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

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.

publication date

  • January 1, 2007