Mathematical Surveys and Monographs Volume 231 Fourier and Fourier- Stieltjes Algebras on Locally Compact Groups Eberhard Kaniuth Anthony To-Ming Lau Fourier and Fourier- Stieltjes Algebras on Locally Compact Groups Mathematical Surveys and Monographs Volume 231 Fourier and Fourier- Stieltjes Algebras on Locally Compact Groups Eberhard Kaniuth Anthony To-Ming Lau EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman Michael I. Weinstein 2010 Mathematics Subject Classification. Primary 43-02, 43A10, 43A20, 43A30, 43A25, 46-02, 22-02. For additional information and updates on this book, visit www.ams.org/bookpages/surv-231 Library of Congress Cataloging-in-Publication Data Names: Kaniuth, Eberhard, author. | Lau, Anthony To-Ming, author. Title: Fourier and Fourier-Stieltjes algebras on locally compact groups / Eberhard Kaniuth, An- thony To-Ming Lau. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Mathe- matical surveys and monographs; volume 231 | Includes bibliographical references and index. Identifiers: LCCN 2017052436 | ISBN 9780821853658 (alk. paper) Subjects: LCSH: Topological groups. | Group algebras. | Fourier analysis. | Stieltjes transform. | Locally compact groups. | AMS: Abstract harmonic analysis – Research exposition (mono- graphs, survey articles). msc | Abstract harmonic analysis – Abstract harmonic analysis – Measure algebras on groups, semigroups, etc. msc | Abstract harmonic analysis – Abstract 1 harmonic analysis – L -algebras on groups, semigroups, etc. msc | Abstract harmonic analysis – Abstract harmonic analysis – Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. msc | Abstract harmonic analysis – Abstract harmonic analysis – Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups. msc | Functional analysis – Research exposition (monographs, survey articles). msc | Topological groups, Lie groups – Research exposition (monographs, survey articles). msc Classification: LCC QA387 .K354 2018 | DDC 515/.2433–dc23 LC record available at https://lccn.loc.gov/2017052436 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to We dedicate this book to our wives, Ulla and Alice, for their lifetime of support and exceptional patience during the preparation of the manuscript. Contents Preface ix Acknowledgments xi Chapter 1. Preliminaries 1 1.1. Banach algebras and Gelfand theory of commutative Banach algebras 1 1.2. Locally compact groups and examples 6 1.3. Haar measure and group algebra 12 1.4. Unitary representations and positive definite functions 18 1.5. Abelian locally compact groups 24 1.6. Representations and positive definite functionals 28 1.7. Weak containment of representations 30 1.8. Amenable locally compact groups 33 Chapter 2. Basic Theory of Fourier and Fourier-Stieltjes Algebras 37 2.1. The Fourier-Stieltjes algebra BpGq 38 2.2. Functorial properties of BpGq 46 2.3. The Fourier algebra ApGq, its spectrum and its dual space 50 2.4. Functorial properties and a description of ApGq 57 2.5. The support of operators in V NpGq 60 2.6. The restriction map from ApGq onto ApHq 66 2.7. Existence of bounded approximate identities 72 2.8. The subspaces AπpGq of BpGq 78 2.9. Some examples 83 2.10. Notes and references 86 Chapter 3. Miscellaneous Further Topics 91 3.1. Host’s idempotent theorem 91 3.2. Isometric isomorphisms between Fourier-Stieltjes algebras 96 3.3. Homomorphisms between Fourier and Fourier-Stieltjes algebras 101 3.4. Invariant subalgebras of V NpGq and subgroups of G 107 3.5. Invariant subalgebras of ApGq and BpGq 113 3.6. Comparison of ApG1q bp ApG2q and ApG1 ˆ G2q 117 ˚ 3.7. The w -topology and other topologies on BpGq 121 3.8. Notes and references 127 Chapter 4. Amenability Properties of ApGq and BpGq 129 4.1. ApGq as a completely contractive Banach algebra 129 4.2. Operator amenability of ApGq 132 vii viii CONTENTS 4.3. Operator weak amenability of ApGq 138 4.4. The flip map and the antidiagonal 140 1 4.5. Amenability and weak amenability of ApGq and of L pGq 144 4.6. Notes and references 152 Chapter 5. Multiplier Algebras of Fourier Algebras 153 5.1. Multipliers of ApGq 153 5.2. MpApGqq “ BpGq implies amenability of G: The discrete case 160 5.3. MpApGqq “ BpGq implies amenability of G: The