property (A)—that is equivalent to the amenability of A(E) ([31, Corollary 3.3]). As Blanco and Grønbæk themselves remark in the introduction of [31], however, that condition is difficult to verify in concrete cases. Already in [153], a Banach space E was identified as a possible candidate of a space lacking property (A), but with A(E) amenable (see also [31, Example 3.6]). However, as Blanco was able to show in [29], this particular space actually enjoys property (A). The question of whether the amenability of A(E) and E having property (A) for a Banach space E are equivalent is thus wide open again. (For a survey that is accurate up to the date of its publication, see [30].) The question of whether the Banach algebra B(E) can be amenable for any infinite-dimensional Banach space E also goes back to Johnson ([188]). For a long time it was generally believed that no such space could exist until the solution to the “Scalar-plus-Compact Problem” ([6]) proved otherwise. In [250], it was shown that, for each countable, compact, metric space Ω, there is a Banach space E with E ∗ = 1 —thus having property (A)—such that B(E)/K(E) ∼ = C(Ω); as K(E) and C(Ω) are amenable, so is B(E). This provides examples of Banach spaces E with B(E) amenable and dim B(E)/K(E) = ∞. On the other hand, it had become clear fairly soon after the publication of [188], that B(H) cannot be amenable for any infinite-dimensional Hilbert space: this follows from the equivalence of amenability and nuclearity for C ∗ -algebras and the fact that nuclear von Neumann algebras have to be subhomogeneous ([352]). (We will discuss this in detail in Chapter 7.) Theorem 3.4.3 is stated in [152], where it is attributed to G. A. Willis. The first value of p other than 2 for which B(p ) was shown to be non-amenable was p = 1: this was accomplished by C. J. Read through ingenious use of random hypergraphs ([280]). Even though [280] appeared in 2006, the manuscript had been in circulation for a couple of years prior to publication. As a result, a simplification of Read’s proof by G. Pisier, in which the random hypergraphs are replaced by expanders ([272]), had already come out by the time Read’s paper finally appeared. In [257], N. Ozawa picked up Pisier’s ideas and simplified the proof even further; in fact, he gave a simultaneous proof for the non-amenability of B(p ) for p = 1, 2, ∞. Even though expanders still play a pivotal rˆ ole in Ozawa’s approach, they do so in the background (via Kazhdan’s property (T ); see [21]): one can understand the proof of the main result

148

3 Examples

of [257] without even knowing what an expander is. Our proof of Theorem 3.4.12 closely follows [257]. Theorem 3.4.13 is from [306], but the proof we present here is different. The proof in [306] is more involved and requires some reduction steps from [86] (see also [87]), but on the other hand, also yields the non-amenability of B(Lp ([0, 1])) for p ∈ (1, ∞). The question of whether a (commutative) radical, amenable Banach algebra exists was apparently raised for the first time by P. C. Curtis, Jr., in 1989. A first example of a radical, amenable Banach algebra was given in [293]. That example is a quotient of L1 (G) for a certain solvable, 4-dimensional Lie group G. Such a quotient algebra is, of course, amenable, but to show that, for certain G, a radical quotient of L1 (G) can be found depends on deep results from abstract harmonic analysis due to J. Boidol, H. Leptin, and D. Poguntke. Another drawback of the example from [293] is that it cannot be commutative. The commutative example we present—the first of its kind—is due to C. J. Read ([279]). Unlike amenable, radical Banach algebras, amenable Banach algebras with nonzero radical are not difficult to obtain. They arise naturally, for instance, as quotients of L1 (G) for locally compact, but not compact, abelian groups G: by Malliavin’s Theorem ([291, 7.6.1, Theorem]), there are sets of nonsynthesis for L1 (G) for such G, i.e., there are closed ideals I of L1 (G) such that rad(L1 (G)/I) = {0}. We would like to mention two more classes of Banach algebras—and the results characterizing their amenable members—without going into the details.

Semigroup algebras The notions of left and right invariant means extend natural from groups to (discrete) semigroups. As in the group case, one can then define left and right amenable semigroups S through the existence of left and right invariant means on ∞ (S); note that, unlike in the group case, left amenability need not necessarily imply right amenability and vice versa. For more on amenable semigroups, see [262, Chapter 1]. If S is a semigroup, the Banach space 1 (S) becomes a Banach algebra through the convolution product f (r)g(s) (f, g ∈ 1 (S), t ∈ S); (f ∗ g)(t) = rs=t

it is not difficult to see that S must be both left and right amenable if 1 (S) is an amenable Banach algebra. The first paper to investigate the consequence of the amenability of 1 (S) for the semigroup S was [100]. An intrinsic characterization of those semigroups S for which 1 (S) is amenable—not nearly as elegant as Theorem 2.1.10—was conjectured in [101] and finally proven by

Notes and Comments

149

Dales, A. T.-M. Lau, and D. Strauss ([74, Theorem 10.12]). As the result is rather technical, we refrain from stating it here.

Weighted group algebras A weight on a locally compact group G is a continuous function ω : G → (0, ∞) satisfying ω(xy) ≤ ω(x)ω(y) (x, y ∈ G). The weighted L1 -algebra L1 (G, ω) consists of those (equivalence classes of) measurable functions f on G such that f ω ∈ L1 (G); equipped with the norm  · ω given by f ω := f ω1 for f ∈ L1 (G) and convolution as product, it is indeed a Banach algebra. In [149]—see [131] for an alternative proof—N. Grønbæk characterized the amenable weighted L1 -algebras. In particular, he showed that, if G is a locally compact group and ω is a weight on G, then L1 (G, ω) is amenable if and only if G is amenable and sup{ω(x)ω(x−1 ) : x ∈ G} < ∞. In fact, in this case, there is an isomorphism L1 (G, ω) ∼ = L1 (G) ([357, Corollary 2]; the hypothesis from [357] that ω(eG ) = 1 is not necessary by [131, Remark 8.13]).

Chapter 4

Amenability-Like Properties

A Banach algebra A is amenable if every derivation D from A into a dual Banach A-bimodule is inner. There are various ways to tweak this definition. What happens if we require derivations from A into all Banach Abimodules—not just the dual ones—to be inner? What happens if we weaken the definition by only demanding that derivations D : A → A∗ be inner? For any class E of Banach A-bimodules, one can define a corresponding amenability-like property in this fashion: require, for every E ∈ E, every derivation D : A → E to be inner. The question here is, of course, whether such a definition is mathematically meaningful: Do we get both strong theorems and interesting examples? We discuss three such amenability-like notions in this chapter: contractibility, weak amenability, and character amenability. There are other ways to generalize the notion of an amenable Banach algebra. What happens if we require derivations from A into dual Banach A-algebra no longer to be inner, but to be approximable—in a sense to be made precise—by inner derivations? Also, A is amenable if and only if it has a bounded approximate diagonal. What happens if we drop the demand that the approximate diagonal be bounded? These two questions lead to the related notions of approximately amenable and pseudo-amenable Banach algebras, respectively, which we also deal with in this chapter. Finally, A is amenable if and only if it has a bounded approximate identity and if a certain short exact sequence splits (Exercise 2.3.9). What happens if we drop the condition that A have a bounded approximate identity? This leads to the class of biflat Banach algebras. A related, but stronger splitting condition characterizes the biprojective Banach algebras: they overlap with, but neither contain nor are contained in the class of amenable Banach algebras. We conclude this chapter with a look at biprojectivity and biflatness.

© Springer Science+Business Media, LLC, part of Springer Nature 2020 V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-1-0716-0351-2 4

151

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4 Amenability-Like Properties

4.1 Contractibility In Definition 2.1.11, we defined the class of amenable Banach algebras in terms derivations into dual Banach bimodules. Consider the following variant: Definition 4.1.1. A Banach algebra A is called contractible if, for every Banach A-bimodule E, every derivation D : A → E is inner. In terms of Hochschild cohomology, a Banach algebra A is contractible if and only if H1 (A, E) = {0} for every Banach A-bimodule. Trivially, contractibility is a stronger condition than amenability. Many of the basic results for amenable Banach algebras have analogues for contractible Banach algebras. The proofs are often similar and, in fact, are even easier in the contractible situation, so that we leave them to the reader: see Exercises 4.1.1 to 4.1.6 below. At first glance, Definition 4.1.1 seems to be more natural than Definition 2.1.11, and one might expect that there is a rich theory of contractible Banach algebras paralleling that of amenable Banach algebras. The problem is the lack of nontrivial examples: to this day, finite direct sums of full matrix algebras (see Exercise 4.1.3 below) are the only known examples of contractible Banach algebras. In fact, if a nontrivial contractible Banach existed, the geometry of its underlying Banach space would have to be somewhat unpleasant: Theorem 4.1.2. Let A be a contractible Banach algebra, and suppose that the underlying Banach space has the approximation property. Then there are N1 , . . . , Nn ∈ N such that A∼ = MN1 ⊕ · · · ⊕ MNn . ∞ Proof. Let d ∈ A ⊗γ A be a diagonal for A. Let (an )∞ n=1 and (bn )n=1 be ∞ sequences in A such that n=1 an bn  < ∞ and

d=

∞ 

an ⊗ bn ;

n=1

∞ without loss of generality, suppose that n=1 an  < ∞ and bn  → 0. Let (Tα )α∈A be a net in F(A) that converges to idA uniformly on the compact subsets of A (see Definition A.2.1(a)). In particular, since {bn : n ∈ N} ∪ {0} is compact, we have limα supn∈N Tα bn − bn  = 0 and, consequently, (idA ⊗ Tα )d − d ≤

∞ 

an Tα bn − bn  → 0

(4.1)

n=1

For α ∈ A, set cα := ΔA ((idA ⊗ Tα )d); in view of (2.8) and (4.1), it is clear that cα → eA . For α ∈ A, define

4.1 Contractibility

153

Sα : A → A,

x →

∞ 

an Tα (bn x);

n=1

it is immediate that Sα is a norm limit of finite rank operators and thus compact. Note that Sα x = ΔA ((idA ⊗ Tα )(d · x))

(x ∈ A, α ∈ A),

so that by (2.9), we have Sα x = ΔA ((idA ⊗ Tα )(x · d)) = xcα

(x ∈ A, α ∈ A).

This, in turn, means that Sα → idA in the operator norm. Therefore, idA is compact and thus dim A < ∞. It follows that rad(A) is complemented and thus has an identity by Exercise 4.1.4(b). As in the proof of Theorem 2.3.25, this implies that A is semisimple. An application of Wedderburn’s Theorem completes the proof.  Corollary 4.1.3. Let G be a locally compact group. Then L1 (G) is contractible if and only if G is finite. A related result is: Theorem 4.1.4. Let A be a contractible Banach algebra, and suppose that A/L has the approximation property for each maximal left ideal L of A. Then there are N1 , . . . , Nn ∈ N such that A∼ = MN1 ⊕ · · · ⊕ MNn . Proof. We first suppose that A is semisimple. As in the proof of Theorem 4.1.4, let d ∈ A ⊗γ A be  a diagonal for A, ∞ ∞ and (b ) be sequences in A such that and let (an )∞ n n=1 n=1 n=1 an bn  < ∞ a ⊗ b . Again, suppose without loss of generality that ∞ and d = n n n=1 ∞ a  < ∞ and b  → 0. n n n=1 Let L be a maximal left ideal of A, let πL : A → A/L be the quotient map and let (Tα )α∈A be a net in F(A/L) that converges to idA/L uniformly on the compact subsets of A/L (see Definition A.2.1(a)). In particular, since {πL (bn ) : n ∈ N}∪{0} is compact in A/L, we have limα supn∈N Tα (πL (bn ))− πL (bn ) = 0 and, consequently, (idA ⊗ Tα )(idA ⊗ πL )d − (idA ⊗ πL )d ∞  ≤ an Tα πL (bn ) − πL (bn ) → 0. n=1

As ΔA ((idA ⊗ πL )d) = πL (eA ) = 0 and thus (idA ⊗ πL )d = 0, this means that (idA ⊗ Tα0 )(idA ⊗ πL )d = 0 for some α0 ∈ A. Let x1 , . . . , xm ∈ A/L and φ1 , . . . ,

154

4 Amenability-Like Properties

φm ∈ (A/L)∗ be such that Tα0 = x1 φ1 + · · · + xm φm . It follows that there is φ ∈ {φ1 , . . . , φm } such that (idA ⊗ φ)(idA ⊗ πL )d = 0. Define ρ : A → A,

∗ x → (idA ⊗ πL φ)(d · x).

From the choice of φ, it is clear that L ⊂ ker ρ whereas ρ(eA ) = 0. By (2.9), we have ∗ ∗ xρ(eA ) = (idA ⊗ πL φ)(x · d) = (idA ⊗ πL φ)(d · x) = 0

(x ∈ L),

so that A is a modular annihilator algebra and, therefore, finite-dimensional by Corollary B.5.8. If A is arbitrary, A/rad(A) is a contractible Banach algebra (see Exercise 4.1.4(a)). Let π : A → A/rad(A) be the quotient map, and L be a maximal left ideal of A/rad(A). Then π −1 (L) is a maximal left ideal of A such that A/π −1 (L) ∼ = (A/rad(A))/L, so that (A/rad(A))/L has the approximation property. Hence, A/rad(A) is semisimple and satisfies the hypotheses of the theorem. From our treatment of the semisimple case, we conclude that dim A/rad(A) < ∞. As in the proof of Theorem 2.3.25, we obtain that rad(A) = {0}. Again, Wedderburn’s Theorem completes the proof.  Corollary 4.1.5. Let A be a contractible, commutative Banach algebra. Then there is N ∈ N such that A ∼ = CN . Corollary 4.1.6. Let A be a contractible C ∗ -algebra. Then there are N1 , . . . , Nn ∈ N such that A∼ = MN1 ⊕∞ · · · ⊕∞ MNn , where ∼ = denotes ∗ -isomorphism. Proof. Let L be a closed maximal left ideal of A. By Theorem C.4.1, there is a pure state φ of A such that L = Lφ = {a ∈ A : a∗ a, φ = 0}; moreover A/L is a Hilbert space and thus has the approximation property. By Theorem 4.1.4, A is thus finite-dimensional, and by Theorem C.4.9, this means that there are N1 , . . . , Nn ∈ N as required. 

Exercises Exercise 4.1.1. Show that every contractible Banach algebra is unital. Exercise 4.1.2. Let A be a contractible Banach algebra. Show that, for every Banach A-bimodule E and each n ∈ N, we have Hn (A, E) = 0.

4.1 Contractibility

155

Exercise 4.1.3. Show that a Banach algebra is contractible if and only if it has a diagonal. Conclude that, for N1 , . . . , Nn ∈ N, the algebra MN1 ⊕ · · · ⊕ MNn is contractible. Exercise 4.1.4. Prove the following hereditary properties for contractibility: (a) If A is a contractible Banach algebra, B is a Banach algebra, and θ : A → B is a homomorphism with dense range, then B is contractible. (b) If A is a contractible Banach algebra, and I is a closed ideal of A, then I is contractible if and only if it has an identity and if and only if it is complemented in A. (c) If A is a Banach algebra and I is a closed ideal of A such that both I and A/I are contractible, then A is contractible. (d) If A and B are contractible Banach algebra, then A ⊗γ B is contractible. Exercise 4.1.5. Let A be a contractible Banach algebra. Show that every admissible, short, exact sequence {0} −→ F −→ E −→ E/F −→ {0} of Banach A-modules (left, right or bi-) splits. Exercise 4.1.6. Let A be a unital Banach algebra. Show that: (a) the short, exact sequence Δ

ˆ −→ A −→ {0} {0} −→ ker Δ −→ A⊗A

(4.2)

of Banach A-bimodules is admissible. (b) the following are equivalent: (i) A is contractible. (ii) (4.2) splits. Exercise 4.1.7. Let G be a locally compact group. Show that L10 (G) has a right identity if and only if G is finite. (Hint: First, show that L1 (G) has an identity, so that G has to be discrete. Then proceed as in the proof of Theorem 1.1.19 to construct a left invariant mean on ∞ (G) that belongs to 1 (G). Finally, apply Exercise 1.1.3.)

4.2 Weak Amenability As the name suggests, weak amenability is weaker than amenability: Definition 4.2.1. A Banach algebra A is called weakly amenable if every derivation D : A → A∗ is inner.

156

4 Amenability-Like Properties

Of course, in terms of Hochschild cohomology Definition 4.2.1 becomes: A is weakly amenable if and only if H1 (A, A∗ ) = {0}. How much weaker than amenability is weak amenability? We start with look at the case of L1 (G) for a locally compact group G. The following lemma is standard, but, nevertheless, we give a proof. Lemma 4.2.2. Let (X, S, μ) be a measure space such that L1 (X, S, μ)∗ = L∞ (X, S, μ), i.e., the canonical isometry from L∞ (X, S, μ) into L1 (X, S, μ) is onto. Then L∞ R (X, S, μ), i.e., the real Banach space of those elements of L∞ (X, S, μ) with real-valued representatives, is order complete, i.e., each nonempty, bounded subset Φ of L∞ R (X, S, μ) has a supremum. Proof. Let L1+ (X, S, μ) denote the cone of those elements of L1 (X, S, μ) that have a positive representative. ∞ Let ∅ = Φ ⊂ L∞ R (X, S, μ) be bounded, i.e., there is φ0 ∈ LR (X, S, μ) 1 such that φ ≤ φ0 for all φ ∈ Φ. For any f, f1 , . . . , fn ∈ L+ (X, S, μ) with f1 + · · · + fn = f and φ1 , . . . , φn ∈ Φ, we have n  j=1

fj , φj  ≤

n 

fj , φ0  = f, φ0 .

j=1

Consequently, the supremum ⎧ n ⎨ sup fj , φj  : n ∈ N, f1 , . . . , fn ∈ L1+ (X, S, μ), ⎩ j=1

 f1 + · · · + fn = f, φ1 , . . . , φn ∈ Φ

(4.3)

is finite for each f ∈ L1+ (X, S, μ). Define ψ : L1+ (X, S, μ) → R by letting f, ψ be the supremum (4.3). It is then immediate from this definition that ψ is positively homogeneous, i.e., tf, ψ = tf, ψ for all f ∈ L1+ (X, S, μ) and t ∈ [0, ∞). Equally obvious is the superadditivity of ψ, i.e., f, ψ + g, ψ ≤ f + g, ψ for f, g ∈ L1+ (X, S, μ). To prove the reversed inequality, let f, g ∈ L1+ (X, S, μ), and let h1 , . . . , hn ∈ L1+ (X, S, μ) be such that h1 + · · · + hn = f + g. For j = 1, . . . , n, define fj ∈ L1+ (X, S, μ) through ⎧ ⎨ f (x)hj (x) , if f (x) + g(x) = 0, (x ∈ X) fj (x) = f (x) + g(x) ⎩ 0, otherwise. Likewise, define g1 , . . . , gn ∈ L1+ (X, S, μ). Then f1 + · · · + fn = f and g1 + · · · + gn = g, so that, for any φ1 , . . . , φn ∈ Φ, we have

4.2

Weak Amenability n 

hj , φj  =

j=1

157 n  j=1

fj , φj  +

n 

gj , φj  ≤ f, ψ + g, ψ.

j=1

It follows that f + g, ψ ≤ f, ψ + g, ψ. All in all, ψ is positively homogeneous and additive on L1+ (X, S, μ) and thus extends to L1R (X, S, μ) as a real-linear functional through f − g, ψ := f, ψ − g, ψ

(f, g ∈ L1+ (X, S, μ)).

The identity L1 (X, S, μ)∗ = L∞ (X, S, μ) then implies that we can identify ψ with an element of L∞ R (X, S, μ); it is easily seen to be required supremum of Φ.  Remark 4.2.3. The hypothesis L1 (X, S, μ)∗ = L∞ (X, S, μ) is not true for every measure space (X, S, μ). It is well known to hold if μ is σ-finite ([56, Theorem 4.5.1]), but it also true if X is a locally compact group, S is its Borel σ-algebra, and μ is Haar measure ([56, Theorem 9.4.8]). Lemma 4.2.4. Let G be a locally compact group, let Φ be a nonempty, bounded subset of L∞ R (G), and let φ := sup Φ. Then we have sup(δx · Φ · δx−1 ) =δx · φ · δx−1 and sup(ψ + Φ) = ψ + φ

(x ∈ G, ψ ∈ L∞ R (G)). 

Proof. Straightforward.

Theorem 4.2.5. Let G be a locally compact group. Then L1 (G) is weakly amenable. Proof. We identify L1 (G)∗ with L∞ (G). Let D : L1 (G) → L∞ (G) be a derivation. Since L1 (G) is neo-unital, D has ˜ : M (G) → L∞ (G) according to Proposition 2.1.6. Note that, an extension D for x, y ∈ G, we have ˜ y ) · δy−1 = (D(δ ˜ x ∗ δx−1 y )) · δy−1 (Dδ ˜ x−1 y + (Dδ ˜ x ) · δx−1 y ) · δy−1 = (δx · Dδ

(4.4)

˜ x−1 y ) · δ(x−1 y)−1 ) · δy−1 + (Dδ ˜ x ) · δx−1 . = δx · ((Dδ ˜ y ) · δy−1 : y ∈ G} is bounded in L∞ (G) by the constant The set Φ := {Re (Dδ R ˜ function D and thus, by Lemma 4.2.2, has a supremum φ1 ∈ L∞ R (G). From (4.4) and Lemma 4.2.4, it is evident that ˜ x ) · δx−1 φ1 = δx · φ1 · δx−1 + Re (Dδ

(x ∈ G)

or, equivalently, ˜ x = φ1 · δ x − δ x · φ1 Re Dδ

(x ∈ G).

(4.5)

158

4 Amenability-Like Properties

Similarly, we obtain φ2 ∈ L∞ R (G) such that ˜ x = φ2 · δ x − δ x · φ2 Im Dδ

(x ∈ G).

Letting φ := −(φ1 + i φ2 ), (4.5) and (4.6) imply that D = adφ .

(4.6) 

Next, we shall focus on C ∗ -algebras. Like group algebras, they will turn out to be weakly amenable. In fact, we shall prove the following, more general result: Theorem 4.2.6. Let A be a C ∗ -algebra, and let I be a closed ideal of A. Then every derivation D : A → I ∗ is inner. Proof. Let D : A → I ∗ be a derivation. We first deal with the case where A is unital. As in Example 2.3.4, consider  f (u)u, θ : 1 (U(A)) → A, f → u∈U(A)

and note that θ is a contractive algebra homomorphism onto A. We can thus turn I—and, consequently, I ∗ —into a (neo-)unital Banach 1 (U(A))bimodule via f · x := θ(f )x and

x · f := xθ(f )

(f ∈ 1 (U(A)), x ∈ I).

˜ := D ◦ θ : 1 (U(A)) → I ∗ is a derivation, and it is obvious that Then D ˜ the hypotheses of Proposition 2.1.2(b) are satisfied, so that D—and, consequently, D—is inner. Let A be non-unital. In this case, we can turn the unitization A# into a C ∗ -algebra (Corollary C.4.7). It is obvious that I is also a closed ideal of I. Extending D from A to A# by setting DeA# := 0, we obtain a derivation from A# to I ∗ , which is inner by the unital case. Hence, D is also inner.  Corollary 4.2.7. Let A be a C ∗ -algebra. Then A is weakly amenable. With Theorem 4.2.5 and Corollary 4.2.7 proven, we shall now focus on Banach algebras that are not weakly amenable. Given a Banach algebra A, and φ ∈ ΦA , we turn C into a Banach Abimodule by letting a · z := a, φz =: z · a

(a ∈ A, z ∈ C);

we denote this module by Cφ . Definition 4.2.8. Let A be a Banach algebra, and let φ ∈ ΦA . Then we call a derivation dφ : A → Cφ a point derivation of A at φ. The relevance of Definition 4.2.8 with regards to Definition 4.2.1 becomes clear in the following proposition:

4.2

Weak Amenability

159

Proposition 4.2.9. Let A be a weakly amenable Banach algebra and let φ ∈ ΦA . Then there are no nonzero point derivations of A at φ. Proof. First note that a · φ, x = xa, φ = x, φa, φ = a, φφ, x

(a, x ∈ A)

and thus a · φ = a, φφ

(a ∈ A).

(4.7)

φ · a = a, φφ

(a ∈ A).

(4.8)

Similarly, one obtains

Let dφ ∈ A∗ be a point derivation at φ, and define D : A → A∗ ,

a → a, dφ φ.

For a, b ∈ A, we obtain D(ab) = ab, dφ φ = (a · b, dφ  + a, dφ  · b)φ = a, φb, dφ φ + a, dφ b, φφ = a · Db + (Da) · b by (4.7), (4.8), and the definition of D. Consequently, D is inner and thus—because a · z = z · a for a ∈ A and  z ∈ C—zero. This, of course, means that dφ = 0. Example 4.2.10. Let D := ball(C). The disc algebra is defined as A(D) := {f ∈ C(D) : f |D is holomorphic}. It is well known that A(D) is a commutative Banach algebra such that ΦA(D) can be canonically identified with D: each z ∈ D corresponds uniquely to a character φz ∈ ΦA(D) via A(D)  f → f (z) ([260, 3.2.13]). For each z ∈ D, the functional A(D) → C, f → f  (z) is a point derivation at φz . Consequently, A(D) is not weakly amenable. We conclude this section with the characterization of those locally compact groups G for which M (G) is weakly amenable. As preparation, we prove two lemmas, the first of which is elementary and entirely Banach algebraic. Lemma 4.2.11. Let A be a Banach algebra, let B and I be a closed subalgebra and a closed ideal of A, respectively, such that A = B  I, let Π : A → B be the projection onto B along I, let φ ∈ ΦB , and let d ∈ A∗ be such that

160

4 Amenability-Like Properties

b + xy, d = 0

(b ∈ B, x, y ∈ I)

(4.9)

and ab, d = b, φa, d = ba, d

(a ∈ A, b ∈ B).

(4.10)

˜ Then φ˜ := φ ◦ Π ∈ ΦA , and d is a point derivation at φ. Proof. It is routinely checked that Π is a surjective homomorphism of Banach algebras; consequently, φ˜ is multiplicative and nonzero. It is equally routine ˜ to verify that d is a point derivation at φ.  Let G be a non-discrete, locally compact group, let A := M (G), B := Md (G), and I := Mc (G). By Proposition 3.1.2, there is d ∈ M (G)∗ satisfying (4.10). To be able to apply Lemma 4.2.11, we need the following: Lemma 4.2.12. Let G be a non-discrete, locally compact group. Then there is d ∈ M (G)∗ \ {0} such that δx + μ ∗ ν, d = 0

(x ∈ G, μ, ν ∈ Mc (G))

(4.11)

and δx ∗ μ, d = μ, d = μ ∗ δx 

(x ∈ G, μ ∈ M (G)).

(4.12)

Proof. First note that M (G)∗ , being the second dual of C0 (G), is a von Neu˜) for a suitable measure mann algebra and thus ∗ -isomorphic to L∞ (X, S, μ ˜) (Theorem C.5.9). We can thus space (X, S, μ ˜) such that M (G) ∼ = L1 (X, S, μ identify the probability measure μ0 ∈ Mc (G) from Proposition 3.1.2 with an ˜) and F0 ∈ M (G)∗ from the same proposition with element f0 of L1 (X, S, μ ∞ ˜); as μ0 and F0 are positive, we do, in fact, have an element φ0 of L (X, S, μ ˜) and φ0 ∈ L∞ ˜). f0 ∈ L1+ (X, S, μ R (X, S, μ ˜), so that, Clearly, Φ := {δx · φ0 · δy : x, y ∈ G} is bounded in L∞ R (X, S, μ ∗ (X, S, μ ˜ ); let d ∈ M (G) be the by Lemma 4.2.11, φ := sup Φ exists in L∞ R linear functional corresponding to φ. It is immediate that μ0 , d = f0 , φ ≥ f0 , φ0  = μ0 , F0  = 1, so that d = 0, and

δ x · d = d = d · δx

(x ∈ G),

so that (4.12) holds. To see that (4.11) holds as well, we need an alternative description of φ. Let F(G) denote the collection of all finite subsets of G ordered by set inclusion; for F ∈ F(G), let φF := sup{δx · φ0 · δy : x, y ∈ F } ∈ L1R (X, S, μ ˜) and let dF ∈ M (G)∗ correspond to φF . Then (φF )F ∈F(G) is a (norm) bounded increasing limit in L∞ (X, S, μ) so that the weak∗ limit of (φF )F ∈F(G) exists; it is immediate that this limit is φ. Consequently, (dF )F ∈F(G) is weak∗ convergent to d.

4.2

Weak Amenability

161

For x ∈ G, note that   μ, δy · F0 · δz  = δzxy , F0  = 0 δx , dF  ≤ y,z∈F

(F ∈ F(G)); (4.13)

x,y∈F

it follows that δx , d = 0. Let μ, ν ∈ Mc (G); we need to show that μ∗ν, d = 0. Jordan’s Decomposition Theorem allows us to express both μ and ν each as a linear combination of four positive elements of M (G), which—by the definition of Mc (G)—must belong to Mc (G) as well. Hence, we may suppose that μ and ν are positive. As a consequence, μ ∗ ν is also positive. An estimate just like (4.13) then yields μ ∗ ν, dF  = 0 for F ∈ F(G), so that μ ∗ ν, d = 0. All in all, (4.11) holds.  We can now prove: Theorem 4.2.13. Let G be a locally compact group. Then the following are equivalent: (i) G is discrete; (ii) M (G) is weakly amenable; (iii) M (G) has no nonzero point derivations. Proof. (i) =⇒ (ii): If G is discrete, then M (G) = 1 (G) = L1 (G) is weakly amenable by Theorem 4.2.5. (ii) =⇒ (iii): This is clear by Proposition 4.2.9. (iii) =⇒ (i): Assume that G is not discrete, and let d ∈ M (G)∗ \ {0} be as specified in Lemma 4.2.12. Set B := Md (G) and I := Mc (G), and define  φ : B → C, μ → μ({x}); x∈G

it is immediate that φ ∈ ΦB . By (4.11) and (4.12), d satisfies (4.9) and (4.10), respectively. By Lemma 4.2.11, d is thus a nonzero point derivation. This contradicts (iii). 

Exercises Exercise 4.2.1. Let M be a von Neumann algebra, and let D : M → M∗ be a derivation. Show that D is inner. Exercise 4.2.2. Let A be a Banach algebra, and let φ ∈ ΦA such that the linear span of {ab : a, b ∈ ker φ} is dense in ker φ. Show that there are no nonzero point derivations at φ.

162

4 Amenability-Like Properties

Exercise 4.2.3. Let z ∈ ∂D. Show that A(D) has no nonzero point derivations at φz . (Hint: Use the Maximum Modulus Principle to show that ker φz has a bounded approximate identity.) Exercise 4.2.4. Let A be a weakly amenable Banach algebra. Show that the closed linear span of {ab : a, b ∈ A} is dense in A. (Hint: Choose φ ∈ A∗ be such that ab, φ = 0 for a, b ∈ A, and consider A  a → a, φφ.) Exercise 4.2.5. We call a Banach bimodule E over a Banach algebra A symmetric if a·x=x·a (a ∈ A, x ∈ E). Show that a commutative Banach algebra A is weakly amenable if and only if, for every symmetric Banach A-bimodule E, every derivation D : A → E is zero. (Hint: Given a symmetric Banach A-bimodule E and a nonzero derivation D : A → E, use Exercise 4.2.4 to construct a bounded homomorphism θ : E → A∗ of Banach A-bimodules such that θ ◦ D is non-zero.)

4.3 Character Amenability We begin with a definition: Definition 4.3.1. Let A be a Banach algebra, and let φ ∈ ΦA ∪ {0}. Then a Banach A-bimodule E is called (a) left φ-linked if a · x := a, φx

(a ∈ A, x ∈ E);

x · a := a, φx

(a ∈ A, x ∈ E).

(b) right φ-linked if

We can use these definitions to define amenability-like properties: Definition 4.3.2. Let A be a Banach algebra, and let φ ∈ ΦA ∪ {0}. Then A is called: (a) left φ-amenable if, for every left φ-linked Banach A-bimodule E every derivation D : A → E ∗ is inner; (b) right φ-amenable if, for every right φ-linked Banach A-bimodule E, every derivation D : A → E ∗ is inner; (c) φ-amenable if it is both left and right φ-amenable. Remark 4.3.3. It is clear from this definition that a Banach algebra A that is left or right φ-amenable for some φ ∈ ΦA cannot have nonzero point derivations at φ.

4.3

Character Amenability

163

If φ = 0, the left or right or simply φ-amenable Banach algebras are easily characterized: Proposition 4.3.4. Let A be a Banach algebra. Then: (i) A is left 0-amenable if and only if A has a bounded right approximate identity; (ii) A is right 0-amenable if and only if A has a bounded left approximate identity; (iii) A is 0-amenable if and only if A has a bounded approximate identity. Proof. The “if” parts of (i) and (ii) are Proposition 2.1.1 (and its obvious analog for Banach algebras with a bounded left approximate identity), and the “only if” parts are established along the lines of the proof of Proposition 2.2.1. By (i) and (ii), A is 0-amenable if and only if it has both a left- and a right-bounded approximate identity, which is the case if and only if A has a bounded approximate identity (see Proposition B.2.3).  For nonzero φ, we have the following (we focus on left φ-amenability; it is obvious that a “right version” also holds): Theorem 4.3.5. The following are equivalent for a Banach algebra A and φ ∈ ΦA : (i) A is left φ-amenable; (ii) there is M ∈ A∗∗ with φ, M  = 1 such that a · M = a, φM

(a ∈ A);

(4.14)

(iii) there is a bounded net (mα )α∈A in A such that a, mα  = 1 for α ∈ A such that (a ∈ A). amα − a, φmα → 0 Proof. (i) =⇒ (ii): The following is easily seen via weak∗ continuity: a · X, φ = a, φφ, X

(a ∈ A, X ∈ A∗∗ ).

(4.15)

Let · denote the canonical A-module actions of A on A∗ . Define new module actions ◦ on A∗ by letting a ◦ ψ := a, φψ

and ψ ◦ a := ψ · a

(a ∈ A, ψ ∈ A∗ ).

This turns A∗ into a left φ-linked Banach A-bimodule. Let M0 ∈ A∗∗ be such that φ, M0  = 1, and define D := adM0 with respect to the module actions of A on A∗∗ dual to ◦. It follows that φ,Da = φ ◦ a − a ◦ φ, M0  = φ · a − φ, aφ, M0  = a · M0 , φ − a, φφ, M0  = 0

(a ∈ A)

164

4 Amenability-Like Properties

where the last inequality is due to (4.15) and the fact that φ, M0  = 1. Hence, D attains its values in A∗∗ /(Cφ)⊥ . As A∗ is left φ-linked, so is its submodule Cφ, so that A∗∗ /(Cφ)⊥ is the dual of a left φ-linked Banach A-bimodule. By the definition of left φ-amenability there is thus M1 ∈ A∗∗ with φ, M1  = 0 and D = adM1 . Set M := M0 − M1 . Then φ, M  = 1 holds, and we have ψ · a, M  = ψ ◦ a, M  = a ◦ ψ, M  = a, φψ, M 

(a ∈ A, ψ ∈ A∗ ).

This proves (ii). (ii) =⇒ (iii): Using the Bipolar Theorem, we can find a net (mα )α∈A bounded by M  such that M = σ(A∗∗ , A∗ )- limα mα . It follows that σ(A,A∗ )

amα − a, φmα −→ 0

(a ∈ A).

(4.16)

Passing to convex combinations as many times before, we can replace the weak limit (4.16) with a norm limit. (iii) =⇒ (i): Let E be a left φ-linked Banach A-bimodule, and let D : A → E ∗ be a derivation, and let ψ ∈ E ∗ be a σ(E ∗ , E) accumulation point of the net (Dmα )α . Passing to a subnet, we can suppose that ψ = σ(E ∗ , E)- limα Dmα . It follows that x, a · ψ = limx, a · Dmα  α

= limx, D(amα ) − (Da) · mα ) α

= lim(x, D(amα − a, φmα ) + D(a, φmα ) − mα · x, Da) α

= lim(a, φx, Dmα  − mα , φx, Da), α

= lima · x, Dmα  − x, Da, α

= x, ψ · a − x, Da

(a ∈ A, x ∈ E),

so that D = ad−ψ .



Remark 4.3.6. In view of the equivalence of (i) and (iii) of Theorem 4.3.5 (and the corresponding “right version”), it is clear that, if A is commutative, it is left φ-amenable if and only if it is right φ-amenable (and thus φ-amenable). Next, we relate (left) φ-amenability for nonzero φ to the existence of certain bounded (right) approximate identities: Proposition 4.3.7. Let A be a Banach algebra, and let φ ∈ ΦA . Then the following are equivalent: (i) A is left φ-amenable and has a bounded right approximate identity; (ii) ker φ has a bounded right approximate identity. Proof. (i) =⇒ (ii): Let (mα )α∈A be a net as specified in Theorem 4.3.5(iii), and let (eβ )β∈B be a bounded right approximate identity for A. As eβ , φ →

4.3

Character Amenability

165

1, we can suppose without loss of generality—by passing to a subnet, if necessary, and replacing eβ by eβ1,φ eβ for β ∈ B—that eβ , φ = 1 for β ∈ B. For α ∈ A and β ∈ B, set fα,β := eβ − mα . It is obvious that the net (fα,β )α,β lies in ker φ, and moreover, lim afα,β = lim(aeβ − amα ) = a − limφ, amα = a α α,β α,β 



(a ∈ ker φ),

=0

i.e., (fα,β )α,β is a bounded right approximate identity for ker φ. (ii) =⇒ (i): Fix x ∈ A such that x, φ = 1, and let (eα )α be a bounded right approximate identity for ker φ. Since a − ax ∈ ker φ for a ∈ A, we obtain that a(x + eα − xeα ) − a = (a − ax)eα − (a − ax) → 0

(a ∈ A),

i.e., (x + eα − xeα )α is a bounded right approximate identity for A. Let M ∈ A∗∗ be a σ(A∗∗ , A∗ ) accumulation point of the bounded net (x − xeα )α , and suppose without loss of generality that M = σ(A∗∗ , A∗ )- limα (x − xeα ). Note that ψ, a · M  = limax − axeα , ψ = 0 α

(a ∈ ker φ, ψ ∈ A∗ ),

so that a · M = 0 for a ∈ ker φ. This means that the map A → A∗∗ ,

a → a · M

(4.17)

has one-dimensional range because it vanishes on ker φ. On the other hand, we have that 0 = lim(x(x + eα − xeα ) − x) α

= σ(A∗∗ , A∗ )- lim x(x − xeα ) − σ(A∗∗ , A∗ )- lim(x − xeα ) α

α

= x · M − M, so that M lies in the range of (4.17), which implies that the range of (4.17) ˜ is CM . Hence, there is φ˜ ∈ A∗ such that a · M = a, φM for a ∈ A. Since φ˜ vanishes on ker φ, it must be a scalar multiple of φ, and since ˜ x, φM = M = x · M = x, φM, we conclude that φ˜ = φ. Therefore, M satisfies (4.14), and Theorem 4.3.5 guarantees the left φ-amenability of A.  For convenience, we define: Definition 4.3.8. Let A be a Banach algebra. Then A is called: (a) left character amenable if it is left φ-amenable for every φ ∈ ΦA ∪ {0};

166

4 Amenability-Like Properties

(b) right character amenable if it is right φ-amenable for every φ ∈ ΦA ∪ {0}; (c) character amenable if it is both left and right character amenable. We conclude this section with a characterization of those locally compact groups G for which L1 (G) and A(G), respectively, are character amenable. Theorem 4.3.9. The following are equivalent for a locally compact group G: (i) G is amenable; (ii) L1 (G) is amenable; (iii) L1 (G) is character amenable; (iv) L1 (G) is 1-amenable (where 1 ∈ L∞ (G) is canonically identified with an element of ΦL1 (G) ); (v) L1 (G) is left or right 1-amenable; (vi) there is φ ∈ ΦL1 (G) such that L1 (G) is left or right φ-amenable. Proof. (i) =⇒ (ii) is one implication of Theorem 2.1.10, and (ii) =⇒ (iii) =⇒ (iv) =⇒ (v) =⇒ (vi) are obvious. Also, (v) =⇒ (i)—in the left 1-amenable case—follows from Theorems 1.1.13 and 4.3.5; the right 1-amenable case is established analogously. (vi) =⇒ (v): Let φ ∈ ΦL1 (G) be such that L1 (G) is left φ-amenable. (The right φ-amenable case is completely analogous.) Fix g ∈ L1 (G) such that g, φ = 1, and define γ : G → C by letting γ(x) := δx ∗ g, φ for x ∈ G. Then γ is bounded and continuous by Proposition D.2.1 such that f (x)γ(x) dx G

 = f (x)Lx−1 g, φ dx = f (x)Lx−1 g dx, φ G

G

= f ∗ g, φ = f, φg, φ = f, φ

(f ∈ L1 (G)),

where the last integral is an L1 (G)-valued Bochner integral (see [56, Appendix E]); in particular, γ ∈ L∞ (G) implements φ via the canonical duality between L1 (G) and L∞ (G). It is obvious that γ(eG ) = 1. Let (eα )α be a bounded approximate identity for L1 (G), and note that δx ∗ (g ∗ δy ∗ g − δy ∗ g), φ = limδx ∗ eα ∗ (g ∗ δy ∗ g − δy ∗ g), φ α

= limδx ∗ eα , φg ∗ δy ∗ g − δy ∗ g, φ = 0 α

(x, y ∈ G).

We conclude that γ(x)γ(y) = δx ∗ g, φδy ∗ g, φ = δx ∗ g ∗ δy ∗ g, φ = δxy ∗ g + δx ∗ (g ∗ δy ∗ g − δy ∗ g), φ = δxy ∗ g, φ = γ(xy)

(x, y ∈ G).

4.3

Character Amenability

167

Hence, γ is a continuous group homomorphism into the multiplicative group C \ {0}, and since γ is bounded, it follows that γ(G) is contained in the unit circle. Let θγ : L1 (G) → L1 (G) be pointwise multiplication with γ. As γ is a group homomorphism, it is routinely checked θγ is an algebra homomorphism and, in fact, an automorphism. It is immediate that θ∗ 1 = φ. The left 1 amenability of L1 (G) thus follows from Exercise 4.3.2 below. In view of Theorems 3.2.10, the following analog of Theorem 4.3.9 for Fourier algebras may come as a surprise: Theorem 4.3.10. The following are equivalent for a locally compact group G: (i) G is amenable; (ii) A(G) is 0-amenable; (iii) A(G) is character amenable. Proof. Note that, due to the commutativity of A(G), we need not distinguish between left and right φ-amenability (see Remark 4.3.6). (iii) =⇒ (ii) is trivial, and in view of Proposition 4.3.4 and Theorem 1.4.1, (ii) =⇒ (i) is also clear. (i) =⇒ (iii): Let φ ∈ ΦA(G) ∪ {0}. If φ = 0, then again Theorem 1.4.1 and Proposition 4.3.4 immediately establish the φ-amenability of A(G). Suppose therefore that φ ∈ ΦA(G) . By Theorem F.2.2(ii), there is a unique x ∈ G such that φ = λ(x), i.e., f, φ = f (x) for f ∈ A(G). Set S := {f ∈ A(G) : f  ≤ 1, f (x) = 1}. We claim that S is not empty. To see this, let ξ ∈ L2 (G) be unit vector, and let g : G → C, y → λ(y)λ(x−1 )ξ|ξ. It is immediate that g ∈ A(G) with g ≤ 1 and g(e) = ξ22 = 1, i.e., g ∈ S. It is straightforward that S is an abelian semigroup with respect to pointwise multiplication in A(G). Set C := {F ∈ A(G)∗∗ : F  ≤ 1, φ, F  = 1}. Then C is a convex, weak∗ compact subset of A(G)∗∗ containing S. Moreover, S acts affinely on C via the canonical module action of A(G) on A(G)∗∗ . By the Markov–Kakutani Fixed Point Theorem ([102, Theorem V.10.6]), there is thus M ∈ C such that f ·M =M

(f ∈ S).

Since A(G) is the linar span of S (see Exercise 4.3.7 below), this means that M is as required by Theorem 4.3.5(ii). Hence, A(G) is φ-amenable. 

