The modern theory of Cuntz semigroups of C*-algebras

Eusebio Gardella; Francesc Perera

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arXiv · 2212.02290

The modern theory of Cuntz semigroups of C*-algebras

Eusebio Gardella·Francesc Perera

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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, Rørdam'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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Eusebio Gardella, Francesc Perera. 2022-12-13. The modern theory of Cuntz semigroups of C*-algebras. https://arxiv.org/abs/2212.02290

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