C_0(X)-algebras, stability and strongly self-absorbing C*-algebras

By Ilan Hirshberg, M. Rørdam, W. Winter. 2006

62 citations recorded by the source corpus.

We study permanence properties of the classes of stable and so-called D-stable C*-algebras, respectively. More precisely, we show that a C_0(X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for C*-algebras absorbing the Jiang--Su algebra Z tensorially). Furthermore, we prove that if D is a K_1-injective strongly self-absorbing C*-algebra, then A absorbs D tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a C*-algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of Z-absorbing C*-algebras.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/182985d0c73559429016ee04ef534cfd6358679e

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In Algebra and Number Theory

1. F. Perera
2. M. Rørdam
3. Marius Dadarlat
4. E. Ortega
5. U. Pennig
6. W. Winter
7. Leonel Robert
8. Andrew S. Toms

In Mathematical Physics

1. F. Perera
2. M. Rørdam
3. Marius Dadarlat
4. E. Ortega
5. W. Winter
6. U. Pennig
7. Leonel Robert
8. Andrew S. Toms

Citations in the corpus

• A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group (2014, background): “In order to justify (a) we observe that if D is quasidiagonal, then every projection p ∈ C(X,MN (D)) has a multiple p(m) := p⊗1Mm(C) such that p(m) is Murray-Von Neumann equivalent to a projection in C(X,MNm(C))⊗ 1D ⊂ C(X,MNm(C))⊗D and that Ai⊗D ∼= Ai by [9].”
• The Corona Factorization property, Stability, and the Cuntz semigroup of a C*-algebra (2009, background): “We then show that property (S) is equivalent to stability provided that the Cuntz semigroup of the C∗-algebra satisfies another (also very weak) comparability property, that we call the ω-comparison property.”