168

4 Amenability-Like Properties

Exercises Exercise 4.3.1. Let A be a Banach algebra, and let φ ∈ Φ. Show that A is φ-amenable if and only if it is both left and right φ-amenable. (Hint: Look at the proof of Proposition 1.1.9.) Exercise 4.3.2. Let A and B be Banach algebra, let φ ∈ ΦB , and let θ : A → B be an algebra homomorphism with dense range. Then, if A is left θ∗ φ-amenable, B is left φ-amenable. Exercise 4.3.3. Let A and B be Banach algebras, and let φ ∈ ΦA and ψ ∈ ΦB such that A is left φ-amenable and B is left ψ-amenable. Show that A ⊗γ B is left φ ⊗ ψ-amenable. Exercise 4.3.4. Let A be a Banach algebra, let φ ∈ ΦA such that A is left φ-amenable, and let I be a closed ideal of A such that φ|I = 0. Show that I is left φ|I -amenable. Exercise 4.3.5. Let G be a locally compact group. Show that the following are equivalent: (i) G is amenable; (ii) for each x ∈ G, the ideal {f ∈ A(G) : f (x) = 0} has a bounded approximate identity; (iii) the ideal {f ∈ A(G) : f (eG ) = 0} has a bounded approximate identity; (iv) there is x ∈ G such that the ideal {f ∈ A(G) : f (x) = 0} has a bounded approximate identity. Also, show that the bounded approximate identities in (ii), (iii), and (iv) can be chosen to be of bound at most 2. Exercise 4.3.6. Give an example of a commutative Banach algebra A which is φ-amenable for every φ ∈ ΦA , but not character amenable. Exercise 4.3.7. Let G be a locally compact group, let x ∈ G, and let S := {f ∈ A(G) : f  ≤ 1, f (x) = 1}. Show that A(G) is the linear span of S.

4.4 Pseudo- and Approximate Amenability In Theorem 2.2.5, we characterized the amenable Banach algebras through the existence of bounded approximate diagonals. Slightly tweaking this characterization, we define: Definition 4.4.1. A Banach algebra A is called pseudo-amenable if there is an approximate diagonal for A.

4.4

Pseudo- and Approximate Amenability

169

Are there pseudo-amenable Banach algebras that fail to be amenable? The question is surprisingly easy to answer. A commutative Banach algebra A is called Tauberian if {a ∈ A : supp a ˆ is compact} is dense in A. We have: Proposition 4.4.2. Let A be a semisimple, commutative, Tauberian Banach algebra with an approximate identity such that ΦA is discrete. Then A is pseudo-amenable. Proof. Let (eα )α∈A be an approximate identity for A; as A is Tauberian, there is no loss of generality to suppose that supp eˆα is compact, i.e., finite, for ˇ Idempotent Theorem ([71, Theorem each α ∈ A. Since ΦA is discrete, Silov’s 2.4.33]) yields for each φ ∈ ΦA an idempotent pφ ∈ A such that pˆφ = χ{φ} . We define a net (dα )α∈A in A ⊗ A by letting  dα := eˆα (φ)pφ ⊗ pφ (α ∈ A). φ∈supp eˆα

We claim that (dα )α∈A is an approximate diagonal for A. pφ = a ˆ(φ)ˆ pφ To see this, first note that, for a ∈ A and φ ∈ ΦA , we have a and so, by semisimplicity, apφ = a ˆ(φ)pφ . Hence, we have  a · dα = eˆα (φ)apφ ⊗ pφ φ∈supp eˆα



=

eˆα (φ)ˆ a(φ)pφ ⊗ pφ

φ∈supp eˆα



=

eˆα (φ)pφ ⊗ a ˆ(φ)pφ

φ∈supp eˆα



=

eˆα (φ)pφ ⊗ pφ a

φ∈supp eˆα

= dα · a

(a ∈ A, α ∈ A).

Furthermore,  Δ A dα =



eˆα (φ)χ{φ} = eˆα

(α ∈ A)

φ∈supp eˆα

holds, and therefore, again by semisimplicity, ΔA dα = eα for α ∈ A. As (eα )α∈A is an approximate identity for A this means that aΔA dα → a for a ∈ A. Hence, (dα )α∈A is indeed an approximate diagonal for A.  Proposition 4.4.2 immediately provides many natural examples of nonamenable, but pseudo-amenable Banach algebras: Example 4.4.3. Let p ∈ [1, ∞), and let p be equipped with coordinatewise multiplication. Then Φ p is canonically identified with N, so that Φ p is discrete. Moreover, p is obviously Tauberian and has an approximate identity,

170

4 Amenability-Like Properties

p namely (χ{1,...,n} )∞ n=1 . Hence,  is pseudo-amenable, but not amenable (see Exercise 2.2.1).

Example 4.4.4. Let G be a locally compact group. Then G can be canonically identified with ΦA(G) (Theorem F.2.2(ii)); so, A(G) is Tauberian (Theorem F.2.2(iii)), and ΦA(G) is discrete whenever G is discrete. If G is thus a discrete group such that A(G) has a—not necessarily bounded—approximate identity, then the hypotheses of Proposition 4.4.2 are satisfied, so that A(G) is pseudo-amenable. By Leptin’s Theorem, A(G) is thus pseudo-amenable for discrete, amenable G. Hence, every discrete, amenable group that is not almost abelian, e.g., the Heisenberg group (see Exercise 1.2.3) equipped with the discrete topology, has a pseudo-amenable, but non-amenable Fourier algebra. Also, A(F2 ) has a—necessarily unbounded—approximate identity by [158, Theorem 2.1]. Hence, A(F2 ) is also pseudo-amenable, but definitely not amenable. A Banach algebra A is defined to be amenable if every derivation from A into a dual Banach A-bimodule is inner. One might adopt the point of view that, in the framework of analysis, it is more natural to require that certain derivations can be approximated by inner derivations: Definition 4.4.5. Let A be a Banach algebra, and let E be a Banach Abimodule. Then a derivation D : A → E is called approximately inner if there is a net (xα )α in E such that Da = lim adxα a α

(a ∈ A).

Definition 4.4.6. A Banach algebra A is called approximately amenable if, for every Banach A-bimodule E, every derivation D : A → E ∗ is approximately inner. The following proposition shows that approximate amenability is also characterized by other, formally both weaker and stronger, conditions: Proposition 4.4.7. Let A be a Banach algebra. Then the following are equivalent: (i) A is approximately amenable; (ii) for every Banach A-bimodule E, every derivation D : A → E is approximately inner; (iii) for every Banach A-bimodule E, every derivation D : A → E ∗ is weak∗ approximately inner, i.e., there is a net (φα )α in E such that Da = σ(E ∗ , E)- lim adφα a α

(a ∈ A).

Proof. (ii) =⇒ (i) =⇒ (iii) is clear. (iii) =⇒ (ii): Let E be a Banach A-bimodule, and let D : A → E be a derivation. We need to show that D lies in the closure of B 1 (A, E) in B(A, E) with respect to the strong operator topology.

4.4

Pseudo- and Approximate Amenability

171

View D as a derivation into the Banach A-bimodule E ∗∗ . By (iii), there is thus a net (Xα )α∈A in E ∗∗ such that Da = σ(E ∗∗ , E ∗ )- limα adXα a for a ∈ A. Let B := A × F(E ∗ ) × (0, ∞) be ordered as follows: for βj = (αj , Fj , j ) with j = 1, 2 define β1 ≺ β2

:⇐⇒

α1 ≺ α, F1 ⊂ F2 , and 1 ≥ 2 .

It is clear that this order turns B into a directed set. For β = (α, F, ) ∈ B, set Xβ := Xα . It is then clear that Da = σ(E ∗∗ , E ∗ )- lim adXβ a β

(a ∈ A),

(4.18)

As E is σ(E ∗∗ , E ∗ )-dense in E ∗∗ , there is, for each β = (α, F, ) ∈ B, an element xβ ∈ E such that |φ, Xβ  − xβ , φ| < 

(φ ∈ F ).

It follows that σ(E ∗∗ , E ∗ )- limβ (Xβ − xβ ) = 0. Consequently, in (4.18), we can replace (Xβ )β∈B by (xβ )β∈B . Since the restriction of σ(E ∗∗ , E ∗ ) to E is σ(E, E ∗ ), we therefore have that Da = weak- lim adxβ a β

(a ∈ A),

i.e., D lies in the closure of B 1 (A, E) in B(A, E) with respect to the weak operator topology. As a consequence of Exercise 1.3.1, a convex subset of B(A, E) is closed in the strong operator topology if and only if it is closed in the weak operator topology. In particular, the closures of B 1 (A, E) in B(A, E) with respect to the weak and the strong operator topology, respectively, coincide. This completes the proof.  Remark 4.4.8. Let A be a Banach algebra with the following property: for every Banach A-bimodule E and every derivation D : A → E ∗ , there is a ∗ sequence (φn )∞ n=1 in E such that D = limn→∞ adφn in the operator norm. This property is certainly stronger than approximate amenability, and is— somewhat surprisingly—already equivalent to amenability ([131, Theorem 3.1]). The question of how pseudo-amenability and approximate amenability are related arises immediately. It appears to be open if an approximately amenable Banach algebra is always pseudo-amenable whereas the converse implication is false as we shall see below. Still, the two notions are not entirely unrelated. We require a lemma: Lemma 4.4.9. Let A be a pseudo-amenable Banach algebra, let E be a Banach A-bimodule that is neo-unital as a right Banach A-module, and let D : A → E ∗ be a derivation. Then D is weak∗ approximately inner.

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4 Amenability-Like Properties

Proof. Let (dα )α∈A be an approximate diagonal for A. As (dα )α∈A is an approximate identity for A and E is neo-unital as a right Banach A-module, we have (x ∈ E). (4.19) lim x · ΔA dα = x α

Suppose that

∞ 

dα =

(α) a(α) n ⊗ bn

(α ∈ A)

n=1

∞ (α) ∞ (α) (α) with supα∈A n=1 an bn  < ∞. For α ∈ A, set φα := n=1 an · (α) Dbn ∈ E ∗ . For a ∈ A and x ∈ E, we then have x, Da − adφα a = x, Da − a · φα + φα · a   ∞ ∞   (α) (α) aa(α) a(α) = x, Da − n · Dbn + n · Dbn · a  =

x, Da − 

=

x, Da − 

=

x, Da −

n=1 ∞  n=1 ∞  n=1 ∞ 

n=1

 ∞   (α) (α) (α) (α) an · D bn a + an · Dbn · a 

n=1 ∞ 

(α) a(α) n bn · Da −

 (α) a(α) n bn

(α) a(α) n · Dbn · a +

n=1

∞ 

 (α) a(α) n · Dbn · a

n=1

· Da

n=1

= x − x · ΔA dα , Da → 0,

by (4.19).

This means that D is weak∗ approximately inner as claimed.



The following theorem relates pseudo- and approximate amenability: Theorem 4.4.10. The following are equivalent for a Banach algebra A: (i) A is approximately amenable; (ii) A# is pseudo-amenable. If A has a bounded approximate identity, then (i) and (ii) imply: (iii) A is pseudo-amenable. Moreover, if A is unital, then (i), (ii), and (iii) are equivalent. Proof. (i) =⇒ (ii): Define D : A → A# ⊗γ A# to be ade⊗e . As DA ⊂ ker ΔA# , Proposition 4.4.7 yields a net (eα )α∈A in ker ΔA# such that Da = limα ade α a for a ∈ A. For α ∈ A, set dα := e ⊗ e − eα . It is then straightforward to verify that (dα )α∈A is an approximate diagonal for A# . (ii) =⇒ (i): Let E be a Banach A-bimodule, and let D : A → E ∗ be a derivation. Turn E into a Banach A# -bimodule by letting

4.4

Pseudo- and Approximate Amenability

(a + λe) · x = a · x + λx

and

173

x · (a + λe) := x · a + λx

(a ∈ A, λ ∈ C, x ∈ E),

and extend D to a derivation D# : A# → E ∗ by setting D# e := 0. As E is trivially (neo-)unital as a right Banach A# -module, Lemma 4.4.9 yields that D# is weak∗ approximately inner; consequently, so is D. By Proposition 4.4.7, this means that A is approximately amenable. Suppose now that A has a bounded approximate identity and that A# is pseudo-amenable. Since A is a closed ideal of A# it follows from Exercise 4.4.2 below that A is pseudo-amenable. Finally, suppose that A is unital and pseudo-amenable; we denote the identity of A be eA whereas e stands for the adjoined identity of A# . Let E be a Banach A-bimodule, and let D : A → E ∗ be a derivation. Extend the module action of A on E to A# , and set E1 := E · eA

and

E2 := E · (e − eA ).

Then E1 and E2 are closed submodules of E such that E = E1 ⊕ E2 . Consequently, we have a canonical isomorphism E ∗ ∼ = E1∗ ⊕ E2∗ where E1∗ ∼ = eA · E ∗ ∗ ∼ ∗ ∗ ∗ and E2 = (e − eA ) · E . Define D1 : A → E1 and D2 : A → E2 by letting D1 a := eA · Da and

D2 a := (e − eA ) · Da

(a ∈ A).

Clearly, it is sufficient to show that both D1 and D2 are weak∗ approximately inner. As E1 is (neo-)unital as a right Banach A-module by construction, it follows from Lemma 4.4.9, that D1 is indeed weak∗ approximately inner, and since x · a = 0 for all x ∈ E2 and a ∈ A, Proposition 2.1.1 yields that D2 is even inner. This completes the proof.  Under certain circumstances, pseudo- and approximate amenability are indeed equivalent: Proposition 4.4.11. Let A be a Banach algebra with an approximate diagonal (dα )α∈A such that (ΔA dα )α∈A is a bounded approximate identity for A. Then A is approximately amenable. Proof. In view of Theorem 4.4.10, it is sufficient to show that A# is pseudoamenable. For α ∈ A, set eα := ΔA dα , and let C ≥ 0 be a bound for (eα )α∈A . Define a net (d# α,β )α,β by letting d# α,β := (e − eα ) ⊗ (e ⊗ eβ ) + dβ

(α, β ∈ A).

# We claim that a subnet of (d# α,β )α,β is an approximate diagonal for A . To see this, first, note that, for a ∈ A and α, β ∈ A, we have # a · d# α,β − dα,β · a ≤ (C + 1)(a − aeα  + a − eβ a) + a · dβ − dβ · a,

so that, with α ∈ A fixed,

174

4 Amenability-Like Properties # lim sup a · d# α,β − dα,β · a ≤ (C + 1)a − aeα 

(a ∈ A).

(4.20)

β

Furthermore, we have ΔA# d# α,β = e − eα − eβ + eα eβ + eβ = e + (eα eβ − eα )

(α, β ∈ A)

and thus, with α ∈ A fixed, lim ΔA# d# α,β = e.

(4.21)

β

Since (eα )α∈A is a bounded approximate identity for A, the right-hand side of (4.20) tends to zero for each a ∈ A. Thus (4.20) and (4.21) imply the # existence of a subset of (d# α,β )α,β that is an approximate diagonal for A .  Combining Proposition 4.4.11 with an inspection of the proof of Proposition 4.4.2, we obtain: Corollary 4.4.12. Let A be a semisimple, commutative, Tauberian Banach algebra with a bounded approximate identity such that ΦA is discrete. Then A is approximately amenable. Example 4.4.13. In Example 4.4.4, we had seen that the Fourier algebra A(G) is pseudo-amenable for any discrete, amenable group G. From Leptin’s Theorem and Corollary 4.4.12, we conclude that A(G) is then also approximately amenable. We shall now establish a criterion for Banach algebras to not be approximately amenable and, as a consequence, come up with examples of pseudoamenable Banach algebras that fail to be approximately amenable. We begin with a simple lemma: Lemma 4.4.14. Let A be an approximately amenable Banach algebra. Then, for any a1 , . . . , an ∈ A and  > 0, there are x ∈ A ⊗γ A and x, y ∈ A such that aj · x − x·aj − aj ⊗ y + x ⊗ aj  < , aj x − aj  < ,

and

yaj − aj  < 

(j = 1, . . . , n).

Proof. By Theorem 4.4.10, A# is pseudo-amenable and thus has an approximate diagonal, say (dα )α∈A ; an inspection of the proof of Theorem 4.4.10 shows that we can suppose without loss of generality that ΔA# dα = e for α ∈ A. Consequently, there are a net (xα )α∈A in A ⊗γ A and nets (xα )α∈A and (yα )α∈A in A such that dα = xα − xα ⊗ e − e ⊗ yα + e ⊗ e Since

(α ∈ A).

4.4

Pseudo- and Approximate Amenability

175

a · dα − dα · a = a · xα − xα · a − a ⊗ yα + xα ⊗ a − (axα ⊗ e − a ⊗ e) + (e ⊗ yα a − e ⊗ a) → 0

(a ∈ A),

we conclude that a · xα − xα · a − a ⊗ yα + xα ⊗ a → 0, axα − a → 0,

and

yα a − a → 0

(a ∈ A).

Given a1 , . . . , an ∈ A and  > 0, the existence of x, x, and y as claimed then follows.  Our next lemma is the technical heart of the argument. The multiplier norm  · M on a Banach algebra A (compare Exercise 2.4.3) is defined through aM := sup{ax, xa : x ∈ Ball(A)}

(a ∈ A).

Obviously,  · M is dominated by  · . Lemma 4.4.15. Let A be a Banach algebra containing two sequences (pn )∞ n=1 and (qn )∞ n=1 with the following properties: (a) pn qn = qn pn = pn for n ∈ N; (b) pn qm = qm pn = 0 for n, m ∈ N with n = m; (c) limn→∞ pn  = ∞; (d) supn∈N pn M < ∞ and supn∈N qn M < ∞. Then A is not approximately amenable. Proof. Passing to a subsequence, if necessary, we can suppose without loss of generality that pn  ≥ (n + 1)4 for n ∈ N. Set a :=

∞ 

1 p2n−1 (2n − 1)2 p2n−1  n=1

and

b :=

∞  n=1

1 (2n)2 p

2n 

p2n

Since pn pm = pn qm pm = 0 for n = m, the following identities hold for n ∈ N: ap2n = 0, 1 p2n−1 , q2n−1 a = (2n − 1)2 p2n−1  p2n−1 b = 0. 1 p2n . bq2n = (2n)2 p2n 

(4.22) (4.23) (4.24) (4.25)

176

4 Amenability-Like Properties

Fix C ≥ 0 such that supn∈N pn M ≤ C and supn∈N pn M ≤ C, and assume toward a contradiction that A is approximately amenable. Let  > 0. By Lemma 4.4.14, there are thus x ∈ A ⊗γ A and x, y ∈ A such that a · x − x · a − a ⊗ y + x ⊗ a < ,

(4.26)

b · x − x · b − b ⊗ y + x ⊗ b < 

(4.27)

ax − a < .

(4.28)

and

We obtain for n, m ∈ N: C 2  ≥ 0 = q2n−1 · (a · x − x · a − a ⊗ y + x ⊗ a) · p2m , by (4.26), = q2n−1 a · x · p2m − q2n−1 a ⊗ yp2m , by (4.22), =

p2n−1 · x · p2m − p2n−1 ⊗ yp2m  , (2n − 1)2 p2n−1 

by (4.23).

This means that p2n−1 · x · p2m − p2n−1 ⊗ yp2m  (2n − 1)2 < C 2 p2n−1 p2m  p2m 

(n, m ∈ N). (4.29)

A similar argument—invoking (4.27), (4.24), and (4.25) instead of (4.26), (4.22), and (4.23)—yields  − p2n−1 · x · p2m + p2n−1 x ⊗ p2m  (2m)2 < C 2 p2n−1 p2m  p2n−1 

(n, m ∈ N). (4.30)

Together with the triangle inequality, (4.29) and (4.30) yield   (2n − 1)2 (2m)2 p2n−1 x ⊗ p2m − p2n−1 ⊗ yp2m  C 2 + ≥ p2m  p2n−1  p2n−1 p2m  |p2n−1 x ⊗ p2m  − p2n−1 ⊗ yp2m | ≥ p2n−1 p2m     p2n−1 x yp2m     (n, m ∈ N), = − p2n−1  p2m   which, in turn, implies that     2 2  |p2n−1 x yp2m     ≤ C 2  (2n − 1) + (2m) −  p2n−1  p2m   (2m + 1)4 (2n)4

(n, m ∈ N)

(4.31) (because pn  ≥ (n + 1)4 for n ∈ N). Setting n = m in (4.31) and n = m + 1, respectively, yields

4.4

Pseudo- and Approximate Amenability

   p2m−1 x yp2m     −  p2m−1  p2m   2

≤C 



177

(2m − 1)2 (2m)2 + (2m + 1)4 (2m)4

 ≤

2C 2  (2m)2

and    p2m+1 x yp2m     −  p2m+1  p2m     (2m + 1)2 (2m)2 2C 2 ≤ C 2 +  ≤ 4 4 (2m + 1) (2m + 2) (2m + 1)2 Combining these two estimates, we obtain      p2m−1 x p2m+1 x  1 1  ≤ 2C 2   − +  p2m−1  p2m+1   (2m)2 (2m + 1)2

(m ∈ N)

(m ∈ N).

(m ∈ N). (4.32)

Since

pn x pn 

≤

Cx pn 

→ 0, we conclude that

   p1 x pn x  p1 x   = lim  − n→∞ p1  p1  pn     n  p2k−1 x p2k+1 x   = lim  −  n→∞  p2k−1  p2k+1   k=1  ∞   1 1 2 ≤ 2C  + , (2n)2 (2n + 1)2 n=1

by (4.32),

≤ 2C 2 . On the other hand, (4.28) and (4.23) yield        p1 x  1 1  .  ≥ p p − 1 x − C ≥ q1 (ax − a) =  1    p1  1 p1  p1  It follows that

  p1 x  p1 x  ≤ 2C 2  + C. 1≤ + 1 − p1  p1   

As  > 0 is arbitrary, this is impossible.

Theorem 4.4.16. Let A be a Banach algebra containing a sequence (en )∞ n=1 such that supn∈N en  = ∞, supn∈N en M < ∞, and en en+1 = en = en+1 en Then A is not approximately amenable.

(n ∈ N).

(4.33)

178

4 Amenability-Like Properties

Proof. First, note that en em = en = en em for a all n, m ∈ N with n < m. Passing to a subsequence, if necessary, we can thus suppose that (en n )∞ n=1 is strictly increasing and that lim (en+1  − en ) = ∞.

(4.34)

n→∞

Set pn := e3n − e3n−1

and qn := e3n+1 − e3n−2

(n ∈ N).

∞ We claim that (pn )∞ n=1 and (qn )n=1 satisfy the hypotheses of Lemma 4.4.15. Since pn  ≥ e3n  − e3n−1  for n ∈ N, it follows from (4.34) that Lemma 4.4.15(c) is satisfied. Also, as supn∈N en M < ∞, it is clear that supn∈N pn M < ∞ and supn∈N qn M < ∞ as well, i.e., Lemma 4.4.15(d) holds. To verify Lemma 4.4.15(a) and (b), observe that pn qm = qm pn for all n, m ∈ N. We thus have

pn qn = qn pn = (e3n+1 − e3n−2 )(e3n − e3n−1 ) = e3n − e3n−1 = pn

(n ∈ N),

which establishes Lemma 4.4.15(a). To show that Lemma 4.4.15(b) is also satisfied, let n, m ∈ N with n < m. As 3m − 2 > 3n and 3m − 1 > 3n + 1, we have pn qm = qm pn = (e3m+1 − e3m−2 )(e3n − e3n−1 ) = 0, and pm qn = qn pm = (e3n+1 − e3n−2 )(e3m − e3m−1 ) = 0 as required. It now follows from Lemma 4.4.15 that A is not approximately amenable.  Example 4.4.17. Let p ∈ [1, ∞), and let p be equipped with coordinatewise 1 multiplication. For n ∈ N, set en := χ{1,...,n} , and note that en p = n p whereas en M = 1. It follows that supn∈N en p = ∞ and supn∈N en M = 1, and obviously, (en )∞ n=1 satisfies (4.33). Hence, Theorem 4.4.16 applies and yields that the pseudo-amenable Banach algebra p is not approximately amenable. Example 4.4.18. A subset L of a discrete group G is called a Leinert set if χL A(G) ∼ = 2 (L): they are named after M. Leinert, who proved that F2 contains infinite Leinert sets ([222]). Assume that A(F2 ) is approximately amenable. Let L ⊂ F2 be an infinite Leinert set. Then, by Exercise 4.4.1 below, χL A(G) ∼ = 2 (L) is approximately amenable, which contradicts Example 4.4.17. Hence, A(F2 ) is pseudo-amenable, but not approximately amenable. We conclude this section with an extension of Theorem 2.1.10, which asserts that, for group algebras, both pseudo- and approximate amenability are nothing but amenability:

4.4

Pseudo- and Approximate Amenability

179

Theorem 4.4.19. For a locally compact group G, the following are equivalent: (i) G is amenable; (ii) L1 (G) is amenable; (iii) L1 (G) is approximately amenable; (iv) L1 (G) is pseudo-amenable. Proof. In view of Theorems 2.1.10 and 4.4.10, it is enough to prove (iv) =⇒ (i). Let (dα )α∈A be an approximate diagonal for L1 (G). Viewing the constant function 1 ∈ L∞ (G) as a (multiplicative) linear functional on L1 (G), we define fα := (id ⊗ 1)dα ∈ L1 (G) for α ∈ A. Let g ∈ L1 (G) be such that g, 1 = 1. Then we have g ∗ fα − fα 1 = g ∗ fα − fα g, 11 = (id ⊗ 1)(g · dα ) − (id ⊗ 1)(dα · g)1 ≤ g · dα − dα · g1 → 0. Fix g ∈ L1 (G) with g, 1 = 1. Noting that δx ∗ g, 1 = g, 1 = 1 for all x ∈ G, we obtain δx ∗ fα − fα 1 ≤ δx ∗ fα − δx ∗ g ∗ fα 1 + δx ∗ g ∗ fα − fα 1 = g ∗ fα − fα 1 + δx ∗ g ∗ fα − fα 1 → 0

(x ∈ G).

Since fα , 1 = dα , 1 ⊗ 1 = ΔL1 (G) dα , 1 → 1, we can suppose—passing to a subnet if necessary—that inf α fα 1 > 0, so that supα fα1 1 < ∞. For α ∈ A, set mα := fα1 1 |fα |; clearly, mα 1 = 1, and δx ∗ mα = fα1 1 |δx ∗ fα | for x ∈ G. It follows that δx ∗ mα − mα 1 =

1 |δx ∗ fα | − |fα |1 fα 1 1 ≤ δx ∗ fα − fα 1 → 0 fα 1

By Lemma 1.1.8, this means that G is amenable.

(x ∈ G). 

Exercises Exercise 4.4.1. Let A be a Banach algebra, and let I be a closed ideal in A. Show that: (a) if A is pseudo-amenable, then so is A/I; (b) if A is approximately amenable, then so is A/I.

180

4 Amenability-Like Properties

Exercise 4.4.2. Let A be a pseudo-amenable Banach algebra, and let I be a closed ideal of A with a bounded approximate identity. Show that I is pseudo-amenable. Exercise 4.4.3. Let A be an approximately amenable Banach algebra. Show that A has both a left and a right approximate identity. Exercise 4.4.4. Let A be an approximately amenable, commutative Banach algebra. Show that A is weakly amenable. Exercise 4.4.5. Let A be a unital, pseudo-amenable Banach algebra. Show that A has an approximate diagonal (dα )α∈A such that Δdα = eA for all α ∈ A. Conclude that B(p ) is not pseudo-amenable—and thus not approximately amenable—for p ∈ [1, ∞].

4.5 Biflatness and Biprojectivity We deal with biprojective Banach algebras first: Definition 4.5.1. A Banach algebra A is called biprojective if ΔA : A⊗γ A → A has a bounded right inverse which is an A-bimodule homomorphism. The reason for the adjective “biprojective” will become clear in Chapter 6. In view of Exercise 4.1.6, every contractible Banach algebra is biprojective. Exercise 4.5.1 below specifies the precise relations between contractibility and biprojectivity. Contractibility is such a strong property that there are no known nontrivial examples. As we shall see, the class of biprojective Banach algebras is much richer. Example 4.5.2. Let (E, F, ·, ·) be a dual pair of Banach spaces, i.e., a pair of Banach spaces (E, F ) with a bounded, bilinear map ·, · : E × F → C that is nondegenerate: for each x ∈ E \ {0}, there is y ∈ F such that x, y = 0, and for each y ∈ F \ {0}, there is x ∈ E such that x, y = 0. Examples of dual pairs of Banach spaces are, for instance, (E, E ∗ , ·, ·) and (E ∗ , E, ·, ·), where E is an arbitrary Banach space, and ·, · is the usual duality. Define (x ⊗ y)(u ⊗ v) := u, yx ⊗ v

(x, u ∈ E, y, v ∈ F ).

(4.35)

It is routinely checked that (4.35) defines a product on E ⊗γ F turning it into a Banach algebra. Fix (x0 , y0 ) ∈ E × F such that x0 , y0  = 1. Define ρ : E ⊗γ F → (E ⊗γ F ) ⊗γ (E ⊗γ F ) through ρ(x ⊗ y) = (x ⊗ y0 ) ⊗ (x0 ⊗ y)

(x ∈ E, y ∈ F ).

It is obvious that ρ is a right inverse of ΔE⊗γ F , and it is routinely checked that it is a E ⊗γ F bimodule homomorphism. Hence, E ⊗γ F is biprojective.

4.5

Biflatness and Biprojectivity

181

Example 4.5.3. Let G be a compact group, and identify L1 (G) ⊗γ L1 (G) with L1 (G × G). Define ρ : L1 (G) → L1 (G × G) through ρ(f )(x, y) := f (xy) If f, g ∈ L1 (G), then ΔL1 (G) (f ⊗ g)(x) :=



(f ∈ L1 (G), x, y ∈ G).

f (xy −1 )g(y) dy =

G

so that

(f ⊗ g)(xy −1 , y) dy

(y ∈ G),

G



ΔL1 (G) (f )(x) =



f (xy −1 , y) dy

(f ∈ L1 (G × G), x ∈ G).

G

It follows that ΔL1 (G) ◦ ρ = idL1 (G) . It is equally easy to see that ρ is an L1 (G)-bimodule homomorphism. The algebras in Example 4.5.3 are amenable, but what about Example 4.5.2? To answer this question, we define: Definition 4.5.4. Let (E, F, ·, ·) be a dual pair of Banach spaces. Then ∞ (a) a linear operator T on E is called F -nuclear if there ∞ are sequences (xn )n=1 ∞ and (yn )n=1 in E and F , respectively, with n=1 xn yn  < ∞ such that ∞  Tx = x, yn xn (x ∈ E); (4.36) n=1

(b) the F -nuclear norm of an F -nuclear operator T on E is the infimum over ∞ ∞ all n=1 xn yn  with n=1 xn yn  < ∞ for which (4.36) holds. The collection of all F -nuclear operators on E is denoted by NF (E). If F = E ∗ , we call the F -nuclear operators nuclear operators and denote NF (E) by N (E). Remark 4.5.5. Given a dual pair of Banach spaces (E, F, ·, ·), there is a canonical, contractive map E ⊗γ F → B(E),

x ⊗ y → x y,

(4.37)

where we view F as a subspace of E ∗ . It is immediate that the range of (4.37) is NF (E) and that the F -nuclear norm on NF (E) is the quotient norm induced by (4.37). It is routinely checked that NF (E) is a subalgebra of B(E) and a Banach algebra with respect to the F -nuclear norm; if E ⊗γ F is equipped with the product from Example 4.5.2, (4.37) becomes an algebra homomorphism. Theorem 4.5.6. For a dual pair (E, F, ·, ·) of Banach spaces, the following are equivalent:

182

4 Amenability-Like Properties

(i) E ⊗γ F is contractible; (ii) E ⊗γ F is amenable; (iii) E ⊗γ F has a bounded approximate identity; (iv) E ⊗γ F has a bounded left approximate identity; (v) NF (E) has a bounded left approximate identity; (vi) dim E = dim F < ∞. Proof. The implications (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) =⇒ (v) are clear, and for (iv) =⇒ (i), just observe that E ⊗γ F ∼ = Mdim E . For (v) =⇒ (vi), first note that B(E)—and thus A(E)—is canonically a Banach NF (E)-bimodule. Suppose that NF (E) has a bounded left approximate identity, say (Tα )α . As (Tα )α is bounded in the F -nuclear norm, it is also bounded in the operator norm, and since F(E) is dense in NF (E), it follows that (Tα )α converges to idE in the strong operator topology. As in the proof of Theorem 3.3.2, we then see that limα Tα S = S for all S ∈ K(E). By Cohen’s Factorization Theorem, each operator in A(E) is thus the product of an F -nuclear operator and another operator in A(E). Since NF (E) ⊂ N (E), and since N (E) is an ideal in B(E), this means that A(E) = N (E). Since (Tα )α is bounded in the operator norm, E has the (bounded) approximation property, so that N (E) ∼ = E ⊗γ E ∗ by Exercise 4.5.6 below. On the other hand, it is standard that the canonical map E ⊗ E ∗  x ⊗ φ → x φ into B(E) extends to an isometry from E ⊗λ E ∗ onto A(E) ([71, Theorem 2.5.3]). All in all, we have canonically E ⊗γ E ∗ ∼ = N (E) = A(E) ∼ = E ⊗λ E ∗ .

(4.38)

Define Tr : E ⊗ E ∗ → C through Tr(x ⊗ φ) = x, φ

(x ∈ E, φ ∈ E ∗ ).

Clearly, Tr is continuous on E ⊗ E ∗ with respect to the projective norm and thus extends to a continuous linear functional on E ⊗γ E ∗ . By (4.38), this means that Tr canonically induces a bounded linear functional on A(E), which we denote likewise by Tr. It is easy to see that either both E and F are infinite-dimensional or that both E and F have the same finite dimension. Assume toward a contradiction that E is infinite-dimensional. Then, for each N ∈ N, there is a projection √ PN ∈ F(E) onto some N -dimensional subspace of E such that PN  ≤ N : the existence of such projections is guaranteed by Theorem A.4.1. It is clear that 1 √ |Tr(PN )| ≤ Tr (N ∈ N). (4.39) N On the otherhand, for each   N ∈ N, there is afinite, biorthogonal system (N ) (N ) (N ) (N ) N xj , φj : j = 1, . . . N such that PN = j=1 xj φj . It follows that

4.5

Biflatness and Biprojectivity N 1 1   (N ) (N )  √ √ Tr(Pn ) = √ xj , φj = N N N j=1

183

(N ∈ N),

which contradicts (4.39).



As a consequence of Theorem 4.5.6 and Exercise 4.5.6 below, we obtain: Corollary 4.5.7. Let (E, F, ·, ·) be a dual pair of Banach spaces. Then: (i) NF (E) is amenable if and only if dim E = dim F < ∞; (ii) NF (E) is biprojective whenever E or F has the approximation property. Hence, for a wide range of dual pairs (E, F, ·, ·) of Banach spaces, the algebras NF (E) are biprojective. As we shall see toward the end of this section, these algebras are the smallest “building blocks” out of which more general biprojective Banach algebras (with a few, relatively mild, restrictions) are constructed. To prepare the ground for this structure theorem, we first have to consider other algebras of operators associated with dual pairs of Banach spaces: Definition 4.5.8. Let (E, F, ·, ·) be a dual pair of Banach spaces. An F bounded operator on E is an operator T ∈ B(E) such that T ∗ F ⊂ F , where F is canonically identified with a subspace of E ∗ . The collection of all F bounded operators on E is denoted by BF (E). We leave it to the reader to verify some elementary properties of F bounded operators in Exercise 4.5.7 below. Definition 4.5.9. Let (E, F, ·, ·) be a dual pair of Banach spaces, and let FF (E) := F(E) ∩ BF (E). A subalgebra A of BF (E) that is a Banach algebra under some norm is called a standard Banach F -operator algebra if: (a) FF (E) ⊂ A; (b) the inclusion A ⊂ BF (E) is continuous. If F = E ∗ , we call A is a standard Banach operator algebra on E. For the definition of the socle soc(A) of a (semiprime) algebra A, see Definition B.5.4 and Proposition B.5.5. Proposition 4.5.10. For a Banach algebra A, the following are equivalent: (i) A is primitive with soc(A) = {0}; (ii) there is a dual pair (E, F, ·, ·) of Banach spaces such that A is algebraically isomorphic to a standard Banach F -operator algebra on E with the induced inclusion A ⊂ BF (E) being a contraction. Proof. (i) =⇒ (ii): Due to the fact that soc(A) = {0}, there is a minimal idempotent p ∈ A. Set E := Ap and F := pA. As pAp = Cp, there is, for each x ∈ E and y ∈ F , a unique λx,y ∈ C such that yx = λx,y p. Define

184

4 Amenability-Like Properties

·, · : E × F → C,

(x, y) → λx,y .

Then it is obvious that ·, · is bilinear. We claim that that ·, · is nondegenerate. To see this, let x ∈ E \ {0}. Since p is a minimal idempotent, Ap is a minimal left ideal of A (Proposition B.5.3(i)). This means that either Ax = {0} or Ax = Ap. Assume that Ax = {0}. Then Cx is a nonzero left ideal contained in Ap, and thus all of Ap. As p ∈ Cp = pAp ⊂ Ap, this means that x is a nonzero scalar multiple of p, and therefore px = x = 0, which is a contradiction. Thus, Ax = Ap must hold. In particular, there is a ∈ A such that ax = p. With y := pa, we obtain x, y = 0. Analogously, we see that, for any y ∈ F \ {0}, there is x ∈ E such that x, y = 0. Equip E and F with norms  · E and  · F , respectively, via xE := inf{a : a ∈ A, ap = x}

(x ∈ E)

yF := inf{a : a ∈ A, pa = y}

(y ∈ F ).

and It is straightforward to verify that ·, · is bounded with respect to  · E and  · F . All in all, (E, F, ·, ·) is a dual pair of Banach spaces. Letting A act on E via left multiplication yields a representation π of A on E. From the definitions of ·E and ·F , it is immediate that π(A) ⊂ BF (E) such that π : A → BF (E) is also contractive. We claim that π is faithful. To see this, note that ker π = {a ∈ A : abp = 0 for b ∈ A}. Define I to be the span of {apb : a, b ∈ A}. Then I is an ideal of A. By definition, {ab : a ∈ ker π, b ∈ I} = {0} holds. Since A is primitive, Proposition B.4.8(i) and Definition B.4.1 imply that I = {0} or ker π = {0}. As p ∈ I, this means that ker π = {0}, i.e., π is faithful. Consequently, A is algebraically isomorphic to a subalgebra of BF (E). It remains to be shown that FF (E) ⊂ π(A). Let x ∈ E and y ∈ F , and set a := xy. Then we have π(a)z = xyz = xz, yp = z, yx = (x y)(z)

(z ∈ E),

i.e., π(a) = x y and thus x y ∈ π(A). From Exercise 4.5.8(i) below, it follows that FF (E) is indeed contained in A. (ii) =⇒ (i): It is straightforward that there is no nontrivial subspace of E that is invariant under FF (E), i.e., E is an irreducible left FF (E)-module (see Remark B.4.4). Since FF (E) ⊂ A, it follows that A acts irreducibly on E as well and, consequently, is primitive. As each rank one projection in FF (E) is a minimal idempotent in A, we conclude that soc(A) = {0}.  In view of Proposition 4.5.10, our first goal is to establish that (certain) biprojective Banach algebras have nonzero socle. Toward that goal, we proceed through a series of lemmas and propositions. If A is a Banach algebra and E is any Banach space, the space A ⊗γ E is canonically a left Banach A-module via

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185

a · (b ⊗ x) := ab ⊗ x

(a, b ∈ A, x ∈ E),

If E is also a left Banach A-module, then ΔA,E : A ⊗γ E → E,

a ⊗ x → a · x

is a left A-module homomorphism: we require this for the following lemma. Lemma 4.5.11. Let A be a biprojective Banach algebra, and let L be a closed left ideal of A which is essential as a left Banach A-module. Then the module map ΔA,A/L has a bounded right inverse which is also a left A-module homomorphism. Proof. Let ρ : A → A ⊗γ A be a bounded right inverse to ΔA which is also an A-bimodule homomorphism, and let π : A → A/L denote the quotient map. Then we have (idA ⊗ π)(ρ(ab)) = (idA ⊗ π)(ρ(a)b) = 0

(a ∈ A, b ∈ L).

Since L is essential as a left Banach A-module, this means that (idA ⊗ π) ◦ ρ vanishes on L and thus induces a bounded linear map ρ˜ : A/L → A ⊗γ (A/L). It is routinely checked that ρ˜ is both a right inverse of ΔA,A/L and a left A-module homomorphism.  With the help of Lemma 4.5.11, we can prove: Proposition 4.5.12. Let A be a biprojective Banach algebra with the approximation property, and let L be a closed left ideal of A which is essential as a left Banach A-module. Then, for each x ∈ A \ L, there is a left A-module homomorphism θ : A/L → A such that θ(x + L) = 0. Proof. Let x ∈ A \ L, and let ρ : A/L → A ⊗γ (A/L) be as specified in Lemma 4.5.11. As ρ is injective and x ∈ / L, it is clear that ρ(x + L) = 0. Since A has the approximation property, the canonical map from A ⊗γ (A/L) into A ⊗λ (A/L) is injective (see, e.g., [91, 5.6., Theorem]; compare also Exercise 4.5.6 below). From the definition of the injective norm (see, e.g., [91, 4.1]), it follows that there are φ ∈ A∗ and ψ ∈ (A/L)∗ such that ρ(x + L), φ ⊗ ψ = 0 and thus (idA ⊗ ψ)(ρ(x + L)) = 0. Let θ := (idA ⊗ ψ) ◦ ρ. Then θ is easily seen to be a left A-module homomorphism, and by definition, θ(x + L) = 0 holds.  Given an algebra A and a subset S of A, we define the left and right annihilator lan(S) and ran(S) of S in A, respectively, as lan(S) := {a ∈ A : aS = {0}}

and

ran(S) := {a ∈ A : Sa = {0}}.

It is immediate that, if I is an ideal of A, the so are lan(I) and ran(I).

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Lemma 4.5.13. Let A be a semiprime, biprojective Banach algebra with the approximation property, and let L be a closed left ideal. Then L is essential as a left Banach A-module. ˜ be the closed linear span of {ab : a ∈ A, b ∈ L}, so that L ˜ Proof. Let L is an essential left Banach A-module contained in L. We need to show that ˜ = L. Assume toward a contradiction that there is x ∈ L \ L. ˜ Then, by L ˜ → A Proposition 4.5.12, there is a left A-module homomorphism θ : A/L ˜ ˜ with θ(x + L) = 0. As ax ∈ L for a ∈ A, we have ˜ = θ(ax + L) ˜ 0 = aθ(x + L)

(a ∈ A),

˜ ∈ ran(A). By the definition of the right annihilator, {ab : a, b ∈ i.e., θ(x + L) ran(A)} = {0} holds, and as A is semiprime, this implies that ran(A) = {0} ˜ = 0. This contradicts the choice of θ. and thus θ(x + L)  Proposition 4.5.14. Let A be a semiprime, biprojective Banach algebra with the approximation property, and let E be an irreducible left Banach A-module. Then there is a minimal idempotent p ∈ A such that E ∼ = Ap. Proof. By Theorem B.4.12, we may suppose that E ∼ = A/L, where L is a maximal modular left ideal of A. By Lemma 4.5.13, L is essential as a left Banach A-module, and thus, by Proposition 4.5.12, ΔA,A/L has a bounded right inverse θ which is also a left A-module homomorphism. As ker θ  A/L is a submodule of A/L, and since A/L is irreducible, it follows that θ is injective, so that θ(A/L) ∼ = A/L is an irreducible left submodule of A, i.e., a minimal left ideal of A. By Proposition B.5.3(ii), there thus is a minimal idempotent p ∈ A such that θ(A/L) = Ap.  Corollary 4.5.15. Let A be a semiprime, biprojective Banach algebra with the approximation property, and let E be an irreducible left Banach A-module. Then E has the approximation property. Proof. By Proposition 4.5.14, E is (topologically) isomorphic to a complemented subspace of A and thus has the approximation property (see Remark A.2.5).  Proposition 4.5.16. Let A be a semisimple, biprojective Banach algebra with the approximation property. Then soc(A) is dense in A. Proof. Let I be the closure of soc(A) in A. Then I is a closed ideal in A and thus, by Lemma 4.5.13, is essential as a left Banach A-module. Assume toward a contradiction that there is a ∈ A \ I. Then, by Proposition 4.5.12, there is a left A-module homomorphism θ : A/I → A such that θ(a + I) = 0. As in the proof of Lemma 4.5.13, we see that θ(a + I) ∈ ran(I), so that, in particular, ran(I) = {0}. By Exercise 4.5.11 below, ran(I) = lan(I) holds, so that lan(I) = {0} as well.

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187

We will obtain a contradiction by showing that lan(I) ⊂ rad(A) = {0}. Let a ∈ lan(soc(A)) = lan(I), and let E be an irreducible left A-module. By Proposition 4.5.14, we may suppose that E = Ap for some minimal idempotent p ∈ A. Since Ap ⊂ soc(A), it follows that aE = {0}. Since E was an arbitrary irreducible left Banach A-module, Definition B.4.9 yields that a ∈ rad(A).  Let A be a primitive (and hence semisimple), biprojective Banach algebra with the approximation property. Then, by Proposition 4.5.16, we know that soc(A) is dense in A and thus—except in the obvious trivial situation—is nonzero. Hence, by Proposition 4.5.10, there is a dual pair (E, F, ·, ·) of Banach spaces such that A is isomorphic to a standard Banach F -operator algebra on E. Our next goal is, therefore, to identify the standard Banach F operator algebras on E that are biprojective; to achieve it, we require another lemma: Lemma 4.5.17. Let (E, F, ·, ·) be a dual pair of Banach spaces, let A be a standard Banach F -operator algebra, and let I be a nonzero ideal of A which is a Banach algebra under some norm such that the inclusion I ⊂ A is continuous. Then NF (E) is contained in I such that the inclusion NF (E) ⊂ I is continuous. Proof. Let T ∈ I \ {0}. Then there are x0 ∈ E and y0 ∈ F with T x0 , y0  = 0. Define S := x0 y0 , so that S ∈ FF (E) ⊂ A and ST x0 = 0. Since both I and FF (E) are ideals of A, it follows that ST ∈ FF (E) ∩ I, so that, in particular, FF (E)∩I = {0}. As FF (E) is simple by Exercise 4.5.8(ii) below, we conclude that FF (E) ⊂ I. Consider the bilinear map V : E × F → I,

(x, y) → x y.

