A characterization of $C^*$-simplicity of countable groups via Poisson boundaries

Abstract

We characterize $C^*$-simplicity for countable groups by means of the following dichotomy. If a group is $C^*$-simple, then the action on the Poisson boundary is essentially free for a generic measure on the group. If a group is not $C^*$-simple, then the action on the Poisson boundary is not essentially free for a generic measure on the group.