The modern theory of Cuntz semigroups of $C^{\ast }$-algebras

Abstract

We give a detailed introduction to the theory of Cuntz semigroups for $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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 Cu.