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

Let Ω be a compact subset of 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 α: Cu(C(Ω))→ Cu(A) with α( 1Ω)≤ 1A, there exists a homomorphism φ: C(Ω)→ A such that Cu(φ)=α and φ 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.