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The modern theory of Cuntz semigroups of C*-algebras

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arxiv 2212.02290 v2 pith:27QVOATV submitted 2022-12-05 math.OA math.RA

Eusebio Gardella , Francesc Perera This is my paper · ORCID
classification math.OAmath.RA
keywords cuntzalgebrastheoryexamplegivemathcalmodernresults
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abstract

We give a detailed introduction to the theory of Cuntz semigroups for C*-algebras. Beginning with the most basic definitions and technical lemmas, we present several results of historical importance, such as Cuntz's theorem on the existence of quasitraces, R{\o}rdam's proof that $\mathcal{Z}$-stability implies strict comparison, and Toms' example of a non $\mathcal{Z}$-stable simple, nuclear C*-algebra. We also give the reader an extensive overview of the state of the art and the modern approach to the theory, including the recent results for C*-algebras of stable rank one (for example, the Blackadar-Handelman conjecture and the realization of ranks), as well as the abstract study of the Cuntz category $\mathbf{Cu}$.

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Cited by 3 Pith papers

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

  1. Strict comparison in reduced group $C^*$-algebras

    math.OA 2024-12 conditional novelty 8.0 of 10

    The reduced C*-algebras of free groups F_n for n ≥ 2, and of finitely generated acylindrically hyperbolic groups with trivial finite radical and rapid decay, have strict comparison and are selfless.

  2. On the Nielsen-Thomsen sequence

    math.OA 2024-12 unverdicted novelty 6.0 of 10

    New bases, rotation maps, and a metric on the Hausdorffized algebraic K1-group distinguish C*-algebras and *-homomorphisms that agree on the Elliott invariant, the Cuntz semigroup, and its unitary version.

  3. Uniform property $\Gamma$ for Crossed products by group actions with the Rokhlin-type properties

    math.OA 2024-12 conditional novelty 5.0 of 10

    Finite group actions with the weak tracial Rokhlin property and compact group actions with the tracial Rokhlin property with comparison preserve uniform property Gamma in crossed products and fixed-point algebras.

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