COVERING DIMENSION FOR CUNTZ SEMIGROUPS

In this talk, I will present a notion of covering dimension for Cuntz semigroups and give an overview of the results found thus far. This dimension is always bounded by the nuclear dimension of the associated C*-algebra and, in the case of subhomogeneous C*-algebras, the two dimensions agree. For separable, simple, Z-stable C*-algebras, the Cuntz semigroup has dimension zero if and only if the algebra has real rank zero or is stably projectionless. I will also introduce a notion of approximation for abstract Cuntz semigroups and show that the property of having the covering dimension at most n is preserved by approximation. In particular, the covering dimension of the Cuntz semigroup of a C*-algebra is determined by the covering dimensions of the Cuntz semigroups of its separable C*-subalgebras. The talk is based on joint work with Hannes Thiel.