Fix x ∈ E, and let (yn )∞ n=1 be a sequence in F such that yn → 0 and x yn → T ∈ I with respect to the norm topology on I. As the inclusion I ⊂ B(E) is continuous, we have T z = lim (x yn )(z) = lim z, yn x = 0 n→∞

n→∞

(z ∈ E),

i.e., T = 0. From the Closed Graph Theorem, we conclude that V is continuous in its second variable, and an analogous argument shows that V is also continuous in its first variable. The Uniform Boundedness Principle then yields that V is bounded. From the universal property of the projective tensor product (see, e.g., [91, 3.2., Proposition]), it follows at once that V induces a continuous linear map from E ⊗γ F into I; obviously, the range of this linear map is NF (E). Consequently, NF (E) is contained in I, and by the definition of the F -nuclear norm, the inclusion is continuous. 

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4 Amenability-Like Properties

Theorem 4.5.18. Let (E, F, ·, ·) be a dual pair of Banach spaces such that E or F has the approximation property. Then, for a standard Banach F operator algebra A on E, the following are equivalent: (i) A is biprojective; (ii) A = NF (E) (with an equivalent, but not necessarily identical norm). Proof. By Corollary 4.5.7, (ii) =⇒ (i) is clear. For the converse, let ρ : A → A ⊗γ A be a bounded right inverse of ΔA as required in the definition of biprojectivity. For x ∈ E and y ∈ F , define x : A → E,

T → T x

T → T ∗ y.

y : A → F,

and

We will show that, for suitable x ∈ E and y ∈ F , a scalar multiple of (x ⊗ y ) ◦ ρ is the desired isomorphism between A and E ⊗γ F ∼ = NF (E). Consider the diagram: NF (E)

/A

ρ

/ A ⊗γ A

x ⊗ y

Δop A

/ E ⊗γ F ·,·

 A

.

(4.40)

 /C

Here, the arrow from NF (E) to A is the canonical inclusion guaranteed by γ Lemma 4.5.17, Δop A : A ⊗ A → A is the reversed multiplication map given by op ΔA (S ⊗ T ) = T S for S, T ∈ A, the second vertical arrow is induced by the given duality between E and F , and the horizontal arrow in the second row is given by A  T → T x, y. A routine diagram chase, shows that (4.40) is commutative. Fix x0 ∈ E and y0 ∈ F such that x0 , y0  = 1, and set P := x0 y0 ; that Δop then P is an idempotent in FF (E). We will show A (ρ(P )) = 0. Let ∞ ∞ ∞ and (Tn )n=1 be sequences in A such that n=1 Sn Tn  < ∞ and (Sn )n=1 ∞ ρ(P ) = n=1 Sn ⊗ Tn . As P 3 = P , it follows that ρ(P ) = ρ(P 3 ) = P · ρ(P ) · P =

∞ 

P Sn ⊗ Tn P.

n=1

For n ∈ N, set xn := Tn x0 and yn := Sn∗ y0 , so that P Sn = x0 yn and Tn P = xn y0 . It follows that ρ(P ) =

∞  n=1

(x0 yn ) ⊗ (xn y0 ) ∈ NF (E) ⊗γ NF (E)

∞ = n=1 xn yn . Since Since E or F the approxiand thus has ∞ x mation property, we may identify the F -nuclear operator n=1 n yn with ∞ the element n=1 xn ⊗ yn of ∈ E ⊗γ F (Exercise 4.5.6 below). As Δop A (ρ(P ))

4.5

Biflatness and Biprojectivity ∞ 

xn , yn  =

n=1

∞ 

Tn x0 , Sn∗ y0  =

n=1

189 ∞ 

Sn Tn x0 , y0  = P x0 , y0  = 1,

n=1

we obtain that Δop A (ρ(P )) = 0 as claimed. For x ∈ E and y ∈ F , define θx,y : NF (E) → E ⊗γ F,

T → (x ⊗ y )(ρ(T )),

i.e., θx,y is the composition of all horizontal arrows in the first row of (4.40). It is routinely checked that θx,y is an NF (E)-bimodule homomorphism on NF (E) ∼ = E ⊗γ F . As P AP ∼ = CP . It follows that there is λx,y ∈ C such that θx,y (P ) = θx,y (P 3 ) = P θx,y (P )P = λx,y P. Since

θx,y (u v) = θx,y ((u y0 )P (x0 v)) = (u y0 )θx,y (P )(x0 v) = λx,y (u y0 )P (x0 v) = λx,y (u v) (u ∈ E, v ∈ F ),

it follows that θx,y (T ) = λx,y T

(T ∈ NF (E)).

(4.41)

op As Δop A (ρ(P )) = 0, we can choose x ∈ E and y ∈ F with ΔA (ρ(P ))x, y = 0. From the commutativity of (4.40), it follows that θx,y (P ) = 0, and consequently, λx,y ∈ C as in (4.41) cannot be zero. Define 1 (x ⊗ y )(ρ(T )) θ : A → E ⊗γ F, T → λx,y

It follows immediately from (4.41), that θ is the identity on NF (E) and thus, in particular, is surjective. We claim that θ is also injective. Being a scalar multiple of a composition of A-bimodule homomorphisms, θ is also an A-bimodule homomorphism, so that I := ker θ is a closed ideal of A. By Lemma 4.5.17, I = {0} implies that NF (E) ⊂ I, which is impossible because θ|NF (E) = idNF (E) . It follows that I = {0}, so that θ is the desired isomorphism.  Corollary 4.5.19. For a primitive Banach algebra A with the approximation property the following are equivalent: (i) A is biprojective; (ii) there is a dual pair (E, F, ·, ·) of Banach spaces such that A ∼ = NF (E). Proof. If A ∼ = NF (E) has the approximation property, then both E and F have it as well. Hence, A ∼ = E ⊗γ F is biprojective.

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4 Amenability-Like Properties

Conversely, if A is biprojective, then soc(A) = {0} by Proposition 4.5.16. By Proposition 4.5.10, there is thus a dual pair (E, F, ·, ·) of Banach spaces such that A is isomorphic to a standard Banach F -operator algebra on E.  Finally, Theorem 4.5.18 yields that A ∼ = NF (E). With a concrete description at hand of all primitive, biprojective Banach algebras with the approximation property, we move on toward a general structure theorem for not necessarily primitive, semisimple, biprojective Banach algebras with the approximation property. We require one more lemma: By a minimal closed (two-sided) ideal of a Banach algebra A, we mean a nonzero, closed ideal I of A such that any other closed ideal of A contained in I is either {0} or I itself. As two minimal closed ideals either coincide or have zero intersection, it is straightforward that the sum in A (see Appendix B) of all such ideals is isomorphic to their algebraic direct sum. We can thus speak of the algebraic direct sum in A of the minimal closed ideals of a Banach algebra A. Lemma 4.5.20. Let A be a semiprime Banach algebra such that soc(A) is dense in A. Then: (i) the algebraic direct sum of the minimal closed ideals of A is dense in A.; (ii) each minimal closed ideal of A is generated as a closed ideal by a minimal left ideal. Proof. We first prove that every minimal left ideal of A is contained in a minimal closed ideal. To see this, let L be a minimal left ideal of A, and let I be the closed, two-sided ideal of A generated by L. Let J be a closed ideal of A which is contained in I. If L ∩ J = {0} the minimality of L yields L ∩ J = L, so that J = I. Suppose therefore that L ∩ J = {0}; it follows that L ⊂ ran(J). As ran(J) is an ideal of A and necessarily closed, it follows that I ⊂ ran(J) and thus {ab : a, b ∈ J} = {0}. Since A is semiprime, this means J = {0}. Hence, I is a minimal closed ideal of A. In view of this, it follows from the definition of soc(A) that soc(A) is contained in the sum of the minimal closed ideals of A. As soc(A) is dense in A by hypothesis, this proves (i). For (ii), let I be a minimal closed ideal of A, and assume that I contains no minimal left ideal. Hence, for any minimal left ideal L of A, we have I ∩ L = {0} and thus aL = {0} for a ∈ I, i.e., I ⊂ lan(L). By the definition of soc(A), this means that I ⊂ lan(soc(A)). Since soc(A) is dense in A, we have lan(soc(A)) = lan(A) and so, in particular, {ab : a, b ∈ I} = {0}. As A is semiprime, this means that I = {0}, which is a contradiction.  Theorem 4.5.21. Let A be a semisimple, biprojective Banach algebra with the approximation property. Then the following hold: (i) the algebraic direct sum of the minimal closed ideals of A is dense in A.;

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191

(ii) for each minimal closed ideal I of A, the left annihilator lan(I) is a primitive ideal of A such that A = I ⊕ lan(I), and every primitive ideal of A arises in this fashion; (iii) for each minimal closed ideal I of A, there is a dual pair (E, F, ·, ·) of Banach spaces such that I ∼ = NF (E). Proof. Proposition 4.5.16 and Lemma 4.5.20(i) yield (i). For (ii) and (iii), let I be a minimal closed ideal of A. By Lemma 4.5.20(ii), I is generated by a minimal left ideal L of A. Since L is an irreducible left Banach A-module, lan(L) = lan(I) is a primitive ideal of A. Let π : A → A/lan(I) be the quotient map. We claim that the restriction of π to I is an isomorphism onto A/lan(I). Since I is a minimal closed ideal of A, it is clear that π|I is either zero or injective. As A is semisimple, and thus semiprime, it is not possible that π(I) = {0}, i.e., I ⊂ lan(I), so that π|I is injective. Let p ∈ A be a minimal idempotent such that L = Ap. Then π(p) = 0 is a minimal idempotent in A/lan(I), so that (A/lan(I))π(p) is a minimal left ideal in A/lan(I); in particular, soc(A/lan(I)) is nonzero. By Proposition 4.5.10, this means that there is a dual pair (E, F, ·, ·) of Banach spaces such that A/lan(I) is isomorphic to a standard Banach F -operator algebra on E. By the definition of a standard Banach F -operator algebra, A acts irreducibly on E. Since any irreducible left Banach A/lan(I)-module is also an irreducible left Banach A-module, Corollary 4.5.15 implies that E has the approximation property. By Lemma 4.5.13 and Exercise 4.5.9, A/lan(I) is again biprojective. It follows from Theorem 4.5.18 that A/lan(I) ∼ = NF (E). Since π|I maps I onto a nonzero ideal of A/lan(I), Lemma 4.5.17 implies (with the proper identifications made) that NF (E) is contained in π(I) meaning that π|I induces a (necessarily) topological isomorphism of I and A/lan(I). This establishes the first part of (ii) as well as (iii). To prove the second part of (ii), let P be a primitive ideal of A. By (i), there is at least one minimal closed ideal I of A such that I ⊂ P . Since {ab : a ∈ I, b ∈ lan(I)} = {0} ⊂ P , it follows from Proposition B.4.8(i) and Definition B.4.1 that lan(I) ⊂ P . As I ⊂ P , the minimality of I yields that I ∩ P = {0} and thus P ⊂ lan(I) as well.  At the end of this section, we turn to biprojectivity’s “little brother”: Definition 4.5.22. A Banach algebra A is biflat if Δ∗∗ : A∗ → (A ⊗γ A)∗ has a bounded left inverse which is an A-bimodule homomorphism. As in the case of biprojectivity, the reason for this choice of terminology will become clear in Chapter 6. Taking adjoints, one sees immediately that every biprojective Banach algebra is biflat, but so is every amenable Banach algebra by Exercise 2.3.9. Exercise 4.5.14 below specifies the precise relations between amenability and biflatness.

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4 Amenability-Like Properties

It is not difficult to come up with biflat Banach algebras that are neither amenable nor biprojective: take a Banach algebra that is amenable, but not biprojective, and another one that is biprojective, but not amenable, and then form their direct sum or their projective tensor product. Much more subtle is the question if there are examples of such Banach algebras that occur “naturally”. We first prove a lemma: Lemma 4.5.23. For a Banach algebra A, the following are equivalent: (i) A is biflat. (ii) there is an A-bimodule homomorphism ρ : A → (A ⊗γ A)∗∗ such that ∗∗ Δ∗∗ A ◦ ρ is the canonical embedding of A into A . Proof. (i) =⇒ (ii) follows immediately from Definition 4.5.22 through taking adjoints. (ii) =⇒ (i): Let ρ be as specified in (ii), and define ρ˜ : (A⊗γ A)∗ → A∗ to be the restriction of ρ∗ to (A⊗γ A)∗ . Clearly, ρ˜ is an A-bimodule homomorphism. To see that ρ˜ is a left inverse of Δ∗ , just observe that a, ρ˜(Δ∗ φ) = Δ∗ φ, ρ(a) = φ, Δ∗∗ (ρ(a)) = a, φ

(a ∈ A, φ ∈ A∗ ). 

This completes the proof.

The following proposition establishes a necessary condition for a Banach algebra to be biflat: Proposition 4.5.24. Let A be a Banach algebra, and let B be a closed subalgebra of A with the following properties: (a) B is amenable; (b) B is a left ideal of A; (c) B has a bounded approximate identity which is also a bounded left approximate identity for A. Then A is biflat. Proof. Since B is amenable, Exercise 2.3.9(b) yields the existence of a γ ∗∗ bounded right inverse ρB of Δ∗∗ → B∗∗ which is also a BB : (B ⊗ B) bimodule homomorphism. Let ι : B → A be the inclusion map, and set ρ˜ := (ι ⊗ ι)∗∗ ◦ ρB . Let (eα )α∈A be a bounded approximate identity for B which is also a bounded left approximate identity for A, and let U be an ultrafilter over A that dominates the order filter. Define ρA : A → (A ⊗γ A)∗∗ ,

a → weak∗ - lim ρ˜(eα ) · a. α→U

It is immediate that ρA is a right A-module homomorphism. Since (eα )α∈A is a bounded left approximate identity for A, the fact that ρB is a right inverse of Δ∗∗ B implies that

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Biflatness and Biprojectivity

193

∗ ∗∗ Δ∗∗ ρ(eα )) · a A (ρA (a)) = weak - lim ΔA (˜ α→U

= weak∗ - lim Δ∗∗ B (ρB (eα )) · a = lim eα a = a α→U

α→U

(a ∈ A),

∗∗ so that Δ∗∗ A ◦ ρA is the canonical embedding of A into A . It remains to be shown that ρA is a left A-module homomorphism. First, note that

ρA (axb) = weak∗ - lim ρ˜(eα ) · axb α→U

= weak∗ - lim axb · ρ˜(eα ), α→U

= weak∗ - lim a · ρ˜(eα ) · xb, α→U

= a · ρA (x) · b

because axb ∈ B, because xb ∈ B,

(4.42)

(a, x ∈ A, b ∈ B).

Let a, x, b ∈ A. Cohen’s Factorization Theorem yields c ∈ B and d ∈ A such that b = cd. From (4.42) and the fact that ρA is a right A-module homomorphism, we conclude that ρA (axb) = ρA (axcd) = ρA (axc) · d = a · ρA (x) · cd = a · ρA (x) · b.

(4.43)

Finally, let a, x ∈ A. Again, by Cohen’s Factorization Theorem, we obtain y ∈ B and z ∈ A such that x = yz; from (4.43), we obtain ρA (ax) = ρA (ayz) = a · ρA (y) · z = a · ρA (x). Hence, ρA is an A-bimodule homomorphism, and Lemma 4.5.23 implies that A is biflat.  In order to obtain a biflat Banach algebra which is not amenable, we should therefore search for Banach algebras which are not amenable themselves, but contain amenable Banach algebras as left ideals. Theorem 4.5.25. Let E be a Banach space with property (A) such that E ∗∗ does not have the bounded approximation property. Then A(E ∗ ) is a biflat Banach algebra which is neither amenable nor biprojective. Proof. As E has property (A), Theorem 3.3.9 ascertains that A(E) is amenable; the same is true for B := {T ∗ : T ∈ A(E)} ∼ = A(E)op . By Exercise 4.5.15, B is a left ideal in A(E ∗ ). Since A(E) is amenable, it has a bounded approximate identity, say (Sα )α . Consequently, (Sα∗ )α is a bounded approximate identity for B; this implies that Sα∗ → idE ∗ in the strong operator topology and thus uniformly on compact subsets of E ∗ . An inspection of the proof of Theorem 3.3.2 shows that this implies that (Sα∗ )α is a bounded left approximate identity for A(E ∗ ). Hence, by Proposition 4.5.24, A(E ∗ ) is biflat.

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4 Amenability-Like Properties

Assume that A(E ∗ ) is amenable. Then A(E ∗ ) has a bounded approximate identity. By Theorem 3.3.4, this means that E ∗∗ has the bounded approximation property—contrary to our hypothesis. Assume that A(E ∗ ) is biprojective. Clearly, A(E ∗ ) is a standard Banach operator algebra on E ∗ . As E ∗ has the approximation property, Theorem 4.5.18 implies that A(E ∗ ) = N (E ∗ ). In particular, N (E ∗ ) has a bounded left approximate identity. This, however, is impossible by Theorem 4.5.6  Example 4.5.26. As observed in Remark 3.3.17, 2 ⊗λ 2 has property (A), whereas (2 ⊗λ 2 )∗∗ ∼ = B(2 ) lacks the approximation property. By Theorem 4.5.25, A((2 ⊗λ 2 )∗ ) = A(2 ⊗γ 2 ) is therefore biflat, but neither amenable nor biprojective.

Exercises Exercise 4.5.1. Show that a biprojective Banach algebra A is contractible if and only if it has an identity and if and only if A# is biprojective. Exercise 4.5.2. Let 1 and 2 be equipped with coordinatewise multiplication. Show that 1 is biprojective whereas 2 is not. Exercise 4.5.3. Let A and B be biprojective Banach algebras. Show that A ⊗γ B is biprojective as well. Exercise 4.5.4. Let A be a Banach algebra with compact multiplication, i.e., for each a ∈ A, the operators A  x → ax and A  x → xa are compact. Show that, if A is amenable, it is biprojective. (Hint: Let (dα )α∈A be a bounded approximate diagonal for A, let U be an ultrafilter over A that dominates the order filter, and define ρ : A → A ⊗γ A through ρ(a) := limα→U a · dα for a ∈ A.) Exercise 4.5.5. Let A be a commutative Banach algebra. Show that: (a) if A is biprojective, then ΦA is discrete; (b) if A is a C ∗ -algebra such that ΦA is discrete, then A is biprojective. (Hint for (a): Use the fact ([71, Proposition 2.4.7]) that ΦA⊗γ A ∼ = ΦA × ΦA , and show that {(φ, φ) : φ ∈ ΦA } is open in ΦA⊗γ A .) Exercise 4.5.6. Let (E, F, ·, ·) be a dual pair of Banach spaces. Show that, if E or F has the approximation property, then the canonical quotient map from E ⊗γ F onto NF (E) is an isometric isomorphism. Exercise 4.5.7. Let (E, F, ·, ·) be a dual pair of Banach spaces. Show that: (a) T ∗ |F ∈ B(F ) for any T ∈ BF (E);

4.5

Biflatness and Biprojectivity

195

(b) BF (E) equipped with the norm T BF (E) := max{T B(E) , T ∗ |F B(F ) }

(T ∈ BF (E))

is a Banach algebra; (c) NF (E) ⊂ BF (E) such that the inclusion is a contraction if NF (E) is equipped with the F -nuclear norm; (d) NF (E) is an ideal of BF (E). Exercise 4.5.8. Let (E, F, ·, ·) be a dual pair of Banach spaces. Show that: (i) for each T∈ FF (E), there are x1 , . . . , xn ∈ E and y1 , . . . , yn ∈ F such n that T = j=1 xj yj ; (ii) FF (E) is simple; (iii) NF (E) is topologically simple. Exercise 4.5.9. Let A be a biprojective Banach algebra, and let I be a closed ideal which is essential as a left Banach A-module. Show that A/I is biprojective. Exercise 4.5.10. Let G be a locally compact group. Show that L1 (G) is biprojective if and only if G is compact. (Hint: For the “only if” part, apply Lemma 4.5.11 with L = L10 (G).) Exercise 4.5.11. Let A be a semiprime algebra, and let I be an ideal of A. Show that lan(I) = ran(I). Exercise 4.5.12. Let A be a semisimple, amenable Banach algebra with the approximation property. Show that A has compact multiplication if and only if there is a family (Nι )ι∈I of positive integers  such that A contains a dense ideal isomorphic to the algebraic direct sum ι∈I MNι with continuous projections onto the coordinates. Exercise 4.5.13. Let A be a biflat Banach algebra. Show that the linear span of {ab : a, b ∈ A} is dense in A. Exercise 4.5.14. Show that a biflat Banach algebra A is amenable if and only if it has a bounded approximate identity and if and only if A# is biflat. Exercise 4.5.15. Let E be a Banach space. Show that {T ∗ : T ∈ A(E)} is a closed left ideal of B(E ∗ ).

Notes and Comments The use of the adjective “contractible” in Definition 4.1.1 is, by all likelihood, due to a formal analogy with a basic result from algebraic topology (see [241,

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4 Amenability-Like Properties

p. 168]): a topological space is contractible if its nth singular homology group vanishes for all n ∈ N. Consequently, a Banach algebra is dubbed contractible if its nth Hochschild cohomology groups vanish for all n ∈ N (see Exercise 4.1.2). This choice of terminology is not entirely unproblematic because, in operator algebra Ktheory, a C ∗ -algebra is called contractible if idA is homotopic to the zero map ([354, Definition 6.4.1]), which is quite different from contractibility in our sense (see Corollary 4.1.6). Contractible Banach algebras also go by other names in the literature. In [342], they are called algebras of bidimension zero, and [224] refers to them as separable algebras (both [342] and [224] consider not only Banach, but more general topological algebras). In [152], it was mentioned that contractible Banach algebras are also known as super-amenable: this is how they were called in [297], for example. But somehow the term “contractible Banach algebra” seems to be the established one by now, and therefore, we use it. Theorem 4.1.2 is due to J. L. Taylor ([342, Proposition 5.11]) and appears to be the first result asserting the triviality of contractible Banach algebras under certain conditions; at the time it was proven, it wasn’t clear yet if there were any Banach spaces lacking the approximation property. Theorem 4.1.4 is due to Yu. V. Selivanov ([316]). Further results that establish the triviality of contractible Banach algebras under hypotheses weaker, but more technical than those of Theorems 4.1.2 and 4.1.4 can be found in [293] and [364], for instance; see also [297, Section 4.1]. A Banach space is said to have the uniform approximation property if each of its ultrapowers has the approximation property. In [258], N. Ozawa shows that every contractible Banach algebra acting on a Banach space with the uniform approximation property is trivial; this generalizes an older result due to V. I. Paulsen and R. R. Smith for contractible Banach algebras acting on a Hilbert space ([265]). Another amenability-like property, which lies properly between contractibility and amenability, was introduced by M. Daws in [81]. Daws defines a Banach algebra to be ultra-amenable if each of its ultrapowers is amenable. Considering fixed ultrafilters, one realizes immediately that ultra-amenable Banach algebras are amenable, and it is not difficult to see that each contractible Banach algebra is ultra-amenable. Also, if G is a discrete group, then 1 (G) is ultra-amenable if and only if G is finite, and a C ∗ -algebra is ultra-amenable if and only if it is subhomogeneous. For more details, see [81]. Weakly amenable Banach algebras were introduced by W. G. Bade, P. C. Curtis, Jr., and H. G. Dales in [14]. They only consider commutative Banach algebras and use the condition stated in Exercise 4.2.5 to define weak amenability; in the same paper, Exercise 4.2.5 is proven. They also present examples of commutative Banach algebras without nonzero point derivations that fail to be weakly amenable. Of course, due to the potential lack of symmetric modules over noncommutative Banach algebras, the definition of weak amenability from [14] is not meaningful outside the commutative framework. It was B. E. Johnson who suggested the use of Definition 4.2.1 to define weak amenability for arbitrary Banach algebras ([190]). (Outside

Notes and Comments

197

Banach algebras, the term “weakly amenable” occurs also in at least two more contexts, which are, however, entirely unrelated to that of Definition 4.2.1: see [64] and [93].) Theorem 4.2.5 was first proven by Johnson ([191]), but the proof we present is due to M. Despi´c and F. Ghahramani ([94]). Corollary 4.2.7 was first obtained by U. Haagerup ([159]), i.e., it was proven before the notion of weak amenability was even formally introduced. Subsequently, Haagerup and N. J. Laustsen gave a simpler proof ([162]). Both proofs require the so-called noncommutative Grothendieck inequality. The Grothendieck-free proof we give—using Theorem 1.5.6—is indicated in [17]. Theorem 4.2.13 is from [73]. In [121], it was conjectured that A(G) for a locally compact group G is weakly amenable if and only if the component of the identity of G is abelian. The “if” part was proven in [121], and the converse was confirmed for compact G in [124] and for connected Lie groups in [221]; in general, the “only if” part is still open. In [26] and [27], A. Blanco investigates the weak amenability of Banach algebras of approximable operators on Banach spaces (see also [30]). The weak amenability of B(E) for Banach spaces E is studied in [28]. In view of Theorem 2.3.24, it is a natural question to ask if amenability in its hypotheses can be replaced by weak amenability: it has been open to this day (see [116] for a related result). The hereditary properties of weak amenability are studied in [148] for commutative Banach algebras and in [151] in the general situation. They are not nearly as nice as those for amenability: for instance, the codimension one ideal L10 (SL(2, R)) of the weakly amenable Banach algebra L1 (SL(2, R)) is not weakly amenable whereas its unitization L10 (SL(2, R))# is again weakly amenable ([198]). In the literature, several amenability-like properties are studied that lie between amenability and weak amenability. For instance, if n ∈ N, a Banach algebra A is defined to be n-weakly amenable if H1 (A, An∗ ) = {0} where An∗ is the nth dual of A ([72]); if A is n-weakly amenable for each n ∈ N, it is called permanently weakly amenable. For example, C ∗ -algebras are permanently weakly amenable ([72]), as are group algebras ([51] and, independently, [232]). In [142], M. E. Gorgi and T. Yazdanpanah call a Banach algebra A ideally amenable if H1 (A, I ∗ ) = {0} for each closed ideal of A. In the terminology of [142], Theorem 4.2.6 thus asserts that every C ∗ -algebra is ideally amenable. (In [142], Theorem 4.2.6 is derived from Corollary 4.2.7 by elementary means.) It appears to be open if L1 (G) is ideally amenable for every locally compact group G; in fact, it seems to be unknown if 1 (F2 ) is ideally amenable or not. In [214], A. T.-M. Lau considered a class of Banach algebras he called F -algebras: these are Banach algebras that are preduals of a von Neumann algebra such that the identity element of that von Neumann algebra is a character. He defined a notion of left amenability with respect to that character. It took considerable time until Lau’s ideas were picked up again and extended to the context of general Banach algebras ([204], [203], and [245]). Our definition of a (left) φ-amenable Banach algebra appeared in [204] for

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4 Amenability-Like Properties

the first time. The notion of character amenability, i.e., the simultaneous consideration of all characters plus the zero functional, is due to M. S. Monfared ([245]). In our exposition, we mostly follow [204] whereas Theorems 4.3.9 and 4.3.10 are from [245]. The topic of character amenability is studied further in [183]. In particular, the authors prove a result in the spirit of Theorem 2.3.24 for character amenability: if K ⊂ C is compact, then the uniform closure of the polynomials on K is character amenable if and only if it equals C(K) ([183, Theorem 5.4]). Pseudo- and approximate amenability fall into the area of “generalized notions of amenability”, whose investigation was initiated by F. Ghahramani and R. J. Loy in [128]; the notion of an approximately amenable Banach algebra was introduced in that paper. If the nets of inner derivations required by Definition 4.4.6 can be chosen to be bounded, one speaks of boundedly approximately amenable Banach algebras ([131]; compare Exercise 2.1.7). Until not very long ago, it was unknown if there is any approximately amenable Banach algebra that fails to be boundedly approximately amenable; an example was eventually constructed in [134]. Pseudo-amenability was defined in [138]. Further papers devoted to the “generalized notions of amenability” are, among others, [50, 51, 75, 76, 131–133, 137, 268], where an example of a Banach space E is constructed such that A(E) is not amenable, but approximately amenable, and [129]. For a comprehensive account of the subject (up to about 2009), see Y. Zhang’s survey article [365]. Example 4.4.4 is from [137], where also the idea for Proposition 4.4.2 is indicated. Proposition 4.4.7 is taken from [131] whereas Theorem 4.4.10 and Proposition 4.4.11 are based on results in [138]. The failure of approximate amenability for p with p ∈ [1, ∞) was first established in [76]; the approach we present, i.e., via Lemma 4.4.15 and Theorem 4.4.16, is taken from [50]. The equivalence of (i) and (iii) in Theorem 4.4.19 appeared in [128] whereas the equivalence of (i) and (iv) is from [138]. In view of Proposition 2.2.1 and Exercise 4.4.3, the question of whether or not an approximately amenable Banach has an approximate identity arises naturally; it seems to be still open. “Generalized notions of amenability” allow for almost limitless combinations with other amenability-like properties. Already in [128], approximate weak amenability is considered; in [252], character pseudo-amenable Banach algebras are studied; and in [244], for instance, O. T. Mewomo, combines approximate amenability with ultra-amenability to define approximate ultraamenability (in the obvious way). It remains to be seen which “generalized notions of amenability” will turn out to be viable. The notions of biprojectivity and biflatness arise naturally in A. Ya. Helemski˘ı’s Banach homological algebra. Definitions 4.5.1 and 4.5.22 are not the original ones, but equivalent characterizations. We shall put biprojectivity and biflatness in their proper context when we discuss Banach homology in Chapter 6. Implication (iii) =⇒ (vi) of Theorem 4.5.6 is proven in [150] (com-

Notes and Comments

199

pare also [174, p. 194]). Example 4.5.3 and Exercise 4.5.10 are [171, Theorem 51]. Biprojectivity and biflatness of the Fourier algebra A(G) of a locally compact group G are investigated in [305]; a dichotomy holds: if A(G) is biflat, then either G is almost abelian or is non-amenable without containing F2 as a closed subgroup. The structure theory for semisimple biprojective Banach algebras that culminates in Theorem 4.5.21 is due to Yu. V. Selivanov ([317]); it was extended to not necessarily semisimple Banach algebras by O. Yu. Aristov ([9]). It appears to be unknown if there is a biprojective radical Banach algebra. Example 4.5.26 along with the results on biflat Banach algebras leading up to it are based on unpublished work by Selivanov ([320]). Like amenable, contractible, and weakly amenable Banach algebras, biprojective and biflat Banach algebras can be characterized through the vanishing of certain cohomology groups ([319]). In view of Exercise 4.5.3, it should be noted that the tensor product of two biflat Banach algebras is also biflat ([237]). Just as amenability, biprojectivity allows for an approximate version: in [363] and [8], (different) notions of approximate biprojectivity are defined and studied.

Chapter 5

Dual Banach Algebras

A dual Banach algebra is a Banach algebra that is also a dual Banach space such that multiplication is separately weak∗ continuous. Examples of dual Banach algebras are, among others, von Neumann algebras, the measure algebra M (G), and the Fourier–Stieltjes algebra B(G) of a locally compact group G, or the algebras B(E) of all bounded linear operators on a reflexive Banach space E. As it turns out, amenability in the sense of Definition 2.1.11 is quite a restrictive condition for these examples: • a von Neumann algebra is amenable if and only if it is subhomogeneous (Theorem 7.1.9 below); • for a locally compact group G, its measure algebra M (G) is amenable only in the obvious case where G is discrete (Theorem 3.1.1), and B(G) is amenable only in the—equally obvious—case where G has a compact abelian subgroup of finite index (Corollary 3.2.11); • no reflexive, infinite-dimensional Banach space E is known for which B(E) is amenable whereas, on the other hand, B(p ) fails to be amenable for every p ∈ (1, ∞) (Theorem 3.4.13). In order to develop a rich amenability theory for dual Banach algebras, it is necessary to modify Definition 2.1.11 in a way that takes the dual space structures involved into account: module actions are required to be separately weak∗ continuous as are all derivations. This leads to the notion of Connesamenability. In this chapter, we begin with a general introduction to Connes-amenability for dual Banach algebras and end with a discussion of the representation theory of dual Banach algebras as developed by M. Daws in [80]; in particular, we give a characterization of Connes-amenable dual Banach algebras in terms of their representation theory.

© Springer Science+Business Media, LLC, part of Springer Nature 2020 V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-1-0716-0351-2 5

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5 Dual Banach Algebras

5.1 Connes-Amenability for Dual Banach Algebras We begin with the definition of a dual Banach algebra: Definition 5.1.1. A dual Banach algebra is a pair (A, A∗ ) of Banach spaces such that: (a) A is a Banach algebra; (b) A = (A∗ )∗ ; (c) multiplication in A is separately σ(A, A∗ ) continuous. Remark 5.1.2. The seemingly pedantic formulation of Definition 5.1.1 is due to the fact that the predual A∗ need not be unique: take, for instance, a dual Banach space with more than one predual and turn it into a dual Banach algebra by defining the product of two elements to be zero. However, for the sake of simplicity, we shall mostly write: “Let A be a dual Banach algebra”; it is then understood that A comes with a specified predual A∗ . In many cases, there is a canonical (even though not necessarily unique) predual. Example 5.1.3. Let G be a locally compact group. By Theorem D.3.2, M (G) is a Banach algebra, and by Riesz’ Representation Theorem, ([56, Theorem 7.3.5]) we can identify M (G) with C0 (G)∗ . Let ν ∈ M (G), and let f ∈ C0 (G). Define  f (xy) dν(y) g : G → C, x → G

We claim that g ∈ C0 (G). As C0 (G) ⊂ UC(G) (Proposition D.4.2(iii)), it is clear that g is continuous. Let  > 0, and let K ⊂ G and L ⊂ G be compact  for x ∈ G \ K and |ν|(G \ L) < 2(f ∞ +1) . (The such that |f (x)| < 2(ν+1) existence of L is guaranteed by the inner regularity of |ν|; see [56, p. 206].) As multiplication and inversion in G are continuous, the set C := {xy −1 : x ∈ K, y ∈ L} is compact. Let x ∈ G \ C, and note that xy ∈ / K for all y ∈ L; we therefore obtain      |f (x)| ≤ |f (xy)| d|ν|(y) < |f (xy)| d|ν|(y) + < + = . 2 2 2 G L It follows that g ∈ C0 (G) as claimed. Let (μα )α be a net in M (G) such that σ(M (G),C0 (G))

→

0. Then we have    f, μα ∗ ν = f (xy) dμα (x) dν(y) = g(x) dμα (x) = g, μα → 0.

μα

G

G

G

Since f ∈ C0 (G) was arbitrary, this means that μα ∗ ν σ(M (G),C0 (G))

σ(M (G),C0 (G))

→

0; anal-

→ 0. Consequently, multiplication in ogously, we see that ν ∗ μα M (G) is separately σ(M (G), C0 (G)) continuous, so that (M (G), C0 (G)) is a dual Banach algebra.

5.1 Connes-Amenability for Dual Banach Algebras

203

Example 5.1.4. Let E be a Banach space. Then B(E ∗ ) can be canonically identified with (E ⊗γ E ∗ )∗ via x ⊗ φ, T := x, T φ

(x ∈ E, φ ∈ E ∗ , T ∈ B(E ∗ )).

In particular, if E is reflexive, we have a canonical duality between B(E) and E ∗ ⊗γ E; in this case, it is easy to verify that (B(E), E ∗ ⊗γ E) is a dual Banach algebra. (Compare Exercise 5.1.4 below.) In view of Exercise 5.1.3 below, Example 5.1.4 yields that every weak∗ closed subalgebra of B(E) for some reflexive Banach space E is a dual Banach algebra. In Section 5.4 below, we shall prove the remarkable result that every dual Banach algebra arises in this fashion. Example 5.1.5. Let H be a Hilbert space, so that, by Example 5.1.4, B(H) is a dual Banach algebra in a canonical fashion. (The weak∗ topology on B(H) is often called the ultraweak topology by von Neumann algebraists.) If M be a von Neumann algebra acting on H, it is ultraweakly closed in B(H) (Theorem C.5.1) and thus a dual Banach algebra by Exercise 5.1.3 below. (Much more remarkably, whenever a C ∗ -algebra is a dual Banach space, then it already is a von Neumann algebra, and its predual is necessarily unique; see Theorem C.5.7.) Example 5.1.6. Let G be a locally compact group, and let p ∈ (1, ∞). Then the Banach algebra PMp (G) is weak∗ closed in B(Lp (G)) and thus a dual Banach algebra. Before we begin our discussion of Connes-amenability for dual Banach algebras, we would like to have a look at another class of examples. Definition 5.1.7. Let A be a Banach algebra, and E be a closed subspace of A∗ . Then we call E left A-introverted (or simply: left introverted ) if the following are satisfied: (a) E is a right A-submodule of A∗ ; (b) for every A ∈ E ∗ and every φ ∈ E, the functional A  φ ∈ A∗ defined by a, A  φ := φ · a, A

(a ∈ A)

lies in E. Example 5.1.8. Let A be Banach algebra. Then, trivially, A∗ is left introverted. Example 5.1.9. Let A be a Banach algebra, and let E be the closed linear span of {φ · a : φ ∈ A∗ , a ∈ A} in A∗ . Then it is straightforward to check that Definition 5.1.7(a) and (b) are satisfied. In view of Proposition D.4.2(ii), this means in particular that LUC(G) is a left L1 (G)-introverted subspace of L∞ (G) for any locally compact group G.

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Example 5.1.10. Let A be a Banach algebra. We call φ ∈ A∗ weakly almost periodic if the map A → A∗ , a → a · φ is weakly compact. It follows from Grothendieck’s Double Limit Criterion ([155]) that φ is weakly almost periodic if and only if A a → φ · a is weakly compact. We define WAP(A) := {φ ∈ A∗ : φ is weakly almost periodic}. It is immediate that WAP(A) is a closed right A-submodule of A. Let A ∈ WAP(A)∗ , let φ ∈ WAP(A), and let a ∈ A. Since x, (A  φ) · a = ax, (A  φ)

= φ · ax, A = (φ · a) · x, A = x, A  (φ · a)

(a ∈ A),

we have (A  φ) · a = A  (φ · a). As A a → φ · a is weakly compact, so is A a → (A  φ) · a, i.e., A  φ ∈ WAP(A). Given a Banach algebra A and a left introverted subspace E of A∗ , we define (5.1) φ, AB := B  φ, A

(A, B ∈ E ∗ ). It is routinely verified that  turns E ∗ into a Banach algebra. Define ι : A → E ∗ via φ, ι(a) = a, φ

(a ∈ A, φ ∈ E); (5.2) its equally routinely verified that ι is an algebra homomorphism. We summarize: Proposition 5.1.11. Let A be a Banach algebra, let E be a left introverted subspace of A∗ , and let  be defined as in (5.1). Then: (i)  turns E ∗ into a Banach algebra; (ii) the canonical contraction ι : A → E ∗ is an algebra homomorphism; (iii) for every B ∈ E ∗ , the right multiplication map E ∗ A → AB is weak∗ continuous; (iv) E ∗ is a dual Banach algebra if and only if E ⊂ WAP(A). Proof. As we already noted, the verification of (i) and (ii) is routine; (iii) is immediate from (5.1). Suppose that E ∗ is a dual Banach algebra, and let φ ∈ E. Let B ∈ E ∗ weak∗ be arbitrary, and let (Aα )α be a net in E ∗ such that Aα → 0. Since E ∗ A → BA is continuous, we have Aα  φ, B = φ, BAα → 0,

5.1 Connes-Amenability for Dual Banach Algebras

i.e., the map

E ∗ → E,

205

A → A  φ

is weak∗ -weakly continuous. Since Ball(E ∗ ) is weak∗ compact, this means that {A  φ : A ∈ Ball(E ∗ )} is weakly compact in E. Consequently {a · φ : a ∈ Ball(A)} = {ι(a)  φ : a ∈ Ball(A)} ⊂ {A  φ : A ∈ Ball(E ∗ )} is relatively weakly compact, which means that φ ∈ WAP(A). Conversely, suppose that E ⊂ WAP(A). Let φ ∈ E, and let C be the weak closure in A∗ of {a · φ : a ∈ Ball(A)}. Then C is weakly compact, and consequently, the weak and the weak∗ topology on A∗ coincide on C. We claim that {A  φ : A ∈ Ball(E ∗ )} ⊂ C. ˜ E = A. To see this let A ∈ Ball(E ∗ ), and let A˜ ∈ Ball(A∗∗ ) be such that A| σ(A∗∗ ,A∗ ) ˜ For b ∈ A, we then Choose a net (aα )α in Ball(A) such that aα → A. have ˜ = φ · b, A = A  φ, b. aα · φ, b = ι(aα )  φ, b = φ · b, ι(aα ) → φ · b, A

As C is weak∗ closed in A∗ , this proves the claim. Let B ∈ E ∗ , and let σ(E ∗ ,E)

(Aα )α be a net in Ball(E ∗ ) such that Aα → 0. It is straightforward that Aα φ → 0 in the weak∗ topology of A∗ and therefore—because this topology coincides with the weak topology on C—in the weak topology of A∗ . Hence, we have φ, BAα = Aα  φ, B → 0. Since φ ∈ E ∗ is arbitrary, this means that E∗ → E∗,

A → BA

(5.3)

ˇ is σ(E ∗ , E) continuous on Ball(A) for any B ∈ E ∗ . From the Kre˘ın–Smulian ∗ Theorem, we conclude that (5.3) is σ(E , E) continuous for any B∈E ∗ .  Before we begin our discussion of Connes-amenability for dual Banach algebras, we would like to have a closer look at an example that will be of particular interest in Section 5.3: Example 5.1.12. Let G be a locally compact group. Then we can identify the C ∗ -algebra WAP(G) introduced in Definition D.4.3 with a subspace of L∞ (G) = L1 (G)∗ . We will now show that this subspace is precisely WAP(L1 (G)), so that we can use the symbols WAP(G) and WAP(L1 (G)) interchangeably. To prove that WAP(G) ⊂ WAP(L1 (G)), let φ ∈ WAP(G), and let C be the weakly closed, absolutely convex hull of {Lx φ : x ∈ G} in C(G). Then C is weakly compact in C(G) ([102, Theorem V.6.4]) and thus in L∞ (G); consequently, the weak and the weak∗ topology of L∞ (G) coincide on C. The module action of L1 (G) on L∞ (G) canonically extends to M (G):

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this is due to the fact that M (G) contains L1 (G) as a closed ideal. We will show that {φ · μ : μ ∈ Ball(M (G))} is contained in C. Since Lx φ = φ · δx for x ∈ G, it is clear that φ · μ ∈ C if μ ∈ Ball(1 (G)). Let μ ∈ Ball(M (G)) be arbitrary, and note that we can—via integration—isometrically embed M (G) into C(G)∗ . A standard Hahn–Banach argument yields a net (μα )α in Ball(1 (G)) such that limf, μα = f, μ

α

(f ∈ C(G)).

Let g ∈ L1 (G) be arbitrary, and recall (Proposition D.4.2(ii)) that g · φ ∈ UC(G) ⊂ C(G); we obtain g, φ · μα = μα ∗ g, φ = g · φ, μα → g · φ, μ = g, φ · μ . As g ∈ L1 (G) was arbitrary, it follows that φ · μ lies in the σ(L∞ (G), L1 (G)) closure of C in L∞ (G), but since C is compact in that topology, we do, in fact, have φ · μ ∈ C. This proves that WAP(G) ⊂ WAP(L1 (G)). For the reversed inclusion, let φ ∈ WAP(L1 (G)), and let K be the weak (= norm) closure of {φ · f : f ∈ Ball(L1 (G))}. Since φ ∈ WAP(L1 (G)), the set K is weakly compact, so that the weak topology and σ(L∞ (G), L1 (G)) coincide on K; in particular, K is σ(L∞ (G), L1 (G)) compact and thus σ(L∞ (G), L1 (G)) closed in L∞ (G). Let (eα )α be an approximate identity for L1 (G) in Ball(L1 (G)), and note that g, φ · eα = eα ∗ g, φ → g, φ

(g ∈ L1 (G)),

i.e., φ ∈ K. From Cohen’s Factorization Theorem, we conclude that there are ψ ∈ L∞ (G) and g ∈ Ball(L1 (G)) such that φ = ψ · g; in particular, φ lies in LUC(G) ⊂ C(G). Finally, we note that {Lx φ : x ∈ G} = {ψ · (g ∗ δx ) : x ∈ G} ⊂ {ψ · h : h ∈ Ball(L1 (G))}, so that φ belongs to WAP(G). We now take a look at dual Banach algebras with an eye on their amenability: Example 5.1.13. Let G be locally compact group. By Theorem 3.1.1, the dual Banach algebra M (G) is amenable if and only if G is discrete and amenable. Example 5.1.14. Already in Section 3.4, we had seen that the von Neumann algebra B(2 ) is not amenable. In fact, amenability is a surprisingly restrictive condition to impose on a von Neumann algebra (see Section 7.1 below): if M is an amenable von Neumann algebra, then there are commutative von Neumann algebras A1 , . . . , An as well as N1 , . . . , Nn ∈ N such that M∼ = MN1 (A1 ) ⊕∞ · · · ⊕∞ MNn (An ).

