Pullbacks, $C(X)$-algebras, and their Cuntz semigroup

Abstract

In this paper we analyse the structure of the Cuntz semigroup of certain $C(X)$-algebras, for compact spaces of low dimension, that have no ${\mathrm{K}}_1$-obstruction in their fibres in a strong sense. The techniques developed yield computations of the Cuntz semigroup of some surjective pullbacks of C${}^{\ast }$-algebras. As a consequence, this allows us to give a complete description, in terms of semigroup valued lower semicontinuous functions, of the Cuntz semigroup of $C(X,A)$, where $A$ is a not necessarily simple C${}^{\ast }$-algebra of stable rank one and vanishing ${\mathrm{K}}_1$ for each closed, two sided ideal. We apply our results to study a variety of examples.