Strict comparison in reduced group C*-algebras

Journal article

Abstract:: We prove that for every n ∈ N such that n ≥ 2, the reduced group C*-algebras of the countable free groups C ∗ r (Fn) have strict comparison. Our method works in a general setting: for every finitely generated acylindrically hyperbolic group G with trivial finite radical and the rapid decay property, we have C ∗ r (G) have strict comparison. This work also has several applications in the theory of C ∗ -algebras including: resolving Leonel Robert’s selflessness problem for C ∗ r (G); uniqueness of embeddings of the Jiang-Su algebra Z up to approximate unitary equivalence into C ∗ r (G); full computations of the Cuntz semigroup of C ∗ r (G) and future directions in the C ∗ -classification program.

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Copyright holder:: Amrutam et al
Rights statement:: © The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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