Continuous fields of C*-algebras over finite dimensional spaces

Abstract

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 $\mathcal{O}_n$, $n\in\{2,3,...,\infty\}$ are locally trivial. They are trivial if $n=2$ or $n=\infty$. For $n\geq 3$ finite, such a field is trivial if and only if $(n-1)[1_A]=0$ in $K_0(A)$, 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. In a more general context, we show that a separable unital continuous field over $X$ with fibers isomorphic to a $KK$-semiprojective is trivial if and only if it satisfies a K-theoretical Fell type condition.