Covering Dimension for Nuclear C * -algebras

We introduce the completely positive rank, a notion of covering dimension for nuclear C *-algebras and analyze some of its properties. The completely positive rank behaves nicely with respect to direct sums, quotients, ideals and inductive limits. For abelian C *-algebras it coincides with covering dimension of the spectrum and there are similar results for continuous trace algebras. As it turns out, a C *-algebra is zero-dimensional precisely if it is AF. We consider various examples, particularly of one-dimensional C *-algebras, like the irrational rotation algebras, the Bunce-Deddens algebras or Blackadar's simple unital projectionless C *-algebra. Finally, we compare the completely positive rank to other concepts of noncommutative covering dimension, such as stable or real rank. The theory of C *-algebras is often regarded as noncommutative topology, mainly because any abelian C *-algebra is completely determined by its spectrum, a locally compact space. So C *-algebras might be thought of as noncommutative topological spaces and this point of view has been very fruitful, especially since the introduction of methods from algebraic topology (such as Ext or K-theory) to C *-algebra theory. It is thus natural to look for invariants of topological spaces which can be defined analogously for C *-algebras. A good candidate for such an invariant 1