Actions of discrete amenable groups on von neumann algebras
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1138 Adrian Ocneanu
Actions of Discrete Amenable Groups on von Neumann Algebras
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Adrian Ocneanu Department of Mathematics, University of California Berkeley, California 94720, USA
Mathematics Subject Classification (1980): 20F29, 46L40, 46L55 ISBN 3-540-15663-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15663-1 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
We study the actions of discrete amenable groups on factor von Neumann algebras.
We give the
classification up to outer conjugacy of the actions of amenable groups on the type II hyperfinite factors. A main result is the unicity up to outer conjugacy of the free action of an amenable group on the hyperfinite type III factor.
ACKNOWLEDGMENTS.
The results of this paper were mainly obtained during
my activity as a research fellow at INCREST-Bucharest, Romania, and were announced in [34J.
The writing up, as well as several improve-
ments in the proof, were done at the University of Warwick, England. I would like to thank Ciprian
as well as my INCREST
colleagues Grigore Arsene, Zoia Ceaurescu, Mihai Pimsner, Sorin Popa, Dan Voiculescu, and especially
for their support.
I received further support from David Evans, Klaus Schmidt and Christopher Zeeman at Warwick. Benjamin Weiss was very kind to send me a description of the results in [36J prior to their pUblication. I am grateful to Vaughan Jones for useful discussions and for the generosity with which he helped make the results in this paper known. My wife Ileana generously helped me carryon my work. Deborah Craig typed and edited the manuscript with remarkable skill and accuracy.
TABLE OF CONTENTS Introduction . . . . . . . . . . . .
...
1
Chapter 1
Main Results
. . . . . . . .
5
Chapter 2
Invariants and Classification
8
Chapter 3
Amenable Groups..
Chapter 4
The Model Action
Chapter 5
.
. . . . . 16
. . . . . . . . 23
Ultraproduct Algebras
31
Chapter 6
The Rohlin Theorem
41
Chapter 7
Cohomology Vanishing
Chapter 8
Model Action Splitting
Chapter 9
Model Action Isomorphism
. . . . . . 59 . . . . . 77 . . 95
References Notation Index Subject Index
112
. . ..
.
114 115
INTRODUCTION In this paper we study automorphic actions of discrete groups on von Neumann algebras. THEOREM.
The main result is the following.
G be a c oun t ab l e d i s c r e t:e amenable group and let
Let
be the hyperfinite III faotor. outer conjugate. An ac
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