Simplicity and pure infiniteness for ${\text{C}}^{\ast }$-algebras associated with stabilizers of boundary actions

Abstract

Given a discrete group $\Gamma$ acting on a compact space $X$, and a stabilizer subgroup $\Lambda \le \Gamma$ of $X$, we use the Rieffel induction of covariant $(\Lambda ,X)$-representations to study classes of representations induced by certain characters on $\Lambda$ and the ${\text{C}}^{\ast }$-algebras they generate. When $X$ is a $\Gamma$-boundary, we obtain new classes of simple, traceless group ${\text{C}}^{\ast }$-algebras. When the $\Gamma$-boundary $X$ is an extreme boundary, we show that the associated group ${\text{C}}^{\ast }$-algebras are also purely infinite, answering part of a question posed by Kalantar and Scarparo on ${\text{C}}^{\ast }$-algebras associated with Thompson's groups. The latter ${\text{C}}^{\ast }$-algebras are shown to be selfless in the sense of Robert.

14 pages. Comments welcome!