The Cuntz semigroup and domain theory

Abstract

Domain theory has its origins in Mathematics and Theoretical Computer Science. Mathematically it combines order and topology. Its central concepts have their origin in the idea of approximating ideal objects by their relatively finite or, more generally, relatively compact parts. The development of domain theory in recent years was mainly motivated by questions in denotational semantics and the theory of computation. But since 2008, domain theoretical notions and methods are used in the theory of $$\hbox {C}^*$$Cź-algebras in connection with the Cuntz semigroup. This paper is largely expository. It presents those notions of domain theory that seem to be relevant for the theory of Cuntz semigroups and have sometimes been developed independently in both communities. It also contains a new aspect in presenting results of Elliott, Ivanescu and Santiago on the cone of traces of a $$\hbox {C}^*$$Cź-algebra as a particular case of the dual of a Cuntz semigroup.