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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
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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  • [1] W. B. Arveson, On groups of automorphisms of operator algebras, J. Functional Analysis 15 (1974), 217-243. MR 50 #1016. MR 0348518 (50:1016)
  • [2] R. R. Kallman, A generalization of free action, Duke Math. J. 36 (1969), 781-789. MR 41 #838. MR 0256181 (41:838)
  • [3] W. L. Paschke, Inner product modules over $ {B^\ast}$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468. MR 50 #8087. MR 0355613 (50:8087)
  • [4] M. A. Rieffel, Induced representations of $ {C^\ast}$-algebras, Advances in Math. 13 (1974), 176-257. MR 50 #5489. MR 0353003 (50:5489)
  • [5] -, Morita equivalence for $ {C^\ast}$-algebras and $ {W^\ast}$-algebras, J. Pure Appl. Algebra 5 (1974), 51-96. MR 0367670 (51:3912)
  • [6] S. Sakai, $ {C^\ast}$-algebras and $ {W^\ast}$-algebras, Springer, New York, 1971. MR 0442701 (56:1082)
  • [7] J. T. Schwartz, $ {W^\ast}$-algebras, Gordon and Breach, New York, 1967. MR 38 #547. MR 0232221 (38:547)
  • [8] M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249-310. MR 0438149 (55:11068)

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Article copyright: © Copyright 1976 American Mathematical Society

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