Approximate uniqueness for maps from C(X) into simple real rank 0 C∗-algebras

Abstract

Let $X$ be a finite CW-complex, and let $\mathcal{A}$ be a unital separable simple finite $\mathcal{Z}$-stable C∗-algebra with real rank $0$. We prove an approximate uniqueness theorem for almost multiplicative contractive completely positive linear maps from $C\left(X\right)$ into $\mathcal{A}$. We also give conditions for when such a map can, within a certain “error,” be approximated by a finite-dimensional ∗-homomorphism.