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

TL;DR

This paper investigates the structure of the Cuntz semigroup for certain $C(X)$-algebras, providing explicit descriptions and computations, especially for low-dimensional spaces with specific fiber conditions.

Contribution

It introduces techniques to compute the Cuntz semigroup of pullback $C^*$-algebras and offers a complete description for $C(X,A)$ in terms of lower semicontinuous functions.

Findings

Computed Cuntz semigroup for specific pullback $C^*$-algebras.

Provided a semigroup-valued lower semicontinuous function description for $C(X,A)$.

Applied results to various examples.

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$^*$-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$^*$-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.