Covering Dimension for Nuclear C*-algebras

TL;DR

This paper introduces the completely positive rank as a new notion of covering dimension for nuclear C*-algebras, analyzing its properties and relationships with existing concepts.

Contribution

It defines the completely positive rank, explores its properties, and compares it with other noncommutative dimension theories, providing new insights into the structure of nuclear C*-algebras.

Findings

Completely positive rank aligns with classical covering dimension for abelian C*-algebras.

Zero-dimensional C*-algebras are exactly the AF algebras.

The new rank behaves well under algebraic operations and relates to existing dimension concepts.

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 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.