nondiscrete case 167 5.4. Completely bounded multipliers 179 5.5. Uniformly bounded representations and multipliers 186 5.6. Multiplier bounded approximate identities in ApGq 191 5.7. Examples: Free groups and SLp2, Rq 195 5.8. Notes and references 202 Chapter 6. Spectral Synthesis and Ideal Theory 205 6.1. Sets of synthesis and Ditkin sets 206 6.2. Malliavin’s theorem for ApGq 210 6.3. Injection theorems for spectral sets and Ditkin sets 211 6.4. A projection theorem for local spectral sets 214 6.5. Bounded approximate identities I: Ideals 220 6.6. Bounded approximate identities II 228 6.7. Notes and references 234 Chapter 7. Extension and Separation Properties of Positive Definite Functions 237 7.1. The extension property: Basic facts 238 7.2. Extending from normal subgroups 242 7.3. Connected groups and SIN-groups 246 7.4. Nilpotent groups and 2-step solvable examples 250 7.5. The separation property: Basic facts and examples 257 7.6. The separation property: Nilpotent Groups 264 7.7. The separation property: Almost connected groups 268 7.8. Notes and references 273 Appendix A 277 A.1. The closed coset ring 277 A.2. Amenability and weak amenability of Banach algebras 280 A.3. Operator spaces 282 A.4. Operator amenability 284 A.5. Operator weak amenability 287 Bibliography 291 Index 303 Preface b ˚ Let G be a locally compact group. Let C pGq be the C -algebra of bounded continuous complex-valued functions on G with the supremum norm, and let C0pGq b be the closed ˚-subalgebra of C pGq that consists of functions vanishing at infinity. If G is abelian, let Gp be the dual group of G, and let ApGq be all fp(Fourier transform of f), f P L1pGpq (the group algebra of the dual group Gp); and let BpGq be all μp (the Fourier-Stieltjes transform of μ), μ P MpGpq (the measure algebra of Gp). Then ApGq b is a subalgebra of C0pGq, and BpGq is a subalgebra of C pGq. Furthermore, ApGq (respectively, BpGq) with norm from L1pGpq (respectively, MpGpq) is a commutative Banach algebra called the Fourier (respectively, Fourier-Stieltjes) algebra of G. In Chapter 2, we shall introduce and study some basic properties of Fourier and Fourier-Stieltjes algebras, ApGq and BpGq, associated to a locally compact group G based on the fundamental paper of Eymard [73]. BpGq will be identified as ˚ ˚ the Banach space dual of the group C -algebra C pGq and a fair number of basic functorial properties will be presented. Similarly, for the Fourier algebra ApGq, the 2 elements will be shown to be precisely the convolution products of L -functions on G. In Chapter 3, we shall study some further topics of ApGq and BpGq. Generaliz- ing the classical description of idempotents in the measure algebra of a locally com- pact abelian group, Host [129] has identified the integer-valued functions in BpGq. Host’s idempotent theorem, which has numerous applications, will be shown in this chapter. A natural question is whether either of the Banach algebras ApGq and BpGq determines G as a topological group. This question has been affirmatively answered by Walter [280]. If G1 and G2 are locally compact groups and BpG1q and BpG2q (respectively, ApG1q and ApG2q) are isometrically isomorphic, then G1 and G2 are topologically isomorphic or anti-isomorphic. Amenable Banach algebras were introduced by B. E. Johnson. He showed the fundamental result that a locally compact group is amenable if and only if the group 1 algebra L pGq is amenable. We present a proof of the “only if” part of Johnson’s result in Chapter 4. In particular, if G is abelian, then ApGq, being isometrically isomorphic to the L1-algebra of the dual group Gp, is amenable. However, when G is nonabelian, then ApGq need not be weakly amenable, even when G is compact. In Chapter 4, we will also consider the completely bounded cohomology theory of the Fourier algebra ApGq and of the Fourier-Stieltjes algebra BpGq. We will show that ApGq, equipped with the operator space structure inherited from being ˚ embedded into V NpGq , is a completely contractive Banach algebra. Using this, we establish in this chapter the fundamental result, due to Ruan [245], that a locally compact group G is amenable precisely when ApGq is operator amenable. ix