This page intentionally left blank Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles Volume 2 Banach *-Algebraic Bundles, Induced Representations, and the Generalized Mackey Analysis J. M. G. Fell Department of Mathematics Universiry of Pennsylvania Philadelphia, Pennsylvania R. S. Doran Department of Mathematics Texas Christian University Fort Worth. Texas ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York Berkeley London Sydney Tokyo Toronto Copyright 1988 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101 United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. 24-28 Oval Road, London NW 1 7DX Library of Congress Cataloging-in-Publication Data Fell, J. M. G. (James Michael Gardner), Date Representations of +-algehras, locally compact groups, and Banach *-algebraic bundles. (Pure and applied mathematics; 125-126) Includes bibliographies and indexes. Contents: v. 1. Basic representation theory of groups and algebras v. 2. Banach *-algebraic bundles, induced representations. and the generalized Mackey analysis. I. Representations of algebras. 2. Banach algebras. 3. Locally compact groups. 4. Fiber bundles (Mathematics) 1. Doran, Robert S., Date 11. Title. 111. Series: Pure and applied mathematics (Academic Press); 125-1 26. QA3.P8 VOI. 125-126 [QA326J 510 s [512'.55] 86-30222 ISBN 0-12-252721-6 (v. I: alk. paper) ISBN 0-12-252722-4 (v. 2: alk. paper) Trottsfrrred to drgiml prnirrrtg 2005 This page intentionally left blank Contents Introduction to Volume 2 (Chapters VIII to XII) 747 Chapter VIII. Banach *-Algebraic Bundles and Their Representations 77 1 1. Multipliers of Algebras 773 2. Banach Algebraic Bundles 783 3. Banach *-Algebraic Bundles 792 4. Semidirect Product Bundles and Central Extension Bundles 798 5. The Cross-Sectional Algebra 803 6. Partial Cross-Sectional Bundles 81 1 7. Transformation Bundles 818 8. Locally Convex Representations of Banach Algebraic Bundles 823 9. *-Representations of Banach *-Algebraic Bundles and Unitary Representations of Groups 827 10. Projective Representations of Groups 833 I I. 1 ntegrable Locally Convex Representations 837 12. The Recovery of a FrCchet Representation of 93 from its Integrated Form 848 u vi Chit ents 13. The Recovery of a *-Representation of W from its Integrated Form 852 14. The Existence of Irreducible *-Representations of Banach *-Algebraic Bundles 860 15. The Extension of *-Representations to Multipliers; Application to Semidirect Products and Partial Cross- Sectional Bundles 864 16. C*-Algebraic Bundles; the Bundle C*-Completion 87 1 17. Bundle Structures for C*-Algebras; the Glimm Algebras 885 18. Transformation Bundles and Systems of Imprimitivity 889 19. Ergodic Measure Transformation Spaces and Certain Irreducible Systems of Imprimitivity 90 1 20. Functionals of Positive Type on Banach *-Algebraic Bundles 906 21. The Regional Topology of *-Representations of Banach *-Algebraic Bundles 913 22. Exercises for Chapter VIII 925 Notes and Remarks 929 Chapter IX. Compact Groups 93 1 1. Preliminary Remarks on Function Spaces 932 2. The Peter-Weyl Theorem 934 3. Finite-Dimensional Representations of G 937 4. Orthogonality Relations 943 5. Characters and Central Functions 947 6. The Stone-Weierstrass Theorem for Representations 9s 1 7. a-Subspaces and Multiplicity for Locally Convex Representations of G 954 8. Weak Fejer Summation 959 9. The Structure of the Regular Representation and the Plancherel Formula 964 10, Induced Representations and the Frobenius Reciprocity Theorem 967 11. The Representations of SU(2) 979 12. The Representations of SO(3) 985 13. Exercises for Chapter IX 988 Notes and Remarks 995 Contents vii Chapter X. Abelian Groups and Commutative Banach *-Algebraic Bundles 999 1. The Character Group 1001 2. The Theorems of Stone, Bochner and Plancherel 1007 3. The Fourier Inversion Theorem, Regularity, and Pontryagin Duality 