5.1 Connes-Amenability for Dual Banach Algebras

207

The stock of amenable dual Banach algebras appears to be surprisingly small, and it seems that a notion of amenability that takes the given weak∗ topology into account is more appropriate for dual Banach algebras. Definition 5.1.15. Let A be a dual Banach algebra. Then: (a) a dual left Banach A-module is called normal if, for each x ∈ E, the map A → E,

a→a·x

is weak∗ continuous; (b) a dual right Banach A-module is called normal if, for each x ∈ E, the map A → E, a → x · a is weak∗ continuous; (c) a dual Banach A-bimodule is called normal if it is normal both as left and a right Banach A-module. Remark 5.1.16. As for dual Banach algebras, we also simply speak of the weak∗ topology of a Banach module: there is no need for it to be unique, but all dual Banach modules are to be understood to come with a fixed (and often canonical) predual. Definition 5.1.17. A dual Banach algebra A is called Connes-amenable if, for every normal, dual Banach A-bimodule E, every weak∗ continuous derivation D : A → E is inner. We shall see in the remainder of this chapter, as well as in Chapter 7 that Connes-amenability is indeed the “right” notion of amenability for dual Banach algebras in the sense that it includes a multitude of interesting examples, but at the same time still allows for a substantial general theory. The notion of Arens regularity is standard in Banach algebra theory (see [71] and [260] for detailed discussions). Our definition below is not the usual one, but easily seen to be equivalent: Definition 5.1.18. A Banach algebra A is called Arens regular if WAP(A) = A∗ . Example 5.1.19. All C ∗ -algebras are Arens regular ([71, Corollary 3.2.27]) whereas L1 (G) for a locally compact group G is Arens regular if and only if G is finite ([71, Theorem 3.3.28]). Moreover, if E is a Banach space, then A(E) is Arens regular if and only if E is reflexive ([71, Theorem 2.6.23]). Suppose that A is an amenable, Arens regular Banach algebra. Then A∗∗ is a dual Banach algebra, which is—as an immediate consequence of Exercise 5.1.9(a) below—Connes-amenable. Does the converse hold, and does the Connes-amenability of A∗∗ force A to be amenable? As we shall see in Chapter 7, this is indeed true if A is a C ∗ -algebra. For general Banach algebras, we have the following:

208

5 Dual Banach Algebras

Theorem 5.1.20. Let A be an Arens regular Banach algebra such that A is an ideal in A∗∗ . Then the following are equivalent: (i) A is amenable; (ii) A∗∗ is Connes-amenable. Proof. Only (ii) =⇒ (i) needs proof. We first claim that A has a bounded approximate identity. Since A∗∗ is Connes-amenable, it has an identity by Exercise 5.1.8 below. Let (eα )α be a bounded net in A converging to that identity in the weak∗ topology of A∗∗ . As multiplication in A∗∗ is separately weak∗ continuous, and since σ(A∗∗ , A∗ ) restricted to A is just σ(A, A∗ ), we have aeα

σ(A,A∗ )

→

a and eα a

σ(A,A∗ )

→

a

(a ∈ A).

(5.4)

Passing to convex combinations, we can replace the weak limits in (5.4) by norm limits and conclude that A has indeed a bounded approximate identity. By Proposition 2.1.9, it is thus sufficient for A to be amenable that every derivation D : A → E ∗ is inner whenever E is a neo-unital Banach Abimodule. Let E be a neo-unital Banach A-bimodule, and let D : A → E ∗ be a derivation. By Proposition 2.1.6, the bimodule action of A on E ∗ extends ˜ : A∗∗ → E ∗ . canonically to A∗∗ , and D extends uniquely to a derivation D ∗ ∗∗ We claim that E is a normal, dual Banach A -bimodule. To see this, let weak∗ (Aα )α∈A be a net in A∗∗ such that Aα → 0, let φ ∈ E ∗ , and let x ∈ E be arbitrary. Since E is neo-unital, there are b ∈ A and y ∈ E such that x = y · b. As A is an ideal in A∗∗ , we have bAα ∈ A for α ∈ A. Since the weak∗ topology weakly of A∗∗ restricted to A is σ(A, A∗ ), we have bAα → 0, so that x · Aα = y · bAα

weakly

→ 0,

and consequently x, Aα · φ = x · Aα , φ → 0. weak∗

Since x ∈ E was arbitrary, this means that Aα · φ → 0. Proving that weak∗ φ · Aα → 0 is done analogously. ˜ is w∗ -continuous, let again (Aα )α be a net in A∗∗ such that To see that D weak∗

Aα → 0, let x ∈ E, and let b ∈ A and y ∈ E be such that x = b · y. We obtain that ˜ α = b · y, DA ˜ α = y, (DA ˜ α ) · b = y, D(Aα b) − Aα · Db → 0 x, DA because D is weakly continuous and E ∗ is a normal, dual Banach A∗∗ ˜ bimodule. From the Connes-amenability of A∗∗ we conclude that D—and hence D—is inner. 

5.1 Connes-Amenability for Dual Banach Algebras

209

If E is a reflexive Banach space that also has the approximation property, then the Banach algebra A(E)∗∗ can be canonically identified with B(E) ([260, 1.7.13, Corollary]); in particular, A(E) is an ideal in A(E)∗∗ . The easy direction of Theorem 5.1.20 together with Theorem 3.3.9 thus yields: Corollary 5.1.21. Let E be a reflexive Banach space with property (A). Then B(E) is Connes-amenable. On the other hand, the non-trivial direction of Theorem 5.1.20 shows that there are reflexive Banach spaces for which B(E) is not Connes-amenable: Example 5.1.22. If p, q ∈ (1, ∞) \ {2} be such that p = q. In Example 3.3.21, we had seen that the Banach algebra A(p ⊕ q ) is not amenable. Since p ⊕ q is reflexive and has the approximation property, A(p ⊕ q ) is Arens regular with A(p ⊕ q )∗∗ ∼ = B(p ⊕ q ). It, therefore, follows from Theorem 5.1.20 p q that B( ⊕  ) is not Connes-amenable. Definition 5.1.23. Let A be a Banach algebra, and let B be a closed subalgebra of A. A quasi-expectation Q : A → B is a projection from A onto B satisfying Q(axb) = a(Qx)b (a, b ∈ B, x ∈ A). If S is any subset of an algebra A, we use ZA (S) to denote the centralizer of S in A, i.e., ZA (S) := {a ∈ A : as = sa for all s ∈ S}. In case A = B(E) for some Banach space E, we also write S  instead of ZB(E) (S). Theorem 5.1.24. Let A be a Banach algebra, let B be a dual Banach algebra, let θ : A → B be an algebra homomorphism, and suppose that one of the following holds: (a) A is amenable; (b) A is a Connes-amenable, dual Banach algebra, and θ is weak∗ -weak∗ continuous. Then there is a quasi-expectation Q : B → ZB (θ(A)). Proof. Let B∗ be the given predual of B, and let E := B ⊗γ B∗ be equipped with the A-bimodule action given by a · (b ⊗ φ) := b ⊗ θ(a) · φ

and

(b ⊗ φ) · a := b ⊗ φ · θ(a)

(a ∈ A, φ ∈ B∗ b ∈ B).

Identifying E ∗ with B(B) as in Example 5.1.4, we see that the corresponding dual B-bimodule action on B(B) is given by (a · T )(b) = θ(a)(T b)

and

(T · a)(b) = (T b)θ(a) (a ∈ A, T ∈ B(B), b ∈ B).

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5 Dual Banach Algebras

If (b) holds, then it is straightforward to check that E ∗ is a normal, dual Banach A-bimodule. Let F be the closed linear span in E of all elements of the form ⎧ ⎫ ⎨ zb ⊗ φ − b ⊗ φ · z ⎬ bz ⊗ φ − b ⊗ z · φ (b ∈ B, φ ∈ B∗ , z ∈ ZB (θ(A)). ⎩ ⎭ z⊗φ The annihilator F ⊥ of F in E ∗ is then a weak∗ closed A-submodule of E ∗ and thus a dual Banach A-module in its own right; also, F is normal if (b) holds. Define D := adidB . If (b) holds, then D is weak∗ continuous. We claim that D attains its values in F ⊥ . To see this, let a ∈ A, b ∈ B, φ ∈ B∗ , and z ∈ ZB (θ(A)). First, note that zb ⊗ φ − b ⊗ φ · z, adidB a

= zb · θ(a) ⊗ φ − b ⊗ φ · zθ(a), idB − zb ⊗ θ(a) · φ − b ⊗ θ(a) · φ · z, idB

= φ, θ(a)zb − φ, zθ(a)b − φ, zbθ(a) + φ, zbθ(a)

= 0. Analogously, one obtains bz ⊗ φ − b ⊗ z · φ, adidB a = 0. Finally, we have z ⊗ φ, adidB a = φ, θ(a)z − zθ(a) = 0. If (a) or (b) holds, the definitions of amenability and Connes-amenability, respectively, yield P ∈ F ⊥ such that D = adP . We claim that Q := idB −P is the desired quasi-expectation. First, it is immediate that Q(B) ⊂ ZB (θ(A)). Secondly, as z ⊗ φ, P = 0

(z ∈ ZB (θ(A)), φ ∈ B∗ ),

it follows that Q is the identity on ZB (θ(A)) and thus a projection onto ZB (θ(A)). Finally, for b ∈ B, φ ∈ B∗ , and z ∈ ZB (θ(A)), we have 0 = zb ⊗ φ − b ⊗ φ · z, P = φ, P(zb) − z(Pb) , so that P(zb) = z(Pb); analogously, we see that P(bz) = (Pb)z. Hence, Q is a quasi-expectation onto ZB (θ(A)) as claimed.  We apply Theorem 5.1.24 to relate the amenability of locally compact groups to the Connes-amenability dual Banach algebras of pseudo-measures. Definition 5.1.25. A locally compact group G is inner amenable if there is a mean M on L∞ (G) such that

5.1 Connes-Amenability for Dual Banach Algebras

δx · φ · δx−1 , M = φ, M

211

(x ∈ G, φ ∈ L∞ (G)).

Example 5.1.26. By Theorem 1.1.9, every amenable, locally compact group is inner amenable, but so is every discrete group: chose M = δe . Theorem 5.1.27. For a locally compact group G, consider the following statements: (i) G is amenable; (ii) M (G) is Connes-amenable; (iii) PMp (G) is Connes-amenable for all p ∈ (1, ∞); (iv) VN(G) is Connes-amenable; (v) there is p ∈ (1, ∞) such that PMp (G) is Connes-amenable. Then: (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) =⇒ (v). Furthermore, if G is inner amenable, then (v) =⇒ (i) holds, too. Proof. (i) =⇒ (ii) is a consequence of Theorem 2.1.10 and Exercise 5.1.9(a) below, and (ii) =⇒ (iii) follows from Exercise 5.1.9(b) below; (iii) =⇒ (iv) =⇒ (v) are trivial. In view of this, only the furthermore part still needs proof. Suppose that G is inner amenable. By Exercise 5.1.11, there is a net (mα )α∈A of non-negative norm one functions in L1 (G) such that δx ∗ mα ∗ δx−1 − mα 1 → 0

(x ∈ G),

or, equivalently, λ1 (x)mα − ρ1 (x−1 )mα 1 → 0 Let q ∈ (1, ∞) be such that 1

1 p

+

1 q

(x ∈ G).

(5.5) 1

= 1. For α ∈ A, set ξα := mαp and

ηα := mαq , so that ξα ∈ Lp (G) and ηα ∈ Lq (G). It follows from (5.5) and Exercise 1.1.7 that λp (x)ξα −ρp (x−1 )ξα p → 0 and λq (x)ηα − ρq (x−1 )ηα q → 0

(x ∈ G). (5.6)

For φ ∈ L∞ (G), let Mφ ∈ B(Lp (G)) be the corresponding multiplication operator, i.e., Mφ ξ := φξ for all ξ ∈ Lp (G). By Theorem 5.1.24—with A = PMp (G), B = B(Lp (G)), and θ the inclusion of PMp (G) in B(Lp (G))— there is a quasi-expectation Q : B(Lp (G)) → PMp (G) . For α ∈ A, define Mα ∈ L∞ (G)∗ by letting φ, Mα := Q(Mφ )ξα , ηα

(φ ∈ L∞ (G)).

Let U be an ultrafilter over A that dominates the order filter, and define

212

5 Dual Banach Algebras

(φ ∈ L∞ (G));

φ, M := lim φ, Mα

α→U

it is clear that 1, M = 1. It is routinely verified that ρp (x)Mφ ρp (x−1 ) = MRx φ

(x ∈ G, φ ∈ L∞ (G)).

(5.7)

For x ∈ G and φ ∈ L∞ (G), we obtain: Rx φ, M = lim Rx φ, Mα

α→U

= lim Q(MRx φ )ξα , ηα

α→U

= lim Q(ρp (x)Mφ ρp (x−1 ))ξα , ηα ,

by (5.7),

= lim ρp (x)(QMφ )ρp (x−1 )ξα , ηα ,

because ρp (G) ⊂ PMp (G) ,

α→U α→U

= lim (QMφ )ρp (x−1 )ξα , ρq (x−1 )ηα

α→U

= lim (QMφ )λp (x)ξα , λq (x)ηα , α→U

by (5.6),

= lim λp (x−1 )(QMφ )λp (x)ξα , ηα

α→U

= lim (QMφ )ξα , ηα

α→U

= φ, M . Even though M need not be a mean, the existence of a right invariant mean on L∞ (G) now follows from the “right version” of Lemma 1.1.18. By Theorem 1.1.9, G is amenable.  Remark 5.1.28. As we shall see in the next section, (i) and (ii) in Theorem 5.1.27 are equivalent even without the hypothesis of inner amenability. Remark 5.1.29. For (v) =⇒ (i) in Theorem 5.1.27, the hypothesis of inner amenability cannot be dropped: the group SL(2, R) is not amenable, but VN(SL(2, R)) is Connes-amenable ([262, p. 46]). We conclude this section with an analog of Theorem 2.2.5, i.e., a characterization of Connes-amenability in terms of diagonal type elements. For any dual Banach algebra A, let Cσ2 (A, C) denote the separately weak∗ continuous elements of C 2 (A, C) ∼ = (A⊗γ A)∗ . Clearly, Cσ2 (A, C) is a closed sub2 module of C (A, C); as a consequence, Cσ2 (A, C)∗ carries a canonical Banach A-bimodule structure, which makes it a quotient module of (A ⊗γ A)∗∗ . Since multiplication in a dual Banach algebra is separately weak∗ continuous, it is clear that Δ∗A A∗ ⊂ Cσ2 (A, C), so that Δ∗∗ A induces a Banach A-bimodule homomorphism Δσ : Cσ2 (A, C)∗ → A. Definition 5.1.30. Let A be a dual Banach algebra. Then D ∈ Cσ2 (A, C)∗ is called a normal, virtual diagonal for A if a · D = D · a and aΔσ D = a

(a ∈ A).

5.1 Connes-Amenability for Dual Banach Algebras

213

It is not too difficult to show that the existence of a normal, virtual diagonal implies Connes-amenability (see Corollary 5.1.35 below), and for many classes of dual Banach algebras, the converse is true as well (see Sections 5.2 and 7.5 below). As we shall see, however, in Section 5.3 below, there are Connes-amenable, dual Banach algebras that lack a normal, virtual diagonal. In order to obtain an analog of Theorem 2.2.5, we, therefore, need to consider a different kind of diagonal type element. We begin with a definition: Definition 5.1.31. Let A be a dual Banach algebra, and let E be a Banach A-bimodule. An element x ∈ E is called σ-weakly continuous if the maps a·x A → E, a → x·a are weak∗ -weakly continuous. We denote the collection of all σ-weakly continuous elements of E by Cσ,w (E). We refer to Exercise 5.1.12 below for some elementary properties of σweakly continuous elements. Let A be a dual Banach algebra. Then A is a normal, dual Banach Abimodule, so that A∗ = Cσ,w (A∗ ) ⊂ Cσ,w (A∗ ) by Exercise 5.1.12(d) below. From Exercise 5.1.12(b), we conclude that Δ∗ maps A∗ into Cσ,w ((A ⊗γ A)∗ ). Consequently, Δ∗∗ induces a homomorphism Δσ,w : Cσ,w ((A ⊗γ A)∗ )∗ → A. We define: Definition 5.1.32. Let A be a dual Banach algebra. Then D ∈ Cσ,w ((A ⊗γ A)∗ )∗ is called a Cσ,w -virtual diagonal for A if a · D = D · a and aΔσ,w D = a

(a ∈ A).

As it turns out, the existence of a Cσ,w -virtual diagonal characterizes the Connes-amenable dual Banach algebras: Theorem 5.1.33. The following are equivalent for a dual Banach algebra A: (i) A is Connes-amenable; (ii) there is a Cσ,w -virtual diagonal for A. Proof. (i) =⇒ (ii): First, note that A⊗γ A embeds canonically into Cσ,w ((A⊗γ A)∗ )∗ . Define a derivation D : A → Cσ,w ((A ⊗γ A)∗ )∗ ,

a → a ⊗ e − e ⊗ a (= a · (e ⊗ e) − (e ⊗ e) · a).

By Exercise 5.1.12(c), the dual Banach A-bimodule Cσ,w ((A ⊗γ A)∗ )∗ is normal, so that, in particular, D is weak∗ continuous. Obviously, D attains its values in the weak∗ closed submodule ker Δσ,w of Cσ,w ((A ⊗γ A)∗ )∗ , which is thus a normal, dual Banach A-bimodule in its own right. Hence, there is

214

5 Dual Banach Algebras

E ∈ ker Δσ,w such that D = adE . Setting D := eA ⊗ eA − E, we obtain a Cσ,w -diagonal for A. (ii) =⇒ (i): Clearly, (ii) implies that A has an identity. Let E be Banach A-bimodule such that E ∗ is normal; without loss of generality, suppose that E is unital (see Exercise 5.1.10 below). For a weak∗ continuous derivation D : A → E ∗ , define θD : A ⊗γ A → E ∗ ,

a ⊗ b → a · Db.

∗ By Lemma 5.1.34 below, θD maps E into Cσ,w ((A ⊗γ A)∗ ), and, consequently, ∗ ∗ ΘD := (θD |E ) maps Cσ,w ((A ⊗γ A)∗ )∗ into E ∗ . Let D ∈ Cσ,w ((A ⊗γ A)∗ )∗ be a Cσ,w -virtual diagonal for A, and set φ := ΘD (D). We claim that D = adφ . To see this, first note that

θD (x · a) = Δx · Da + θD (x) · a

(a ∈ A, x ∈ A ⊗γ A),

which can easily be checked if x is an elementary tensor and then follows for general x by linearity and continuity. This, in turn, entails that ΘD (X · a) = Δσ,w X · Da + ΘD (X) · a (a ∈ A, X ∈ Cσ,w ((A ⊗γ A)∗ )∗ ) (5.8) We thus obtain x, φ · a = x, ΘD (D) · a

= x, ΘD (D · a) − Da ,

by (5.8) and because E ∗ is unital,

= x, ΘD (a · D) − x, Da

= x, a · ΘD (D) − x, Da

= x, a · φ − x, Da

(a ∈ A, x ∈ E), 

i.e., indeed D = adφ .

To complete the proof of Theorem 5.1.33, we require the technical Lemma 5.1.34 below. To make its proof more transparent, we introduce new notation: given a dual Banach algebra A and a left Banach A-module E, we define LC σ,w (E) := {x ∈ E : A a → a · x is weak

∗

-weakly continuous };

similarly, we define RC σ,w (E) for a right Banach A-module E. It is obvious that LC σ,w (E) ∩ RC σ,w (E) equals Cσ,w (E) if E is a Banach A-bimodule. Lemma 5.1.34. Let A be a dual Banach algebra with identity, let E be a unital Banach A-bimodule such that E ∗ is normal, and let D : A → E ∗ be a weak∗ continuous derivation. Then the adjoint of θD : A ⊗γ A → E ∗ ,

a ⊗ b → a · Db.

5.1 Connes-Amenability for Dual Banach Algebras

215

maps E into Cσ,w ((A ⊗γ A)∗ )∗ . Proof. As E ∗ is normal, we have E = RC σ,w (E) by Exercise 5.1.12(d). As θD ∗ is obviously a homomorphism of left Banach A-modules, θD is a homomorphism of right Banach A-modules. It is immediate that analogs of Exercise 5.1.12(b) hold for left and right Banach modules, respectively. Hence, we obtain that ∗ ∗ θD (E) ⊂ θD (RC σ,w (E ∗∗ )) ⊂ RC σ,w ((A ⊗γ A)∗ ). ∗ It remains to be shown that θD (E) ⊂ LC σ,w ((A ⊗γ A)∗ ) as well. Set R := eA ⊗ A. Then R is a closed submodule of the right Banach A-module A ⊗γ A such that we have a direct sum decomposition A ⊗γ A = ker Δ ⊕ R of right Banach A-modules. Consequently, there is a direct sum decomposition (A ⊗γ A)∗ = (ker Δ)∗ ⊕ R∗ of left Banach A-modules. Set θ1 := θD |ker Δ and ∗ = θ1∗ ⊕ θ2∗ . We will θ2 := θD |R , so that θD = θ1 ⊕ θ2 and, consequently, θD ∗ ∗ γ ∗ show that both θ1 and θ2 map E into LC σ,w ((A ⊗ A) ). We claim that θ1∗ is a homomorphism of left Banach A-modules; of course, this is equivalent to the saying that θ1 is a homomorphism of right Banach ∞ A modules. To see this,

let a ∈ A and x ∈ ker Δ. Choose sequences

∞ (an )n=1 ∞ ∞ and (bn )n=1 in A with n=1 an bn  < ∞ such that x = n=1 an ⊗ bn . We obtain that

θ1 (x · a) = θD (x · a) ∞  = an · D(bn a) = =

n=1 ∞  n=1 ∞ 

an · (bn Da + (Dbn ) · a) an · Dbn · a

n=1

= θD (x) · a = θ1 (x) · a. ∗ Arguing the way we did to obtain the inclusion θD (E) ⊂ RC σ,w ((A⊗γ A)∗ ), ∗ γ ∗ we see that θ1 (E) ⊂ LC σ,w ((A ⊗ A) ). To see that θ2∗ (E) ⊂ LC σ,w ((A ⊗γ A)∗ ) as well, denote the map A a → eA ⊗ a ∈ R by ρ; obviously, ρ is an isomorphism of right Banach Amodules. It is equally obvious that θD ◦ ρ = D, i.e., θ2 = D ◦ ρ−1 , and thus θ2∗ = (ρ−1 )∗ ◦D∗ . As D is weak∗ continuous, we have D∗ E ⊂ A∗ ⊂ LC σ,w (A∗ ) (see Exercise 5.1.13 below). Since (ρ−1 )∗ is an isomorphism of left Banach A-modules, this means that

216

5 Dual Banach Algebras

θ2∗ (E) = (ρ−1 )∗ (DE) ⊂ (ρ−1 )∗ (LC σ,w (A∗ )) = LC σ,w (R∗ ) ⊂ LC σ,w ((A ⊗γ A)∗ ) 

as claimed.

Corollary 5.1.35. Let A be a dual Banach with a normal, virtual diagonal. Then A is Connes-amenable. Proof. We claim that Cσ,w ((A ⊗γ A)∗ ) ⊂ Cσ2 (A, C) (with the canonical identifications in place, of course). Fix b ∈ A, and define ρb : A → A ⊗ A,

a → b ⊗ a.

Then ρb is a homomorphism of right Banach A-modules, so that ρ∗b is a homomorphism of left Banach A-modules and, consequently, ρ∗b (LC σ,w ((A ⊗γ A)∗ ) ⊂ LC σ,w (A∗ ); it is immediate from Definition 5.1.30 that A has an identity, so that LC σ,w (A∗ ) = A∗ by Exercise 5.1.13 below. In view of the definition of ρb and the fact that b was arbitrary, this means that every element of Cσ,w ((A ⊗γ A)∗ ) is weak∗ continuous in the second variable. Analogously, one sees that the same is true for the first variable. Hence, if D ∈ Cσ2 (A, C)∗ is a normal, virtual diagonal for A, its restriction to Cσ,w ((A ⊗γ A)∗ ) is a Cσ,w -virtual diagonal. By Theorem 5.1.33, this means that A is Connes-amenable.  Even though the existence of a normal, virtual diagonal is a condition stronger than Connes-amenability, there is an analog of Theorem 5.1.33 for normal, virtual diagonals, i.e., a characterization of their existence through the innerness of certain derivations: see Exercise 5.1.15 below.

Exercises Exercise 5.1.1. (a) Let (A, A∗ ) be a dual Banach algebra. Show that (the canonical image of) A∗ is a submodule of A∗ . (b) Let A be a Banach algebra, and suppose that there is a submodule E of A∗ such that the canonical map from A into E ∗ is an isometric isomorphism. Show that (A, E) is a dual Banach algebra. Exercise 5.1.2. Let G be a locally compact group. Show that (B(G), C ∗ (G)) and (Br (G), Cr∗ (G)) are dual Banach algebras. Exercise 5.1.3. Let (A, A∗ ) be a dual Banach algebra, let B a σ(A, A∗ )closed subalgebra of A, and let ⊥ B := {φ ∈ A∗ : φ|B ≡ 0}. Show that (B, ⊥ B) is a dual Banach algebra.

5.1 Connes-Amenability for Dual Banach Algebras

217

Exercise 5.1.4. Let E be a Banach space. Show that (B(E ∗ ), E ⊗γ E ∗ ) is a dual Banach algebra if and only if E is reflexive. Exercise 5.1.5. Let A be a Banach algebra, and let E be a closed right A-submodule of A∗ . Show that E is left introverted if and only if, for each φ ∈ E, the σ(A∗ , A) closure of {a · φ : a ∈ Ball(A)} in A∗ is contained in E. Conclude that E is left introverted if it is σ(A∗ , A) closed or contained in WAP(A). Exercise 5.1.6. Let A be a Banach algebra, and let E and F be left Aintroverted subspaces of A∗ such that F ⊂ E. Show that the restriction map from E ∗ onto F ∗ is an algebra homomorphism. Exercise 5.1.7. Let A be a Banach algebra, and let ι : A → WAP(A)∗ be the canonical map (5.2). Show that, for any dual Banach algebra B and for any algebra homomorphism θ : A → B there is a weak∗ continuous algebra homomorphism Θ : WAP(A)∗ → B, such that the diagram ι / WAP(A)∗ A II II II II II Θ θ I$  B

commutes. Exercise 5.1.8. Let A be a Connes-amenable dual Banach algebra. Show that A is unital. Exercise 5.1.9. Let A be a Banach algebra, let B be a dual Banach algebra, and let θ : A → B be a continuous homomorphism with weak∗ dense range. Show that: (a) if A is amenable, then B is Connes-amenable; (b) if A is dual and Connes-amenable, and if θ is weak∗ continuous, then B is Connes-amenable. Exercise 5.1.10. Let A be a dual Banach algebra with identity. Show that A is Connes-amenable if and only if, for every unital, normal, dual Banach A-bimodule E, every weak∗ -continuous derivation D : A → E is inner. Exercise 5.1.11. Let G be a locally compact group. Show that G is inner amenable if and only if there is a net (mα )α of non-negative functions of norm one in L1 (G) such that δx ∗ mα ∗ δx−1 − mα 1 → 0 for all x ∈ G. Exercise 5.1.12. Let A be a dual Banach algebra, and let E be a Banach A-bimodule. Show that: (a) Cσ,w (E) is a closed submodule of E;

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5 Dual Banach Algebras

(b) if F is another Banach A-bimodule and θ : E → F is a bounded Abimodule homomorphism, then θ(Cσ,w (E)) ⊂ Cσ,w (F ) holds; (c) Cσ,w (E)∗ is a normal, dual Banach A-bimodule; (d) E ∗ is normal if and only if E = Cσ,w (E). Exercise 5.1.13. Let (A, A∗ ) be a dual Banach algebra. Show that A∗ ⊂ Cσ,w (A∗ ) and that A∗ = LC σ,w (A∗ ) = RC σ,w (A∗ ) if A has an identity. Also, give an example of a dual Banach algebra (A, A∗ ) for which A∗  Cσ,w (A∗ ). Exercise 5.1.14. Let A be a dual Banach algebra. Show that Cσ,w (A∗ ) ⊂ WAP(A), and give an example showing that this inclusion can be proper. (Hint: What can we say about Cσ,w (M (G)∗ ) and WAP(M (G)) for a locally compact group G?) Exercise 5.1.15. Show that the following are equivalent for a dual Banach algebra A: (i) A has a normal, virtual diagonal; (ii) for every Banach A-bimodule E, every weak∗ continuous derivation D : A → E ∗ is inner.

5.2 The Case of the Measure Algebra By Theorem 3.1.1, the measure algebra M (G) of a locally compact group G is amenable if and only if G is discrete and amenable, i.e., the amenability of M (G) imposes rather strong restrictions on G. In this section, we shall see that the situation looks quite different if we do not require M (G) to be amenable, but Connes-amenable. We shall prove: Theorem 5.2.1. The following are equivalent for a locally compact group G: (i) G is amenable; (ii) M (G) is Connes-amenable; (iii) M (G) has a normal, virtual diagonal. The implication (i) =⇒ (ii) was already observed as part of Theorem 5.1.27, and (iii) =⇒ (ii) is immediate by Corollary 5.1.35. We shall prove that not only (i) =⇒ (ii), but already (i) =⇒ (iii), and that (ii) =⇒ (i). If G is a locally compact group such that M (G) has a normal, virtual diagonal, then this normal, virtual diagonal is, by definition, an element of Cσ2 (M (G), C)∗ . Our first step towards a proof of Theorem 5.2.1 is thus to identify Cσ2 (M (G), C) as a space of functions on G × G. Definition 5.2.2. Let X and Y be locally compact Hausdorff spaces. A bounded function f : X × Y → C is called separately C0 if:

5.2 The Case of the Measure Algebra

219

(a) for each x ∈ X, the function Y → C,

y → f (x, y)

belongs to C0 (Y ); (b) for each y ∈ Y , the function X → C,

x → f (x, y)

belongs to C0 (X). We denote the collection of all separately C0 -functions by SC 0 (X × Y ). It is straightforward to verify that SC 0 (X × Y ) is a C ∗ -subalgebra of  (X × Y ). The following lemma shows that Definition 5.2.2 is actually stronger than it appears at first glance: ∞

Lemma 5.2.3. Let X and Y be locally compact Hausdorff spaces, and let f ∈ SC 0 (X × Y ). Then the following hold: (i) for each μ ∈ M (X), the function Y → C,

y →

belongs to C0 (Y ); (ii) for each ν ∈ M (Y ), the function X → C,

x →

 f (x, y) dμ(x) X

 f (x, y) dν(y) Y

belongs to C0 (X). Proof. We only prove (i). Let μ ∈ M (X). As μ is inner regular, there is no loss of generality if we suppose that X is compact. Suppose that Y is not compact, and let Y∞ be its one-point compactification. Extend f to X × Y∞ by setting f (x, ∞) = 0 for x ∈ X, so that f is separately continuous on X × Y∞ . Let τ be the topology of pointwise convergence on C(X). As the map Y∞ → C(X),

y → f (·, y)

is continuous with respect to the given topology on Y∞ and to τ on C(X), the set K := {f (·, y) : y ∈ Y∞ } is τ -compact. By [40, Theorem A.18], this means that K is weakly compact, so that the weak topology and τ coincide on K. Let (yα )α be a convergent τ net in Y∞ with limit y. Since f (·, yα ) → f (·, y), it follows that

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5 Dual Banach Algebras



 lim α

X

f (x, yα ) dμ(x) =

This means that the function Y∞ → C,

f (x, y) dμ(x). X

 y →

f (x, y) dμ(x) X

is continuous on Y∞ ; since it vanishes at ∞ by definition, this establishes (i). The case where Y is compact can be dealt with analogously (and is, in fact, slightly easier).  Given two locally compact Hausdorff spaces X and Y , we use the symbol Cσ (M (X), M (Y ); C) for the space of bounded, separately weak∗ continuous bilinear maps from M (X)×M (Y ) to C. As it turns out, it can be canonically identified with SC 0 (X × Y ): Proposition 5.2.4. Let X and Y be locally compact Hausdorff spaces. Then: (i) for each f ∈ SC 0 (X × Y ), the bilinear map   f (x, y) dμ(x) dν(y) Θf : M (X) × M (Y ) → C, (μ, ν) → Y

(5.9)

X

belongs to Cσ (M (X), M (Y ); C); (ii) the map SC 0 (X × Y ) → Cσ (M (X), M (Y ); C),

f → Θf

(5.10)

is an isometric isomorphism. Proof. To prove (i), let f ∈ SC 0 (X × Y ). By Lemma 5.2.3(i), Θf is well defined, and it is obviously bounded; it also follows from Lemma 5.2.3(i) that Θf is weak∗ continuous in the second variable. Since f is separately continuous, it is μ × ν-measurable for all μ ∈ M (X) and ν ∈ M (Y ) by Theorem D.3.1. By Fubini’s Theorem ([56, Theorem 7.6.7 and Lemma 9.4.2]), the integral in (5.9) is, therefore, independent of the order of integration, i.e.,   f (x, y) dν(y) dμ(x) (μ ∈ M (X), ν ∈ M (Y )). Θf (μ, ν) = X

Y

It then follows from Lemma 5.2.3(ii) that Θf is also weak∗ continuous in the first variable. For the proof of (ii), first note that, clearly, Θf  ≤ f ∞ for all f ∈ SC 0 (X × Y ). On the other hand, Θf  ≥ sup{|Θf (δx , δy )| : x ∈ X, y ∈ Y } = sup{|f (x, y)| : x ∈ X, y ∈ Y } = f ∞

(f ∈ SC 0 (X × Y )),

5.2 The Case of the Measure Algebra

221

so that (5.10) is an isometry. Let Θ ∈ Cσ (M (X), M (Y ); C) be arbitrary, and define f : X × Y → C, (x, y) → Φ(δx , δy ). It is immediate that f ∈ SC 0 (X × Y ) such that Θf (δx , δy ) = Θ(δx , δy ) for all  x ∈ X and y ∈ Y . Separate weak∗ continuity yields that Θ = Θf . Let G be a locally compact group. Then SC 0 (G × G) becomes a Banach M (G)-bimodule through the following convolution formulae for f ∈ SC 0 (G × G) and μ ∈ M (G) (compare Example 5.1.3):  f (x, yz) dμ(z) (x, y ∈ G) (5.11) (μ · f )(x, y) := G



and (f · μ)(x, y) :=

(x, y ∈ G).

f (zx, y) dμ(z) G

(5.12)

In this context, we can extend Proposition 5.2.4 such that it makes reference to the M (G)-bimodule structures on SC 0 (G × G) and Cσ2 (M (G); C): Proposition 5.2.5. Let G be a locally compact group. Then SC 0 (G × G) → Cσ2 (M (G); C),

f → Θf ,

as defined in Proposition 5.2.4, is an isometric isomorphism of Banach M (G)-bimodules. Proof. The M (G)-bimodule action on SC 0 (G×G) induces an M (G)-bimodule action on SC 0 (G × G)∗ . Invoking Theorem D.3.1, we can isometrically embed M (G) ⊗γ M (G) into SC 0 (G × G)∗ via  f (x, y) dμ(x) dν(y) (f ∈ SC 0 (G × G), μ, ν ∈ M (G)). f, μ ⊗ ν = G

We are done if we succeed to show that the image of M (G) ⊗γ M (G) under this embedding is an M (G)-submodule of SC 0 (G × G)∗ such that the module actions on this submodule are the canonical ones on M (G) ⊗γ M (G). Let λ, μ, ν ∈ M (G). We need to show that λ · (μ ⊗ ν) = (λ ∗ μ) ⊗ ν

(5.13)

(μ ⊗ ν) · λ = μ ⊗ (ν ∗ λ).

(5.14)

as well as Indeed, we have

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5 Dual Banach Algebras

f, λ · (μ ⊗ ν) = f · λ, μ ⊗ ν

   = f (zx, y) dλ(z) dμ(x) dν(y) G G G   f (x, y) d(λ ∗ μ)(x) dν(y) = G

G

= f, (λ ∗ μ) ⊗ ν

(f ∈ SC 0 (G × G)),

i.e., (5.13) holds; analogously, we see that (5.14) holds as well.



We are now already in a position to prove one of the open implications of Theorem 5.2.1: Proof (of Theorem 5.2.1, (i) =⇒ (iii)). Let (mα )α∈A be a net of non-negative norm one functions in L1 (G), as required by Reiter’s property (P1 ). Define a net (mα )α∈A in M (G × G) by letting  f (x, x−1 )mα (x) dx (f ∈ C0 (G × G), α ∈ A). f, mα := G

For x ∈ G, f ∈ C0 (G × G), and α ∈ A, we have  f (xt, t−1 )mα (y) dy f, (δx ⊗ δe ) ∗ mα = G f (y, y −1 x)mα (x−1 y) dy, = G  = f (y, y −1 x)(δx ∗ mα )(y) dy G



as well as f, mα ∗ (δe ⊗ δx ) =

G

f (y, y −1 x)mα (y) dy.

From the definition of Reiter’s property (P1 ), we thus conclude that sup (δx ⊗ δe ) ∗ mα − mα ∗ (δe ⊗ δx ) → 0

x∈K

(5.15)

for all compact subsets K of G. Let U be an ultrafilter over A that dominates the order filter. Define D ∈ SC 0 (G × G)∗ by letting  f (x, y) dmα (x, y) (f ∈ SC 0 (G × G)). (5.16) f, D := lim α→U

G×G

(By Theorem D.3.1, the integrals in (5.16) do exist.) Note first that

5.2 The Case of the Measure Algebra

223

 f, Δσ D = lim

f (xy) dmα (x, y)  = lim f (xx−1 )mα (x) dx = f (e)

α→U

G×G

α→U

G×G

(f ∈ C0 (G)),

i.e., Δσ D = δe . Also, we have for μ ∈ M (G) and f ∈ SC 0 (G × G) that |f, μ · D − D · μ | = |f · μ − μ · f, D | 

  = lim (f (zx, y) − f (x, yz)) dμ(z) dmα (x, y) α→U G×G G    = lim (f (zx, y) − f (x, yz)) dmα (x, y) dμ(z) , α→U G

G×G

by Fubini’s Theorem,   ≤ lim (f (zx, y) − f (x, yz)) dmα (x, y) d|μ|(z) α→U G G×G   = lim f (x, y) d((δz ⊗ δe ) ∗ mα − mα ∗ (δe ⊗ δz ))(x, y) d|μ|(z) α→U G G×G  ≤ lim f (δz ⊗ δe ) ∗ mα − mα ∗ (δe ⊗ δz ) d|μ|(z) α→U

→ 0,

G

by (5.15) and the inner regularity of|μ|.

As μ ∈ M (G) and f ∈ SC 0 (G × G) were arbitrary, this means that μ·D =D·μ

(μ ∈ M (G)).

Identifying SC 0 (G×G) and Cσ2 (M (G), C) via Proposition 5.2.5—and thus SC 0 (G × G)∗ and Cσ2 (M (G), C)∗ —we conclude that D ∈ Cσ2 (M (G), C)∗ is a normal, virtual diagonal for M (G).  We now begin with the preparations for the proof of Theorem 5.2.1, (ii) =⇒ (i). For a locally compact group G, let GLU C denote the character space of the commutative C ∗ -algebra LUC(G)—so that LUC(G) ∼ = C(GLU C ) via the Gelfand transform. For the properties of GLU C that are interesting to us, we refer to Exercises 5.2.1 and 5.2.2 below. Recall from the previous section that LUC(G) is left introverted in L∞ (G); in particular, for each f ∈ LUC(G) and φ ∈ LUC(G)∗ , the functional φ  f ∈ L1 (G)∗ ∼ = L∞ (G) introduced in Definition 5.1.7(b) lies already in LUC(G). Definition 5.2.6. Let G and H be locally compact groups. Define LUCSC 0 (G × H) :={f ∈ LUC(G × H) : ω  f ∈ SC 0 (G × H)for allω ∈ (G × H)LU C }.

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5 Dual Banach Algebras

We require two lemmas: Lemma 5.2.7. Let G and H be locally compact groups. Then, for every f ∈ LUCSC 0 (G × H), we have: (i) {L(x,y) f : x ∈ G} is relatively weakly compact in LUC(G × H) for each y ∈ H; (ii) {L(x,y) f : y ∈ H} is relatively weakly compact in LUC(G × H) for each x ∈ G. Proof. We only prove (i) in the case where G is not compact. (As in the proof of Lemma 5.2.3, the compact case is similar but easier.) Fix y ∈ H. By [40, Theorem A.18], it is sufficient to show that {L(x,y) f : x ∈ G} is relatively compact in LUC(G × H) with respect to the topology τ of pointwise convergence on (G × H)LU C . (This is due to the fact that LUC(G × H) ∼ = C((G × H)LU C ).) ˆ Let f ∈ C((G × H)LU C ) denote the Gelfand transform of f . As (G × H)LU C is a right topological semigroup, the map G → C,

x → fˆ(δx,y ω) = L (x,y) f (ω)

is continuous for each ω ∈ (G × H)LU C , i.e., G → LUC(G × H),

x → L(x,y) f

(5.17)

is continuous if LUC(G × H) is equipped with τ . Let G∞ denote the one-point compactification of G, and let (xα )α be a net in G with xα → ∞. As ω  f ∈ SC 0 (G × H) for any ω ∈ (G × H)LU C by the definition of LUCSC 0 (G × H), we have ˆ L (xα ,y) f (ω) = f ((δxα ,y ω) = (ω  f )(xα , y) → 0

(ω ∈ (G × H)LU C ),

i.e., (5.17) has an extension to G∞ that is continuous with respect to the given topology on G∞ and τ on LUC(G × H). As G∞ is compact, the image of that continuous extension of (5.17) to G∞ must be τ compact. Consequently, the range of (5.17) is relatively compact in LUC(G × H) with respect to τ .  Lemma 5.2.8. Let G and H be locally compact groups, and suppose that f ∈ LUCSC 0 (G × H) and φ ∈ LUC(G × H)∗ . Then φ  f ∈ LUC(G × H) belongs to LUCSC 0 (G × H). Proof. As in the proof of Lemma 5.2.7, we will only focus on the case where G is not compact. Let ψ ∈ LUC(G × H)∗ be arbitrary. Fix y ∈ H; we claim that the function G → C,

x → L(x,y) f, ψ

(5.18)

belongs to C0 (G). As (5.18) is clearly continuous, all we need to show is that it vanishes at ∞. Let (xα )α be a net in G such that xα → ∞, and let τ

5.2 The Case of the Measure Algebra

225

denote the topology of pointwise convergence on G×G; it is routinely checked τ that that L(xα ,y) f → 0. As {L(x,y) f : x ∈ G} is relatively weakly compact by Lemma 5.2.7(i), the weak topology and the coarser Hausdorff topology τ must coincide on the weak closure of {L(x,y) f : x ∈ G}. This means, in particular, that L(xα ,y) f, ψ → 0, i.e., the function (5.18) belongs indeed to C0 (G). Analogously—using Lemma 5.2.7(ii) instead of (i)—we see that H → C,

y → L(x,y) f, ψ

lies in C0 (H) for each x ∈ G. All in all, this means that G × H → C,

(x, y) → L(x,y) f, ψ

(5.19)

belongs to SC 0 (G × H). Let ω ∈ (G × H)LU C , and note that (ω  (φ  f ))(x, y) = ((ωφ)  f )(x, y) = L(x,y) f, ωφ

(x ∈ G, y ∈ H)

by Exercise 5.2.3 below. Setting ψ := ωφ in (5.19) then yields that ω  (φ  f ) ∈ SC 0 (G × H). As ω ∈ (G × H)LU C was arbitrary, this means that  φ  f ∈ LUCSC 0 (G × H). Let G be a locally compact group. Then we denote by Gop the locally compact group we obtain by reversing the multiplication of G. The following proposition summarizes the important properties of LUC(G × H) in the case where H = Gop : Proposition 5.2.9. Let G be a locally compact group. Then: (i) Δ∗M (G) C0 (G) ⊂ LUCSC 0 (G × Gop ); (ii) LUCSC 0 (G × Gop ) is a closed M (G)-submodule of SC 0 (G × G); (iii) LUCSC 0 (G × Gop )∗ is a normal, dual Banach M (G)-bimodule. Proof. To prove (i), let f ∈ C0 (G), and note that L(x,y) Δ∗ f = Δ∗ (Lx Ry f )

((x, y) ∈ G × Gop ).

