Abstract :
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 Blackadarʹs 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.
