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

Abstract

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 τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φ and ψ induce the same element in KL(C(X), A) and the two lineal functionals τ∘φ and τ∘ψ 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.