Duality for Crossed Products of von Neumann Algebras
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731
Yoshiomi Nakagami Masamichi Takesaki
Duality for Crossed Products of von Neumann Algebras
Springer-Verlag Berlin Heidelberg New York 1979
Authors Yoshiomi Nakagami Department of Mathematics Yokohama City University Yokohama Japan Masamichi Takesaki Department of Mathematics University of California Los Angeles, CA 90024 U.S.A.
AMS Subject Classifications (1970): 46 L10 ISBN 3-540-09522-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09522-5 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloomq In Publication Data Nakagami, Yoshiomi, 1940Duality for crossed products of von Neumann algebras. (Lecture notes in mathematics: 731) Bibliography: p. Includes index, 1. Von Neumann algebras--Crossed products. 2. Duality theory (Mathematics) I. Takosaki, Masamichi, 1933- II. Title. III, Series: Lecture notes in Mathematics (Berlin) 731. OA3.L28 no, 731 [OA326] 510',8s [512'.55J 7Q-17038 ISBN 0-387-09522-5
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 1979 Printed in Germany
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INTRODUCTION The recent development in the theory of operator algebras showed the importance of the stUdy of automorphism groups of von Neumann algebras and their crossed products.
The main tool here is duality theory for locally compact groups. Let
be a von Neumann algebra equipped with a continuous action
locally compact group
G.
For a unitary representation
be the aweakly closed subspace of ators
T
from
V)
into
for any pair
where
of
is contained in
of unitary representations of
U means the conjugate representation of
basis for the entire duality mechanism.
let
spanned by the range of all intertwining oper-
It is easily seen that U,V
of a
G,
U.
G,
=
and that
This simple fact is the
At this point, one s.hou.ld recall the form-
ulation of the TannakaTatsuuma duality theorem. In spite of the above simple basis, the absence of the dual group in the noncommutative case forces us to employ the notationally (if not mathematically) complicated Ropfvon Neumann algebra approach to the duality pr-Lncd p.Le ,
I L should
however be pointed out that the Ropf von Neumann algebra approach simply means a 2(G systematic usage of the unitary W on L x G) given by (WGS)(s,t) = S(s,ts). G This operator W is nothing else but the operator version of the group multiplicaG tion table. In this sense, W is a very natural object whose importance can not G be overestimated. For example, the TannakaTatsuuma duality theorem simply asserts 2(G» that a nonzero x E £(L is
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