Covering dimension of Cuntz semigroups

Abstract

We introduce a notion of covering dimension for Cuntz semigroups of C * -algebras. This dimension is always bounded by the nuclear dimension of the C * -algebra, and for subhomogeneous C * -algebras both dimensions agree.Cuntz semigroups of Z-stable C * -algebras have dimension at most one. Further, the Cuntz semigroup of a simple, Z-stable C * -algebra is zero-dimensional if and only if the C * -algebra has real rank zero or is stably projectionless.

Covering dimension of Cuntz semigroups II

“…In this section, we show that the dimension of a Cu-semigroup is determined by its countably based sub-Cu-semigroups; see Theorem 5.7. We then generalize some results from [TV21] by dropping the countably based assumption; see Propositions 5.8 and 5.9.…”

“…As an application of Theorem D, we generalize some results from [TV21] by removing the assumption of countable basedness; see Propositions 5.8 and 5.9.…”

“…The permanence properties from [TV21] together with Theorems A and B show that associating to each C * -algebra the dimension of its Cuntz semigroup is a wellbehaved invariant that satisfies all but one property of a noncommutative dimension theory; see Paragraph 6.4. The failing property is the compatibility with minimal unitizations; see Example 6.5.…”

“…In [TV21], we introduced a notion of covering dimension (see Definition 2.8) for Cuntz semigroups and their abstract counterparts, the Cu-semigroups as introduced in [CEI08] and extensively studied in [APT18,APT20]. Among other results, we proved the expected permanence properties (recalled in Proposition 2.9), studied the relation between the dimension of Cu(A) and the nuclear dimension of A, and computed the dimension of Cuntz semigroups of simple, Z-stable C * -algebras.…”

“…The goal of this paper is to further develop the results from [TV21] and provide additional tools to compute the dimension of Cuntz semigroups and Cu-semigroups.…”