A general family of selfless inclusions is established for reduced amalgamated free products of C*-algebras, with applications to new HNN extensions and selflessness for graph products over suitable graphs.
Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions
3 Pith papers cite this work. Polarity classification is still indexing.
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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 $n\geq2$.
verdicts
UNVERDICTED 3representative citing papers
Commensurator groups of torsion-free hyperbolic groups are C*-selfless.
Reduced twisted group C*-algebras of selfless groups with rapid decay are selfless, implying that those of acylindrically hyperbolic groups with rapid decay are pure and have strict comparison.
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Selfless reduced amalgamated free products and HNN extensions
A general family of selfless inclusions is established for reduced amalgamated free products of C*-algebras, with applications to new HNN extensions and selflessness for graph products over suitable graphs.
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Selfless inclusions arising from commensurator groups of hyperbolic groups
Commensurator groups of torsion-free hyperbolic groups are C*-selfless.
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Strict comparison for twisted group C*-algebras
Reduced twisted group C*-algebras of selfless groups with rapid decay are selfless, implying that those of acylindrically hyperbolic groups with rapid decay are pure and have strict comparison.