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 $\mathrm{PSL}(n\ge 2,\mathbb{R})$, 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 $\mathrm{SL}(n,\mathbb{R})$ for $n\ge 2$.

v3: Major changes. Most of the article was rewritten. New results include sufficient conditions to obtain selfless inclusions from crossed products