$C^{\ast }$-irreducibility of commensurated subgroups

Given a commensurated subgroup $\mathrm{\Lambda }$ of a group $\mathrm{\Gamma }$, we completely characterize when the inclusion $\mathrm{\Lambda }\le \mathrm{\Gamma }$ is $C^{\ast }$-irreducible and provide new examples of such inclusions. In particular, we obtain that $\mathrm{P}\mathrm{S}\mathrm{L}(n,\mathbb{Z})\le \mathrm{P}\mathrm{G}\mathrm{L}(n,\mathbb{Q})$ is $C^{\ast }$-irreducible for any $n\in \mathbb{N}$, and that the inclusion of a $C^{\ast }$-simple group into its abstract commensurator is $C^{\ast }$-irreducible. The main ingredient that we use is the fact that the action of a commensurated subgroup $\mathrm{\Lambda }\le \mathrm{\Gamma }$ on its Furstenberg boundary ${\mathrm{\partial }}_F\mathrm{\Lambda }$ can be extended in a unique way to an action of $\mathrm{\Gamma }$ on ${\mathrm{\partial }}_F\mathrm{\Lambda }$. Finally, we also investigate the counterpart of this extension result for the universal minimal proximal space of a group.