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.