The range of approximate unitary equivalence classes of homomorphisms from AH-algebras

Abstract

Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms $${\phi, \psi: C\to A}$$ are approximately unitarily equivalent if and only if $$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$$ where T(A) is the tracial state space of A. In this paper we prove the following: Given $${\kappa\in KL(C,A)}$$ with $${\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}$$ and with κ([1 C ]) = [1 A ] and a continuous affine map $${\lambda: T(A)\to T_{\mathfrak f}(C)}$$ which is compatible with κ, where $${T_{\mathfrak f}(C)}$$ is the convex set of all faithful tracial states, there exists a unital monomorphism $${\phi: C\to A}$$ such that $$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$$ for all $${c\in C_{s.a.}}$$ and $${\tau\in T(A).}$$ Denote by $${{\rm Mon}_{au}^e(C,A)}$$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$$ where KLT(C, A)++ is the set of compatible pairs of elements in KL(C, A)++ and continuous affine maps from T(A) to $${T_{\mathfrak f}(C).}$$ Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and $${\kappa\in KL(C(X), A)}$$ with $${\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}$$ for which there is no homomorphism h: C(X) → A so that [h] = κ.