As C0 (G) ⊂ UC(G) by Proposition D.4.2(iii), the norm continuity of Δ∗ thus yields that Δ∗ f ∈ LUC(G × Gop ). To show that Δ∗ f ∈ LUCSC 0 (G × Gop ), let ω ∈ (G × Gop )LU C . Since G × Gop —identified with its canonical image in (G×Gop )LU C —is dense in (G×Gop )LU C , there is a net ((xα , yα ))α in G×Gop such that (xα , yα ) → ω. Passing to a subnet, we may suppose that (xα yα )α converges to some z ∈ G or tends to ∞ (which, of course, can only occur if G is not compact). In the first case, we have (ω  Δ∗ f )(x, y) = lim Δ∗ f (xxα , yα y) α

= lim f (xxα yα y) = f (xzy) α

((x, y) ∈ G × Gop )

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5 Dual Banach Algebras

and in the second one (ω  Δ∗ f )(x, y) = lim Δ∗ f (xxα , yα y) α

= lim f (xxα yα y) = 0 α

((x, y) ∈ G × Gop ).

In either case, we ω  Δ∗ f ∈ SC 0 (G × Gop ) holds, which proves (i). For the proof of (ii), let μ ∈ M (G), and let f ∈ LUCSC 0 (G × Gop ) ⊂ LUC(G × Gop ) ∩ SC 0 (G × G). With · standing for the bimodule action of M (G) on SC 0 (G × G) defined in (5.11) and (5.12), it is routinely checked that μ · f, f · ν ∈ LUC(G × Gop ) ∩ SC 0 (G × Gop ). Let ω ∈ (G × Gop )LU C . As ω  (μ · f )(x, y) = L(x,y) (μ · f ), ω = μ · L(x,y) f, ω

((x, y) ∈ G × Gop )

by Exercise 5.2.3 below, an application of Lemma 5.2.7 as in the proof of Lemma 5.2.8 yields that ω  (μ · f ) ∈ SC 0 (G × Gop ). A similar, but easier argument yields that ω  (f · μ) ∈ SC 0 (G × Gop ) as well. This proves (ii). For (iii), first observe that the canonical embedding of M (G × Gop ) into LUC(G × Gop )∗ via integration is an algebra homomorphism (see Exercise 5.2.4(a) below). It is routinely verified that M (G×Gop )—or rather: its image in LUCSC 0 (G×Gop )∗ —is an M (G)-submodule of LUCSC 0 (G×Gop )∗ ; equally routinely, we verify that μ · ν = (μ × δe )ν|LU CSC 0 (G×H)

(μ ∈ M (G), ν ∈ M (G × Gop )). (5.20)

By Exercise 5.2.4(b) below, the map LUC(G × Gop )∗ → LUC(G × Gop )∗ ,

φ → (μ × δe )φ

(5.21)

is weak∗ continuous for any μ ∈ M (G). Let φ ∈ LUC(G × Gop )∗ be arbitrary. By Exercise 5.2.4(a), there is a net (να )α in M (G × Gop ) that converges to φ in the weak∗ topology of LUC(G × Gop )∗ . From (5.20) and the weak∗ continuity of (5.21) for any μ ∈ M (G), we conclude that μ · φ = weak∗ - lim μ · να = weak∗ - lim(μ × δe ) ∗ να = (μ ⊗ δe )φ α

α

(μ ∈ M (G))

Let f ∈ LUCSC 0 (G × Gop ) be arbitrary, and note that f, μ · φ = f, (μ × δe )φ = φ  f, μ × δe

(μ ∈ M (G)).

Further note that φ  f ∈ LUCSC 0 (G × Gop ) by Lemma 5.2.8, so that, in particular, (φ  f )(·, e) ∈ C0 (G). Let (μα )α be a net in M (G) such that μα

σ(M (G),C0 (G))

→

0. Then we have

limf, μα · φ = limφ  f, μα × δe = lim(φ  f )(·, e), μα = 0 α

α

α

5.2 The Case of the Measure Algebra

227

It follows that M (G) → LUCSC 0 (G × H op )∗ ,

μ → μ · φ

is weak∗ continuous. Noting that φ · μ = φ(δe × μ)

(μ ∈ M (G)),

we see analogously that M (G) → LUCSC 0 (G × Gop )∗ ,

μ → φ · μ

is also weak∗ -continuous. As φ ∈ LUCSC 0 (G × Gop )∗ was arbitrary, this proves (iii).  Let G be a locally compact group. By Proposition 5.2.9(i), Δ∗ maps C0 (G) into LUCSC 0 (G × Gop ). Consequently, Δ0,σ := (Δ∗ |C0 (G) )∗ is a weak∗ continuous homomorphism of Banach M (G)-bimodules from LUCSC 0 (G × Gop )∗ onto M (G). Proposition 5.2.10. Let G be a locally compact group such that M (G) is Connes-amenable. Then there is D ∈ LUCSC 0 (G × Gop )∗ such that μ·D =D·μ

(μ ∈ M (G))

and

Δ0,σ D = δe .

Proof. Define a derivation D : M (G) → LUCSC 0 (G × Gop )∗ ,

μ → μ × δe − δe × μ.

It is straightforward to check that D is weak∗ continuous and attains its values in ker Δ0,σ . Being the kernel of a weak∗ continuous bimodule homomorphism, ker Δ0,σ is a weak∗ closed submodule of the normal, dual Banach M (G)module LUCSC 0 (G × Gop )∗ and thus a normal, dual Banach M (G)-module in its own right. As M (G) is Connes-amenable, there is thus E ∈ ker Δ0,σ such that D = adE . The element D := δe × δe − E then has the desired properties.



Remark 5.2.11. Since LUCSC 0 (G × G)∗ need not be equal to SC 0 (G × G)∗ , Proposition 5.2.10 does not (yet) allow us to conclude that M (G) has a normal, virtual diagonal. Before we can close the circle, we require one last lemma. (For the notion of an essential ideal in a C ∗ -algebra, see Definition C.6.4.) Lemma 5.2.12. Let G and H be locally compact groups. Then LUCSC 0 (G × H) is an essential, closed ideal of LUC(G × H).

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5 Dual Banach Algebras

Proof. Let f ∈ LUCSC 0 (G × H), let g ∈ LUC(G × H); obviously, f g ∈ LUC (G × H) holds. Let ω ∈ (G × H)LU C , and recall that (G × H)LU C ⊂ LUC(G × G)∗ is a semigroup under  and is defined as ΦLU C(G×H) . We thus have (ω  f g)(x, y) = f g, δx,y ω = f, δxy ω g, δxy ω

= (ω  f )(x, y)(ω  g)(x, y)

(x ∈ G, y ∈ H),

i.e., ω  f g = (ω  f )(ω  g). Since ω  f, ω  g ∈ SC 0 (G × H), this means that ω  f g ∈ SC 0 (G × H). As ω ∈ (G × H)LU C was arbitrary, this means that f g ∈ LUCSC 0 (G × H). Consequently, LUCSC 0 (G × H) is an ideal of LUC(G × H). As C0 (G × H) ⊂ LUCSC 0 (G × H), it is clear that LUCSC 0 (G × H) is even an essential ideal of LUC(G × H).  We are finally in a position to complete the proof of Theorem 5.2.1. Proof (of Theorem 5.2.1, (ii) =⇒ (i)). Let D ∈ LUCSC 0 (G × Gop )∗ be as in Proposition 5.2.10. Via Riesz’s Representation Theorem ([56, Theorem 7.3.5]), we can identify D with a regular Borel measure M on the character space of the commutative C ∗ -algebra LUCSC 0 (G × Gop ); we can then define the variation |M | ∈ LUCSC 0 (G × Gop )∗ of M in the measure theoretic sense ([56, p. 126]). It is routinely checked that δx · |M | = |M | · δx

(x ∈ G).

(5.22)

By Lemma 5.2.12, LUCSC 0 (G × Gop ) is an essential, closed ideal of LUC(G × Gop ). By Theorem C.6.7, we may, therefore, view LUC(G × Gop ) as a C ∗ subalgebra of the multiplier algebra M(LUCSC 0 (G×Gop )). By Remark C.6.9, M(LUCSC 0 (G × Gop )), embeds canonically into LUCSC 0 (G × Gop )∗∗ ; we may, therefore, view M(LUCSC 0 (G × Gop ))—and thus LUC(G × Gop )—as a C ∗ -subalgebra of LUCSC 0 (G × Gop )∗∗ ; in particular, it makes sense to speak of f, |M | in a canonical way for any f ∈ LUC(G × Gop ). Define M : LUC(G) → C,

f → f ⊗ 1, |M | .

As f ⊗ 1 ∈ LUC(G × Gop ) for each f ∈ LUC(G), it is clear that M = 0 is a well-defined, positive, linear functional. Noting that the canonical embedding from LUC(G × Gop ) into LUCSC 0 (G × Gop )∗∗ respects left translations, we get

5.2 The Case of the Measure Algebra

229

Lx f, M = L(x,e) (f ⊗ 1), |M |

= f ⊗ 1, δx · |M |

= f ⊗ 1, |M | · δx ,

by (5.22),

= L(e,x) (f ⊗ 1), |M |

= f ⊗ 1, |M |

= f, M

(f ∈ LUC(G), x ∈ G).

Normalizing M , we eventually obtain a left invariant mean on LUC(G). Hence, by Theorem 1.1.13, G is amenable.  We conclude this section with an observation in the discrete case: Corollary 5.2.13. The following are equivalent for a discrete group G: (i) G is amenable; (ii) 1 (G) is amenable; (iii) 1 (G) is Connes-amenable.

Exercises Exercise 5.2.1. Let G be a locally compact group, and let GLU C := ΦLU C(G) . Show that: (a) if LUC(G)∗ is equipped with the weak∗ topology, then GLU C is a compact subsemigroup of (LUC(G)∗ , ); (b) the canonical embedding ι : G → GLU C is a semigroup homomorphism that is a homeomorphism onto a dense subsemigroup of GLU C . Exercise 5.2.2. A right topological semigroup is a semigroup S equipped with a Hausdorff topology such that S x → xy is continuous for each y ∈ S. Show the following for a locally compact group G: (a) GLU C is a compact, right topological semigroup; (b) if K is a compact, right topological semigroup, and θ : G → K is a continuous semigroup homomorphism, then θ∗ : C(K) → C(G),

f → f ◦ θ

attains its values in LUC(G); (c) if K is a compact, right topological semigroup, and θ : G → K is a continuous semigroup homomorphism, then there is a unique continuous semigroup homomorphism Θ : G → GLU C such that the diagram ι

/ GLU C GE EE EE EE EE  Θ θ " K

230

5 Dual Banach Algebras

commutes. (Hint for (b): For f ∈ C(K), first use [40, Theorem A.18], to conclude that G x → Lx (θ∗ f ) is continuous with respect to the weak topology on C(G), and then invoke Lemma 1.3.2 to conclude that θ∗ f ∈ LUC(G).) Exercise 5.2.3. Let G be a locally compact group, let f ∈ LUC(G), and let φ ∈ LUC(G)∗ . Show that (φ  f )(x) = Lx f, φ

(x ∈ G).

Exercise 5.2.4. Let G be a locally compact. Define θ : M (G) → LUC(G)∗ via  f (x) dμ(x) (f ∈ LUC(G)). f, θ(μ) := G

Show that: (a) θ is an isometric algebra homomorphism with weak∗ dense range; (b) for each μ ∈ M (G), the map LUC(G)∗ φ → θ(μ)φ is weak∗ continuous.

5.3 Connes-Amenability without a Normal, Virtual Diagonal By Corollary 5.1.35, the existence of a normal, virtual diagonal for a dual Banach algebra implies the Connes-amenability of the algebra. In this section, we shall see that the converse fails. Our counterexamples, will be of the form WAP(L1 (G))∗ for certain locally compact groups G: these are dual Banach algebras by Proposition 5.1.11(iv). By Example 5.1.12, we can identify WAP(L1 (G)) with WAP(G) for any locally compact group G; we shall thus simply write WAP(G)∗ instead of WAP(L1 (G))∗ . It is not difficult to see that WAP(G)∗ is Connes-amenable if and only if G is amenable (Exercise 5.3.1 below). The remainder of this section is devoted to showing that, for a large class of locally compact groups G— among them all non-compact abelian groups and all infinite discrete groups— WAP(G)∗ fails to have a normal, virtual diagonal. For a locally compact group G, let GWAP denote the character space of the commutative C ∗ -algebra WAP(G); we refer to Exercise 5.3.2 below for more information about GWAP . The Gelfand transform of WAP(G) induces an isometric isomorphism between the Banach spaces M (GWAP ) and WAP(G)∗ . However, both spaces are also Banach algebras in a canonical way (Proposition 5.1.11 and Theorem D.3.2). Our first result shows, that both multiplicative structures are compatible— along with some additional information. (By an ideal of a semigroup S, we mean a subset I = ∅ of S such that sI ∪ Is ⊂ I for s ∈ S.)

5.3 Connes-Amenability without a Normal, Virtual Diagonal

231

Proposition 5.3.1. Let G be a non-compact, locally compact group G. Then: ∗ (i) GWAP(G) : M (GWAP ) → WAP(G)∗ is an isometric isomorphism of Banach algebras; (ii) GWAP \ G is a closed ideal of GWAP ; (iii) C0 (G)⊥ is a weak∗ closed ideal of WAP(G)∗ , which is isometrically isomorphic to M (GWAP \ G) as a Banach algebra.

Proof. (i): Certainly, G ∗ is multiplicative if restricted to the linear span of {δs : s ∈ GWAP }. As this is a weak∗ dense subalgebra of M (GWAP ), the weak∗ continuity of G ∗ immediately yields its multiplicativity. (ii): As GWAP \ G = {s ∈ GWAP : f, s = 0 for f ∈ C0 (G)}, it is clear that GWAP \ G is closed in GWAP . To see that GWAP \G is an ideal, let s ∈ GWAP \G and t ∈ GWAP . Assume that st ∈ G. Then there is a function f ∈ C0 (G) such that (Gf )(st) = 0. As G is dense in GWAP , there are nets (xα )α and (yβ )β∈B in G such that s = limα xα and y = limβ yβ . For every β ∈ B, the function G x → f (xyβ ) / G, it follows that belongs to C0 (G). As s ∈ lim f (xα yβ ) = (Gf )(syβ ) = 0 α

(β ∈ B)

and, consequently, (Gf )(st) = lim lim f (xα yβ ) = 0, β

α

which is a contradiction, so that st ∈ GWAP \ G. Analogously, we see that ts ∈ GWAP \ G. (iii): The restriction map WAP(G)∗ → M (G),

φ → φ|C0 (G)

(5.23)

is weak∗ continuous and an algebra homomorphism by Exercise 5.1.6. Consequently, its kernel is a weak∗ closed ideal of WAP(G)∗ . The restriction map from C(GWAP ) onto GWAP \ G is onto—and, in fact, a quotient map by Tietze’s Extension Theorem ([301, Theorem 4.1.13]). The adjoint of that map is then a weak∗ continuous isometry from M (G \ GWAP ) into M (GWAP ); an argument as in the proof of (i) shows that this isometry is multiplicative. We can thus view M (GWAP \ G) as a closed subalgebra of WAP(G)∗ . As the inclusion of M (GWAP \ G) into WAP(G)∗ is weak∗ continuous, the unit ball of M (GWAP \ G) is weak∗ compact in ˇ Theorem ([102, Theorem V.5.7]), this WAP(G)∗ . By the Kre˘ın–Smulian means that M (GWAP \ G) is a weak∗ closed subspace of WAP(G)∗ , and a moment’s thought reveals that M (G \ GWAP ) is the weak∗ closed linear span of {δs : s ∈ GWAP \ G} in WAP(G)∗ . It is thus clear that

232

5 Dual Banach Algebras

M (GWAP \ G) ⊂ C0 (G)⊥ . Assume that there is φ ∈ C0 (G)⊥ \ M (GWAP \ G). The Hahn–Banach Theorem, then yields f ∈ WAP(G) such that (Gf )(s) = 0 (s ∈ GWAP \ G)

and

f, φ = 0.

Let (xα )α be a net in G such that xα → ∞ in G∞ , the one-point compactification of G. Passing to a subnet, we can suppose that there is s ∈ GWAP —necessarily not in G—such that xα → s in GWAP . It follows that limα f (xα ) = (Gf )(s) = 0. Hence, f lies in C0 (G), so that f, φ = 0. This is a contradiction.  Before we continue with our study of WAP(G)∗ for a locally compact group G, we will take a closer look at the measure algebra M (S) for a general locally compact, semitopological semigroup S. If S has an identity element eS , then δeS is trivially an identity for the measure algebra M (S). The converse is easily seen to be false: Example 5.3.2. Let S := {s0 , s1 , . . . , sN }, and turn it into a semigroup by defining for j, k ∈ {0, 1, . . . , N }: sj , j = k, sj sk := s0 , j = k. It is then routinely checked that S is a commutative semigroup that has an identity if and only if N = 1. On the other hand, it is straightforward that δs1 + · · · + δsN − (N − 1)δs0 is an identity for 1 (S). This begets the question for which locally compact, semitopological semigroups S precisely the measure algebra M (S) has an identity. We shall answer the “left version” of this question in Theorem 5.3.4 below. We first require a definition: Definition 5.3.3. Let S be a semigroup. A set U ⊂ S is called a set of local left units for S if, for each s ∈ S, there is u ∈ U with us = s. A set of local left units for S is called minimal if none of its proper subsets is a set of local left units for S. Theorem 5.3.4. The following are equivalent for a locally compact, semitopological semigroup: (i) M (S) has a left identity; (ii) S contains a minimal, finite set of local left units. To prove Theorem 5.3.4, we proceed through a series of lemmas, the first of which is entirely algebraic: Lemma 5.3.5. Let S be a semigroup, and let U ⊂ S be a set of local left units for S. Then the following are equivalent: (i) U is minimal;

5.3 Connes-Amenability without a Normal, Virtual Diagonal

233

(ii) if u, v ∈ U and vu = u, then v = u. Proof. (i) =⇒ (ii): Let u, v ∈ U be such that vu = u, and assume that v = u. For s ∈ S such that us = s, note that vs = v(us) = (vu)x = us = s. Hence, U \ {u} is a set of local left units for S, contradicting the minimality of U . (ii) =⇒ (i): Assume that U is not minimal, i.e., there is u ∈ U such that U \ {u} is a set of local left units for S. Then there is v ∈ U \ {u} such that vu = u. By (ii), this means that v = u, which is impossible.  Corollary 5.3.6. Let S be a semigroup, and let U ⊂ S be a minimal set of local left units for S. Then U consists of idempotents. Proof. Let u ∈ U . Then there is v ∈ U such that vu = u. By Lemma 5.3.5, this means v = u, so that u2 = u.  We now turn to the locally compact situation: Lemma 5.3.7. Let S be a locally compact, semitopological semigroup, and let K ⊂ S be a compact set of local left units for S. Then K contains a minimal set of local left units for S. Proof. Let s ∈ S. As K is a set of local left units for S, there is u1 ∈ S such that u1 s = s. Inductively, define a sequence (un )∞ n=1 in K such that un um = um

(n, m ∈ N, n > m)

and

un s = s (n ∈ N).

(5.24)

Since K is compact, the sequence (un )∞ n=1 has an accumulation point u ∈ K, which—due to (5.24)—satisfies u2 = u and us = s. This means that E := {u ∈ K : u2 = u} is a set of local left units for S. Define a partial order on E by letting u≺v

:⇐⇒

vu = uv = u.

Let C be the collection of all totally ordered subsets C of S that are maximal, i.e., no proper superset of C is totally ordered. As each C ∈ C is totally ordered, it is directed, so that (s)s∈C is a net. Let C ∈ C, and let uC ∈ K be an accumulation point of (s)s∈C . It is straightforward that u2C = uC , i.e., uC ∈ E, and that s ≺ uC for all s ∈ C. From the maximality of C, it follows that uC ∈ C. We claim that U := {uC : C ∈ C} is a minimal set of local left units for S. To see this, let s ∈ S, and let u ∈ K0 be such that ux = x. Then there is C ∈ C such that u ∈ C and thus u ≺ uC . We conclude that uC s = uC us = us = s. Hence, U is indeed a set of local units for S. Let C1 , C1 ∈ C be such that uC1 uC2 = uC2 , i.e., uC1 ≺ uC2 . It follows that s ≺ uC2 for all s ∈ C1 . Hence,

234

5 Dual Banach Algebras

C1 ∪{uC2 } is totally ordered, and the maximality of C1 entails that uC2 ∈ C1 , so that uC2 ≺ uC1 and thus uC1 = uC2 . Hence, Lemma 5.3.5(ii) is satisfied, so that U is minimal.  Lemma 5.3.8. Let S be a locally compact, semitopological semigroup such that M (S) has a left identity λ. Then λ({t ∈ S : ts = s}) = 1 holds for each s ∈ S. Proof. Let s ∈ S, and note that 1 = δs ({s}) = (λ ∗ δs )({s})    = χ{s} (tr) dλ(t) dδs (r) = χ{s} (ts)dλ(t) = λ({t ∈ S : ts = s}). S

S

S



This proves the claim.

Proof (of Theorem 5.3.4). (i) =⇒ (ii): Suppose that M (S) has a left identity λ. As the variation |λ| of λ is regular, there is a compact set K ⊂ S such that |λ|(G \ K) < 1. For s ∈ S, set Xs := {t ∈ S : ts = s}, and note that 1 ≤ |λ|(Xs ) = |λ|(Xs ∩ K) + |λ|(Xs \ K) by Lemma 5.3.8. As |λ|(Xs \K) ≤ |λ|(G\K) < 1, it follows that |λ|(Xs ∩K) > 0, so that, in particular, Xs ∩ K = ∅. This means that K is a set of local left units for S. By Lemma 5.3.7, K contains a minimal set U of local left units for S. Let u, v ∈ U be such that Xu ∩ Xv = ∅, i.e., there is t ∈ S such that tu = u and tv = v. As U is a set of local left units for S, there is w ∈ U such that wt = t; it follows that wu = wtu = tu = u and, similarly, wv = v. Since U is minimal, Lemma 5.3.5 yields that u = w = v. This means that, for u, v ∈ U such that u = v, we have Xu ∩ Xv = ∅. As λ(Xs ) = 1 for s ∈ S by Lemma 5.3.8, this is possible only if U is finite. (ii) =⇒ (i): Let {u1 , . . . , uN } be a finite, minimal set of local left units for S. Then it is routine (albeit somewhat tedious) to verify that N  n=1

(−1)n+1

 1≤j1 (1−)a. By Lemma 5.4.4(ii), there are x ∈ Eμ and φ ∈ Eμ∗ with xφ = 1 such that πμ (a)x, φ = μ, a . Consequently, we have π(a) ≥ πμ (a) ≥ |πμ (a)x, φ | = |μ, a | > (1 − )a. As  > 0 was arbitrary, this means that π(a) = a.



In view of Proposition 5.4.5, our next step towards proving Theorem 5.4.1 is to show that Proposition 5.4.5(b) is always satisfied. Lemma 5.4.6. Let A be a dual Banach algebra, and let μ ∈ A∗ be such that μ = 1. Then there is an admissible norm for μ. Proof. For n ∈ N, apply the construction of Exercise 5.4.1 with E = A, F = A∗ , and T given by A x → x · μ. It follows from Exercise 5.4.1, that n

n

2− 2 φ ≤ φn ≤ 2 2 φ

(n ∈ N, φ ∈ A∗ ).

(5.31)

Let the space FT and the norm  · T from Exercise 5.4.2 be denoted by Eμ and  · μ , respectively. Then A · μ ⊂ Eμ holds by Exercise 5.4.2(a). As μ ∈ A∗ = Cσ,w (A∗ ), the operator A x → x · μ is, in particular, weakly compact. Hence, Exercise 5.4.3 yields that Eμ is reflexive. Replacing Eμ by the closure of A·μ in it, if necessary, we can suppose without loss of generality that A · μ is dense in Eμ . The first inequality in (5.31) yields a · μ2 =

∞ 

2−n a · μ2 ≤

n=1

∞ 

a · μ2n = a · μ2μ

(a ∈ A),

(5.32)

n=1

so that  · μ satisfies (5.28) on A · μ. From the second inequality in (5.31) and (5.32), we conclude that n

2− 2 ab · μn ≤ ab · μ ≤ ab · μn

(a, b ∈ A)

and thus ab · μ2μ ≤

∞ 

a2 b · μ2n = a2 b · μ2μ

(a, b ∈ A),

n=1

i.e.,  · μ satisfies (5.27) on A · μ as well. By construction, A · μ ⊂ A∗ extends injectively to Eμ .  All in all,  · μ is an admissible norm for μ. Putting everything together is now easy:

244

5 Dual Banach Algebras

Proof (of Theorem 5.4.1). Let (A, A∗ ) be a dual Banach algebra. Let the unitization A# of A be normed as A ⊕1 C. Then (A# , A ⊕∞ C) is a dual Banach algebra satisfying Proposition 5.4.5(a). By Lemma 5.4.6, it satisfies Proposition 5.4.5(b) as well. Hence, there are a reflexive Banach space E and an isometric, weak∗ -weak∗ continuous representation π : A# → B(E). Restricting π to A yields the desired representation of A. 

Exercises Exercise 5.4.1. Let E and F be Banach spaces, let T ∈ B(E, F ) be a contraction, and for n ∈ N, define  n  n yn := inf 2− 2 x + 2 2 y − T x : x ∈ E (y ∈ F ). Show that  · n is a norm on F such that n

n

2− 2 y ≤ yn ≤ 2 2 y

(y ∈ F ).

Exercise 5.4.2. In the situation of Exercise 5.4.1, define   ∞  2 FT := y ∈ F : yn < ∞ n=1

and set

 yT :=

∞ 

 12 y2n

(y ∈ F ).

n=1

Show that (FT ,  · T ) is a Banach space with the following properties: (a) T maps E contractively into FT ; (b) the inclusion map ιT : FT → F is contractive; ∗∗ ∗∗ −1 is injective such that (ι∗∗ (F ) = FT . (c) ι∗∗ T : FT → F T ) Exercise 5.4.3. In the situation of Exercise 5.4.2, show that FT is reflexive if and only if T is weakly compact. (Hint: For the “if” part, use the weak compactness of T to show that n

∗∗ −2 ι∗∗ Ball(F ∗∗ ) T (Ball(FT )) ⊂ F + 2

(n ∈ N),

and then apply Exercise 5.4.2(c) to conclude that FT∗∗ = FT .) Exercise 5.4.4. Let E and F be Banach spaces, and let T ∈ B(E, F ). Show that T is weakly compact if and only if there are a reflexive Banach space X and operators S ∈ B(E, X) and R ∈ B(X, F ) such that T = RS.

5.5 Connes-Amenability and Connes-Injectivity

245

5.5 Connes-Amenability and Connes-Injectivity We shall now give a characterization of Connes-amenable dual Banach algebras in terms of their representation theory. For our convenience (and with Chapter 7 in mind), we define: Definition 5.5.1. Let A be a dual Banach algebra with identity. We call A Connes-injective if, for every reflexive Banach space E and for every unital weak∗ -weak∗ continuous representation π : A → B(E), there is a quasiexpectation Q : B(E) → π(A) Remark 5.5.2. Note that, in Definition 5.5.1, no weak∗ -weak∗ continuity is required for Q. We shall devote the remainder of this section to prove: Theorem 5.5.3. The following are equivalent for a dual Banach algebra A with identity: (i) A is Connes-amenable; (ii) A is Connes-injective. Remark 5.5.4. In order for (ii) =⇒ (i) of Theorem 5.5.3 to hold true, it is important that Definition 5.5.1 is made in terms of all weak∗ -weak∗ continuous representation of A on reflexive Banach spaces: it is not enough to consider just one weak∗ -weak∗ continuous (even isometric) representation of A on some reflexive Banach space. To see this, let p, q ∈ (1, ∞) \ {2} be such that p = q. Then there is a canonical representation π—the identity—of B(p ⊕ q ) on p ⊕ q . As trivially π(B(p ⊕ q )) = C idp ⊕q , there is a quasi-expectation from B(p ⊕ q )) onto π(B(p ⊕ q )) . However, by Example 5.1.22, B(p ⊕ q ) is not Connes-amenable. ThisstandsinmarkedcontrasttothevonNeumannalgebrasituation(seeSection 7.2 below). Recall that, for Banach spaces E and F , we can isometrically isomorphically identify (E ⊗γ F )∗ and B(F, E ∗ ) via the duality x ⊗ y, T = x, T y

(x ∈ E, y ∈ F, T ∈ B(E, F ∗ )).

(5.33)

If A is a Banach algebra, E is a left Banach A-module, and F is a right Banach A-module, then E ⊗γ F is a Banach A-bimodule via a · (x ⊗ y) := a · x ⊗ y

and

(x ⊗ y) · a := x ⊗ y · a

(a ∈ A, x ∈ E, y ∈ F ).

The dual module action on (E ⊗γ F )∗ ∼ = B(F, E ∗ ) is thus given by (a · T )x :=T (x · a) and (T · a)(x) := a · T x

(a ∈ A, x ∈ E, T ∈ B(F, E ∗ )).

246

5 Dual Banach Algebras

Given a dual Banach algebra (A, A∗ ), the following proposition gives a characterization of Cσ,w ((A ⊗γ A)∗ ) in terms of (5.33). Proposition 5.5.5. Let (A, A∗ ) be a dual Banach algebra, let T ∈ B(A, A∗ ), and define τl , τr : A ⊗γ A → A∗ via τl (a ⊗ b) := a · T b

and

τr (a ⊗ b) := (T ∗ a) · b

(a, b ∈ A).

Then the following are equivalent: (i) T ∈ Cσ,w (B(A, A∗ )); (ii) τl and τr are weakly compact with ranges in A∗ . Proof. Define Tl , Tr : A → B(A, A∗ ) via Tl a := T · a and Tr a := a · T

(a ∈ A).

Trivially, T ∈ Cσ,w (B(A, A∗ )) if and only if Tl and Tr are σ(A, A∗ )-weakly continuous. By Exercise 5.5.2 below, this is the case if and only if there are weakly compact (Tl )∗ , (Tr )∗ : A⊗γ A → A∗ such that Tl = ((Tl )∗ )∗ and Tr = ((Tr )∗ )∗ . For a, b, c ∈ A, we see that (Tl )∗ (a ⊗ b), c = a ⊗ b, Tl c

= a ⊗ b, T · c = ca, T b = c, a · T b = c, τl (a ⊗ b)

as well as (Tr )∗ (a ⊗ b), c = a ⊗ b, Tr c = a ⊗ b, c · T

= a, T (bc) = bc, T ∗ a = c, (T ∗ a) · b = c, τr (a ⊗ b) . so that τl = (Tl )∗ and τr = (Tr )∗ . This yields the equivalence of (i) and (ii).



The proof of our next proposition uses techniques already employed for the proof of Lemma 5.4.6. We also require a result from the theory of real interpolation spaces; we refer to [20] for this. Proposition 5.5.6. Let (A, A∗ ) be a unital dual Banach algebra such that eA  = 1, and let T ∈ Cσ,w (B(A, A∗ )). Then there are a constant C > 0, a contractive left Banach A-module E and a contractive right Banach Amodule F such that E and F are normal and reflexive, as well as, μ0 ∈ E ∗ and λ0 ∈ F ∗ with μ0  = λ0  = 1 such that |a, T b | ≤ CT μ0 · aE ∗ b · λ0 F ∗

(a, b ∈ A).

Furthermore, μ0 · A and A · λ0 are dense in E ∗ and F ∗ , respectively.

(5.34)

5.5 Connes-Amenability and Connes-Injectivity

247

Proof. Throughout this proof, we suppose that T  = 1, which we can do without loss of generality. Let τl be as in Proposition 5.5.5, and define, for n ∈ N, a norm  · n on A∗ by letting  n  n μn = inf 2− 2 x + 2 2 μ − τl (x) : x ∈ A ⊗γ A (μ ∈ A∗ ). As in the proof of Lemma 5.4.6, we check that  · n is equivalent to the given norm on A∗ . Define ⎧ ⎫ ∞  12 ⎨ ⎬  E := μ ∈ A∗ : μE := μ2n 0 such that C0−1 ||| · ||| ≤  · F ∗ ≤ C0 ||| · |||. Note that, for a, b ∈ A, we have τl∗ (a)(b) = sup{|ac, T b | : c ≤ 1} = sup{|ac, τr∗ (b)(eA ) | : c ≤ 1} ≤ τr∗ (b)a. (5.35) ∞ ∞ For a,

b∞∈ A and  > 0, choose sequences (an )n=1 and (bn )n=1 in A such that a = n=1 an as well as







−n 2

max 2

an , 2

n 2

2 τl∗ (an )

 12 ≤ aE ∗ + ,

(5.36)

≤ |||b||| + .

(5.37)

n=1

and 

∞  

−n 2

2

b − bn  + 2

n 2

2 τr∗ (bn )

 12

n=1

We then obtain |a, T b | = |eA , τl∗ (a)(b) | ≤ τl∗ (a)(b) ∞  τl∗ (an )(b) ≤ ≤ ≤ ≤

n=1 ∞  n=1 ∞  n=1 ∞ 

τl∗ (an )(bn − b) + τl∗ (an )(bn ) τl∗ (an )(bn − b) + τr∗ (bn )an , n

n

n

by (5.35) n

2 2 τl∗ (an )2− 2 bn − b + 2 2 τr∗ (bn )2− 2 an 

n=1

≤ 2(aE ∗ + )(|||b||| + ),

by Cauchy–Schwarz, (5.37), and (5.37).

5.5 Connes-Amenability and Connes-Injectivity

249

As  > 0 was arbitrary, we see that |a, T b | ≤ 2C0 aE ∗ bF ∗ . Set C := 2C0 , μ0 := ι∗E (eA ), and λ0 := ι∗F (eA ). Then (5.34) holds, and μ0 · A and A · λ0 are dense in E ∗ and F ∗ , respectively. It is clear that μ0 E ∗ ≤ 1. Conversely, we have ι∗E (eA )E ∗ = sup{|eA , τl (x) | : x ∈ A ⊗γ A, τl (x)E ≤ 1} ≥ sup{|eA , τl (a ⊗ b) : a, b ∈ Ball(A)} ≥ sup{|a, T b | : a, b ∈ Ball(A)} = T  = 1, so that μ0 E ∗ = 1. Similarly, we see that λ0 F ∗ = 1.



With the help of Propositions 5.5.5 and 5.5.6, we can now prove: Theorem 5.5.7. Let (A, A∗ ) be a unital dual Banach algebra with eA  = 1, and let T ∈ B(A, A∗ ). Then the following are equivalent: (i) T ∈ Cσ,w (B(A, A∗ )); (ii) there are a normal, reflexive, contractive Banach left A-module E, x ∈ E, μ ∈ E ∗ and S ∈ B(E) such that a, T b = S(b · x), μ · a

(a, b ∈ A).

Proof. (i) =⇒ (ii): Suppose T ∈ Cσ,w (B(A, A∗ )), and let X, Y , μ0 ∈ X ∗ , λ0 ∈ Y ∗ , and C > 0 be as in Proposition 5.5.5 (our X here is the E and our Y here the F from Proposition 5.5.5). Define R ∈ (E ∗ ⊗γ F ∗ )∗ = B(F ∗ , E) by letting (a, b ∈ A). μ0 · a ⊗ b · λ0 , R = a, T b

This is, of course, only defined on μ0 · A ⊗ A · λ0 , but as μ0 · A and A · λ0 are dense in E ∗ and F ∗ , respectively, it is immediate from Proposition 5.5.5 that R extends to E ∗ ⊗γ F ∗ such that R ≤ CT . Set E := X ⊕2 Y ∗ , so that E is a normal, reflexive left Banach A-module and E ∗ = X ∗ ⊕2 Y . Define S ∈ B(E) by letting S(x, b · λ0 ) = (R(b · λ0 ), 0)

(x ∈ X, b · λ0 ∈ F ∗ ),

and let x = (0, λ0 ) ∈ E and μ := (μ0 , 0) ∈ E ∗ . It follows for a, b ∈ A that S(b · x), μ · a = R(b · λ0 , 0), μ0 · a0

= μ0 · a ⊗ b · λ0 , R

= a, T b , which establishes (i). (ii) =⇒ (i): Let π : A → B(E) be the weak∗ -weak∗ continuous representation associated with the left module action of A on E, and let

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5 Dual Banach Algebras

π∗ : E ∗ ⊗γ E → A∗ be its preadjoint. For a, b ∈ A, we have a, T b = π(a)Sπ(b)x, μ

= π∗ (Sπ(b)(x) ⊗ μ), a

= π∗ (x ⊗ S ∗ π(a)∗ μ), b , so that T A ⊂ A∗ and T ∗ A ⊂ A∗ . To see that T ∈ Cσ,w (B(A, A∗ )), we shall show that τl and τr —as defined in Proposition 5.5.5—are weakly compact. We shall only deal with τl (τr is dealt with analogously). As the closed unit ball of A ⊗γ A is the closed convex hull of the set {a ⊗ b : a, b ∈ Ball(A)} it is sufficient to show that {a · T b : a, b ∈ Ball(A)} is relatively weakly sequentially compact by [102, ∞ Theorems V.6.1 and V.6.3]. Thus, let (an )∞ n=1 and (bn )n=1 be sequences in A bounded by 1. As E is reflexive, every closed ball of E is weakly sequentially ∞ ∞ compact. Therefore, there are subsequences (ank )∞ k=1 and (bnk )k=1 of (an )n=1 ∞ and (bn )n=1 , respectively, as well as, y ∈ E such that π(ank )Sπ(bnk )x, ν → y, ν

(ν ∈ E ∗ ).

For c ∈ A, we thus obtain lim ank · T (bnk ), c = lim cank , T bnk

k→∞

k→∞

= lim π(cank )Sπ(bnk )x, μ

k→∞

= lim π(ank )Sπ(bnk )x, π(c)∗ μ

k→∞

= y, π(c)∗ μ

= π∗ (y ⊗ μ), c , so that ank · T (bnk ) → π∗ (y ⊗ μ) weakly. This proves (ii).



Even though we mentioned module homomorphism between Banach modules already a couple of times (with the tacit understanding that everybody knows what they are), we will now introduce them formally along with some notation that we shall use in the sequel: Definition 5.5.8. Let A be a Banach algebra. (a) Let E and F be left Banach A-modules. Then θ ∈ B(E, F ) is called a leftA-module homomorphism if θ(a · x) = a · θ(x)

(a ∈ A, x ∈ E).

We denote the set of all left A-module homomorphisms in B(E, F ) by A B(E, F ); if F = E, we simply write A B(E). (b) Let E and F be right Banach A-modules. Then θ ∈ B(E, F ) is called a rightA-module homomorphism if

5.5 Connes-Amenability and Connes-Injectivity

θ(x · a) = θ(x) · a

251

(a ∈ A, x ∈ E).

We denote the set of all right A-module homomorphisms in B(E, F ) by BA (E, F ); if F = E, we simply write BA (E). (c) Let E and F be Banach A-bimodules. Then θ ∈ B(E, F ) is called an Abimodule homomorphism if it is both a left and a right Banach A-module homomorphism. We denote the set of all A-bimodule homomorphisms in B(E, F ) by A B A (E, F ); if F = E, we simply write A B A (E). Remark 5.5.9. If θ is a left module homomorphism between left Banach modules, then θ∗ is a right module homomorphism (and vice versa); if θ a bimodule homomorphism, then so is θ∗ . Let E be a reflexive Banach space. Then B(B(E)) is a dual Banach space with canonical predual B(E) ⊗γ E ⊗γ E ∗ , which becomes an B(E)-bimodule via S · (T ⊗ x ⊗ μ) := T ⊗ Sx ⊗ μ and

(T ⊗ x ⊗ μ) · S := T ⊗ x ⊗ S ∗ μ (S, T ∈ B(E), x ∈ E, μ ∈ E ∗ ).

The dual module action on B(B(E)) is then given by (S · S)(T ) = SS(T ) and

(S · S)(T ) = S(T )S (S, T ∈ B(E), S ∈ B(B(E))).

(Compare the proof of Theorem 5.1.24.) We note: Lemma 5.5.10. Let E be a reflexive Banach space, and let A be a closed subalgebra of B(E). Then: (ii) A B A (B(E)) is an A-submodule of B(B(E)); (ii) there is a closed A-submodule X of B(E) ⊗γ E ⊗γ E ∗ such that A B A (B(E))

∼ = ((B(E) ⊗γ E ⊗γ E ∗ )/X)∗ ,

(5.38)

so that, in particular, A B A (B(E)) is weak∗ closed in B(B(E)); (iii) the map Ξ : (B(E) ⊗γ E ⊗γ E ∗ )/X → B(A, A∗ ) given by a ⊗ b, Ξ(T ⊗ x ⊗ μ + X)

:= aT bx, μ

(a, b ∈ A, T ∈ B(E), x ∈ E, μ ∈ E ∗ )

is a contractive A-bimodule homomorphism. Proof. (i): This is straightforward to verify. (ii): Define X as the closed linear span in B(E) ⊗γ E ⊗γ E ∗ of the elements of the form

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5 Dual Banach Algebras

aT ⊗ x ⊗ μ − T ⊗ x ⊗ a∗ μ and T a ⊗ x ⊗ μ − T ⊗ ax ⊗ μ (a ∈ A , T ∈ B(E), x ∈ E, μ ∈ E ∗ ). It is routinely verified that X ⊥ = A BA (B(E)). (Compare again the proof of Theorem 5.1.24.) (iii): It is easy to check that Ξ is well defined. The remaining claims are straightforward to verify.  Our next aim is to construct, for a dual Banach algebra A, a normal, reflexive left Banach A-module E such that Ξ defined in Lemma 5.5.10(iii) becomes a surjection onto Cσ,w (B(A, A∗ )). Definition 5.5.11. Let A be a Banach algebra, and let E be a left Banach A-module. Then E is called cyclic if there is x ∈ E such that E = A · x. Remark 5.5.12. Let A, E, and x be as in Definition 5.5.11, and let κ be the cardinality of A. Then the cardinality of A · x is at most κ, and since every element of E is the limit of a sequence in A·x, this means that the cardinality of E is at most κℵ0 (see [164] for background on cardinal arithmetic). Of course, one can also define cyclic right Banach modules; an analog of Remark 5.5.12 applies in this situation. Lemma 5.5.13. Let A be a dual Banach algebra. Then there is a normal, reflexive, contractive left Banach A-module E with the following properties: (i) every cyclic, normal, reflexive, contractive left Banach A-module is isometrically isomorphic to a 1-complemented submodule of E, i.e., for every such module F there are an isometry ι ∈ A B(F, E) and a contraction P ∈ A B(E, F ) such that P ι = idF ; (ii) if x1 , x2 ∈ E, μ1 , μ2 ∈ E ∗ , X(x1 , x2 ) := {a · (x1 , x2 ) : a ∈ A} ⊂ A · x1 ⊕2 A · x2 and Y (μ1 , μ2 ) := {(μ1 , μ2 ) · a : a ∈ A} ⊂ μ1 · A ⊕2 μ2 · A, then Y (μ1 , μ2 )∗ ⊕2 X(x1 , x2 ) is isometrically isomorphic to a 1-complemented submodule of E. Proof. Let Cycl (A) and Cycr (A) denote the classes of cyclic, normal, reflexive, contractive left and right Banach A-modules, respectively. As the members of these classes all have cardinality at most κℵ0 (with κ being the cardinality of A), Cycl (A) and Cycr (A) are actually sets. We can thus define ⎛ ⎛ ⎞ ⎞   E := ⎝2 X ⎠ ⊕2 ⎝2 Y ∗⎠ . X∈Cycl (A)

Evidently, E has the required properties.

