$C^*$-irreducibility of commensurated subgroups

Abstract

Given a commensurated subgroup $Λ$ of a group $Γ$, we completely characterize when the inclusion $Λ\leq Γ$ is $C^*$-irreducible and provide new examples of such inclusions. In particular, we obtain that $\rm{PSL}(n,\mathbb{Z})\leq\rm{PGL}(n,\mathbb{Q})$ is $C^*$-irreducible for any $n\in \mathbb{N}$, and that the inclusion of a $C^*$-simple group into its abstract commensurator is $C^*$-irreducible. The main ingredient that we use is the fact that the action of a commensurated subgroup $Λ\leqΓ$ on its Furstenberg boundary $\partial_FΛ$ can be extended in a unique way to an action of $Γ$ on $\partial_FΛ$. Finally, we also investigate the counterpart of this extension result for the universal minimal proximal space of a group.