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

Abstract

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