Spectral gap actions and invariant states

TL;DR

This paper introduces spectral gap actions of discrete groups on von Neumann algebras and explores their connection to invariant states, characterizing properties like inner amenability and property (T) through invariant states.

Contribution

It establishes new characterizations of inner amenability and property (T) for discrete groups via spectral gap actions and invariant states on von Neumann algebras.

Findings

Inner amenability of ICC groups corresponds to multiple inner invariant states.

Property (T) groups have invariant states approximated by normal invariant states.

Spectral gap actions provide a framework linking group properties to operator algebra 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 $\Gamma$ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra $L(\Gamma)$. Moreover, a countable discrete group $\Gamma$ has property $(T)$ if and only if for any action $\alpha$ of $\Gamma$ on a von Neumann algebra $N$, every $\alpha$-invariant state on $N$ is a weak-$^*$-limit of a net of normal $\alpha$-invariant states.