Covering dimension of Cuntz semigroups II

Abstract

We show that the dimension of the Cuntz semigroup of a [Formula: see text]-algebra is determined by the dimensions of the Cuntz semigroups of its separable sub-[Formula: see text]-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 [Formula: see text]-algebra by sub-[Formula: see text]-algebras. We show that many properties for Cuntz semigroups are preserved by approximation and satisfy a Löwenheim–Skolem condition.

Topics & Concepts