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.
Proximality and selflessness for group C*-algebras
10 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
We prove that the reduced group C*-algebras of infinite countable discrete groups having topologically-free extreme boundaries, or more generally groups that satisfy certain combinatorial property including all acylindrically hyperbolic groups with no nontrivial finite normal subgroups and all Zariski-dense subgroups of PSL(n,R), are selfless in the sense of L. Robert. This generalizes the recent results of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell, and of Vigdorovich. We also prove that selflessness is stable under tensor product among exact C*-algebras and that a C*-probability space is selfless provided that it is either simple and purely infinite or simple, exact, Z-stable, and uniquely tracial.
verdicts
UNVERDICTED 10representative citing papers
Establishes equivalence between topological freeness of limit set actions and mixed identity freeness for weakly hyperbolic groups, with 3-transitivity rigidity and application showing non-affine Kac-Moody groups are MIF.
Uniform amenability at infinity holds for free groups and limit groups, implying uniform strong convergence in the operator algebraic sense for convergent sequences of such groups in the marked group space.
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.
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.
Selflessness of separable tracial C*-algebras is equivalent to approximate selflessness via a finitary condition proved by diagonalization in the tracial ultrapower.
A new Toeplitz exactness theorem provides a general machine to upgrade strong convergence in C*-correspondences.
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.
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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Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups
Establishes equivalence between topological freeness of limit set actions and mixed identity freeness for weakly hyperbolic groups, with 3-transitivity rigidity and application showing non-affine Kac-Moody groups are MIF.
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Uniform amenability at infinity
Uniform amenability at infinity holds for free groups and limit groups, implying uniform strong convergence in the operator algebraic sense for convergent sequences of such groups in the marked group space.
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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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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.
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A finitary criterion for selfless tracial C*-algebras
Selflessness of separable tracial C*-algebras is equivalent to approximate selflessness via a finitary condition proved by diagonalization in the tracial ultrapower.
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Toeplitz exactness for strong convergence
A new Toeplitz exactness theorem provides a general machine to upgrade strong convergence in C*-correspondences.
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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.