The modern theory of Cuntz semigroups of C *-algebras

Journal article, 2026

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, R0rdam's proof that Z-stability implies strict comparison, and Toms' example of a non 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 Cu.