Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions

Abstract

We present a sufficient condition for a (twisted) group action on a $\mathrm{C}^\ast$-algebra such that the corresponding reduced (twisted) group $\mathrm{C}^\ast$-algebra inclusion into the reduced (twisted) crossed product is selfless. This condition is an action-dependent version of Ozawa's $\mathrm{PHP}$ property for groups. We show that topologically free extreme boundary actions, projective actions of lattices in ${\rm PSL}(n\ge 2,\mathbb R)$, as well as certain strongly proximal actions have this property, and thus we provide examples of selfless inclusions from crossed products. As a byproduct, we also obtain that reduced (twisted) group $\mathrm{C}^\ast$-algebras of some group extensions of the form finite-by-$G$, with $G$ having the property $\mathrm{PHP}$, have stable rank one and are pure, and in particular have strict comparison. Examples include all acylindrically hyperbolic groups and all lattices in ${\rm SL}(n,\mathbb R)$ for $n\geq2$.