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Selfless reduced free product $C^*$-algebras

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arxiv 2505.13265 v2 pith:7JXO6AMJ submitted 2025-05-19 math.OA

Ben Hayes , Srivatsav Kunnawalkam Elayavalli , Leonel Robert This is my paper · ORCID
classification math.OA
keywords freealgebrasreducedselflessnessexamplesgeneralproductproducts
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abstract

We study selflessness in the general setting of reduced free products of $C^*$-algebras. Towards this end, we develop a suitable theory of rapid decay for filtrations in arbitrary $C^*$-probability spaces. We provide several natural examples and permanence properties of this phenomenon. By using this framework in combination with von Neumann algebraic techniques involving approximate forms of orthogonality, we are able to prove selflessness for general families of reduced free product $C^*$-algebras. As an instance of our results, we prove selflessness and thus strict comparison for the canonical $C^*$-algebras generated by Voiculescu's free semicircular systems. Our results also provide new examples of purely infinite reduced free products.

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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. Selfless C*-correspondences, operator valued C*-probability spaces and completely positive maps

    math.OA 2026-07 conditional novelty 8.0 of 10

    A unified theory of selfless C*-correspondences is developed and applied to completely positive maps and conditional expectations, yielding new permanence, regularity, and absorption results.

  2. An isomorphism theorem for infinite reduced free products

    math.OA 2026-02 conditional novelty 8.0 of 10

    The infinite reduced free product C^{*r∞} absorbs any 1-NCCW direct limit A with trivial K-theory, so A *r C^{*r∞} ≅ C^{*r∞}; in particular C([0,1])^{*r∞} ≅ Z^{*r∞}.

  3. 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.

  4. Selfless reduced $C^{*}$-algebras of linear groups

    math.OA 2026-02 reject novelty 7.0 of 10

    For nontrivial linear groups, the reduced C*-algebra is selfless exactly when it is simple, i.e., when the group has trivial amenable radical.

  5. Strongly converging unitary representations for extensions by exact groups

    math.OA 2026-07 conditional novelty 6.0 of 10

    Extensions by exact groups—semidirect products, abelian-base wreath products, graph wreath products, and certain free-by-cyclic groups—are shown to admit strongly converging finite-dimensional unitary representations.

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