$G$-finite von Neumann algebras and weakly almost periodic functions

Abstract

Let $M$ be a von Neumann algebra which acts in a standard way on the Hilbert space $H$, let $G$ be a subgroup of the group of all $*$-automorphisms of $M$. We prove that $M$ is $G$-finite (in the sense of I. Kov\'acs and Sz\"ucs, i.e. the set of all normal, $G$-invariant states on $M$ is separating) if and only if, for every $x\in M$ and all $\xi,\eta\in H$, the coefficient function $g\mapsto \langle g(x)\xi|\eta\rangle$ is weakly almost periodic.