Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions
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2026
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| author | Flores, Felipe Klisse, Mario Cobhthaigh, Mícheál Ó Pagliero, Matteo |
| author_facet | Flores, Felipe Klisse, Mario Cobhthaigh, Mícheál Ó Pagliero, Matteo |
| contents | 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 $n\geq2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_19758 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions Flores, Felipe Klisse, Mario Cobhthaigh, Mícheál Ó Pagliero, Matteo Operator Algebras Functional Analysis Group Theory 22D25, 46L05, 19K14, 46L35 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 $n\geq2$. |
| title | Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions |
| topic | Operator Algebras Functional Analysis Group Theory 22D25, 46L05, 19K14, 46L35 |
| url | https://arxiv.org/abs/2601.19758 |