Spectral Gap Actions and Invariant States

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.