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Mathematische Annalen

, Volume 339, Issue 3, pp 695–732 | Cite as

\({\mathcal{C}}_{0}\) (X)-algebras, stability and strongly self-absorbing \({\mathcal{C}}^{*}\) -algebras

  • Ilan Hirshberg
  • Mikael Rørdam
  • Wilhelm Winter
Article

Abstract

We study permanence properties of the classes of stable and so-called \({\mathcal{D}}\)-stable \({\mathcal{C}}^{*}\)-algebras, respectively. More precisely, we show that a \({\mathcal{C}}_{0}\) (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 \({\mathcal{C}}^{*}\)-algebras absorbing the Jiang–Su algebra \({\mathcal{Z}}\) tensorially). Furthermore, we prove that if \({\mathcal{D}}\) is a K 1-injective strongly self-absorbing \({\mathcal{C}}^{*}\)-algebra, then A absorbs \({\mathcal{D}}\) 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 \({\mathcal{C}}^{*}\)-algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of \({\mathcal{Z}}\)-absorbing \({\mathcal{C}}^{*}\) -algebras.

Mathematics Subject Classification (2000)

46L05 47L40 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Ilan Hirshberg
    • 1
  • Mikael Rørdam
    • 2
  • Wilhelm Winter
    • 3
  1. 1.Department of MathematicsBen Gurion University of the NegevBe’er ShevaIsrael
  2. 2.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark
  3. 3.Mathematisches Institut der Universität MünsterMünsterGermany

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