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

Abstract

Given a commensurated subgroup $\Lambda$ of a group $\Gamma$, we completely characterize when the inclusion $\Lambda \le \Gamma$ is $C^{\ast }$-irreducible and provide new examples of such inclusions. In particular, we obtain that $\mathrm{PSL}(\mathrm{n},\mathbb{Z})\le \mathrm{PGL}(\mathrm{n},\mathbb{Q})$ 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 $\Lambda \le \Gamma$ 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.

arXiv comment10 pages. Minor changes. Accepted in Pacific Journal of Mathematics

References in corpus (6)

• Boundary representations of operator spaces, and compact rectangular matrix convex sets
• C*-simplicity and the unique trace property for discrete groups
• Bernoulli disjointness
• Rigidity of amalgamated free products in measure equivalence
• Strong amenability and the infinite conjugacy class property
• C*-irreducibility for reduced twisted group C*-algebras