Continuous fields of C*-algebras over finite dimensional spaces

Let X be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of C(X)-algebras by C(X)subalgebras with controlled complexity. The following applications are given. All unital separable continuous fields of C*-algebras over X with fibers isomorphic to a fixed Cuntz algebra On, n ∈ {2, 3, ...,∞} are locally trivial. They are trivial if n = 2 or n =∞. For n ≥ 3 finite, such a field is trivial if and only if (n− 1)[1A] = 0 in K0(A), where A is the C*-algebra of continuous sections of the field. We give a complete list of the Kirchberg algebras D satisfying the UCT and having finitely generated K-theory groups for which every unital separable continuous field over X with fibers isomorphic to D is automatically locally trivial or trivial. In a more general context, we show that a separable unital continuous field over X with fibers isomorphic to a KK-semiprojective Kirchberg C*-algebra is trivial if and only if it satisfies a K-theoretical Fell