Classification of homomorphisms from $C(\Omega)$ 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 ${\alpha :\mathrm { Cu}(C(\Omega ))\to \mathrm {Cu}(A)}$ with , there exists a homomorphism $\phi : C(\Omega )\to A$ such that $\mathrm {Cu}(\phi )=\alpha $ . Moreover, if $K_1(A)$ is trivial, then $\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.