Nonfaithful selfless C*-probability spaces are purely infinite and simple, so every selfless C*-algebra is either purely infinite or stably finite and hence pure.
Strict comparison for twisted group C*-algebras
6 Pith papers cite this work. Polarity classification is still indexing.
6
Pith papers citing it
abstract
We prove that any reduced twisted group C*-algebra of a selfless group with the rapid decay property is selfless. As an application, we show that twisted group C*-algebras of acylindrically hyperbolic groups (possibly with nontrivial finite radical) and rapid decay are pure, and hence have strict comparison.
verdicts
UNVERDICTED 6representative citing papers
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.
Reduced twisted group C*-algebras of groups with property P_PHP are completely selfless, and those of finite-by-G extensions have stable rank one and are pure.
Commensurator groups of torsion-free hyperbolic groups are C*-selfless.
Separable type III_1 factors with trivial bicentralizer are selfless W*-probability spaces for every faithful normal state.
Twisted reduced group C*-algebras of amenable groups are selfless precisely when the pair satisfies Kleppner's condition, with the same holding for inclusions of normal subgroups under the relative condition.
citing papers explorer
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The Selfless Dichotomy
Nonfaithful selfless C*-probability spaces are purely infinite and simple, so every selfless C*-algebra is either purely infinite or stably finite and hence pure.
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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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Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions
Reduced twisted group C*-algebras of groups with property P_PHP are completely selfless, and those of finite-by-G extensions have stable rank one and are pure.
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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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Selfless W$^*$-probability spaces and Connes' bicentralizer problem
Separable type III_1 factors with trivial bicentralizer are selfless W*-probability spaces for every faithful normal state.
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Selflessness for twisted group C*-algebras of amenable groups and their inclusions
Twisted reduced group C*-algebras of amenable groups are selfless precisely when the pair satisfies Kleppner's condition, with the same holding for inclusions of normal subgroups under the relative condition.