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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Inner product modules arising from compact automorphism groups of von Neumann algebras


Author: William L. Paschke
Journal: Trans. Amer. Math. Soc. 224 (1976), 87-102
MSC: Primary 46L10
DOI: https://doi.org/10.1090/S0002-9947-1976-0420294-7
MathSciNet review: 0420294
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Abstract: Let M be a von Neumann algebra of operators on a separable Hilbert space H, and G a compact, strong-operator continuous group of $ ^\ast$-automorphisms of M. The action of G on M gives rise to a faithful, ultraweakly continuous conditional expectation of M on the subalgebra $ N = \{ A \in M:g(A) = A\forall g \in G\} $, which in turn makes M into an inner product module over N. The inner product module M may be ``completed'' to yield a self-dual inner product module $ \bar M$ over N; our most general result states that the $ {W^\ast}$-algebra $ A(\bar M)$ of bounded N-module maps of $ \bar M$ into itself is isomorphic to a restriction of the crossed product $ M \times G$ of M by G. When G is compact abelian, we give conditions for $ A(\bar M)$ and $ M \times G$ to be isomorphic and show, among other things, that if G acts faithfully on M, then $ M \times G$ is a factor if and only if N is a factor. As an example, we discuss certain compact abelian automorphism groups of group von Neumann algebras.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0420294-7
Article copyright: © Copyright 1976 American Mathematical Society