Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions
Felipe Flores, Mario Klisse, Mícheál Ó Cobhthaigh, Matteo Pagliero
TL;DR
This paper addresses the regularity properties of reduced twisted group C*-algebras beyond the untwisted setting and without assuming rapid decay. It shows that groups with property $P_{\mathrm{PHP}}$ yield completely selfless reduced twisted group C*-algebras, which are simple and have a unique trace/quasitrace, strict comparison, and stable rank one. It then extends these results to finite-by-$G$ extensions, proving that the corresponding twisted C*-algebras are pure with stable rank one by decomposing them into sums of algebras of the form $C^*_r(H,\nu)\otimes M_n(\mathbb{C})\otimes M_{|G/H|}(\mathbb{C})$. The corollaries apply to acylindrically hyperbolic groups and lattices in ${\rm PSL}(n,\mathbb{R})$ (including ${\rm SL}(n,\mathbb{Z})$), leveraging topologically free extreme boundary actions and Zariski-density; a key structural lemma and a Cuntz semigroup description support the conclusions. Overall, the work broadens known regularity results to twisted group C*-algebras in new classes of groups without the rapid decay hypothesis, contributing to the classification program through purity and stable rank considerations.
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. 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 $n\geq2$.