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Pure C*-algebras

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arxiv 2406.11052 v3 pith:RARUMSKS submitted 2024-06-16 math.OA

Ramon Antoine , Francesc Perera , Hannes Thiel , Eduard Vilalta This is my paper · ORCID
classification math.OA
keywords algebraspurecomparisonclassnon-elementaryveryweakadequately
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We demonstrate that pure C*-algebras form a robust class by proving that pureness follows from very weak comparison and divisibility properties. Using this, we show that every simple, non-elementary C*-algebra with a unique quasitrace and with very mild comparison is pure, and, as a result, has strict comparison. Furthermore, sufficiently non-commutative C*-algebras of stable rank one and with weak comparison are likewise pure. We also show that adequately non-elementary C*-algebras with finite nuclear dimension are pure, which leads to the verification of the non-simple Toms-Winter conjecture for a large class of C*-algebras.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Extensions of pure C*-algebras

    math.OA 2025-06 conditional novelty 8.0 of 10

    Pureness of C*-algebras is preserved under extensions: an algebra is pure iff every closed ideal and its quotient are pure.

  2. The Selfless Dichotomy

    math.OA 2026-06 unverdicted novelty 7.0 of 10

    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.

  3. $\mathcal{Z}$-stable Graph Algebras

    math.OA 2025-11 conditional novelty 7.0 of 10

    For acyclic graphs and graph algebras with finitely many ideals, Z-stability is equivalent to a new graph condition ('distinct detours'); with Condition (K) this condition also characterizes purity.

  4. The Global Glimm Property for C*-algebras of topological dimension zero

    math.OA 2025-07 accept novelty 7.0 of 10

    For C*-algebras with topological dimension zero, nowhere scatteredness is exactly the Global Glimm Property, solving the Global Glimm Problem in this class.

  5. Nonnuclear Bunce-Deddens algebras

    math.OA 2026-07 accept novelty 6.0 of 10

    Free odometer crossed products by nonamenable residually finite groups yield simple nonnuclear C*-algebras that frequently have real rank zero, stable rank one, unique trace, and selflessness, with explicit K-theory o...

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