Covering dimension for nuclear C∗-algebras

By W. Winter. 2001

98 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66ded4a88d96ddb055a49707e8f40a48360572db

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In Algebra and Number Theory

1. Stuart White
2. W. Winter
3. Andrew S. Toms
4. Erik Christensen
5. A. Sinclair
6. Roger R. Smith
7. G'abor Szab'o
8. Z. Niu

In Applied Mathematics

1. W. Winter
2. Stuart White
3. Andrew S. Toms
4. A. Sinclair
5. Roger R. Smith
6. Erik Christensen
7. G'abor Szab'o
8. M. Rørdam

In Mathematical Physics

1. Stuart White
2. W. Winter
3. Andrew S. Toms
4. G'abor Szab'o
5. Erik Christensen
6. A. Sinclair
7. Roger R. Smith
8. David Kerr

Citations in the corpus

• The Elliott conjecture for Villadsen algebras of the first type (2006, background): “We shall see later that this failure is dramatic enough to ensure that the algebra also has infinite decomposition rank.”
• Perturbations of nuclear C*-algebras (2009, methodology): “In particular, we will use the following two facts from [53] and [54] (see also [55], where these assertions are established when F is not finite dimensional).”