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.

32 pages; extended introduction, added details on inductive limits and elements with thin boundary, rearranged order of results in Section 4; to appear in Adv. Math