Classification of Homomorphisms from C (X) to Simple C-Algebras of Real Rank Zero

Let A be a unital simple C*-algebra of real zero, stable rank one, with weakly unperforated K 0( A) and unique normalized quasi-trace ., and let X be a compact metric space. We show that two monomorphisms ., . : C(X).A are approximately unitarily equivalent if and only if . and . induce the same element in KL(C(X), A) and the two lineal functionals ... and ... 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.