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.
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.
We say that a C * -algebra is nowhere scattered if none of its quotients contains a minimal projection. We characterize this property in various ways, by topological properties of the spectrum, by divisibility properties in the Cuntz semigroup, by the existence of Haar unitaries for states, and by the absence of nonzero ideal-quotients that are elementary, scattered or type I.Under the additional assumption of real rank zero or stable rank one, we show that nowhere scatteredness implies even stronger divisibility properties of the Cuntz semigroup.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.