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

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 ∈ M : g ( A ) = A ∀ g ∈ G } 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 M ¯ \bar M over N; our most general result states that the W ∗ {W^\ast } -algebra A ( M ¯ ) A(\bar M) of bounded N-module maps of M ¯ \bar M into itself is isomorphic to a restriction of the crossed product M × G M \times G of M by G. When G is compact abelian, we give conditions for A ( M ¯ ) A(\bar M) and M × G M \times G to be isomorphic and show, among other things, that if G acts faithfully on M, then M × G 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.