Y ∈Cycr (A)



5.5 Connes-Amenability and Connes-Injectivity

253

Theorem 5.5.14. Let (A, A∗ ) be a unital dual Banach algebra with eA  = 1. Then there are a reflexive Banach space E and a unital, isometric, weak∗ weak∗ continuous representation π : A → B(E) such that Ξ : (B(E) ⊗γ E ⊗γ E ∗ )/X → B(A, A∗ ) as defined in Lemma 5.5.10(iii)—with respect to the left module action of A on E induced by π—is a bijection onto Cσ,w (B(A, A∗ )). Proof. By Theorem 5.4.1, there are a reflexive Banach space E0 and an isometric, weak∗ -weak∗ continuous representation π0 : A → B(E0 ). Replacing E0 by π(eA )E0 if necessary, we can suppose that π is unital. Let a ∈ A, and let  > 0. Then there is x ∈ Ball(E0 ) such that π0 (a)x > a − . As π0 (A)x ∈ Cycl (A), it follows that, for the left Banach A-module E specified in Lemma 5.5.13, the corresponding weak∗ -weak∗ continuous representation π : A → B(E) is isometric. The surjectivity of Ξ then follows from Theorem 5.5.7 and the properties of E. As a consequence, Ξ ∗ is an isomorphism onto its range. The remainder of the proof will thus consist of showing that Ξ ∗ is surjective. Let S ∈ A BA (E). We define X ∈ Cσ,w (B(A, A∗ ))∗ in the following way. For T ∈ Cσ,w (B(A, A∗ ), there are—by the properties of E—x ∈ E, μ ∈ E ∗ , and S ∈ B(E) such that a ⊗ b, T = S(b · x), μ · a

(a, b ∈ A);

set T, X := S(S)x, μ . With X as in the proof of Lemma 5.5.10, it follows that S ⊗ x ⊗ μ + X, Ξ ∗ (X)

= S ⊗ x ⊗ μ + X, X

= S(S)x, μ

= S ⊗ x ⊗ μ + X, S

(x ∈ E, μ ∈ E ∗ , S ∈ B(E)),

so that Ξ ∗ (X) = S—provided that X is well defined. Let T ∈ Cσ,w (B(A, A∗ )), and suppose that, for j = 1, 2, there are xj ∈ E, μj ∈ E ∗ , and Sj ∈ B(E) such that a ⊗ b, T = S(b · xj ), μj · a

(a, b ∈ A, j = 1, 2);

Let t ∈ (0, 1) be such that tS1  = (1 − t)S2 . It follows that C := tS1  = S1 S2 (S1  + S2 )−1 . We obtain |a ⊗ b, T | = t|S1 (b · x1 ), μ1 · a | + (1 − t)|S2 (b · x2 ), μ2 · a | ≤ tS1 μ1 · ab · x1  + (1 − t)S2 μ2 · ab · x2  1 1 ≤ C(μ1 · a2 + μ2 · a2 2 (b · x1 2 + b · x2 2 ) 2 . Define R ∈ B(X(x1 , x2 ), Y (μ1 , μ2 )∗ ∼ = (Y (μ1 , μ2 ) ⊗γ X(x1 , x2 ))∗ via (μ1 · a, μ2 · a) ⊗ (b · x1 , b · x2 ), R := a ⊗ b, T

(a, b ∈ A).

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5 Dual Banach Algebras

The foregoing calculation shows that R is well defined such that R ≤ C. Set Z := Y (μ1 , μ2 )∗ ⊕2 X(x1 , x2 ), and let PY ∗ and PX be the canonical projections from Z onto Y (μ1 , μ2 )∗ and X(x1 , x2 ), respectively. As Z ∗ = Y (μ1 , μ2 ) ⊕2 X(x1 , X2 )∗ , let PY and PX ∗ be defined analogously, so that ∗ . Set x0 = (0, (x1 , x2 )) ∈ Z, μ0 = ((μ1 , μ2 ), 0) ∈ Z ∗ , PY = PY∗ ∗ and PX ∗ = PX and define S0 ∈ B(Z) by letting S0 (y ∗ , x) = (Rf, 0)

(y ∗ ∈ Y (μ1 , μ2 )∗ , x ∈ X(x1 , x2 )),

so that S0 (b · x0 ),μ0 · a

= R(b · x1 , b · x2 ), (μ1 · a, μ2 · a) = a ⊗ b, T

(a, b ∈ A).

By Lemma 5.5.13(ii), there are an isometry ι ∈ A B(Z, E) and a contraction   P ∈ A B(E, Z) such that P ι = idZ . For j = 1, 2, define Uj ∈ A B Z, A · xj  ∗  and Vj ∈ BA Z , μj · A by letting Uj (y ∗ , a · (x1 , x2 )) = a · xj

and Vj ((μ1 , μ2 ) · a, x∗ ) = μj · a (a ∈ A, x∗ ∈ X(x1 , x2 )∗ , y ∗ ∈ Y (μ1 , μ2 )∗ ).

Note that a ⊗ b, T = Sj (b · xi ), μj · a

= Vj∗ Sj Uj (b · x0 ), μ0 · a

= PY ∗ Vj∗ Sj Uj PX (b · x0 ), μ0 · a

= PY ∗ S0 PX (b · x0 ), μ0 · a

(j = 1, 2, a, b ∈ A).

As A · x0 is dense in X(x1 , x2 ) and μ0 · A is dense in Y (μ1 , μ2 ), we conclude that PY ∗ Vj∗ Sj Uj PX = PY ∗ S0 PX . As S ∈ A BA (B(E), we obtain S(Sj )xj , μj = S(Sj )Uj PX x0 ), Vj PY μ0

= P ιPY ∗ Vj∗ S(Sj )Uj PX P ι(x0 ), μ0

= P S(ιPY ∗ Vj∗ Sj Uj PX P ι(x0 ), μ0

= P S(ιPY ∗ R0 PX P )ι(x0 ), μ0

(j = 1, 2),

so that S(S1 )x1 , μ1 = S(S2 )x2 , μ2 . This means that X is indeed well defined.  Proof (of Theorem 5.5.3). Applying Theorem 5.1.24, immediately yields (i) =⇒ (ii). (ii) =⇒ (i): Renorming A if necessary, we can suppose that eA  = 1. Let E and π : A → B(E) be as in Theorem 5.5.14. By the definition of Connesinjectivity, there is a quasi-expectation Q ∈ A B A (B(E)). Let Ξ be as defined

5.5 Connes-Amenability and Connes-Injectivity

255

in Lemma 5.5.10(iii). As Ξ is a bijection onto Cσ,w (B(A, A∗ )) by Theorem 5.5.14, Ξ ∗ : Cσ,w (B(A, A∗ ))∗ → A BA (B(E)) is also a bijection. Define D := (Ξ ∗ )−1 (Q) ∈ Cσ,w (B(A, A∗ ))∗ . As Ξ is a module homomorphism, so are Ξ ∗ and (Ξ ∗ )−1 ; since Q(B(E)) ⊂ π(A) , it follows that a · D = D · a for a ∈ A. To see that Δσ,w D = eA , recall that the closed unit ball of A ⊗γ A is dense in the closed unit ball of Cσ,w (B(A, A∗ ))∗ . Thus, there is a bounded net (dα )α in A ⊗γ A that converges to D in the weak∗ topology of Cσ,w (B(A, A∗ ))∗ . Let x ∈ E and μ ∈ E ∗ , and note that, with the notation from the proof of Theorem 5.5.14, we have x, μ = Q(idE )x, μ

= Ξ ∗ (D), idE ⊗ x ⊗ μ + X

= limΞ ∗ (dα ), idE ⊗ x ⊗ μ + X

α

= limidE ⊗ x ⊗ μ + X, dα

α

= limπ(ΔA dα )x, μ

α

= π(Δσ,w D)x, μ . This means that π(Δσ,w D) = idE and thus Δσ,w D = eA .



Exercises Exercise 5.5.1 Let (A, A∗ ) be a dual Banach algebra, and let I be a σ(A, A∗ ) closed ideal of A. Show that the following are equivalent: (i) I is unital; (ii) I is Connes-amenable; (iii) there is a σ(A, A∗ ) continuous projection from A onto I ⊥ . Conclude that, if A# is turned into a dual Banach algebra as in the proof of Theorem 5.4.1, then A# is Connes-amenable if and only if A is unital. Exercise 5.5.2 Let E and F be Banach spaces, and let T ∈ B(E ∗ , F ∗ ). Show that the following are equivalent and imply that T is weakly compact: (i) T is σ(E ∗ , E)-σ(F ∗ , F ∗∗ ) continuous; (ii) T ∗ (F ∗∗ ) ⊂ E; (iii) there is weakly compact T∗ ∈ B(F, E) such that T = (T∗ )∗ .

Notes and Comments The name tag “dual Banach algebra” occurs for the first time in [295], but the main idea, i.e., to consider Banach algebras that are also dual Banach spaces

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5 Dual Banach Algebras

with the obvious link between multiplication and the weak∗ topology goes back much further (see, for instance, [62] or, in particular, [196] and [259]). As we observed in Remark 5.1.2, the question of whether the predual of a dual Banach algebra is necessarily unique has a negative answer for elementary reasons. A much more subtle question is whether the canonical predual of a dual Banach algebra occurring “in nature” is unique. The answer to this question is negative, too: in [83] it is shown that the dual Banach algebra 1 (Z), which has the canonical predual c0 (Z), has a continuum of preduals other than c0 (Z) that also turn it into a dual Banach algebra. In certain cases, however, uniqueness assertions hold, e.g., in the additional presence of a Hopf algebra structure or in the case of A∗∗ where A is Arens regular and ideal in A∗∗ ([80], [84], and [85]). Our discussion of left introverted spaces is based on [216]; Example 5.1.12 is from [346]. In [196], Hochschild cohomology with separately weak∗ continuous cochains is considered. In his epochal paper [57], A. Connes suggests the term “amenable” for a von Neumann algebra satisfying Definition 5.1.17 ([57, Remark 5.33]). The moniker “Connes-amenability” seems to originate in [173]. Our discussion of Connes-amenability is mostly based on [295]. In [62], a notion of amenability for a Banach algebra A relative to a certain submodule E of A∗ is discussed: if E = A∗ this yields amenability in the usual sense; if A is a dual Banach algebra, E = A∗ yields Connes-amenability. With hypothesis (b), Theorem 5.1.24 is [39, Theorem 2]. (In [39] only von Neumann algebras are considered, but the proof carries over to the more general setting without modifications.) With hypothesis (a), Theorem 5.1.24 is essentially [63, Proposition 2.2] (the proof in [63] is different from ours, and the hypotheses are somewhat more restrictive). Of course, there are examples of inner amenable, locally compact groups which are neither amenable nor discrete: for instance, all [IN]-groups ([260, 5.1.9, Definition]) are inner amenable. For references to the original literature on inner amenability, see [262]. Implication (iv) =⇒ (i) of Theorem 5.1.27 for inner amenable groups was proven in [218]. Theorem 5.1.33 is from [300] whereas Corollary 5.1.35 is much older: it was first proven for von Neumann algebras in [103], and subsequently, the observation that the proof carries over to general dual Banach algebras was made in [62]. Theorem 5.2.1 combines the main results of [298] and [299]. In view of Corollary 5.2.13 and the characterization of amenable semigroup algebras from [74], it is natural to ask for a characterization of those semigroups S for which 1 (S)—with its canonical predual c0 (S)—is Connes-amenable. The problem here is that unless S is weakly cancellative, i.e., if for all x, y ∈ S, the sets {z ∈ S : xz = y} and {z ∈ S : zx = y} are finite, 1 (S) fails to be a dual Banach algebra from the start. However, if S is weakly cancellative, then 1 (S) is indeed a dual Banach algebra ([79, Proposition 5.1]). Connesamenability for 1 (S) with weakly cancellative S, however, appears to be a rather strong condition: if S is cancellative, i.e., if for each y ∈ S, the maps S x → xy and S x → yx are injective, or if it has an identity, then the Connes-amenability of 1 (S) forces S to be already an amenable group

Notes and Comments

257

([79, Theorem 5.13]). The paper [79] also studies the Connes-amenability of weighted semigroup algebras. In view of Theorems 2.1.10 and 5.2.1, it is a plausible conjecture that, for a locally compact group G, the Connes-amenability of B(G) is equivalent to the amenability of A(G). In view of Theorem 3.2.10, this would mean that B(G) is Connes-amenable if and only if G is almost abelian. This is indeed the case: the “if” part is obvious, and the “only if” part was proven in [310], after some partial results had ready been obtained in [300], e.g., if G is a product of finite groups or discrete and amenable. Theorem 5.3.11 is from [303]; the proof in [303] contains a gap that was closed in [307]. There are non-compact, locally compact groups G for which WAP(G)∗ does have a normal virtual diagonal. For N ≥ 2, let SO(N ) act on RN via matrix multiplication; the semidirect product SO(N )  RN is called the motion group of RN . Based on results from [52], it is shown in [303] that WAP(G)∗ has a normal, virtual diagonal if G = SO(N )  RN . For abelian semigroups, Theorem 5.3.4 was proven in [213]; the discrete, abelian case had previously been dealt with in [179]. The actual formulation of Theorem 5.3.4 for general, not necessarily abelian semigroups is from [307]; the proof is patterned after that from [213]. (Already in [179], it is remarked that Theorem 5.3.4 holds for discrete, not necessarily abelian semigroups.) Theorem 5.4.1 is from M. Daws’ paper [80]. The circle of ideas expounded in Exercises 5.4.1 to 5.4.4 goes back to [78]. Theorem 5.4.1 prompts the question if there is an analog of von Neumann’s Bicommutant Theorem for general dual Banach algebras. A naive attempt to extend the Bicommutant Theorem to general dual Banach algebras even fails in the Hilbert space setting: view the algebra A of upper triangular 2 × 2 matrices as a subalgebra of B(22 ); then A is a dual Banach algebra, but A = B(22 ). Still, there is something like a Bicommutant Theorem for dual Banach algebras ([82]): if A is a unital dual Banach algebra, then there are a reflexive Banach space E and an isometric, weak∗ -weak∗ continuous representation π : A → B(E) such that π(A) = π(A). Connes-injective dual Banach algebras are called “injective” in [80], obviously with an eye on the von Neumann algebra situation (see Section 7.2 below). However, as Definition 5.5.1 does not appear to correspond to the categorical notion of injectivity within a suitable category, and since the adjective “injective” is already used twice—in different contexts—in this book (see Sections 6.3 and 7.2 below), we chose to rechristen them. Theorem 5.5.3 is also from [80] and answers affirmatively a question raised in [63, Section 3]. As Exercise 5.5.1 makes clear, it is pointless to extend Theorem 5.5.3 to non-unital algebras via the obvious unitization. In [330], R. Stokke defines left dual Banach algebras and introduces the notion of left Connes-amenability for them. An example of a left dual Banach algebra is LUC(G)∗ for a locally compact group G, which is a dual Banach algebra only if G is compact; it is left Connes-amenable if and only if G is amenable ([330, Theorem 7.9]).

Chapter 6

Banach Homological Algebra

At the end of Chapter 2, we mentioned that there is an alternative approach to Hochschild cohomology (due to A. Ya. Helemski˘ı’s Moscow school), which is more powerful and sophisticated than the direct one presented in Section 2.4. It adapts the tools of homological algebra to the Banach algebra setting. This chapter is an introduction to the gospel according to Alexander (Helemski˘ı) for the not yet converted. This means that no background in homological algebra is required, and that no use is made the jargon of category theory, i.e., terms like “category”, “functor”, “natural transformation”, or “derived functor” have been avoided. Of course, anyone who has had previous exposure to category theory will immediately recognize how the definitions, theorems, etc., in this chapter, can be reworded in terms of category theory. Although no familiarity with homological algebra is necessary to be able to read this chapter, it is certainly helpful: many of the concepts and results in this chapter are straightforward adaptations of results from (algebraic) homological algebra to the Banach context.

6.1 Projectivity We begin our introduction to Banach homological algebra with what is perhaps its most important concept: projectivity. Let A be a Banach algebra with unitization A# . If E is a Banach space, the projective tensor product A# ⊗γ E becomes a left Banach A-module through a · (b ⊗ x) := ab ⊗ x

(a ∈ A, b ∈ A# , x ∈ E).

Definition 6.1.1. Let A be a Banach algebra, and let E be a Banach space. Then the left Banach A-module A# ⊗γ E is called free.

© Springer Science+Business Media, LLC, part of Springer Nature 2020 V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-1-0716-0351-2 6

259

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6 Banach Homological Algebra

Already in the proof of Theorem 2.3.21, we noted that for any Banach algebra A and for any left Banach A-module E, the module action extends canonically to A# ; the same is true for right modules or bimodules. Therefore, the multiplication map ΔA# ,E : A# ⊗γ E → E,

a ⊗ x → a · x

is well-defined. It is obvious that ΔA# ,E ∈ A B(A# ⊗γ E, E) has a right inverse in B(E, A# ⊗γ E). Definition 6.1.2. Let A be a Banach algebra. A left Banach A-module P is called projective if the multiplication map ΔA# ,P : A# ⊗γ P → P has a right inverse in A B(P, A# ⊗γ P ). The reason why we use ΔA# ,P : A# ⊗γ P → P in Definition 6.1.2 instead of ΔA,P : A ⊗γ P → P is that, in general, ΔA,P is not surjective, and without surjectivity, it makes no sense to require right inverses to exist (compare Exercise 6.1.2). Example 6.1.3. Every free left Banach module is projective as follows easily from Exercise 6.1.1. Example 6.1.4. Let E be a Banach space, and let A be a standard Banach operator algebra on E. Then E is an essential left Banach A-module through T · x := T x

(T ∈ A, x ∈ E).

Fix y ∈ E and φ ∈ E ∗ such that y, φ = 1. Define θ : E → A ⊗γ E through θ(x) := (x  φ) ⊗ y

(x ∈ E).

It is clear from this definition that θ ∈ A B(E, A ⊗γ E) is such that ΔA,E ◦ θ = idE . By Exercise 6.1.2, E is thus projective. Next, we give a characterization of projective left Banach modules, for which we require yet another definition: Definition 6.1.5. Let A be a Banach algebra, and let E and F be left Banach A-modules. Then θ ∈ A B(E, F ) is called admissible if θ(E) and ker θ are closed, complemented subspaces of F and E, respectively. Remark 6.1.6. We previously defined admissible, short, exact sequences of Banach modules (Definition 2.3.20). It is easily checked that a short exact sequence {0} −→ F −→ E −→ E/F −→ {0} of left Banach A-modules is admissible in the sense of Definition 2.3.20 if and only if the quotient map from E onto E/F —or, equivalently, the inclusion of F into E—is admissible.

6.1 Projectivity

261

Proposition 6.1.7. Let A be a Banach algebra, and let P be a left Banach A-module. Then the following are equivalent: (i) P is projective; (ii) there is a left Banach A-module Q such that Q ⊕ P is free; (iii) if E and F are left Banach A-modules, if θ ∈ A B(E, F ) is surjective and admissible, and if σ ∈ A B(P, F ), then there is ρ ∈ A B(P, E) such that σ = θ ◦ ρ, i.e., the diagram P }} } } σ }}  } ~} / F. E ρ

θ

commutes; (iv) every admissible, short, exact sequence {0} −→ F → E −→ P −→ {0} of left Banach A-modules splits. Proof. (i) =⇒ (ii): By definition, ΔA# ,P : A# ⊗γ P → P has a right inverse in A B(P, A# ⊗γ P ), say ρ. Then Q = ker ΔA# ,P satisfies Q ⊕ P ∼ = A# ⊗γ P . (ii) =⇒ (iii): Suppose first that P is free. Let T ∈ B(F, E) be a right inverse of θ, and define R := T ◦ σ. By the definition of a free left Banach module, there is a Banach space X such that P ∼ = A# ⊗γ X, and by Exercise 6.1.1 below, there is a unique ρ ∈ A B(P, E) such that ρ(eA# ⊗ x) = Rx

(x ∈ X).

We thus obtain θ(ρ(a ⊗ x)) = θ(a · ρ(eA# ⊗ x)) = a · θ(Rx) = a · σ(eA# ⊗ x) = σ(a ⊗ x)

(a ∈ A# , x ∈ X),

so that σ = θ ◦ ρ. ˜ := σ ◦ ΔA# ,P . For the general case, replace P by A# ⊗ P and σ by σ ˜ = θ ◦ ρ˜. Let The free case then yields ρ˜ ∈ A B(A# ⊗γ P, E) such that σ τ ∈ A B(P, A# ⊗γ P ) be a right inverse of ΔA# ,P , and set ρ := ρ˜ ◦ τ . (iv) is a particular case of (iii). (iv) =⇒ (i): Apply (iv) to the short, exact sequence ΔA# ,P

{0} −→ ker ΔA# ,P −→ A# ⊗γ P −→ P −→ {0}.

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6 Banach Homological Algebra



This yields the claim. Combining Proposition 6.1.7 and Exercise 4.1.5, we obtain:

Corollary 6.1.8. Let A be a contractible Banach algebra. Then every left Banach A-module is projective. Of course, one can define projective right Banach modules and projective Banach bimodules as well. We will now look at the bimodule situation. Definition 6.1.9. Let A be a Banach algebra. A Banach A-bimodule is called projective if the multiplication map ΔA# ,P,A# : A# ⊗γ P ⊗γ A# → P,

a ⊗ x ⊗ b → a · x · b.

has a right inverse in A B A (P, A# ⊗γ P ⊗γ A# ). In analogy with the left module situation, we call a Banach bimodule over a Banach algebra A free if it is of the form A# ⊗γ E ⊗γ A# for some Banach space E; of course, any free Banach A-bimodule is projective. After Definition 4.5.1, we stated that the reason for our choice of terminology would become clear in this chapter. We will now see that a Banach algebra is biprojective if and only if it is a projective Banach bimodule over itself. Lemma 6.1.10. The following are equivalent for a Banach algebra A: (i) A is a projective Banach A-bimodule; (ii) both multiplication maps ΔA# ,A : A# ⊗γ A → A and ΔA,A# : A ⊗γ A# → A have right inverses in A B A (A, A# ⊗γ A) and A B A (A, A ⊗γ A# ), respectively. Proof. (i) =⇒ (ii): The short exact sequence ΔA# ,A

{0} −→ ker ΔA# ,A −→ A# ⊗γ A −→ A −→ {0} is clearly admissible and thus splits by (the bimodule analog of) Proposition 6.1.7. The same is true for ΔA,A#

{0} −→ ker ΔA,A# −→ A ⊗γ A# −→ A −→ {0}. (ii) =⇒ (i): Let ρ ∈ A BA (A, A# ⊗γ A) be a right inverse of ΔA# ,A , and let σ ∈ A BA (A, A⊗γ A# ) be a right inverse of ΔA,A# . It is routinely checked that (idA# ⊗ σ) ◦ ρ ∈ A BA (A, A# ⊗γ A ⊗γ A# ) is a right inverse of ΔA# ,A,A# .  Theorem 6.1.11. The following are equivalent for a Banach algebra A: (i) A is biprojective; (ii) A is a projective A-bimodule.

6.1 Projectivity

263

Proof. (i) =⇒ (ii): Let ρ ∈ A B A (A, A⊗γ A) be a right inverse of ΔA : A⊗γ A → A. It is obvious that ρ is also a right inverse of both ΔA# ,A : A# ⊗γ A → A and ΔA,A# : A ⊗γ A# → A. By Lemma 6.1.10, A is a projective Banach A-bimodule. (ii) =⇒ (i): By Lemma 6.1.10, there is a right inverse ρ ∈ A B A (A, A# ⊗γ A) of ΔA# ,A . We have ρ(ab) = a · ρ(b) ∈ A ⊗γ A

(a, b ∈ A).

In order to establish that A is biprojective, it is thus sufficient to show that the linear span of {ab : a, b ∈ A} is dense in A. Fix a ∈ A. Then there are b ∈ A and a ∈ A ⊗γ A such that ρ(a) = eA# ⊗ b + a. Since ΔA# ,A ◦ ρ = idA , it follows that a = b + ΔA# ,A a, so that ρ(a) = eA# ⊗ a − eA# ⊗ ΔA# ,A a + a. Consequently, we have ρ(a2 ) = a · ρ(a) = a ⊗ a − a ⊗ ΔA# ,A a + a · a and

(6.1)

ρ(a2 ) = ρ(a) · a = eA# ⊗ a2 − eA# ⊗ (ΔA# ,A a)a + a · a. ∗

Choose φ ∈ A with φ|A2 ≡ 0, and define an extension φ letting eA# , φ#  = 1. Then (6.1) yields

#

(6.2) #

of φ to A

by

ρ(a2 ), φ# ⊗ φ = a, φ# a, φ = a, φ2 whereas, from (6.2), we obtain ρ(a2 ), φ# ⊗ φ = a2 , φ = 0, so that a, φ = 0. As a ∈ A was arbitrary, this means that φ ≡ 0. The Hahn–Banach Theorem implies that A2 is dense in A. 

Exercises Exercise 6.1.1. Let E be a Banach space, let A be a Banach algebra, let F be a left Banach A-module, and let T ∈ B(E, F ). Show that there is a unique θ ∈ A B(A# ⊗γ E, F ) such that θ(eA# ⊗ x) = T x

(x ∈ E).

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6 Banach Homological Algebra

Exercise 6.1.2. Let A be a Banach algebra, and let P be an essential left Banach A-module. Show that P is projective if and only if ΔA,P : A⊗γ P → P has a right inverse in A B(P, A ⊗γ P ). Exercise 6.1.3. Let A be a Banach algebra, let P be a projective left Banach A-module, and let Q be a projective right Banach A-module. Show that P ⊗γ Q is a projective Banach A-bimodule.

6.2 Resolutions and Ext-Groups The reason why projective modules are so important is that they can be used quite effectively to compute the Hochschild cohomology groups of a Banach algebra. We start with the basic definitions: Definition 6.2.1. A cochain complex E = (En , δ n )n∈Z of Banach spaces is a sequence (En )n∈Z of Banach spaces along with bounded, linear maps δn

· · · −→ En −→ En+1 −→ · · · such that δ n ◦ δ n−1 = 0 for n ∈ Z. Example 6.2.2. Let A be a Banach algebra, and let E be a Banach Abimodule. For n ∈ N0 , let En := C n (A, E), and let δ n be the n-coboundary operator. For n ∈ Z with n < 0, let En := {0} and δ n := 0. Then (En , δ n )n∈Z is a cochain complex by Lemma 2.4.3. This puts the name Hochschild cochain complex into perspective. In this example, the spaces En are {0} if n < 0. In such a situation, we shall simply forget about the uninteresting spaces and denote the complex by (En , δ n )∞ n=0 . Definition 6.2.3. Let E = (En , δ n )n∈Z be a complex of Banach spaces, and let n∈Z. For n ∈ Z, let Z n (E):= ker δ n , let B n (E):=δ n−1 (En−1 ), and define Hn (E) := Z n (E)/B n (E). Then Hn (E) is the n-th cohomology group of the complex E. Remark 6.2.4. The cohomology groups of a cochain complex of Banach spaces are in fact linear spaces. Equipped with the quotient topology, each such cohomology group becomes even a topological vector space (albeit not necessarily Hausdorff). Remark 6.2.5. If we choose E as in Example 6.2.2, we have Hn (E) = Hn (A, E) for n ∈ N0 and Hn (E) = {0} for n < 0. If we forget about the uninteresting groups Hn (E) = {0} for n < 0, we obtain Hochschild cohomology as a particular case of Definition 6.2.3.

6.2 Resolutions and Ext-Groups

265

The first results we prove in this section are general results on cohomology in the sense of Definition 6.2.3, which, in particular, apply to Hochschild cohomology. We first deal with the question of when two cochain complexes have the same cohomology. n )n∈Z be cochain Definition 6.2.6. Let E = (En , δEn )n∈Z and F = (Fn , δF complexes of Banach spaces. A morphism φ : E → F is a family (φn )n∈Z of bounded linear maps φn : En → Fn such that n δF ◦ φn = φn+1 ◦ δEn

(n ∈ Z).

n Remark 6.2.7. If E = (En , δEn )n∈Z and F = (Fn , δF )n∈Z are cochain complexes of Banach spaces, then any morphism φ : E → F induces a sequence φ¯ = (φ¯n )n∈Z of group homomorphisms—in fact, linear maps —φ¯n : Hn (E) → Hn (F). n Definition 6.2.8. Let E = (En , δEn )n∈Z and F = (Fn , δF )n∈Z be cochain complexes of Banach spaces. Two morphism φ, ψ : E → F are called homotopic if there is a family τ = (τn )n∈Z of bounded linear maps τn : En+1 → Fn such that n−1 ◦ τn−1 + τn ◦ δEn (n ∈ Z). (6.3) φn − ψ n = δ F

The family τ is called a homotopy of φ and ψ. The reason why homotopy is an important concept when dealing with cohomology is the following theorem: Theorem 6.2.9. Let E and F be cochain complexes of Banach spaces, and ¯ let φ, ψ : E → F be homotopic morphisms. Then φ¯ = ψ. Proof. Fix n ∈ Z, and let x ∈ Z n (E). By (6.3), it is immediate that φn (x) −  ψn (x) ∈ B n (F) and thus φ¯n (x + B n (E)) = ψ¯n (x + B n (E)). Definition 6.2.10. Two cochain complexes E and F of Banach spaces are called homotopically equivalent if there are morphisms φ : E → F and ψ : F → E such that φ ◦ ψ and ψ ◦ φ are homotopic to the identity morphism on F and E, respectively. We didn’t formally define the composition of two morphisms of cochain complexes of Banach spaces, but it should be obvious what we mean. As an immediate consequence of Theorem 6.2.9, we obtain: Corollary 6.2.11. Let E and F be homotopically equivalent cochain complexes of Banach spaces. Then Hn (E) ∼ = Hn (F)

(n ∈ Z).

Definition 6.2.12. Let E = (En , δEn )n∈Z be a cochain complex of Banach spaces. Then:

266

6 Banach Homological Algebra

n (i) a subcomplex of E is a cochain complex F = (Fn , δF )n∈Z of Banach spaces such that, for each n ∈ Z, the space Fn is a closed subspace of n = δEn |Fn ; En and δF (ii) given a subcomplex F of E, the corresponding quotient complex E/F is n n )n∈Z where the maps δE/F are those induced the complex (En /Fn , δE/F n by the maps δE ; (iii) the subcomplex F is called complemented if, for each n ∈ Z, the space Fn is complemented in En .

Theorem 6.2.13 (Long, Exact Sequence). Let E be a cochain complex of Banach spaces, and let F be a complemented subcomplex. Then we have a long, exact sequence · · · −→ Hn−1 (E/F) −→ Hn (F) −→ Hn (E) −→ Hn (E/F) −→ Hn+1 (F) −→ · · ·

(6.4)

of cohomology groups. Proof. For each n ∈ Z, let ρn : En /Fn → En be a bounded, linear right inverse of the quotient map πn : En → En /Fn . For each n ∈ Z, the natural maps ι

π

n n {0} −→ Fn −→ En −→ En /Fn −→ {0}

induce group homomorphisms—in fact: linear maps— ¯ ι

π ¯

n n Hn (F) −→ Hn (E) −→ Hn (E/F).

It is routinely checked that ¯ιn (Hn (F)) ⊂ ker π ¯n . For the converse inclusion, ¯n (x + B n (E)) = 0, i.e., there is y ∈ En−1 /Fn−1 let x ∈ Z n (E) be such that π n−1 such that πn (x) = δE/F y. Set z := x − δEn−1 (ρn−1 (y)). It is immediate that z ∈ Z n (E). Moreover, we have n−1 πn (z) = πn (x) − πn (δEn−1 (ρn−1 (y))) = πn (x) − δE/F y = πn (x) − πn (x) = 0,

so that, in fact, z ∈ Z n (F). From the definition of z, it is clear that x and z belong to the same equivalence class in Hn (E), so that x+B n (E) ∈ ¯ιn (Hn (F)). For n ∈ Z, define σn : En /Fn → En+1 , First, note that

n x → δEn (ρn (x)) − ρn+1 (δE/F x).

6.2 Resolutions and Ext-Groups

267

n πn+1 (σn (x)) = πn+1 (δEn (ρn (x))) − πn+1 (ρn+1 (δE/F x)) n n = δE/F (πn (ρn (x))) − δE/F x n n = δE/F x − δE/F x

= 0, so that σn maps into Fn+1 . Let x ∈ Z n (E/F). Then we obtain n+1 δF (σn (x)) = δEn+1 (δEn (ρn (x))) = 0,

i.e., σn maps Z n (E/F) into Z n+1 (F). If x ∈ B n (E/F), i.e., if there is y ∈ n−1 y = x, we obtain En−1 /Fn−1 such that δE/F n−1 σn (x) = δEn (ρn (δE/F y)) n−1 n = δEn (ρn (δE/F y) − δEn−1 (ρn−1 (y))) = δF (−σn−1 (y)) ∈ Bn+1 (F).

Hence, σn induces a group homomorphism σ ¯n : Hn (E/F) → Hn+1 (F). n ¯n = π ¯n (Hn (E)). We claim that σ ¯n (H (E/F)) = ker ¯ιn+1 and that ker σ ¯n (Hn (E/F)), i.e., there is y ∈ Let x ∈ Fn be such that x + B n (F) ∈ σ Z n (En /Fn ) such that x = σn (y) = δEn (ρn (y)) ∈ B n+1 (E). It follows that ¯n (Hn (E/F)) ⊂ ker ¯ιn+1 holds. Conversely, x + B n (F) ∈ ker ¯ιn+1 . Hence, σ let x ∈ Z n+1 (F) be such that x + B n+1 (F) ∈ ker ¯ιn+1 . This means, there is y ∈ En such that x = δEn y. Let z := y − ρn (πn (y)); it is immediate that πn (z) = 0, so that z ∈ Fn . Furthermore, we have x = δEn (ρn (πn (y))) − ρn+1 (πn (x)) + δEn y − δEn (ρn (πn (y))) n = δEn (ρn (πn (y))) − ρn+1 (δE/F (πn (y))) + δEn z n = σn (πn (y)) + δF z.

¯n (Hn (E/F)). This yields ker ¯ιn+1 ⊂ σ n Let x ∈ Z (E). We have n σn (πn (x)) = δEn (ρn (πn (x))) − ρn+1 (δE/F (πn (x)))

= δEn (ρn (πn (x))) − ρn+1 (πn+1 (δEn x)) = δEn (ρn (πn (x))) = δEn (ρn (πn (x))) − δEn x n = δF (ρn (πn (x)) − x), ∈B

n+1

as ρn (πn (x)) − x ∈ Fn ,

(F).

¯n . Conversely, let x ∈ Z n (E/F) be such It follows that π ¯n (Hn (E)) ⊂ ker σ n+1 n (F). Then there is y ∈ F such that σn (x) = δF y. Let that σn (x) ∈ B z := ρn (x) − y. As

268

6 Banach Homological Algebra n δEn x = δEn (ρn (x)) − δEn y = δEn (ρn (x)) − σn (x) = ρn+1 (δE/F x) = 0,

we have z ∈ Z n (E). It is immediate that πn (z) = x (as y ∈ Fn ). Hence, we ¯n (Hn (E)). conclude that ker σ ¯n ⊂ π This completes the proof.  Remark 6.2.14. The connecting maps in (6.4) are not only group homomorphisms, but linear. If each cohomology group is equipped with its (possibly non-Hausdorff) quotient topology, these maps are even continuous. We now move from cochain complexes of Banach spaces to chain complexes of left Banach modules. Definition 6.2.15. Let A be a Banach algebra. A chain complex E = (En , dn )n∈Z of left Banach A-modules is a sequence (En )n∈Z of left Banach A-modules along with bounded, A-module homomorphisms d

n · · · ←− En ←− En+1 ←− · · ·

such that dn ◦ dn+1 = 0 for n ∈ Z. We make the distinction between the chain and cochain complexes mainly for our convenience: it doesn’t really matter in which direction the arrows point. We have analogs of Definitions 6.2.6, 6.2.8, and 6.2.10: Definition 6.2.16. Let A be a Banach algebra, and let E = (En , dEn )n∈Z and F = (Fn , dF n )n∈Z be chain complexes of left Banach A-modules. A morphism φ : E → F is a family (φn )n∈Z of bounded A-module homomorphisms φn : En → Fn such that E dF n ◦ φn+1 = φn ◦ dn

(n ∈ Z).

Definition 6.2.17. Let A be a Banach algebra, and let E = (En , dEn )n∈Z and F = (Fn , dF n )n∈Z be chain complexes of left Banach A-modules. Two morphism φ, ψ : E → F are called homotopic if there is a family τ = (τn )n∈Z of bounded A-module homomorphisms τn : En → Fn+1 such that E φn − ψ n = d F n ◦ τn + τn−1 ◦ dn−1

(n ∈ Z).

The family τ is called a homotopy of φ and ψ. Definition 6.2.18. Let A be a Banach algebra. Two chain complexes E and F of left Banach A-modules are called homotopically equivalent if there are morphisms φ : E → F and ψ : F → E such that φ ◦ ψ and ψ ◦ φ are homotopic to the identity morphism on F and E, respectively. We still require more definitions. Definition 6.2.19. Let A be a Banach algebra. A chain complex E = (En , dn )n∈Z of left Banach A-modules is called admissible if:

6.2 Resolutions and Ext-Groups

269

(i) ker dn = dn+1 (En+1 ) for n ∈ Z; (ii) each module homomorphism dn is admissible. Definition 6.2.20. Let A be a Banach algebra, and let E be a left Banach A-module. A resolution for E is an admissible chain complex

d

d

0 1 {0} ←− E ←− E0 ←− E1 ←− E2 ←− · · ·

of left Banach A-modules. If the modules E0 , E1 , . . . are projective, the resolution is called projective. Example 6.2.21. Let A be a Banach algebra, and let E be a left Banach Amodule. For n ∈ N0 , let Bn (E) := A# ⊗γ A ⊗γ · · · ⊗γ A ⊗γ E.    n−times

Define : B0 (E) → E,

a ⊗ x → a · x,

and, for n ∈ N0 , let dn : Bn+1 (E) → Bn (E) be given by dn (a ⊗ a1 ⊗ · · · ⊗ an+1 ⊗ x) = aa1 ⊗ · · · ⊗ an+1 ⊗ x n  + (−1)k a ⊗ a1 ⊗ · · · ⊗ ak ak+1 ⊗ · · · ⊗ an+1 ⊗ x k=1

+ (−1)n+1 a ⊗ a1 ⊗ · · · ⊗ an+1 · x (a ∈ A# , a1 , . . . , an+1 ∈ A, x ∈ E). It is immediate that and d0 , d1 , . . . are left A-module homomorphisms, and a dull, albeit tedious calculation yields ◦d0 = 0 and dn ◦dn+1 = 0 for n ∈ N0 . The modules B0 (E), B1 (E), . . . are free and, therefore, projective. Obviously, is admissible. Let φ : A# → C be the character with kernel A. For n ∈ N0 , define τn :Bn (E) → Bn+1 (E), a ⊗ a1 ⊗ · · · ⊗ an ⊗ x → eA# ⊗ (a − φ(a)eA# ) ⊗ a1 ⊗ · · · ⊗ an ⊗ x. It is then routinely checked that dn ◦ τn + τn−1 ◦ dn−1 = idBn (E)

(n ∈ N0 ),

where formally d−1 = . For n ∈ N0 , the map Bn (E) → ker dn−2 ⊕ ker dn−1 ,

x → (dn−1 x, dn (τn (x)))

(6.5)

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6 Banach Homological Algebra

is a topological isomorphism (formally let d−2 := 0): an inverse of (6.5) is given by ker dn−2 ⊕ ker dn−1 → Bn (E),

(x, y) → τn−1 (x) + y.

(6.6)

It follows that ker dn−1 = dn (Bn+1 (E)) for n ∈ N0 and that dn is admissible for n ∈ N0 . Hence, the complex d



d

0 1 {0} ←− E ←− B0 (E) ←− B1 (E) ←− B2 (E) ←− · · ·

is a projective resolution of E, the Bar resolution of E. We use B• (E) to denote the chain complex d

d

0 1 {0} ←− B0 (E) ←− B1 (E) ←− B2 (E) ←− · · ·

As the modules in the Bar resolution are free and thus projective, the following is an immediate consequence of Example 6.2.21: Corollary 6.2.22. Let A be a Banach algebra, and let E be a left Banach A-module. Then E has a projective resolution. Of course, one module can have several projective resolutions: if A is a Banach algebra and P is a projective left Banach A-module, then ΔA# ,P

{0} ←− P ←− A# ⊗γ P ←− ker ΔA# ,P ←− {0} is a projective resolution different from B• (P ). Nevertheless, two resolutions of the same module are not completely unrelated provided that one is projective: Theorem 6.2.23 (Comparison Theorem). Let A be a Banach algebra, let E and F be left Banach A-modules, and let θ ∈ A B(E, F ). Furthermore, let

d

d

0 1 {0} ←− E ←− P0 ←− P1 ←− P2 ←− · · ·

(6.7)

be a projective resolution of E, and let d



d

0 1 {0} ←− F ←− E0 ←− E1 ←− E2 ←− · · ·

(6.8)

be an arbitrary resolution of F . Then there is a morphism φ = (φn )∞ n=0 from the chain complex (6.7) to the chain complex (6.8) such that the diagram {0} o

Eo



θ

{0} o

 F o

P0 o

d0

··· o

dn−1

φ0





 E0 o

Pn o

dn

φn d0

··· o

dn−1

 En o

Pn+1 o

dn+1

···

φn+1 dn

 En+1 o

dn+1

···

6.2 Resolutions and Ext-Groups

271

commutes. Proof. We proceed using induction. Suppose that we have already found φ0 , φ1 , . . . φn with φk ∈ A B(Pk , Ek ) for k = 0, 1, . . . , n such that the diagram {0} o

Eo



θ

{0} o

 F o

P0 o

d0

··· o

dn−1

φ0



 E0 o

Pn o

dn

Pn+1 o

dn+1

··· (6.9)

φn d0

··· o

dn−1

 En o

dn

En+1 o

dn+1

···

commutes. Let Fn := ker dn−1 = dn (En ), and let θ˜ := dn . Then θ˜ : En+1 → Fn is an admissible, surjective homomorphism of left Banach A-modules. Define σ : Pn+1 → En as φn ◦ dn . Since dn−1 ◦ φn ◦ dn = φn−1 ◦ dn−1 ◦ dn = 0, the A-module homomorphism σ attains its values in Fn . By Proposition 6.1.7(iii), there is φn+1 : Pn+1 → En+1 such that σ := θ˜ ◦ φn+1 . Clearly, if we extend the diagram (6.9) by φn+1 , this extended diagram still commutes.  Corollary 6.2.24. Let A be a Banach algebra, and let E be a left Banach A-module. Then any two projective resolutions for E are homotopically equivalent. 

Proof. Apply Theorem 6.2.23 with θ = idE .

We shall now see that projective resolutions can be used efficiently to calculate cohomology groups. Lemma 6.2.25. Let A be a Banach algebra, let F be a left Banach A-module and let E = (En , dn )n∈Z be a chain complex of left Banach A-modules. Define δ n : A B(En , F ) → A B(En+1 , F ),

T → T ◦ dn .

Then (A B(En , F ), δ n )n∈Z is a cochain complex of Banach spaces. Proof. Obvious.



We suggestively denote the cochain complex described in Lemma 6.2.25 by A B(E, F ). The following lemma is equally easy to check: Lemma 6.2.26. Let A be a Banach algebra, let F be a left Banach A-module, and let E1 and E2 be homotopically equivalent chain complexes of left Banach A-modules. Then the cochain complexes of Banach spaces A B(E1 , F ) and A B(E2 , F ) are homotopically equivalent. We can now define Ext-groups:

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Definition 6.2.27. Let A be a Banach algebra, let E and F be left Banach A-modules, let a projective resolution d



d

0 1 {0} ←− E ←− P0 ←− P1 ←− P2 ←− · · · .

for E be given, and let P denote the chain complex of left Banach A-modules d

d

0 1 {0} ←− P0 ←− P1 ←− P2 ←− · · · .

(6.10)

Then, for n ∈ N0 , the n-th Ext-group of E by F is defined as ExtnA (E, F ) := Hn (A B(P, F )). Remark 6.2.28. Let two projective resolutions for E given. By Corollary 6.2.24, these two resolutions are homotopically equivalent, and so are the corresponding chain complexes in (6.10), which we denote by P1 and P2 . By Lemma 6.2.26, the cochain complexes of Banach spaces A B(P1 , F ) and A B(P2 , F ) are also homotopically equivalent. By Corollary 6.2.11, this means that (n ∈ N0 ). Hn (A B(P1 , F )) ∼ = Hn (A B(P2 , F )) Hence, ExtnA (E, F ) is independent of a particular projective resolution for E. The following is extremely useful when it comes to calculating Ext-groups: Theorem 6.2.29. Let A be a Banach algebra, let E and F be left Banach A-modules, and let {0} −→ E2 −→ E1 −→ E1 /E2 −→ {0} and {0} −→ F2 −→ F1 −→ F1 /F2 −→ {0} be admissible, short, exact sequences of left Banach A-modules. Then we have long exact sequences {0} −→ A B(E, F2 ) −→ A B(E, F1 ) −→ A B(E, F1 /F2 ) n −→ Ext1A (E, F2 ) −→ · · · −→ Extn A (E, F2 ) −→ ExtA (E, F1 ) n+1 (E, F2 ) −→ · · · −→ Extn A (E, F1 /F2 ) −→ ExtA

and {0} −→ A B(E1 /E2 , F ) −→ A B(E1 , F ) −→ A B(E2 , F ) n −→ Ext1A (E1 /E2 , F ) −→ · · · −→ Extn A (E1 /E2 , F ) −→ ExtA (E1 , F ) n+1 (E1 /E2 , F ) −→ · · · . −→ Extn A (E2 , F ) −→ ExtA

Proof. Both exact sequences follow from Theorem 6.2.13 as follows.

