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 $N = \{ A \in M:g(A) = A\forall g \in G\}$, 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.