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.
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