Classification of homomorphisms from \(C(X)\) to simple \(C^*\)-algebras of real rank zero (Q1578858)

Let \(A\) be a unital simple \(C^*\)-algebra of real rank zero, stable rank one, with weakly unperforated \(K_0(A)\) and unique normalized quasi-trace \(\tau\), and let \(X\) be a compact metric space. In this paper the authors show that two monomorphisms \(\varphi, \psi:C(X)\to A\) are approximately unitarily equivalent if and only if \(\varphi\) and \(\psi\) induce the same element in \(KL(C(X), A)\) and the two linear functionals \(\tau\circ\varphi\) and \(\tau\circ\psi\) are equal. They show that, with an injectivity condition, an almost multiplicative morphism from \(C(X)\) into \(A\) with vanishing \(KK\)-obstacle is close to a homomorphism.