Uniformly recurrent subgroups and simple $C^*$-algebras

Abstract

We study uniformly recurrent subgroups (URS) introduced by Glasner and Weiss \cite{GW}. Answering their query we show that any URS $Z$ of a finitely generated group is the stability system of a minimal $Z$-proper action. We also show that for any sofic $URS$ $Z$ there is a $Z$-proper action admitting an invariant measure. We prove that for a $URS$ $Z$ all $Z$-proper actions admits an invariant measure if and only if $Z$ is coamenable. In the second part of the paper we study the separable $\C^*$-algebras associated to URS's. We prove that if an URS is generic then its $\C^*$-algebra is simple. We give various examples of generic URS's with exact and nuclear $\C^*$-algebras and an example of a URS $Z$ for which the associated simple $\C^*$-algebra is not exact and not even locally reflexive, in particular, it admits both a uniformly amenable trace and a nonuniformly amenable trace.