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

Abstract

The purpose of this note is to prove two results. First, we observe that discrete groups with property $\mathrm{P}_{\mathrm{PHP}}$ in the sense of Ozawa give rise to completely selfless reduced twisted group $\mathrm{C}^\ast$-algebras, thereby extending a theorem of Ozawa from the untwisted to the twisted case. We also observe that an adaptation of property $\mathrm{P}_{\mathrm{PHP}}$ for an inclusion of groups implies that the associated inclusion of reduced twisted group $\mathrm{C}^\ast$-algebras is selfless in the sense of Hayes-Kunnawalkam Elayavalli-Patchell-Robert. Second, we show that reduced (twisted) $\mathrm{C}^\ast$-algebras of some group extensions of the form finite-by-$G$, with $G$ having the property $\mathrm{P}_{\mathrm{PHP}}$, have stable rank one and are pure, which implies strict comparison. Our results do not assume rapid decay, and extend a theorem of Raum-Thiel-Vilalta. Examples covered by our results include reduced twisted group $\mathrm{C}^\ast$-algebras of all acylindrically hyperbolic groups and all lattices in ${\rm SL}(n,\mathbb R)$ for $n\geq2$.