Continuous C-algebras over topological spaces

We define continuous C*-algebras over a topological space X and establish some basic results. If X is a locally compact Hausdorff space, continuous C*-algebras over X are equivalent to ordinary continuous C_0(X)-algebras. The main purpose of our study is to prove that every continuous, full, separable, nuclear C*-algebra over X is KK(X)-equivalent to a stable Kirchberg algebra over X. (Here a Kirchberg algebra over X is a separable, nuclear, and strongly purely infinite C*-algebra over X with primitive ideal space homeomorphic to X.) In the case that X is a one-point space, this result is known as that every separable nuclear C*-algebra is KK-equivalent to a stable Kirchberg algebra. Moreover, as an intermediate result, we obtain the X-equivariant exact embedding result for continuous C*-algebras over X.