Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra

Abstract

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