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

TL;DR

This paper proves that certain reduced twisted group C*-algebras of specific group extensions are pure, have stable rank one, and satisfy strict comparison, extending previous results without assuming rapid decay.

Contribution

It extends known theorems to twisted cases and group extensions, showing pure and stable rank one properties for a broader class of reduced twisted group C*-algebras.

Findings

Reduced twisted group C*-algebras of some group extensions have stable rank one.

These algebras are pure and satisfy strict comparison.

Results apply to acylindrically hyperbolic groups and lattices in SL(n,R).

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 $n\geq2$.