Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles --- Volume 2: Banach *-Algebraic Bundles, Induced Representations, and the Generalized Mackey Analysis (Pure and Applied Mathematics, 126)

755 Pages·1988·10.64 MB·English

by  J. M.G. Fell

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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
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Copyright 1988 by Academic Press, Inc.
All rights reserved.
No part of this publication may be reproduced or
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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
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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