$C^*$-irreducibility of commensurated subgroups

Given a commensurated subgroup $\Lambda$ of a group $\Gamma$, we completely characterize when the inclusion $\Lambda\leq \Gamma$ 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 $\Lambda\leq\Gamma$ on its Furstenberg boundary $\partial_F\Lambda$ can be extended in a unique way to an action of $\Gamma$ on $\partial_F\Lambda$. Finally, we also investigate the counterpart of this extension result for the universal minimal proximal space of a group.