Classification of homomorphisms from C(X) to simple C*-algebras of real rank zero

Let A be a unital simple C*-algebra of real rank zero, stable rank one, with weakly unperforated K-o(A) and unique normalized quasi-trace tau, and let X be a compact metric space. We show that two monomorphisms phi, psi : C(X) --> A are approximately unitarily equivalent ii, and only if phi and psi, induce the same element in KL(C(X), A) and the two linear functionals tau o phi and tau o psi, are equal. We also show that, with an injectivity condition, an almost multiplicative morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism.