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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva, 84105, Israel

    Ilan Hirshberg

  2. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Mikael Rørdam

  3. Mathematisches Institut der Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    Wilhelm Winter

Authors
  1. Ilan Hirshberg
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  2. Mikael Rørdam
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  3. Wilhelm Winter
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Correspondence to Wilhelm Winter.

Additional information

Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University

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Hirshberg, I., Rørdam, M. & Winter, W. \({\mathcal{C}}_{0}\) (X)-algebras, stability and strongly self-absorbing \({\mathcal{C}}^{*}\) -algebras. Math. Ann. 339, 695–732 (2007). https://doi.org/10.1007/s00208-007-0129-8

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  • DOI: https://doi.org/10.1007/s00208-007-0129-8

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