C∗-irreducibility of commensurated subgroups

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