Covering dimension and quasidiagonality

Abstract

We introduce the decomposition rank, a notion of covering dimension for nuclear C^*-algebras. The decomposition rank generalizes ordinary covering dimension and has nice permanence properties; in particular, it behaves well with respect to direct sums, quotients, inductive limits, unitization and quasidiagonal extensions. Moreover, it passes to hereditary subalgebras and is invariant under stabilization. It turns out that the decomposition rank can be finite only for strongly quasidiagonal C^*-algebras and that it is closely related to the classification program.