Classification of homomorphisms from $C(\Omega )$ to a $C^{\ast }$-algebra

Abstract

Let $\Omega$ be a compact subset of $\mathbb{C}$ and let $A$ be a unital simple, separable $C^{\ast }$-algebra with stable rank one, real rank zero and strict comparison. We show that, given a Cu-morphism $\alpha :\mathrm{Cu}(C(\Omega ))\to \mathrm{Cu}(A)$ with $α(\langle \mathds{1}_Ω\rangle)\leq \langle 1_A\rangle$, there exists a homomorphism $\varphi :C(\Omega )\to A$ such that $\mathrm{Cu}(\varphi )=\alpha$ and $\varphi$ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^{\ast }$-algebras to $A$ in terms of the Cuntz semigroup.