Spectral gap actions and invariant states

Abstract

We define spectral gap actions of discrete groups on von Neumann algebras and study their relations with invariant states. We will show that a finitely generated ICC group $Γ$ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra $L(Γ)$. Moreover, a countable discrete group $Γ$ has property $(T)$ if and only if for any action $α$ of $Γ$ on a von Neumann algebra $N$, every $α$-invariant state on $N$ is a weak-$^*$-limit of a net of normal $α$-invariant states.