The Cuntz semigroup and domain theory

TL;DR

This paper explores the connections between domain theory and the Cuntz semigroup in C*-algebra theory, highlighting how domain-theoretic concepts are applied and presenting new insights into the cone of traces as a dual of the Cuntz semigroup.

Contribution

It provides an expository overview of domain theory relevant to Cuntz semigroups and introduces new results relating the cone of traces to the dual of the Cuntz semigroup.

Findings

Domain theory concepts are relevant for Cuntz semigroup analysis.

The cone of traces can be viewed as a dual of the Cuntz semigroup.

New connections between traces and domain-theoretic structures are established.

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 question in denotational semantics and the theory of computation. But since 2008, domain theoretical notions and methods are used in the theory of C*-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 C*-algebra as a particular case of the dual of a Cuntz semigroup.