6.2 Resolutions and Ext-Groups

273

Since, for n ∈ N0 , each module Bn (E) is projective, we may view B(B • (E), F2 ) as a complemented subcomplex of A B(B• (E), F1 ) such that A the resulting quotient complex is canonically isomorphic to A B(B(E), F1 /F2 ) (Please, check this!). The first exact sequence is thus a consequence of Theorem 6.2.13, Definition 6.2.27, and Exercise 6.2.5. We may canonically view B• (E2 ) as a complemented subcomplex of B• (E1 ) such that the resulting quotient complex is isomorphic to B• (E1 /E2 ). Applying Definition 6.2.27 and Theorem 6.2.13, we obtain the second exact sequence.  We shall now express Hochschild cohomology groups in terms of Extgroups. For a Banach algebra A, let Aenv := A# ⊗γ (Aop )# denote the enveloping algebra of A. Any Banach A-bimodule E is a unital, left Banach Aenv -module in a canonical fashion via (a · b) · x := a · x · b

(a ∈ A# , b ∈ (Aop )# , x ∈ E).

It is routinely verified that E is a projective Banach A-bimodule if and only if it is a projective left Banach Aenv -module. Theorem 6.2.30. Let A be a Banach algebra, and let E be a Banach Abimodule. Then we have Hn (A, E) ∼ = ExtnAenv (A# , E)

(n ∈ N0 ).

Proof. The Bar resolution of the left Banach A# -module A# is in fact a projective resolution of the left Banach Aenv -module A# , so that we can use B• (A# ) to calculate the groups ExtnAenv (A# , E) for n ∈ N. Since Bn (A# ) = A# ⊗γ A ⊗γ · · · ⊗γ A ⊗γ A#   

(n ∈ N0 ),

n−times

the universal property of the projective tensor product produces canonical isomorphisms # Aenv B(Bn (A ), E)

= A# BA# (Bn (A# ), E) ∼ = C n (A, E)

(n ∈ N0 ).

A tedious, but not difficult calculation shows that these isomorphisms induce an isomorphism of the cochain complex Aenv B(B• (A# ), E) and the Hochschild cochain complex.  Together, Theorems 6.2.29 and 6.2.30 yield: Corollary 6.2.31. Let A be a Banach algebra, and let {0} −→ F −→ E −→ E/F −→ {0}

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6 Banach Homological Algebra

be an admissible, short, exact sequence of Banach A-bimodules. Then we have a long exact sequence {0} −→ H0 (A, F ) −→ H0 (A, E) −→ H0 (A, E/F ) −→ H1 (A, F ) −→ · · · −→ Hn (A, F ) −→ Hn (A, E) −→ Hn (A, E/F ) −→ Hn+1 (A, F ) −→ · · · .

One might want to ask what the point is of being able to compute Hochschild cohomology groups using Ext-groups. The answer lies in Definition 6.2.27: Ext-groups can be computed based on any projective resolution. In certain situations, we can perhaps choose a projective resolution for which the Ext-groups—and hence the Hochschild cohomology groups—are particularly easy to compute. We conclude this section with an application of Theorems 6.2.29 and 6.2.30 to biprojective Banach algebras: Theorem 6.2.32. Let A be a biprojective Banach algebra. Then Hn (A, E) = {0} for all n ≥ 3 and for all Banach A-bimodules E. Proof. Consider the following admissible, short, exact sequences of Banach A-bimodule, i.e., left Banach Aenv -modules: {0} −→ A −→ A# −→ A# /A −→ {0}, {0} −→ J −→ Aenv −→ Aenv /J −→ {0}, where J is the closed linear span of (A# ⊗γ A) ∪ (A ⊗γ A# ) in Aenv , and {0} −→ A ⊗γ A −→ P −→ J → {0} with P := (A ⊗γ A# ) ⊕ (A# ⊗γ A), where A ⊗γ A → P,

a → (a, a)

and P → J,

(a, b) → a − b.

We apply the second of the long exact sequences from Theorem 6.2.29 to these short, exact sequences and consider the following segments of the resulting long exact sequences: Ext2Aenv (A, E) −→ Ext3Aenv (A# /A, E) −→ Ext3Aenv (A# , E) −→ Ext3Aenv (A, E), Ext2Aenv (Aenv , E) −→ Ext2Aenv (J, E) −→ Ext3Aenv (Aenv /J, E) −→ Ext3Aenv (Aenv , E), and

6.2 Resolutions and Ext-Groups

275

Ext1Aenv (A ⊗γ A, E) −→ Ext2Aenv (J, E) −→ Ext2Aenv (P, E) −→ Ext2 (A ⊗γ A, E). As the left Banach Aenv -modules A, Aenv , and A⊗γ A are projective, it follows from Exercise 6.2.6 that the endpoints of these segments are all {0}. We thus obtain isomorphisms Ext3Aenv (A# , E) ∼ = Ext3Aenv (A# /A, E) ∼ = Ext3 env (Aenv /J, E), A

∼ = Ext2Aenv (J, E) ∼ Ext2 env (P, E). = A

(6.11)

As the left Aenv -modules A# ⊗γ A and A ⊗γ A# are projective, so is P . It follows, again from Exercise 6.2.6, that Ext2Aenv (P, E) = {0}. By (6.11) and Theorem 6.2.30, this means H3 (A, E) = {0}. The claim for arbitrary n ≥ 3, now follows from Theorem 2.4.9. 

Exercises Exercise 6.2.1. Let E and F be cochain complexes of Banach spaces, and let φ : E → F be a morphism. Show that φ induces a sequence φ¯ = (φ¯n )n∈Z of group homomorphisms—in fact, linear maps —φ¯n : Hn (E) → Hn (F). Exercise 6.2.2. Use Corollary 6.2.11 to derive Theorem 2.4.9. Exercise 6.2.3. Work out Example 6.2.21 in detail. Exercise 6.2.4. Show that any two morphisms as in Theorem 6.2.23 are homotopic. Exercise 6.2.5. Let A be a Banach algebra, and let E and F be left Banach A-modules. Show that Ext0A (E, F ) ∼ = A B(E, F ). canonically. Exercise 6.2.6. Let A be a Banach algebra, and let E be a left Banach A-module. Show that the following are equivalent: (i) E is projective; (ii) Ext1A (E, F ) = {0} for all left Banach A-modules F ; (iii) ExtnA (E, F ) = {0} for all n ∈ N and for all left Banach A-modules F . Exercise 6.2.7. Work out the proof of Theorem 6.2.30 in detail.

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Exercise 6.2.8. Let A be a unital Banach algebra, and let E be a unital Banach A-bimodule. Show that Hn (A, E) ∼ = ExtnA⊗γ Aop (A, E)

(n ∈ N).

Exercise 6.2.9. Let A be a Banach algebra, and let E and F be left Banach A-modules, so that B(E, F ) becomes a Banach A-bimodule via (a · T )(x) := a · T x and

Show that

(T · a)(x) := T (a · x) (a ∈ A, x ∈ E, T ∈ B(E, F )).

ExtnA (E, F ) ∼ = Hn (A, B(E, F ))

(n ∈ N0 ).

(Hint: Consider the Bar resolution of E as a left Banach A# -module and show that ∼ n (n ∈ N0 ) A B(Bn (E), F ) = C (A, B(E, F )) such that the isomorphisms commute with the respective coboundary operators.) Exercise 6.2.10. Let E be a Banach space. Show that Hn (B(E), B(E)) = {0} for n ∈ N.

6.3 Flatness and Injectivity We conclude this chapter with a discussion of two further important properties of Banach modules: flatness and injectivity. The following definition is a prerequisite for the definition of flatness. Definition 6.3.1. Let A be a Banach algebra, let E be a right Banach Amodule, and let F be a left Banach A-module. The projective module tensor product E ⊗γA F of E and F is defined as the quotient of E ⊗γ F by the closed linear span of {x · a ⊗ y − x ⊗ a · y : x ∈ E, y ∈ F, a ∈ A}. There is no need, in general, for E ⊗γA F to again be some sort of Banach module. Proposition 6.3.2. Let A be a Banach algebra, let R be a closed right ideal of A# with a bounded left approximate identity, and let E be a left Banach A-module. Then the multiplication map ΔR,E : R ⊗γ E → E,

r ⊗ x → r · x

induces a topological isomorphism of the Banach spaces R ⊗γA E and R · E := {x · r : x ∈ E, r ∈ R}.

6.3 Flatness and Injectivity

277

Proof. By the right variant of Corollary B.2.5, R·E is indeed a Banach space, ∞ is an open map onto R · E. Let x = so that Δ n=1 rn ⊗ xn ∈ ker ΔR,E ∞ R,E with n=1 rn xn  < ∞; without loss of generality, suppose that xn = 0 for n ∈ N. Let  ∞ ∞   ∞ N R := (sn )n=1 ∈ R : sn xn  < ∞ and sn · xn = 0 n=1

n=1

For (sn )∞ n=1 ∈ R, let |||(sn )∞ n=1 ||| :=

∞ 

sn xn .

n=1

Then (R, ||| · |||) is a left Banach R-mpdule, and any bounded left approximate identity for R is a bounded left approximate identity for R. By definition, (rn )∞ n=1 ∈ R. By Cohen’s Factorization Theorem, there are r ∈ R and (tn )∞ n=1 ∈ R such that rn = rtn for n ∈ N. Consequently, x=

∞  n=1

rtn ⊗ xn =

∞ 

(rtn ⊗ xn − r ⊗ tn · xn )

n=1

lies in the closed linear span of {r · a ⊗ x − r ⊗ a · x : r ∈ R, x ∈ E, a ∈ A}.  Definition 6.3.3. Let A be a Banach algebra. A left Banach A-module F is called flat if for every admissible, short, exact sequence {0} −→ E2 −→ E1 −→ E1 /E2 −→ {0} of right Banach A-modules, the sequence {0} −→ E2 ⊗γA F −→ E1 ⊗γA F −→ (E1 /E2 ) ⊗γA F −→ {0} is exact. Of course, an analogous definition can be made for right Banach modules. Example 6.3.4. Let A be a Banach algebra, and let L be a closed left ideal of A# with a bounded right approximate identity. Let {0} −→ E2 −→ E1 −→ E1 /E2 −→ {0} be an admissible, short, exact sequence of right Banach A-modules. Consider the commutative diagram

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6 Banach Homological Algebra

{0} o {0} o

E2 ⊗γA L o

E1 ⊗γA L o

(E1 /E2 ) ⊗γA L o





 (E1 /E2 ) · L o

E2 · L o

E1 · L o

{0} (6.12) {0},

where the vertical arrows are the maps induced by the respective multiplication operators. By (the right module version of) Proposition 6.3.2, the columns are all isomorphisms. Since the second row of (6.12) is trivially exact, the same is true for the first row. Hence, L is a flat left Banach A-module. We shall soon encounter further examples of flat Banach modules. Let A a Banach algebra, and let E be an arbitrary Banach space. Then B(A# , E) becomes a left Banach A-module through (a · T )(b) := T (ba)

(a ∈ A, b ∈ A# , T ∈ B(A# , E)).

Modules of this type are called cofree. If E itself is a left Banach A-module, # we have a canonical homomorphism ΔA ,E : E → B(A# , E) of left Banach A-modules defined by #

(ΔA

,E

(a ∈ A# , x ∈ E).

x)a := a · x

The following is an analog of Proposition 6.1.7: Proposition 6.3.5. Let A be a Banach algebra, and let I be a left Banach A-module. Then the following are equivalent: #

(i) the canonical homomorphism ΔA ,I : I → B(A# , I) has a left inverse in # A B(B(A , I), I); (ii) there is a left Banach A-module J such that I ⊕ J is cofree; (iii) if E and F are left Banach A-modules, if θ ∈ A B(F, E) is injective and admissible, and if σ ∈ A B(F, I), then there is ρ ∈ A B(E, I) such that ρ ◦ θ = σ, i.e., the diagram ? IO ~~ ~ ~ ρ ~~ ~~ /E F σ

θ

commutes; (iv) every admissible, short, exact sequence {0} −→ I −→ E −→ E/I −→ {0} of left Banach A-modules splits. Definition 6.3.6. Let A be a Banach algebra. A left Banach A-module I satisfying the equivalent conditions of Proposition 6.3.5 is called injective.

6.3 Flatness and Injectivity

279

As for flatness, we can define injectivity for right Banach modules as well. Example 6.3.7. Let A be an amenable Banach algebra, and let {0} −→ F −→ E −→ E/F −→ {0}

(6.13)

be an admissible, short, exact sequence of left or right Banach A-modules, where F is a dual module. By Theorem 2.3.21 (or rather its right module version), (6.13) splits, so that Proposition 6.3.5(iv) is satisfied. Consequently, F is injective. Example 6.3.8. Let A be an arbitrary Banach algebra, and let P be a projective right Banach A-module. With the canonical identification B(A# , P ∗ ) ∼ = # γ ∗ A# ,P ∗ ∗ = ΔP,A# . From the definition of pro(A ⊗ P ) , we see easily that Δ jectivity, it is immediate that Proposition 6.3.5(i) is satisfied. Hence, P ∗ is an injective left Banach A-module. Next, we shall see that flatness and injectivity are dual to each other: Theorem 6.3.9. Let A be a Banach algebra, and let E be a left Banach A-module. Then the following are equivalent: (i) E is flat; (ii) E ∗ is an injective right Banach A-module. Proof. (i) =⇒ (ii): Let F1 and F2 be right Banach A-modules, and let θ ∈ A B(F2 , F1 ) be injective and admissible, so that the short, exact sequence θ

{0} −→ F2 −→ F1 −→ F1 /θ(F2 ) → {0} of right Banach A-modules is admissible. As E is flat, the sequence θ⊗id

{0} −→ F2 ⊗γA E −→E F1 ⊗γA E −→ (F1 /θ(F2 )) ⊗γA E −→ {0} is exact, and so is the dual sequence {0} ←− A B(F2 , E ∗ )

(θ⊗idE )∗

) ←− A B(F1 /θ(F2 ), E ∗ ) ←− {0}. (6.14) Let σ ∈ A B(F2 , E ∗ ). The exactness of (6.14)—more precisely: the surjectivity of (θ ⊗ idE )∗ —yields ρ ∈ A B(F1 , E ∗ ) with σ = (θ ⊗ idE )∗ ρ, i.e., ρ ◦ θ = σ. Hence, E ∗ satisfies (the right module analog of) Proposition 6.3.5(iii). (ii) =⇒ (i): This is proven by reversing the arguments from (i) =⇒ (ii).  ←−

A B(F1 , E

∗

With Theorem 6.3.9 and Examples 6.3.7, we obtain more examples of flat Banach modules: Example 6.3.10. Every left or right Banach module over an amenable Banach algebra is flat.

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Example 6.3.11. Every projective left or right Banach module is flat. So far, we have treated flatness and injectivity for one-sided Banach modules only. It is straightforward how to define these notions for Banach Abimodules such that analogs of Proposition 6.3.5 and Theorem 6.3.9 hold. We leave the details to the reader (Exercise 6.3.5 below). By Theorem 6.1.11, a Banach algebra is biprojective if and only if it is a projective Banach bimodule over itself. An analog of this statement holds for biflatness. We start with an analog of Lemma 6.1.10: Lemma 6.3.12. The following are equivalent for a Banach algebra A: (i) A is a flat Banach A-bimodule; (ii) both maps Δ∗A# ,A : A∗ → (A# ⊗γ A)∗ and Δ∗A,A# : A∗ → (A ⊗γ A# )∗ have left inverses in A B A ((A# ⊗γ A)∗ , A∗ ) and A B A ((A ⊗γ A# )∗ , A∗ ), respectively. Proof. (i) =⇒ (ii): By (the bimodule analog of) Theorem 6.3.9, A∗ is an injective Banach A-bimodule. The claim then follows immediately from (the bimodule analog of) Proposition 6.3.5(iv). (ii) =⇒ (i): Let θ ∈ A B A ((A# ⊗γ A)∗ , A∗ ) and σ ∈ A BA ((A ⊗γ A# )∗ , A∗ ) be left inverses of Δ∗A# ,A and Δ∗A,A# , respectively. For φ ∈ (A# ⊗γ A ⊗γ A# )∗ and c ∈ A# , define φc ∈ (A# ⊗γ A)∗ by letting a ⊗ b, φc  := a ⊗ b ⊗ c, φ

(a ∈ A# , b ∈ A).

Define ρ ∈ A BA ((A# ⊗γ A ⊗γ A# )∗ , (A ⊗γ A# )∗ ) through b ⊗ c, ρ(φ) := b, θ(φc )

(b ∈ A, c ∈ A# ).

Then σ ◦ ρ ∈ A B A ((A# ⊗γ A ⊗γ A# )∗ , A∗ ) is a left inverse of Δ∗A# ,A,A# : A∗ → (A# ⊗γ A ⊗γ A# )∗ , so that A∗ is injective. Hence, A is flat.  Lemma 6.3.13. Let A be a Banach algebra which is a flat Banach Abimodule. Then the linear span of {ab : a, b ∈ A} is dense in A. Proof. Let φ ∈ A∗ be such that ab, φ = 0 for a, b ∈ A. By (the bimodule analog of) Theorem 6.3.9, the Banach A-bimodule A∗ is injective. Consequently—by (the bimodule analog of) Proposition 6.3.5(iv)— the injective A-bimodule homomorphism Δ∗A# ,A : A∗ → (A# ⊗γ A)∗ has a left inverse θ ∈ A BA ((A# ⊗γ A)∗ , A∗ ). Define ψ˜ : A# ⊗γ A → C,

(λeA# + a) ⊗ b → a, φb, φ,

˜ Let a, b ∈ A and c ∈ A# . Since ab, φ = 0, it follows that and let ψ := θ(ψ). ˜ = 0, i.e., b · ψ˜ = 0 and consequently b · ψ = 0 for all b ∈ B. This c ⊗ a, b · ψ means that ab, ψ = 0 for a, b ∈ A as well. On the other hand, we have

6.3 Flatness and Injectivity

281

(λeA# + a) ⊗ c, ψ˜ · b = λb + ba, φc, φ = λb, φc, φ = b, φλc + ac, φ = b, φ(λeA# + a) ⊗ c, Δ∗A# ,A φ It follows that

ψ˜ · b = b, φΔ∗A# ,A φ

(λ ∈ C, a, b, c ∈ A). (b ∈ A)

and thus ψ · b = b, φφ

(b ∈ A).

This, however, means that b, φ2 = b, ψ · b = b2 , ψ = 0

(b ∈ A),

so that φ = 0. The Hahn–Banach Theorem yields that the linear span of {ab : a, b ∈ A}is dense in A as claimed.  Theorem 6.3.14. Let A be a Banach algebra. Then the following are equivalent: (i) A is biflat; (ii) A is a flat Banach A-bimodule. Proof. (i) =⇒ (ii): Suppose that Δ∗A : A∗ → (A ⊗γ A)∗ has a left inverse in γ ∗ ∗ ∗ ∗ # γ ∗ A B A ((A ⊗ A) , A ). It follows easily that then ΔA# ,A : A → (A ⊗ A) ∗ ∗ γ # ∗ # γ ∗ ∗ and ΔA,A# : A → (A ⊗ A ) have left inverses in A B A ((A ⊗ A) , A ) and γ # ∗ ∗ A B A ((A ⊗ A ) , A ), respectively. It follows from Lemma 6.3.12 that A is a flat Banach A-bimodule (ii) =⇒ (i): By Lemma 6.3.12, Δ∗A# ,A has a left inverse θ ∈ A B A ((A# ⊗γ ∗ A) , A∗ ). Let φ ∈ (A# ⊗γ A)∗ be such that φ|A⊗γ A ≡ 0. It follows that φ·a = 0 for all a ∈ A. Hence, θ(φ) vanishes on {ab : a, b ∈ A}. By Lemma 6.3.13, this means that θ(φ) = 0. Hence, θ drops to an A-bimodule homomorphism from  (A ⊗γ A)∗ to A∗ , which is clearly a left inverse of Δ∗A . With the help of Theorem 6.3.14, we see that certain Hochschild cohomology groups of biflat Banach algebras are trivial: Theorem 6.3.15. Let A be a biflat Banach algebra. Then Hn (A, A∗ ) = {0} for all n ∈ N. In particular, A is weakly amenable. Proof. By Theorem 6.3.14, A is a flat Banach A-bimodule, so that A∗ is an injective Banach A-bimodule. It is immediate that then A∗ is also an injective left Banach Aenv -module. From Theorem 6.2.30 and Exercise 6.3.3, we conclude that Hn (A, A∗ ) ∼ = ExtnAenv (A# , A∗ ) = {0} This completes the proof.

(n ∈ N). 

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6 Banach Homological Algebra

Exercises Exercise 6.3.1. Let A be a Banach algebra, let E be a right Banach Amodule, and let F be a left Banach A-module. Show that the isomorphism between (E ⊗γ F )∗ and B(E, F ∗ ) induces an isometric isomorphism of the Banach spaces (E ⊗γA F )∗ and A B(E, F ∗ ). Exercise 6.3.2. Prove Proposition 6.3.5. Exercise 6.3.3. Let A be a Banach algebra, and let F be a left Banach Amodule. Show that F is injective if and only if Ext1A (E, F ) = {0} for all left Banach A-modules E if and only if ExtnA (E, F ) = {0} for all n ∈ N and for all left Banach A-modules E. Exercise 6.3.4. Work out the proof of Theorem 6.3.9 (ii) =⇒ (i) in detail. Exercise 6.3.5. Define flatness and injectivity for Banach bimodules such that analogs of Proposition 6.3.5 and Theorem 6.3.9 hold.

Notes and Comments Banach homological algebra—or more generally: topological homological algebra—was initiated by several people independently of one another, such as A. Ya. Helemski˘ı and J. L. Taylor ([342]). For certain applications, e.g., to several variable spectral theory, it is necessary to not only consider Banach algebras and modules but also more general topological algebras and modules (see [110]). This chapter presents the very basics of Helemski˘ı’s approach. It is intended as a “prequel” to his textbook [172] and his monograph [174]. Essentially all the material in this chapter is from these two sources. Helemski˘ı has also written several survey articles on topological homological algebra, e.g., [171], [167], [169], or [168], which give an overview of the state of the area up to the date of their respective appearance. In [173], Helemski˘ı showed that a von Neumann algebra M is Connesamenable if and only if its predual is an injective Banach M-bimodule. The “if part” of this result extends to general dual Banach algebras ([300]), but the only “only if” part doesn’t: for a locally compact group G, the canonical predual C0 (G) of M (G) is injective if and only if G is finite ([336] and [300]). Comparing Banach homological algebra with purely algebraic homological algebra, as expounded in [42], [212], [238], or [355], one cannot help getting the impression that topological homology is about little more than adding a few functional analytic overtones to the concepts and results from homological algebra. This impression, however, is misleading: the categories of algebraic homological algebra as expounded, e.g., in [238] are abelian whereas

Notes and Comments

283

the typical categories of functional analysis aren’t. One encounters novel and interesting phenomena in Banach homological algebra that have no analog in the purely algebraic setting.

Forbidden values for homological dimensions Let A be a Banach algebra. Then the global homological dimension of A is defined as dg A := inf{n ∈ N0 : Extn+1 A (E, F ) = {0} for all left Banach A-modules E and F } and its homological bidimension of A is defined to be db A := inf{n ∈ N0 : Hn+1 (A, E) = {0} for all Banach A-bimodules E}. It is not hard to see that db A ≥ dg A. Helemski˘ı’s Global Dimension Theorem asserts that dg A ≥ 2 whenever A is commutative with infinite character space ([170]; see [275] for an exposition of this theorem and its proof that should be accessible to someone who has worked through this chapter). Consequently, whenever A is a commutative Banach algebra with infinite character space, there is a Banach A-bimodule E such that H2 (A, E) = {0}. Similar results hold for certain radical Banach algebras ([136]), infinite-dimensional, liminal C ∗ -algebras ([236]), and the Banach algebra B(E) where E is the Banach space constructed by C. J. Read in [278] ([219]).

Additivity formulae for homological dimensions Let A be a unital Banach algebra, and let B be a commutative, biprojective Banach algebra with infinite character space. Then dg A ⊗γ B# = dg A + dg B# = dg A + 2 holds as proven in [318]. The weak homological bidimension of a Banach algebra A is defined as wdb A := inf{n ∈ N0 : Hn+1 (A, E ∗ ) = {0} for all Banach A − bimodules E}, so that wdb A = n − 1 holds if and only if A is n-amenable in the sense of [263], but not (n − 1)-amenable. In [237], it is shown that wdb A ⊗γ B = wdb A + wdb B holds for any two Banach algebras A and B with bounded approximate identities.

Chapter 7

Operator Algebras on Hilbert Spaces

In this chapter, we are dealing with operator algebras on Hilbert spaces, i.e., closed subalgebras of B(H) for some Hilbert space H. The first five sections of this chapter are devoted to the study of selfadjoint operator algebras on Hilbert spaces, i.e., C ∗ -algebras. Over the past few decades, the theory of C ∗ -algebras has thrived and grown, and now is an area of mathematics very much independent of general Banach algebra theory. Amenability brings C ∗ -algebras back into the framework of general Banach algebras. The purely Banach algebraic notion of amenability turns out to be equivalent to an important C ∗ -algebraic property: nuclearity. Most of this chapter is devoted to establishing this equivalence. Very often, the only—or at least the most convenient—way of proving a result for C ∗ -algebras is to make the detour through their enveloping von Neumann algebras. A substantial part of this chapter will thus deal with von Neumann algebras. As it turns out, amenability in the sense of Definition 2.1.11 is too strong to yield an interesting theory for von Neumann algebras. The right notion of amenability for von Neumann algebras is Connesamenability. As we will show in this chapter, it is equivalent to a number of important von Neumann algebraic properties, some of which, at first glance, have little to do with amenability: • injectivity: a von Neumann algebra M acting on some Hilbert space H is injective if its commutant is complemented in B(H), such that the corresponding projection onto M has norm one (compare Section 5.5); • semidiscreteness: this is some kind of approximation property for von Neumann algebras; • existence of a normal, virtual diagonal : for von Neumann algebras, the converse of Corollary 5.1.35 is true (compare Theorem 5.2.1 and Example 5.3.17). For a general C ∗ -algebra A, the Connes-amenability (or injectivity, etc.) of A∗∗ turns out to be equivalent to A being amenable (or nuclear), which © Springer Science+Business Media, LLC, part of Springer Nature 2020 V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-1-0716-0351-2 7

285

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7 Operator Algebras on Hilbert Spaces

is open for general Arens regular Banach algebras except under exceptional circumstances (see Theorem 5.1.20). When it comes to studying amenable, but not necessarily self-adjoint operator algebras on Hilbert spaces, the straightforward way to come up with examples is to pick an amenable C ∗ -algebra A and an invertible, but nonunitary operator T on that Hilbert space and consider T AT −1 . Does every amenable operator algebra on a Hilbert space arise in this fashion? This question was open for several decades and only recently settled in the negative ([49]); in fact, the counterexample is a subalgebra of a 2-subhomogeneous C ∗ algebra. On the other hand, every amenable commutative operator algebra on a Hilbert space is similar to a C ∗ -algebra ([240]). The last two sections of this chapter will present these results.

7.1 Amenable von Neumann Algebras In the previous chapters, we have seen that amenability imposes strong constraints on the structure of dual Banach algebras. The case of von Neumann algebras is no exception. The following is well known: Theorem 7.1.1. The following are equivalent for a von Neumann algebra M: (i) M is subhomogeneous; (ii) there are N1 , . . . , Nn ∈ N and abelian von Neumann algebras A1 , . . . , An such that M∼ = MN1 (A1 ) ⊕ · · · ⊕ MNn (An ). Proof. Of course, only (i) =⇒ (ii) needs consideration. If M is subhomogeneous, it has to be of type I. By Theorem C.7.9, there are von Neumann algebras N1 , . . . , Nn such that Nj is of type INj with Nj ∈ N for j = 1, . . . , n and M∼ = N1 ⊕∞ · · · ⊕∞ Nn , and by Theorem C.7.13, Nj ∼ = MNj ⊗ Z(Nj ) holds for j = 1, . . . , n. This completes the proof.  Of course, this yields: Corollary 7.1.2. Let M be a subhomogeneous von Neumann algebra. Then M is amenable. The remainder of this section is devoted to proving the converse. We start with a definition: Definition 7.1.3. Let A be a Banach ∗ -algebra. We say that A is of type (QE) if, for each ∗ -representation π of A on a Hilbert space H, there is a quasi-expectation Q : B(H) → π(A) .

7.1 Amenable von Neumann Algebras

287

As we shall soon see, amenability forces any Banach ∗ -algebra to be of type (QE), but before we can prove it, we require the following lemma: Lemma 7.1.4. Let M be a von Neumann algebra acting on a Hilbert space H. Then the following are equivalent: (i) there is a quasi-expectation Q : B(H) → M; (ii) for every faithful, normal ∗ -representation π of M on a Hilbert space K, there is a quasi-expectation Q : B(K) → π(M). Proof. Of course, only (i) =⇒ (ii) needs proof. Let π be a faithful, normal ∗ -representation of M on a Hilbert space K, and set N := π(M.). By [98, Theorem 3, I.4.4], there is a Hilbert space L such that M ⊗ idL := {x ⊗ idL : x ∈ M} ⊂ B(H ⊗2 L) and N ⊗ idL ⊂ B(K ⊗2 L) (defined analogously) are spatially isomorphic, i.e., there is a unitary U ∈ B(H ⊗2 L, K ⊗2 L) such that U ∗ (N ⊗ idL )U = M ⊗ idL . Fix ξ0 , η0 ∈ L with ξ0 |η0 = 1, and define P0 : B(H ⊗2 L) → B(H) through (P0 T )ξ|η := T (ξ ⊗ ξ0 )|η ⊗ η0

(T ∈ B(H), ξ, η ∈ H).

(7.1)

Identifying B(H) with B(H) ⊗ idL , we see that P0 is a projection onto B(H). Furthermore, for T ∈ B(H ⊗2 L) and R, S ∈ B(H), we have P0 ((R ⊗ idL )T (S ⊗ idL ))ξ|η = (R ⊗ idL )T (S ⊗ idL )(ξ ⊗ ξ0 )|η ⊗ η0

= T (Sξ ⊗ ξ0 )|R∗ η ⊗ η0

= (P0 T )Sξ|R∗ η

= R(P0 T )Sξ|η

(ξ, η ∈ H),

so that P0 is a quasi-expectation. Let P : B(H) → M be a quasi-expectation, and define P˜ : B(H ⊗2 L) → M ⊗ idL ,

T → (P ◦ P0 )T ⊗ idL ;

it is clear that P˜ is also a quasi-expectation. Let U ∈ B(H ⊗2 L, K ⊗2 L) be unitary such that U ∗ (N ⊗ idL )U = M ⊗ idL , and define ˜ : B(K ⊗2 L) → N ⊗ idL , Q

˜ ∗ T U ))U ∗ T → U (P(U

Fix again ξ0 , η0 ∈ L with ξ0 |η0 = 1, and define Q0 : B(K ⊗2 L) → B(K) as in (7.1). Then Q : B(K) → N, is the desired quasi-expectation.

˜ T → (Q0 ◦ Q)(T ⊗ idL ) 

With Lemma 7.1.4 at hand, we can now bring amenability into the picture.

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Proposition 7.1.5. Let A be an amenable Banach ∗ -algebra. Then A is of type (QE). Proof. Let π be a ∗ -representation of A on a Hilbert space H. Then, by Theorem C.9.4, there is a faithful, normal, semifinite weight φ on the von Neumann algebra π(A) . Applying the GNS-construction with respect to this weight, we obtain a faithful, normal ∗ -representation (πφ , Hφ ) of π(A) (Proposition C.9.3). By Theorem 5.1.24(i)—applied to φφ ◦ π : A → B(Hφ )—there is a quasi-expectation P : B(Hφ ) → (πφ ◦ π)(A) . Let J : Hφ → Hφ be the modular conjugation arising from the construction of (πφ , Hφ ) (Theorem C.9.5), so that and J(πφ ◦ π)(A) J = (πφ ◦ π)(A) . J 2 = idHφ Define

˜ : B(Hφ ) → (πφ ◦ π)(A) , Q

T → JP(JT J)J.

˜ is a quasi-expectation. Since πφ is a faithful, It is straightforward that Q ∗ normal -representation, it follows from Lemma 7.1.4 that there is a quasi expectation Q : B(H) → π(A) . Hence, A is of type (QE). To show that every amenable von Neumann algebra is subhomogeneous, it is thus sufficient to prove that every von Neumann algebra that is not of the form described in Theorem 7.1.1(ii) is not of type (QE). We proceed indirectly and by reduction: assuming that a von Neumann algebra which is not subhomogeneous is of type (QE), we obtain another von Neumann algebra which then should be of type (QE) as well, but for which we can show that it isn’t. For this reason, we establish two hereditary properties. For the first one, we refer to Exercise 7.1.2 below. The second one is harder to prove. Lemma 7.1.6. Let A be a unital C ∗ -algebra of type (QE), and let B be a unital C ∗ -subalgebra of A such that there is a quasi-expectation Q : A → B. Then B is of type (QE). Proof. Let A∗∗ act as a von Neumann algebra on the Hilbert space H, and let π be a ∗ -representation of B on some Hilbert space K. Then there is a projection p ∈ Z(B∗∗ ) such that pB∗∗ ∼ = π(B) . As A is of type (QE), there ∗∗ is a quasi-expectation P : B(H) → A . Define R : B(pH) → B∗∗ ,

T → (Q∗∗ ◦ P)(T p).

As Q∗∗ : A∗∗ → B∗∗ is also a quasi-expectation, we obtain R(T p) = (RT )p ∈ B∗∗ p

(T ∈ B(pH)),

so that R attains its values in B∗∗ p. It is easily checked that R : B(pH) → B∗∗ p is a quasi-expectation. It now follows from Lemma 7.1.4 that there is  a quasi-expectation from B(K) onto π(B) .

7.1 Amenable von Neumann Algebras

289

So far, the only von Neumann algebras for which we definitively know that they are not of type (QE) are the algebras VN(G), where G is a locally compact, inner amenable group G that fails to be amenable (Exercise 7.1.1); for instance, VN(F2 ) is not of type (QE). With the help of Lemma 7.1.6 and Exercise 7.1.2, we shall now exhibit another example. ∞ Example 7.1.7. Let M := ∞ - n=1 Mn . It is clear that M is a finite type I von Neumann algebra with a center isomorphic to ∞ . We claim that M is not of type (QE) (and thus not amenable by Proposition 7.1.5). Assume toward a contradiction that M is of type (QE). As Z(M) ∼ = ∞ , the character ˇ space of Z(M), is isomorphic to βN, the Stone–Cech Compactification of N. Let U be a free ultrafilter over N, so that U corresponds to a point of βN \ N. Define   ∗ ∈ M : lim tr (x x ) = 0 , MU := (xn )∞ n n n n=1 n→U

where trn is the normalized canonical trace on Mn . It is easily seen that MU is a closed ∗ -ideal of M. By Exercise 7.1.2, this means that M/MU is of type (QE) as well. In fact, MU is a maximal ideal of M by [338, Corollary V.4.10]. By [338, Theorem V.5.2], this means that M/MU is a finite factor of which the unique faithful, normal trace (Theorem C.7.4) is given by tr : M/MU → C,

(xn )∞ n=1 + MU → lim trn xn . n→U

There are two mutually exclusive possibilities: • M/MU is a type In factor for some n ∈ N (and thus finite-dimensional); • M/MU is a type II1 factor.  ∞ (n) For each n ∈ N, choose a decreasing sequence pk of projections in Mn k=1 such that n (n) (n) (n, k ∈ N), (7.2) dim pk Mn pk = k 2 (n)

where · denotes the Gauß bracket; note that pk = 0 whenever 2k > n. For k ∈ N, set   (1) (2) (3) pk := pk , pk , pk , . . . ∈ M. Then (pk )∞ n=1 is a decreasing sequence of projections in M. Assume that M/MU is finite-dimensional. Then the sequence (pk + MU )∞ n=1 must become constant eventually, i.e., there is K ∈ N such that     (n) (n) = lim trn pm (k, m ≥ K). (7.3) lim trn pk n→U

n→U

However, it is clear from (7.2) that   1 (n) lim trn pk = k n→∞ 2

(k ∈ N),

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which contradicts (7.3). Hence, M/MU is of type II1 . We now claim that we can choose U in such a fashion that M/MU contains a W ∗ -subalgebra N which is ∗ -isomorphic to VN(F2 ). Choose a strictly decreasing sequence (Nj )∞ j=1 of normal subgroups of F2 ∞ such that nj := [F2 : Nj ] < ∞ for j ∈ N and j=1 Nj = {eF2 }. (The existence of such a sequence follows from [178, (2.9) Theorem].) For j ∈ N, identify Mnj with B(2 (F2 /Nj )). Then the left regular representation of F2 /Nj induces a group homomorphism πj from F2 into the unitaries of Mnj such that ker πj = Nj and trnj πj (x) = 0 for x ∈ F2 \ Nj (see Exercise 7.1.3 below). Choose U such that {n1 , n2 , n3 , . . .} ∈ U, and define a group homomorphism π from F2 into the unitaries of M/MU by letting π(x) := (xn )∞ n=1 + MU with

πj (x), if n = nj , xn := 0, otherwise We claim that π is injective. To see this, let x ∈ F2 \ {eF2 }. Then there is / Nj and thus trnj πj (x) = 0 for all j ≥ j0 . It follows that j0 ∈ N such that x ∈ / MU . Let N be the W ∗ -subalgebra of M/MU generated by the eF2 − π(x) ∈ set {π(x) : x ∈ F2 }. We claim that N and VN(F2 ) are isomorphic as W ∗ -algebras. To see this, we apply the GNS-construction to M/MU and the normal weight φ = tr. We obtain a faithful, normal ∗ -representation (πφ , H) of M/MU , and thus may simply view M/MU as a von Neumann algebra acting on Hφ . Setting ξ := Λφ (eM ), we obtain a cyclic vector (Definition C.4.2) for M/MU that is also separating (Definition C.5.4) for M (because φ is a trace). By construction, we have (7.4) tr x = xξ|ξ

(x ∈ M/MU ). Let K be the closure in H of {xξ : x ∈ N}. From (7.4), it follows at once that (π(x)ξ)x∈F2 is an orthonormal basis for K. We thus have a unitary operator   λx δx → λx π(x)ξ. U : 2 (F2 ) → K, x∈F2

x∈F2

Let P ∈ B(H) be the orthogonal projection onto K. Then P ∈ N holds, and N → NP,

x → xP

(7.5)

is a normal ∗ -homomorphism. Let x ∈ N be such that xP = 0. Since xP ξ = xξ, and since ξ is a separating vector for M/MU , it follows that x = 0. Hence, (7.5) is even an isomorphism. Finally, note that U ∗ (π(x)|K )U δy = U ∗ π(x)π(y)ξ = U ∗ π(xy)ξ = δxy = λ2 (x)δy It follows that

(x, y ∈ F2 ).

7.1 Amenable von Neumann Algebras

B(K) → B(2 (F2 )),

291

T → U ∗ T U

induces a spatial ∗ -isomorphism of NP and VN(F2 ). As we have already observed, M/MU is a finite type II1 factor. Since M/MU is a factor, its center is trivially countably decomposable. Hence, M/MU itself is countably decomposable by Proposition C.7.2. It then follows from [312, Proposition 4.4.23] that there is a (norm one) quasi-expectation Q : M/MU → N. Since M/MU is of type (QE), Lemma 7.1.6 implies that the same is true for N. As N ∼ = VN(F2 ), this is a contradiction. All in all, M is not of type (QE). With Example 7.1.7 established, the general result is now just one (relatively easy) lemma away. ∞ Lemma 7.1.8. Let A be a unital C ∗ -algebra containing ∞ - n=1 Mn as a ∗ unital C -subalgebra. Then A is not of type (QE). Proof. For n ∈ N, let pn ∈ A be the projection corresponding to In ∈ Mn . Then pn Apn is a C ∗ -subalgebra of A with identity pn containing Mn as a unital C ∗ -subalgebra. Let φn be a state on pn Apn , and define ˜ n : Mn ⊗ Zp Ap (Mn ) → Mn , Q n n

a ⊗ b → b, φn a.

˜ n is a quasi-expectation with Identifying Mn with Mn ⊗ Cpn , we see that Q ˜ Qn  = 1. By Exercise 7.1.4 below, Mn ⊗Zpn Apn (Mn ) and pn Apn are isomorphic as C ∗ -algebras, so that we have a quasi-expectation Qn : pn Apn → Mn with Qn  = 1. Define Q : A → ∞ -

∞

Mn ,

a → (Q1 (p1 ap1 ), Q2 (p2 ap2 ), Q3 (p3 ap3 ), . . .).

n=1

∞ Then Q is a quasi-expectation. As we saw in Example 7.1.7, ∞ - n=1 Mn is not of type (QE). Then Lemma 7.1.6 implies that A is not of type (QE) either.  We can now prove the converse of Corollary 7.1.2 (and more): Theorem 7.1.9. The following are equivalent For a W ∗ -algebra M, the following are equivalent: (i) (ii) (iii) (iv)

M is amenable; M is of type (QE); ∞ M does not contain ∞ - n=1 Mn as a C ∗ -subalgebra; there are N1 , . . . , Nn ∈ N and abelian von Neumann algebras A1 , . . . , An such that M∼ = MN1 (A1 ) ⊕∞ · · · ⊕∞ MNn (An ).

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7 Operator Algebras on Hilbert Spaces

Proof. (i) =⇒ (ii) follows from Proposition 7.1.5, and (iv) =⇒ (i) is Corollary 7.1.2. (ii) =⇒ (iii) is Lemma 7.1.8. For (iii) =⇒ (iv), assume that M is not of the form described in (iv). The type decomposition of von Neumann algebras (Theorem C.7.9) then leaves three alternatives: Case 1: There is a strictly increasing sequence (nk )∞ k=1 in N such that M has, for each k ∈ N, a direct summand of type Ink . In this case, it is obvious ∞ that M contains ∞ - n=1 Mn as a C ∗ -subalgebra, which is a contradiction. Case 2: M has a properly infinite direct summand. It then follows from 2 ∗ [312, Proposition 2.2.4] that this summand contains ∞ B( ) as a C ∗ -subalgebra. 2 ∞ It is not hard to show that B( ) contains  - n=1 Mn as a C -subalgebra, which is again a contradiction. Case 3: M has a direct summand of type II1 . Then, by [312, Proposition 2.2.13], there is a sequence (pn )∞ n=1 of nonzero, mutually orthogonal projections in that summand. Consequently, the algebras pn Mpn have no direct summand of type I. Hence, it follows ∞that each subalgebra pn Mpn contains Mn as C ∗ -subalgebra. Again, ∞ - n=1 Mn is contained in M as a  C ∗ -subalgebra, which is once again a contradiction. In Section 3.4, we had already seen that B(2 ) is not amenable. As a consequence of Theorem 7.1.9, this extends to not necessarily separable Hilbert spaces: Corollary 7.1.10. Let H be a Hilbert space. Then B(H) is amenable if and only if dim H < ∞. Remark 7.1.11. As a consequence of [53, Theorem 1.1], the Banach spaces ∞ B(2 ) lacks the approximaB(2 ) and ∞ - n=1 Mn are isomorphic. Since  ∞ ∞ tion property by [334], the same is true for  - n=1 Mn . Let M be a von Neumann algebra which does not satisfy equivalent conditions of Theothe ∞ ∗ -subalgebra. In the rem 7.1.9; in particular, it contains ∞ - n=1 Mn as a C ∞ ∞ proof of Lemma 7.1.8, we saw that then M contains  - n=1 Mn as a complemented subspace, so that M lacks the approximation property as well. On the other hand, it is straightforward that every von Neumann algebra satisfying Theorem 7.1.9(iv) has the approximation property. We can thus add a fifth equivalent statement to Theorem 7.1.9: (v) M has the approximation property.

Exercises Exercise 7.1.1. Let G be a locally compact, inner amenable group which is not amenable. Show that VN(G) is not of type (QE). (Hint: Proceed as in the proof of Theorem 5.1.27, but with the rˆ oles of λ2 and ρ2 interchanged.)

7.1 Amenable von Neumann Algebras

293

Exercise 7.1.2. Let A be a Banach ∗ -algebra of type (QE), and let I be a closed ∗ -ideal of A. Show that A/I is also of type (QE). Exercise 7.1.3. Let G be a finite group, so that B(2 (G)) ∼ = M|G| . Show that tr|G| λ2 (x) = 0 for x ∈ G \ {eG }. Exercise 7.1.4. Let n ∈ N, and let A be a unital C ∗ -algebra containing Mn as a unital ∗ -subalgebra. Show that Mn ⊗ ZA (Mn ) → A,

a ⊗ b → ab

is a ∗ -isomorphism.

