Covering dimension of Cuntz semigroups II

TL;DR

This paper establishes that the dimension of the Cuntz semigroup of a C*-algebra can be determined by its separable subalgebras, removing previous separability constraints through a new approximation approach.

Contribution

Introduces a novel approximation framework for Cuntz semigroups that preserves properties and removes separability assumptions in dimension analysis.

Findings

Dimension of Cuntz semigroup is determined by separable subalgebras.

New approximation notion for Cuntz semigroups compatible with algebra approximations.

Many properties of Cuntz semigroups are preserved under approximation.

Abstract

We show that the dimension of the Cuntz semigroup of a C*-algebra is determined by the dimensions of the Cuntz semigroups of its separable sub-C*-algebras. This allows us to remove separability assumptions from previous results on the dimension of Cuntz semigroups. To obtain these results, we introduce a notion of approximation for abstract Cuntz semigroups that is compatible with the approximation of a C*-algebra by sub-C*-algebras. We show that many properties for Cuntz semigroups are preserved by approximation and satisfy a L\"owenheim-Skolem condition.