Covering dimension of Cuntz semigroups

TL;DR

This paper introduces a new notion of covering dimension for Cuntz semigroups, relates it to nuclear dimension, and characterizes it for Z-stable and simple C*-algebras, providing insights into their structure.

Contribution

It defines the covering dimension for Cuntz semigroups and establishes its bounds and equivalences with other dimensions for specific classes of C*-algebras.

Findings

Covering dimension is bounded by the nuclear dimension.

Cuntz semigroups of Z-stable C*-algebras have dimension at most one.

Zero-dimensional Cuntz semigroups correspond to C*-algebras with real rank zero or stably projectionless.

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.