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

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 .