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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6 Pith papers cite this work. Polarity classification is still indexing.
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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.
Certain crossed products of C(X,D) by minimal homeomorphisms and compact group actions with Rokhlin-type properties are pure C*-algebras with stable rank one.
Factorial tracially complete C*-algebras with CPoU have real rank zero and stable rank one, giving a description of the Cuntz semigroup including for Z-stable cases.
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.
A compilation of 99 open problems in the structure and classification of nuclear C*-algebras.
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The real and stable rank of tracially complete C*-algebras
Factorial tracially complete C*-algebras with CPoU have real rank zero and stable rank one, giving a description of the Cuntz semigroup including for Z-stable cases.