1016 4. Commutative Banach *-Algebraic Bundles 1024 5. Commutative C*-Algebraic Bundles 1035 6. Exercises for Chapter X 1042 Notes and Remarks 1047 Chapter XI. Induced Representations and the Irnprirnitivity Theorem 1 049 1. Operator Inner Products 1055 2. Duality for Operator Inner Products 1064 3. V-bounded Endomorphisms and Modules; Completely Positive Maps 1070 4. Rigged Modules and Abstractly Induced Representations 1078 5. General Properties of Abstractly Induced Representations 1088 6. Imprimitivity Bimodules and the Abstract Imprimitivity Theorem 1095 7. Topological Versions of the Abstract Imprimitivity Theorem 1107 8. Positivity of *-Representations with Respect to a Banach *-Algebraic Bundle 1121 9. Induced Hilbert Bundles and Induced Representations of a Banach *-Algebraic Bundle 1131 10. Mackey’s and Blattner’s Formulations of Induced Representations of Groups 1147 11. a-Positive Representations and the C*-Completion of W 1155 12. Elementary Properties of Induced Representations of Banach *-Algebraic Bundles 1160 13. Restrictions and Tensor Products of Induced Representations of Banach *-Algebraic Bundles 1173 14. The Imprimitivity Theorem for Banach *-Algebraic Bundles 1179 15. A generalized Mackey-Stone-von Neumann Theorem 1198 16. Conjugation of Representations 1204 17. Non-Involutory Induced Representations 1225 18. Exercises for Chapter XI 1234 Notes and Remarks 1240 viii Cont ents Chapter XII. The Generalized Mackey Analysis 1243 1. The Mackey Normal Subgroup Analysis for Finite Groups I246 2. The Mackey Analysis in the General Case; Step I 1264 3. The Mackey Analysis in the Gencral Case; Step I1 1273 4. The Mackey Analysis in the General Casc; Step 111 1284 5. The Mackey Analysis in the General Casc; Final Results 1301 6. Saturated C*-Algebraic Bundles with Unit Fibers of Compact Type 1311 7. Saturated Bundles over Compact Groups 327 8. Examples 34 0 9. Further Examples 356 10. Exercises for Chapter XI1 3 66 Notes and Remarks 368 Bibliography 373 Name Index 45 1 Subject Index 1457 Index of Notation 1477 Topics in Volume 1 Chapter I. Preliminaries Chapter 11. Integration Theory and Raiiach Bundles Chapter IJI. Locally Compact Groups Chapter IV. Algebraic Representation Theory Chapter V. Locally Convex Representations and Banach Algebras Chapter VI. C*-Algebras and Their *-Representations Chapter VII. The Topology of the space of *-Representations It is unworthy of a mathematician to see with other people’s eyes and to accept as true or as proven that for which he himsclf has no proof. M aes tlin Introduction To Volume 2 (Chapters Eight to Twelve) The main purpose of the present Volume 2 is to give a detailed presentation of the Mackey “normal subgroup analysis” for classifying the unitary representations of groups-and this not merely in the customary context of locally compact groups, but in the larger context of Banach *-algebraic bundles. Before proceeding to sketch the background of the Mackey normal subgroup analysis, we should make special mention of Chapters IX and X, which are somewhat disjoint from the rest of the volume. They consist largely of the now classical general results on the representation theory of compact and abelian groups respectively. As regards compact groups, all the principal general results (centering around the Peter-Weyl Theorem) on their repre- sentation theory can be obtained in the context of locally convex representa- tions rather than merely unitary ones; and this is the context in which they are presented in Chapter IX. Combining these classical results with the “Mackey analysis” of Chapter XII, we shall be able (gX11.7) to generalize much of the representation theory of compact groups (including the Froben- ius Reciprocity Theorem) to a substantial class of Banach *-algebraic bundles over compact groups. As for Chapter X, the first part presents the main 747