7.2 Injective von Neumann Algebras To define injective von Neumann algebra, we introduce the notion of a conditional expectation. Definition 7.2.1. Let A be a C ∗ -algebra, and let B be a C ∗ -subalgebra of A. A conditional expectation E : A → B is a norm one projection from A onto B. We had previously introduced the notion of a quasi-expectation (Definition 5.1.23), and in view of this choice of terminology, the question arises if there is a connection between conditional expectations and quasi-expectation. This question is answered by the following theorem, which is [338, Theorem III.3.4] (the claim about complete positivity follows from an inspection of the proof): Theorem 7.2.2. Let A be C ∗ -algebra, let B be a C ∗ -subalgebra of A, and let E : A → B be a conditional expectation. Then E is a completely positive quasi-expectation satisfying (Ea)∗ (Ea) ≤ E(a∗ a)

(a ∈ A).

Definition 7.2.3. Let M be a von Neumann algebra acting on a Hilbert space H. Then M is called injective if there is a conditional expectation E : B(H) → M . Example 7.2.4. Since B(H) = C idH , the von Neumann algebra B(H) is injective for every Hilbert space H. Remark 7.2.5. Even though it is not apparent from the definition, injectivity is a Hilbert space independent property (see Exercise 7.2.1 below). Clearly, the notion of injectivity for von Neumann algebras bears resemblance to that of Connes-injectivity of general dual Banach algebras as introduced in Definition 5.5.1. The remainder of this section will be devoted to showing that Connes-injectivity implies injectivity for von Neumann algebras.

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7 Operator Algebras on Hilbert Spaces

Lemma 7.2.6. Let A be a unital C ∗ -algebra, and let M be a unital C ∗ subalgebra of A which is also a finite, countably decomposable von Neumann algebra. Then the following are equivalent: (i) there is φ ∈ Z 0 (M, A∗ ) such that φ|M is positive and faithful; (ii) there is φ ∈ A∗+ ∩ Z 0 (M, A∗ ) such that φ|M is faithful; (iii) there is a conditional expectation E : A → M. Proof. (i) =⇒ (ii): Without loss of generality, suppose that φ∗ = φ. Let φ = φ+ − φ− be the Jordan Decomposition (Theorem C.3.11) of φ with φ+ , φ− ≥ 0 and φ = φ+  + φ− . For any u ∈ U(A), we have u∗ · φ · u = φ. Uniqueness of the Jordan Decomposition implies that u∗ · φ+ · u = φ+ for all u ∈ U(A), so that φ+ ∈ Z 0 (M, A∗ ). As φ+ |M ≥ φ|M , it follows that φ+ is faithful. (ii) =⇒ (iii): For any functional ψ ∈ M∗ , let ψn denote its normal and ψs its singular part (Definition C.5.16). Let φ ∈ A∗ be as in (ii), and define θ : A → M∗ ,

a → (φ · a|M )n .

For any positive a ∈ A, we have x∗ x, φ · a = ax∗ x, φ = |x|a|x|, φ ≥ 0

(x ∈ M).

As forming the normal or singular part, respectively, of a functional respects positivity, it follows that θ is positive. Set tr := θ(eA ). Then tr is a normal trace on M. We claim that tr is faithful. Let ψ := φ|M , so that ψ = tr + ψs . It suffices show that tr p = 0 for every nonzero projection p ∈ M. Let p ∈ M be such a projection. By [200, 10.5.15 Exercise], there is a nonzero projection q ∈ M with q ≤ p such that q, ψs = 0. It follows that tr p ≥ tr q = q, φ > 0 because φ is faithful. Let a ∈ A be positive. It follows that 0 ≤ θ(a) ≤ atr. The Radon–Nikod´ ym type theorem [312, Theorem 1.24.3] yields a positive Ea ∈ M such that 1

1

θ(a) = (Ea) 2 · tr · (Ea) 2 = (Ea) · tr; since tr is faithful, Ea is uniquely determined. Extend E to a linear map from A into M. The uniqueness of the decomposition of a functional into its normal and its singular part, yields that θ(x) = (φ · x|M )n = (φ|M )n · x = tr · x = x · tr = (Ex) · tr

(x ∈ M).

Hence, E is a projection onto M. Since E is positive and satisfies EeA = eA , it follows that E = 1. (iii) =⇒ (i): Let tr be a faithful, normal trace on M, and let φ := tr ◦ E. 

7.2 Injective von Neumann Algebras

295

For the main theorem of this section, we require the theory of continuous crossed products of von Neumann algebras. A covariant system is a triple (M, G, α), where M is a von Neumann algebra, G is a locally compact group, and α is a homomorphism from G into the ∗ -automorphisms of M such that, for each a ∈ M the map G  g → αx (a) is continuous with respect to the given topology on G and the weak operator topology on M (arguing as in the proof of Lemma 1.3.2, one can see that this already implies continuity with respect to the strong operator topology on M). To each covariant system (M, G, α), one can associate another von Neumann algebra W ∗ (M, G, α), the continuous crossed product of M by G. Our sources on continuous crossed products are [70] and (to a lesser extent) [200] (in [70], the notation N ⊗α R is used instead of W ∗ (N, R, α)). Theorem 7.2.7. For a von Neumann algebra M acting on a Hilbert space H, the following are equivalent: (i) M is injective. (ii) there is a quasi-expectation Q : B(H) → M . Proof. By Theorem 7.2.2, (i) =⇒ (ii) is clear. Let Q : B(H) → M be a quasi-expectation. Suppose first that M is finite and countably decomposable. Let tr be a faithful, normal trace on M , and define φ := tr ◦ Q. Then φ satisfies Lemma 7.2.6(i), so that there is a conditional expectation E : B(H) → M . Next, let M be semifinite. By [200, Theorem 6.3.8] and [199, Exercise 5.7.45], there is an increasing net (pα )α∈A of projections in M such that • pα → idH in the strong operator topology, and • each pα M pα is finite and countably decomposable for each α ∈ A. Since Q(pα T pα ) = pα (QT )pα for T ∈ B(H), restricting Q to pα B(H)pα induces a quasi-expectation Qα : pα B(H)pα → pα M pα . By the foregoing, we then have a conditional expectation Eα : pα B(H)pα → pα M pα . Let U be an ultrafilter over A that dominates the order filter. Define E : B(H) → B(H),

T → weak∗ - lim Eα T. α→U

It is easily seen that E is a norm one projection onto M , i.e., a conditional expectation. We now turn to the general case. Since we have already settled the semifinite case, we may suppose by Exercises 7.2.3 and 7.2.4 that M is of type III. We first suppose that M is also countably decomposable. By [70, II.4.8 Theorem], there is a covariant system (N, R, α) such that M ∼ = W ∗ (N, R, α), where N is a von Neumann algebra ˆ = R on M ([70, Section I.4]). ˆ of R of type II∞ . Consider the dual action α By [70, Proposition I.4.12], the fixed point algebra of this dual action {x ∈ W ∗ (N, R, α) : α ˆ t (x) = x for all t ∈ R}

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7 Operator Algebras on Hilbert Spaces

is (canonically) isomorphic to N. Let m : ∞ (R) → C be an invariant mean. For ξ, η ∈ H define (t ∈ R, x ∈ M ).

φξ,η (t, x) := ˆ αt (x)ξ|η

It is immediate that φξ,η (·, x) ∈ ∞ (R) for all x ∈ M . Define P : M → N by letting (x ∈ M , ξ, η ∈ H). (Px)ξ|η := φξ,η (·, x), m

It is routinely checked that P is a norm one projection onto N. From Theorem 7.2.2, it follows that P ◦ Q : B(H) → N is a quasi-expectation. Since N is of type II∞ and thus semifinite, it follows that there is a conditional expectation from B(H) onto N. Exercise 7.2.2 implies that there is a conditional expectation E˜ : B(H) → N . The construction of W ∗ (N, R, α) ([70, Definition I.2.10]) shows that M is generated by its subalgebra N and a one-parameter subgroup {ut : t ∈ R} of U(M ) such that ut xu∗t = αt (x)

(t ∈ R, x ∈ N).

(7.6)

In particular, it follows that M = M = N ∩ {ut : t ∈ R} .

(7.7)

For ξ, η ∈ H, let ˜ )u∗t ξ|η

ψξ,η (t, T ) := ut (ET

(t ∈ R, T ∈ B(H).

Define E : B(H) → B(H) through (ET )ξ|η := ψξ,η (·, T ), m

(T ∈ B(H), ξ, η ∈ H).

It is immediate that E has norm one. From (7.7), it follows at once that E|M = idM . Since m is an invariant mean, it is easy to see that E attains its values in {ut : t ∈ R} ; with (7.6), it is also straightforward to verify that E(B(H)) ⊂ N . Hence, (7.7) again implies that the range of E is contained in M. Thus, E : B(H) → M is a conditional expectation, and M is injective. By Exercise 7.2.2, M is injective. Finally, suppose that M is of type III and arbitrary. As in our treatment of the semifinite case, we obtain an increasing a net (pα )α of projections in M such that pα → idH in the strong operator topology, and such that each pα M pα is countably decomposable. It is routinely checked that each of the von Neumann algebras pα M pα is of type III as well. As in our treatment of the semifinite case, we conclude that each pα Mpα and eventually M itself is injective.  Remark 7.2.8. In the proof of Theorem 7.2.7, it is not necessary, in the type III case, to treat countably decomposable von Neumann algebras first and then reduce the general situation to the countably decomposable case: [70,

7.2 Injective von Neumann Algebras

297

Theorem II.4.8] can be extended to hold for arbitrary von Neumann algebras of type III. In the general situation, one has to construct the modular automorphism group with respect to a weight instead of a state. The reason why [70], as well as [200], confine themselves to the countably decomposable case is evident: this simply makes the construction much easier. For the general situation, see [332]. In conjunction with Theorem 5.1.23(ii), Theorem 7.2.7 immediately yields a first connection between injectivity and Connes-amenability: Corollary 7.2.9. Let M be a Connes-amenable von Neumann algebra. Then M is injective. We include this section with increasing our stock of examples of injective von Neumann algebras. Example 7.2.10. Let M be a von Neumann algebra of type I. We claim that M is injective. First, note that by Corollary C.7.10 and Exercise 7.2.4 below, we may suppose that M is of type Iα for some cardinal number α. By Theorem ¯ where Hα is a Hilbert space of C.7.13, M is isomorphic to B(Hα )⊗Z(M), dimension α. Since both Z(M) and B(H) ∼ = K(H)∗∗ are Connes-amenable, it follows that M is Connes-amenable, too, and thus injective by Corollary 7.2.9.

Exercises Exercise 7.2.1. Show that a von Neumann algebra M is injective if and only if π(M) is injective for every faithful, normal ∗ -representation of M. (Hint: Proceed as in the proof of Lemma 7.1.4, and utilize the easily verified fact that, if two von Neumann algebras are spatially isomorphic, then so are their commutants.) Exercise 7.2.2. Let M be a von Neumann algebra on a Hilbert space H. Show that M is injective if and only M is. (Hint: Proceed as in the proof of Proposition 7.1.5.) Exercise 7.2.3. Let M be an injective von Neumann algebra, and let p ∈ M be a projection. Show that pMp is injective. Exercise 7.2.4. Let (Mι )ι∈I be a family of injective von Neumann algebras. Show that ∞ - ι∈I Mι is an injective von Neumann algebra. Exercise 7.2.5. Let M and N be Connes-amenable von Neumann algebras. ¯ is also Connes-amenable. Show that M⊗N

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7 Operator Algebras on Hilbert Spaces

7.3 Nuclear C ∗ -Algebras We now introduce the C ∗ -algebraic property that will turn out to be equivalent to amenability: nuclearity. Definition 7.3.1. A C ∗ -algebra A is called nuclear if there is only one C ∗ norm on A ⊗ B for every C ∗ -algebra B. Remark 7.3.2. By Theorem C.8.6, a C ∗ -algebra A is nuclear if and only if the identity from A ⊗max B to A ⊗min B is an isometry. Example 7.3.3. Let N ∈ N, and let B be any C ∗ -algebra. As MN ⊗ B and MN (B) are ∗ -isomorphic as ∗ -algebras, and since MN (B) has a unique C ∗ norm (Example C.4.8), it is clear that MN is nuclear. Invoking the structure theorem Theorem C.4.9, it can easily be seen that all finite-dimensional C ∗ algebras are nuclear. Next, we show that all commutative C ∗ -algebras are nuclear. Proposition 7.3.4. Let A and B be commutative C ∗ -algebras. Then there is only one C ∗ -norm on A ⊗ B. Proof. By Exercise 7.3.1 below, we may suppose without loss of generality ˜ that both A and B are unital. Let  ·  be a C ∗ -norm on A ⊗ B, and let A⊗B denote the completion of (A ⊗ B,  · ). Set K := {(φ, ψ) : φ ∈ ΦA , ψ ∈ ΦB such that φ ⊗ ψ ∈ ΦA⊗B ˜ }. Let φ ∈ ΦA⊗B ˜ , and define φA : A → C and φB : B → C by letting a, φA := a ⊗ eB , φ

(a ∈ A)

and

b, φB := eA ⊗ b, φ

(b ∈ B).

It is clear that φA ∈ ΦA , φB ∈ ΦB , and that φ = φA ⊗ φB . It follows that a = sup{| a, φ ⊗ ψ | : (φ, ψ) ∈ K}

(a ∈ A ⊗ B).

(7.8)

We claim that K = ΦA ×ΦB , and assume otherwise toward a contradiction. It is easily seen that K is closed in ΦA × ΦB , so that there are non-void, open sets UA ⊂ ΦA and UB ⊂ ΦB with (UA × UB ) ∩ K = ∅. Choose nonzero functions f ∈ C(ΦA ) and g ∈ C(ΦB ) with supp(f ) ⊂ UA and supp(g) ⊂ UB . Identifying C(ΦA ) and C(ΦB ) with A and B, respectively, via Theorem C.2.5, ˜ we may view f ⊗ g as an element of A⊗B. According to (7.8) and the choices of f and g, we have f ⊗ g = 0. But this is impossible because both f = 0 and g = 0. It follows that K = ΦA × ΦB and hence a = sup{| a, φ ⊗ ψ | : φ ∈ ΦA , ψ ∈ ΦB }

(a ∈ A ⊗ B).

Obviously, the right-hand side of this equation is independent of  · .



7.3 Nuclear C ∗ -Algebras

299

Lemma 7.3.5. Let A and B be unital C ∗ -algebras, let  ·  be a C ∗ -norm on ˜ denote the completion of (A ⊗ B,  · ). For φ ∈ S(A), A ⊗ B, and let A⊗B define Sφ := {ψ ∈ S(B) : φ ⊗ ψ has norm one on A ⊗ B with respect to  · }. Then: (i) for each ψ ∈ Sφ , the functional φ ⊗ ψ on A ⊗ B has a unique extension ˜ to an element of S(A⊗B); (ii) Sφ is a weak∗ compact, convex subset of S(B); (iii) if φ is pure, and if b ∈ B is self-adjoint such that there is only one C ∗ -norm on A ⊗ C(σ(b)), then b = sup{| b, ψ | : ψ ∈ Sφ }; (iv) if every self-adjoint element b ∈ B has the property that there is only one C ∗ -norm on A ⊗ C(σ(b)), then Sφ = S(B). Proof. (i) and (ii) are routine. To prove (iii) let C ∗ (b) be the unital C ∗ -subalgebra of B generated by b; by Gelfand theory, we have C ∗ (b) ∼ = C(σ(b)). There is χ ∈ ΦC ∗ (b) such that b = | b, χ |. The functional φ ⊗ χ on A ⊗ C ∗ (b) ∼ = A ⊗ C(σ(b)) is continuous with respect to ·min . Since we are supposing that there is only one C ∗ -norm on A ⊗ C ∗ (b), we have that φ ⊗ χ is also continuous with respect to  · ; it ˜ thus extends to an element ψ ∈ S(A⊗B). Define ψB ∈ S(B) through c, ψB := eA ⊗ c, ψ

(c ∈ B).

From the definition, it is immediate that b = | b, ψB |. We claim that ψB ∈ Sφ (which establishes (iii)). Let c ∈ B be such that 0 ≤ c ≤ eB , and define positive functionals φ1 , φ2 ∈ A∗ through a, φ1 := a ⊗ c, ψ

and

a, φ2 := a ⊗ (eB − c), ψ

(a ∈ A).

We have a, φ = a ⊗ eB , ψ = a, φ1 + a, φ2

(a ∈ A).

Since φ is pure, it follows that there is t ≥ 0 such that φ1 = tφ. Since t = t eA , φ = eA , φ1 = eA ⊗ c, ψ = c, ψB , it follows that a ⊗ c, ψ = a, φ1 = t a, φ = a, φ c, ψB

(a ∈ A).

By linearity, we obtain eventually that ψ = φ ⊗ ψB , so that ψB ∈ Sφ .

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7 Operator Algebras on Hilbert Spaces

For the proof of (iv), suppose first again that φ is pure. We claim that Sφ = S(B) in this particular case. Assume toward a contradiction that there is ψ0 ∈ S(B)\Sφ . By (ii), an application of the Hahn–Banach Theorem yields a ∈ A such that sup{Re a, ψ : ψ ∈ Sφ } < Re a, ψ0 . Letting b := aeA + 12 (a + a∗ ), we obtain sup{| b, ψ | : ψ ∈ Sφ } < | b, ψ0 | ≤ b. This contradicts (iii). For ψ ∈ S(B), let S ψ := {φ ∈ S(A) : ψ ∈ Sφ }. It is easily checked that S ψ is a weak∗ compact, convex subset of S(A). By the foregoing, every pure state of A is in S ψ . By the Kre˘ın–Milman Theorem, every element of S(A) is the weak∗ limit of convex combinations of pure  states, so that S(A) = S ψ . In particular, ψ ∈ Sφ holds. Theorem 7.3.6. Let A be a commutative C ∗ -algebra. Then A is nuclear. Proof. Let B be another C ∗ -algebra. By Exercise 7.3.1 below, there is no loss of generality if we suppose that both A and B are unital. ˜ be the completion of Let  ·  be a C ∗ -norm on A ⊗ B, and let A⊗B (A ⊗ B,  · ). Let a ∈ A ⊗ B be self-adjoint. There is χ ∈ ΦC ∗ (a) , i.e., a pure state of C ∗ (a), such that a = | a, χ |. By [251, Theorem 5.1.13], we may ˜ extend χ to a pure state of A⊗B; we denote this extension likewise by χ. Define χA ∈ S(A) and χB ∈ S(B) through a, χA := a ⊗ eB , χ

(a ∈ A)

and

b, χB := eA ⊗ b, χ

(b ∈ B).

By Exercise 7.3.2 below, χA is multiplicative and thus pure. As in the proof of Lemma 7.3.5(iii), we see that χ = χA ⊗ χB , so that, in particular, χ ∈ {φ ⊗ ψ : φ ∈ S(A), ψ ∈ S(B)}. By Proposition 7.3.4, the hypothesis of Lemma 7.3.5(iv) is satisfied, so that ˜ {φ ⊗ ψ : φ ∈ S(A), ψ ∈ S(B)} ⊂ S(A⊗B). It follows that a = | a, χ | = sup{| a, φ ⊗ ψ | : φ ∈ S(A), ψ ∈ S(B)} ≤ a, and hence a = sup{| a, φ ⊗ ψ | : φ ∈ S(A), ψ ∈ S(B)}. Since the right-hand side of the last equality is independent of  · , this completes the proof.  We will present further examples of nuclear C ∗ -algebras later on.

7.3 Nuclear C ∗ -Algebras

301

Exercises Exercise 7.3.1. Let A and B be C ∗ -algebras such that B is non-unital, and let  ·  be a C ∗ -norm on A ⊗ B. Show that  ·  extends to A ⊗ B# as a C ∗ -norm. (Hint: Take a faithful ∗ -representation of A ⊗ B on some Hilbert space, and then use the fact that B has a bounded approximate identity to extend it to A ⊗ B# .) Exercise 7.3.2. Let A be a C ∗ -algebra, and let φ ∈ S(A) be pure. Show that φ|Z(A) is multiplicative.

7.4 Semidiscrete von Neumann Algebras Definition 7.4.1. A von Neumann algebra M is called semidiscrete if there is a net (Sα )α of unit preserving, completely positive, weak∗ continuous maps in F(M) such that φ, x = lim φ, Sα x

α

(x ∈ M, φ ∈ M∗ ).

The definition of semidiscreteness bears some resemblance to Definition A.2.1. For the predual of a semidiscrete von Neumann algebra, we actually obtain the metric approximation property: Proposition 7.4.2. Let M be a semidiscrete von Neumann algebra. Then M∗ has the metric approximation property. Proof. Let (Sα )α be a net as in Definition 7.4.1. Since each Sα is positive and unit preserving, each Sα has norm one. Since each Sα is w∗ -continuous, it follows that (Sα∗ |M∗ )α is a net in F(M∗ ) that converges to idM∗ in the weak operator topology. Passing to convex combinations, we obtain a net of operators in F(M∗ ) bounded by one that converges to idM∗ in the strong operator topology.  Corollary 7.4.3. Let A be a C ∗ -algebra such that A∗∗ is semidiscrete. Then both A∗ and A have the metric approximation property. The aim of this section is to show that injectivity is the same as semidiscreteness. We begin with the proof of semidiscreteness =⇒ injectivity. Both on C ∗ -algebras and on their duals, we have a notion of positivity. It thus makes sense to speak of positive maps between those objects: Definition 7.4.4. Let A and B be C ∗ -algebras. Then a linear map T : B → A∗ is called positive if T B+ ⊂ A∗+ .

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7 Operator Algebras on Hilbert Spaces

Given a C ∗ -algebra A and n ∈ N, we can naturally identify the dual of the C -algebra Mn (A) with Mn (A∗ ). Hence, we can define: ∗

Definition 7.4.5. Let A and B be C ∗ -algebras. Then we call a linear map T : B → A∗ completely positive if T (n) : Mn (B) → Mn (A∗ ) is positive for each n ∈ N. Lemma 7.4.6. Let A and B be unital C ∗ -algebras. For φ ∈ (A ⊗max B)∗ , define Tφ : B → A∗ through a, Tφ b := a ⊗ b, φ

(a ∈ A, b ∈ B).

Then: (i) φ ∈ S(A ⊗max B) if and only if Tφ is completely positive and Tφ eB ∈ S(A); (ii) if T : B → A∗ is completely positive with T eB ∈ S(A), then there is φ ∈ S(A ⊗max B) such that T = Tφ ; (iii) if T ∈ F(B, A∗ ) is completely positive with T eB ∈ S(A), then there is φ in the convex hull of {ψA ⊗ ψB : ψA ∈ S(A), ψB ∈ S(B)} such that T = Tφ ; (iv) φ ∈ S(A ⊗min B) if and only if there is a net of completely positive maps (Tα )α in F(B, A∗ ) with Tα eB ∈ S(A) such that a, Tφ b = lim a, Tα b

α

(a ∈ A, b ∈ B). 

Proof. See Exercise 7.4.4 below.

Definition 7.4.7. Let M be a von Neumann algebra, and let B be a unital C ∗ -algebra. Define Snor (M ⊗ B) := {φ ∈ S(M ⊗max N) : Tφ B ⊂ M∗ }. and xnor := sup



 x∗ • x, φ : φ ∈ Snor (M ⊗ B)

(x ∈ M ⊗ B).

We write M ⊗nor B for M ⊗ B equipped with  · nor and M ⊗nor B for the completion of M ⊗nor B. Remark 7.4.8.  · nor is a C ∗ -norm (see Exercise 7.4.5 below), and consequently, M ⊗nor B is a C ∗ -algebra. Proposition 7.4.9. Let M be a semidiscrete von Neumann algebra. Then, for each unital C ∗ -algebra B, the identity map M ⊗nor B → M ⊗min B is an isometry.

7.4 Semidiscrete von Neumann Algebras

303

Proof. Let φ ∈ Snor (M ⊗ B); it is sufficient to show that φ ∈ S(M ⊗min B). Let Tφ : B → M∗ be defined as in Lemma 7.4.6, and note that Tφ B ⊂ M∗ because φ ∈ Snor (M ⊗ B). Let (Sα )α be a net as in Definition 7.4.1, and let Tα := Sα∗ Tφ . Then (Tα )α is a net of completely positive maps in F(B, M∗ ) with Tα eB ∈ S(M) such that x, Tφ b = lim x, Tα b

α

(x ∈ M, b ∈ B).

(7.9)

By Lemma 7.4.6(iii), there is a net (φα )α in S(A ⊗min B) such that Tα = Tφα . It is then immediate from (7.9) that φ is the weak∗ limit of (φα )α in (M ⊗nor B)∗ . Due to the minimality of  · min , the C ∗ -algebra M ⊗min B is a quotient of the C ∗ -algebra M ⊗nor B; let I denote the kernel of the quotient map. Each  φα vanishes on I, and so does φ. It follows that φ ∈ S(M ⊗min B). As an immediate consequence of Proposition 7.4.9, we obtain a first connection between semidiscreteness and nuclearity: Corollary 7.4.10. Let A be a C ∗ -algebra such that A∗∗ is semidiscrete. Then A is nuclear. Proof. Let B be a C ∗ -algebra. Without loss of generality suppose that both A and B are unital. Let φ ∈ S(A ⊗max B). We claim that φ ∈ S(A ⊗min B). By weak∗ continuity, φ has a unique extension φ˜ ∈ Snor (A∗∗ ⊗ B), so that, by Proposition 7.4.9, φ˜ ∈ S(A∗∗ ⊗min B) holds, which means that φ ∈ S(A ⊗min B). This establishes the nuclearity of A.  A second consequence of Proposition 7.4.9 is even more immediate: Corollary 7.4.11. Let M be a semidiscrete von Neumann algebra. Then, for each unital C ∗ -algebra B and for each unital C ∗ -subalgebra C of B, the inclusion map M ⊗nor C → M ⊗nor B is an isometry. We shall see next that the conclusion of Corollary 7.4.11 characterizes the injective von Neumann algebras (so that, in particular, semidiscrete von Neumann algebras are injective). Lemma 7.4.12. Let A be a C ∗ -algebra acting on a Hilbert space H such that there is a cyclic unit vector ξ ∈ H for A. Let φξ denote the vector state given by ξ, let E be the linear span of {φ ∈ A∗ : 0 ≤ φ ≤ φξ } in A∗ , and let T : A → E be given by a, T x := xaξ|ξ

(x ∈ A , a ∈ A).

Then T is a completely positive bijection with a completely positive inverse.

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7 Operator Algebras on Hilbert Spaces

Proof. It follows immediately from [97, 2.5.1. Proposition] or [266, 3.3.5. Proposition] that T : A → E is a positive bijection. Let x1 , . . . , xn ∈ A and let a1 , . . . , an ∈ A. For Θ := [T (x∗j xk )] j=1,...,n and k=1,...,n

A = [a∗j ak ] j=1,...,n , we then have k=1,...,n

A, Θ = =

n 

a∗j ak , T (x∗j xk )

j,k=1 n 

x∗j xk a∗j ak ξ|ξ

j,k=1

 2  n     = xj aj ξ   j=1  ≥ 0. As a consequence of [264, Lemma 3.13], this is sufficient to conclude that T is completely positive. To show that T −1 is completely positive, let Φ = [φj,k ] j=1,...,n ∈ Mn (E) k=1,...,n

be positive. For a1 , . . . , an ∈ A, set Ξ := (a1 ξ, . . . , an ξ) ∈ Hn . Since T −1 maps into A , we obtain (T

−1 (n)

)

Φ)Ξ, Ξ =

=

=

n 

T −1 (φj,k )aj ξ|ak ξ

j,k=1 n 

T −1 (φj,k )a∗j ak ξ|ξ

j,k=1 n 

a∗j ak , φj,k

j,k=1

≥ 0. Since a1 , . . . , an ∈ A were arbitrary, and since ξ ∈ H is cyclic for A, this  implies that (T −1 )(n) Φ is positive in Mn (A ). Remark 7.4.13. In general, T −1 need not be continuous. Lemma 7.4.14. For a von Neumann algebra M, the following are equivalent: (i) for each unital C ∗ -algebra B and for each unital C ∗ -subalgebra C of B, the inclusion map M ⊗nor C → M ⊗nor B is an isometry;

7.4 Semidiscrete von Neumann Algebras

305

(ii) for each unital C ∗ -algebra B and for each unital C ∗ -subalgebra C of B, the restriction map Snor (M ⊗ B) → Snor (M ⊗ C) is surjective; (iii) for each unital C ∗ -algebra B and for each unital C ∗ -subalgebra C of B, every completely positive map T : C → M∗ with T eB ∈ S(M) has a completely positive extension to B. Proof. The equivalence of (i) and (ii) is straightforward. (ii) =⇒ (iii): Let B be a unital C ∗ -algebra, let C be a unital C ∗ -subalgebra of B, and let T : C → M∗ be completely positive. By Lemma 7.4.6(ii), there is φ ∈ S(M ⊗max C) such that T = Tφ . Since T C ⊂ M∗ , it follows that φ ∈ Snor (M ⊗ C). By (ii), there is ψ ∈ Snor (M ⊗ B) such that ψ|M⊗C = φ. Then Tψ : B → M∗ is completely positive and extends T . (iii) =⇒ (ii): Let B be a unital C ∗ -algebra, let C be a unital C ∗ -subalgebra of B, and let φ ∈ Snor (M ⊗ C). Then Tφ : C → M∗ is completely positive and thus has a completely positive extension T : B → M∗ . Again by Lemma 7.4.6(ii), T is of the form Tψ for some ψ ∈ Snor (M ⊗ B). It is clear that ψ extends φ.  Theorem 7.4.15. Let M be a von Neumann algebra such that, for each unital C ∗ -algebra B and for each unital C ∗ -subalgebra C of B, the inclusion map M ⊗nor C → M ⊗nor B is an isometry. Then M is injective. Proof. Certainly, M satisfies Lemma 7.4.14(i). Fix φ ∈ S(M) ∩ M∗ . Through the GNS-construction, we obtain a Hilbert space Hφ and a normal ∗ representation πφ of M on Hφ . The von Neumann algebra πφ (M) then has a cyclic vector in Hφ by construction. Set B := B(Hφ ), let C := πφ (M) , and define T : C → M∗ be as in Lemma 7.4.12; since φ ∈ M∗ , it follows that θ attains its values in M∗ . Since T is completely positive by Lemma 7.4.12, it has a completely positive extension T˜ : B → M∗ . Let b ∈ B be such that 0 ≤ b ≤ eB . Since T˜ is positive, 0 ≤ T˜b ≤ T˜eB = T eB = φ holds. By Lemma 7.4.12, this means that T˜B = θ(C), so that E := T −1 ◦ T˜ is well defined. Clearly, E is a (completely) positive projection onto πφ (M) and thus a conditional expectation. Consequently, πφ (M) is injective. Let (pα )α be a maximal, orthogonal family of projections in Z(M) with the following property: for each pα , there is φα ∈ S(M) ∩ M∗ such that πφα |Mpα is faithful. Clearly, each von Neumann algebra Mpα satisfies the hypotheses of the theorem, so that, by the first part of the proof πφα (Mpα ) ∼ = Mpα is injective. As M∼ Mpα , = ∞ α

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7 Operator Algebras on Hilbert Spaces

we conclude from Exercise 7.2.4 that M is injective.



Together, Corollary 7.4.11 and Theorem 7.4.15 yield: Corollary 7.4.16. Let M be a semidiscrete von Neumann algebra. Then M is injective. As another consequence of Theorem 7.4.15, we obtain: Corollary 7.4.17. Let A be a nuclear C ∗ -algebra. Then A∗∗ is injective. Proof. Let B be a unital C ∗ -algebra, and let φ ∈ Snor (A∗∗ ⊗ B). It follows that φ is uniquely determined by its values on A ⊗ B. Since A is nuclear, φ|A⊗B is continuous with respect to  · min , so that φ ∈ S(A∗∗ ⊗min B). It follows that the identity map A∗∗ ⊗nor B → A∗∗ ⊗min B is an isometry. The claim then is an immediate consequence of Theorem 7.4.15.  So far, we know that, for a C ∗ -algebra A, the implications A∗∗ semidiscrete =⇒ A nuclear =⇒ A∗∗ injective hold. To close the circle, we now turn to proving injectivity =⇒ semidiscreteness. The following characterization (Theorem 7.4.19 below) will be useful. But first, we need a lemma: Lemma 7.4.18. Let M be a von Neumann algebra, let B be a unital C ∗ algebra, and let φ ∈ S(M ⊗min B) be such that Tφ eB ∈ M∗ . Then there is a net (φα )α in the convex hull of {φM ⊗ φB : φM ∈ S(M) ∩ M∗ , φB ∈ S(B)} converging to φ in the weak∗ topology such that Tφα eB = Tφ for each φα . Proof. Clearly, we may find a net (φα )α in the convex hull of {φM ⊗ φB : φM ∈ S(M) ∩ M∗ , φB ∈ S(B)} such that φ = weak∗ - limα φα . We shall modify this net in such a way that the additional requirement is satisfied. For the sake of simplicity, set ψ := Tφ eB and ψα := Tφα eB . It is immediate that ψ = weak- limα ψα . Passing to convex combinations, we can choose (φα )α in such a way that limα ψα − ψ = 0. Let φB ∈ S(B) be arbitrary, and let φ˜α := φα + |ψα − ψ| ⊗ φB . Then (φ˜α )α satisfies the following: • weak∗ - limα φ˜α = φ; • Tφ˜α eB ≥ ψ;

7.4 Semidiscrete von Neumann Algebras

307

• limα Tφ˜α eB − ψ = 0. Sakai’s Radon–Nikod´ ym Theorem [312, Theorem 1.24.3], yields a net (hα )α in M with 0 ≤ hα ≤ eM such that x, ψ = hα xhα , Tφ˜α eB

(x ∈ M).

Define a net (φ¯α )α through x ⊗ b, φ¯α = hα xhα ⊗ b, φ˜α

(x ∈ M, b ∈ B),

so that Tφ¯α = ψ; also, (φ¯α )α is certainly a net of states contained in the right set. To see that φ = weak∗ - limα φ¯α = φ, note that φ¯α ≤ φ˜α . Let x ∈ M be positive, and let b ∈ B such that 0 ≤ b ≤ eB . Then 0 ≤ x ⊗ b, φ˜α − φ¯α ≤ x ⊗ eB , φ˜α − φ¯α = x, Tφ˜α − ψ → 0, so that indeed φ = weak∗ - limα φ¯α . So, (φ¯α )α has the desired properties.



Theorem 7.4.19. Let M be a von Neumann algebra acting on a Hilbert space H. Then the following are equivalent: (i) M is semidiscrete. (ii) the ∗ -homomorphism π : M ⊗ M → B(H),

x ⊗ y → xy

is continuous with respect to  · min . Proof. (i) =⇒ (ii): Let ξ ∈ H be such that ξ = 1. Then M ⊗ M → C,

x ⊗ y → xyξ|ξ

belongs to Snor (M ⊗ M ). It follows that π is continuous with respect to  · nor . Since  · min =  · nor on M ⊗ M by Corollary 7.4.11, the claim follows. (ii) =⇒ (i): We first treat the case in which there is a cyclic unit vector ξ ∈ H for M. Let φξ ∈ B(H)∗ be the vector state associated with ξ, and let φ := π ∗ φξ ; note that Tφ eM = φξ |M . As π is continuous with respect to  · min , we have φ ∈ S(M ⊗min M ). By Lemma 7.4.18, there is a net (φα )α in the convex hull of {φM ⊗ φM : φM ∈ S(M) ∩ M∗ , φM ∈ S(M )} that converges to φ in the weak∗ topology and satisfies Tφα eM = φξ |M . For the corresponding weak∗ continuous, completely positive maps Tφα : M → M∗ the weak∗ convergence of (φα )α to φ means that Tφ y, x = lim Tφα y, x

α

(x ∈ M y ∈ M ).

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7 Operator Algebras on Hilbert Spaces

As Tφα eM = φξ |M , it follows that all Tφα along with Tφ attain their values in the linear span of {ψ ∈ M∗ : 0 ≤ ψ ≤ φξ |M }. Let T : M → M∗ be as defined in Lemma 7.4.12. It then makes sense to define Sα := T −1 ◦Tφα . Clearly, each Sα is weak∗ continuous, completely positive, and unit preserving. Passing to a subnet, we may suppose that, for all x ∈ M , the weak∗ limit Sx ∈ M of (Sα x)α exists. Since ξ is cyclic for M, it is separating for M . Since (Sx)ξ, ξ = lim (Sα x)ξ|ξ

α

= lim T −1 (Tφα x)ξ|ξ

α

= lim Tφα x, eM

α

= Tφ x, eM

= xξ|ξ

(x ∈ M ), we have S = idM , and therefore M is semidiscrete. By Exercise 7.4.3, M = M is also semidiscrete. Now, let M be arbitrary. Fix ξ ∈ H with ξ = 1. Let P ∈ B(H) be the orthogonal projection onto the closed linear subspace K = Mξ of H, so that P ∈ M . Then ξ ∈ K is cyclic for the von Neumann algebra P M acting on K. The commutant of P M in B(K) is easily seen to be (isomorphic to) P M . It is not hard to see that the ∗ -homomorphism P M ⊗ P M → B(K),

x ⊗ y → xy

is continuous with respect to  · min . Consequently, P M is semidiscrete by the foregoing. For each P ∈ M , the map πP : M → P M,

x → P x

is a ∗ -homomorphism. By the foregoing, the family consisting of those πP such that P M is semidiscrete separates the points of M. Through an argument similar to the one given at the end of the proof of Theorem 7.4.15, we see that M is semidiscrete.  The first steps of the implication from injectivity to semidiscreteness, will be the following two relatively easy lemmas: Lemma 7.4.20. Let M be a finite, injective von Neumann algebra acting on a Hilbert space H, and let tr be a faithful, normal trace on M. Then for any u1 , . . . , un ∈ U(M) and  > 0, there is φ ∈ S(B(H)) ∩ B(H)∗ such that uj · φ − φ · uj  ≤ 

(j = 1, . . . , n)

and φ|M − tr ≤ .

7.4 Semidiscrete von Neumann Algebras

309

Proof. Let  > 0, and let u1 , . . . , un ∈ U(M). Let E : B(H) → M be a conditional expectation, and let ψ := tr ◦ E. Then ψ ∈ S(B(H)) satisfies u∗ · ψ · u = ψ for all u ∈ U(M). As S(B(H)) ∩ B(H)∗ is w∗ -dense in S(B(H)), there is a net (ψα )α in S(B(H)) ∩ B(H)∗ such that ψα → ψ in the weak∗ topology. This means, in particular, that weak- lim(ψα − u∗ · ψα · u) = 0 α

(j = 1, . . . , n)

and weak- lim ψα |M = tr. α

Passing to convex combinations, we obtain φ ∈ S(B(H)) ∩ B(H)∗ with the desired properties.  For any Hilbert space H, we write HS(H) to denote the Hilbert–Schmidt operators on H. Lemma 7.4.21. Let H be a Hilbert space, and let J : H → H be a conjugate linear, isometric isomorphism. Then, for any S1 , T1 , . . . , Sn , Tn ∈ B(H), we have      n  −1   Sj ⊗ JTj J    j=1  ⎧ ⎫  ⎪  ⎪ n ⎨ ⎬    ∗  = sup  Sj HTj  : H ∈ HS(H), HHS(H) ≤ 1 . ⎪ ⎪ ⎩j=1  ⎭ HS(H)

Proof. Identifying H ⊗2 H with H denoting the conjugate linear space of H and HS(H) in the canonical fashion, the operator S ⊗ T ∈ B(H ⊗2 H) with S, T ∈ B(H) becomes HS(H) → HS(H),

H → SHT ∗ .

Since J is an isometry, we obtain for S1 , T1 , . . . , Sn , Tn ∈ B(H) that         n   n   −1    Sj ⊗ JTj J  =  Sj ⊗ T j    j=1  j=1  ⎧ ⎫  ⎪  ⎪ n ⎨ ⎬   ∗ = sup  S HT : H ∈ HS(H), H ≤ 1 . j HS(H) j  ⎪ ⎪ ⎩ j=1  ⎭ HS(H)

This is the claim. We start proving



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7 Operator Algebras on Hilbert Spaces

injectivity =⇒ semidiscreteness with the finite case: Proposition 7.4.22. Let M be a finite, injective von Neumann algebra. Then M is semidiscrete. Proof. A moment’s thought, reveals that we have a decomposition Mι M∼ = ∞ ι∈I

such that each Z(Mι ) has a faithful, normal state φi . For each ι, let trι : Mι → Z(Mι ) be the faithful, Z(Mι )-valued trace that exists according to Theorem C.7.4; setting τι := φι ◦ trι for each ι, we obtain a faithful, normal tracial state τι on each Mι . As M is injective, so is each Mι by Exercise 7.2.3. If each Mι , in turn, is semidiscrete, then so is M by Exercise 7.4.2. We can thus suppose that M has a faithful normal tracial state τ . Applying the GNS-construction with respect to τ , we obtain a Hilbert space H on which M acts as a von Neumann algebra, and a cyclic, separating trace vector ξ for M. The conjugate linear map J : Mξ → Mξ,

xξ → x∗ ξ

is then well defined and extends to a conjugate linear isometric isomorphism of H into itself. We want to apply Theorem 7.4.19. Let x1 , . . . , xn ∈ M, and let y1 , . . . , yn ∈ M . In order to establish that Theorem 7.4.19(ii) holds, it is sufficient to show that           n   n   xj yj ξ|ξ  ≤  x ⊗ y . j j     j=1   j=1 min

Jzj∗ J

Let z1 , . . . , zn ∈ M be such that yj = for j = 1, . . . , n, and suppose without loss of generality that x1 , z1 , . . . , xn , zn ∈ ball(M). Let  ∈ (0, 1). By the Russo–Dye Theorem ([12, Corollary 6.2.14]), there are elements u1 , . . . , um ∈ U(M) such that z1 , . . . , zn lie in the convex hull of {u1 , . . . , um }. By Lemma 7.4.20, there is φ ∈ S(B(H)) ∩ B(H)∗ such that uj · φ − φ · uj  ≤ 2 and

(j = 1, . . . , m)

φ|M − τ  ≤ 2 < .

(7.10) (7.11)

The trace duality between T (H) and B(H) yields a positive operator N ∈ T (H) with Tr N = 1 and φ, T = Tr(T N )

(T ∈ B(H)).

7.4 Semidiscrete von Neumann Algebras

311

The inequalities (7.10) then translate into uj N − N uj  ≤ 2

(j = 1, . . . , m)

1

Let H := N 2 . Then H ∈ HS(H) with HHS(H) = 1, and the Powers– Størmer Inequality (Lemma 3.4.8) implies that uj H − Huj HS(H) ≤ 

(j = 1, . . . , m).

Since each zj is a convex combination of some of the uk ’s, this yields zj H − Hzj HS(H) ≤  Consequently, we obtain      n   xj Hzj     j=1 

HS(H)

(j = 1, . . . , n).

         n     ≥ xj Hzj H    j=1  HS(H)          n    − n. ≥  xj zj H H     j=1

(7.12)

HS(H)

On the other hand, we have  ⎞ ⎛      n    n     ⎠ ⎝ H x z H − τ x z  j j j j     j=1  j=1 HS(H)  ⎛⎛ ⎞ ⎞ ⎛ ⎞   n n      ⎝ ⎝ ⎠ ⎠ ⎝ ⎠ = Tr xj zj N − τ xj zj    j=1 j=1  ⎛ ⎞ ⎛ ⎞   n n     = φ ⎝ xj zj ⎠ − τ ⎝ xj zj ⎠   j=1 j=1    n     ≤  by (7.11), x z j j ,   j=1  ≤ n. Ultimately, this yields

(7.13)

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7 Operator Algebras on Hilbert Spaces

     n   xj ⊗ yj     j=1 

min

     n  ∗   = xj ⊗ Jzj J   j=1  min     n   ≥ xj Hzj  , by Lemma 7.4.21,    j=1  HS(H)  ⎛ ⎞   n     ⎝ ⎠ by (7.12) and (7.13), ≥ τ xj zj  − 2n,   j=1    n     =  xj zj ξ|ξ  − 2n  j=1       n  ∗  =  xj Jzj Jξ|ξ  − 2n  j=1       n  =  xj yj ξ|ξ  − 2n.  j=1 

As  ∈ (0, 1) was arbitrary, we conclude that          n   n   xj yj ξ|ξ  ≤   x ⊗ y j j     j=1  j=1 

,

min

so that M is semidiscrete by Theorem 7.4.19.



With Proposition 7.4.22 proven, the route we have to take lies clearly ahead of us. Next, we tackle the semifinite case: Proposition 7.4.23. Let M be a semifinite, injective von Neumann algebra. Then M is semidiscrete. Proof. Suppose that M acts as a von Neumann algebra on a Hilbert space H. Invoking [200, Theorem 6.3.8] as in the proof of Theorem 7.2.7, we obtain an increasing net (pα )α of projections in M such that pα → eM in the strong operator topology, and each von Neumann algebra pα Mpα is finite. By Exercise 7.2.3, each pα Mpα is also injective and thus semidiscrete by Proposition 7.4.22. Let pα ,  > 0, x1 , . . . , xn ∈ M, and φ1 , . . . , φm ∈ M∗ be arbitrary. Then there is pβ ≥ pα such that | φj , xk − pβ xk pβ |