The modern theory of Cuntz semigroups of C*-algebras

TL;DR

This paper provides a comprehensive overview of the modern theory of Cuntz semigroups in C*-algebras, covering foundational concepts, key historical results, and recent advances including classification and structural properties.

Contribution

It offers an extensive introduction and survey of the current state of Cuntz semigroup theory, highlighting recent developments and the abstract categorical approach.

Findings

Cuntz's theorem on quasitraces

R{ }ordam's proof linking $ ext{Z}$-stability and strict comparison

Toms' example of a non $ ext{Z}$-stable C*-algebra

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 $\mathbf{Cu}$.