On groups of automorphisms of operator algebras

Source abstract

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R , say α( g ) = ad W g with W g ∈ U ( R ). Then there is a weakly continuous unitary representation of G , by unitaries in R , implementing α if and only if the W g 's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now G is a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R . In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G -invariant state on R . This generalizes a result of Hajian and Kakutani on invariant measures (2).