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arXiv · math/0610344

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

Ilan Hirshberg·Mikael Rordam·Wilhelm Winter
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Abstract

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.

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Ilan Hirshberg, Mikael Rordam, Wilhelm Winter. 2006-10-10. C_0(X)-algebras, stability and strongly self-absorbing C*-algebras. https://arxiv.org/abs/math/0610344

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