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

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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.

### publication date

• January 1, 2007