A Groupoid Approach to C*-Algebras
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793 Jean Renault
A Groupoid Approach to C*-Algebras
Springer-Verlag Berlin Heidelberg New York 1980
Author Jean Renault Departement de Mathematiques Faculte des Sciences 45 Orleans - La Source France
AMS Subject Classifications (1980): 22 D 25, 46 L 05, 54 H 15, 54 H 20 ISBN 3-540-09977-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-0997?-8 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
CONTENTS Page Introduction
I Chapter I : LOCALLY COMPACTGROUPOIDS
5
I.
Definitions and Notation
2.
Locally Compact Groupoids and Haar Systems
16
3.
Quasi-lnvariant Measures
22
4.
Continuous Cocycles and Skew-Products
35
Chapter I I : THE C*-ALGEBRA OF A GROUPOID
5
47
1.
The Convolution Algebras Cc(G,~ ) and C*(G,o)
48
2.
Induced Representations
74
3.
Amenable Groupoids
86
4.
The C*-Algebra of an r-Discrete Principal Groupoid
97
5.
Automorphism Groups, KMS States and Crossed Products
Chapter I I I
: SOME EXAMPLES
109
121
1.
Approximately-Finite Groupoids
121
2.
The Groupoids 0
138
n
Appendix : The Dimension Group of the GICAR Algebra
148
References
151
Notation Index
155
Subject Index
157
INTRODUCTION
The interplay between ergodic theory and von Neumann algebra theory goes back to the examples of non-type I factors which Murray and von Neumann obtained by the group measure construction [54].
A natural and probably d e f i n i t i v e point of view which
joins both theories has recently been exposed by P. Hahn [45].
I t uses the notion of
measure groupoid, introduced by G. Mackey "to bring to l i g h t and e x p l o i t certain apparently f a r reaching analogies between group theory and ergodic theory" ([53], p.187).
In p a r t i c u l a r , the group measure algebra may be regarded as the von
Neumann algebra of the regular representation of some principal measure groupoid. Moreover, most of the properties of the algebra may be interpreted in terms of the groupoid. The same standpoint is adopted by J. Feldman and C.Moore [31], in the framework of ergodic equivalence r e l a t i o n s .
Besides, they characterize abstractly
the von Neumann algebras arising from t h e i r construction. I t is natural to expect that topological l o c a l l y compact groupoids play a simil a r role in the theory of C*-algebras. The notions of topological and of Lie groupoid were introduced by Ehresmann for applications to d i f f e r e n t i a l topology and geometry. More recen
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