C₀(X)-algebras, stability and strongly self-absorbing C*-algebras

We study permanence properties of the classes of stable and so-called 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 struck sign tensorially). Furthermore, we prove that if D is a K₁-injective strongly self-absorbing C* -algebra, then A absorbs C* 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 struck sign-absorbing C* -algebras.