Covering dimension of Cuntz semigroups II

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 Löwenheim-Skolem condition.

Nowhere scattered C*-algebras

“…If A is a separable C * -algebra of stable rank one, then Cu(A) is inf-semilattice ordered by [APRT18, Theorem 3.8], and thus satisfies the interval axiom. With the techniques of [TV21b], we can show that this also holds in the nonseparable case.…”

“…For properties of Cu-semigroups, the Löwenheim-Skolem condition was considered in [TV21b], where it was also shown that properties like (O5), (O6) and weak cancellation each satisfy it; see Sections 7 and 9 for definitions. Proposition 4.11.…”

“…In analogy to its definition for C * -algebras (as defined in Section 4), we say that a property P for Cu-semigroups satisfies the Löwenheim-Skolem condition if for every Cu-semigroup with property P there exists a family S of countably based sub-Cu-semigroups of S each having property P, and such that S is σ-complete and cofinal; see [TV21b,Paragraph 5.2]. The next result can be proved with the methods that are used to to prove [TV21b, Proposition 5.3].…”