Homomorphisms from AH-algebras

Original · EN

Let C be a general unital AH-algebra and let A be a unital simple C*-algebra with tracial rank at most one. Suppose that ϕ, ψ: C→ A are two unital monomorphisms. We show that ϕ and ψ are approximately unitarily equivalent if and only if [ϕ]&=&[ψ] in KL(C,A), ϕ&=&ψ ϕ†&=&ψ†, where ϕ and ψ are continuous affine maps from tracial state space T(A) of A to faithful tracial state space T f(C) of C induced by ϕ and ψ, respectively, and ϕ‡ and ψ‡ are induced homomorphisms from K₁(C) into (T(A))/ρₐ(K₀(A)), where (T(A)) is the space of all real affine continuous functions on T(A) and ρₐ(K₀(A)) is the closure of the image of K₀(A) in the affine space (T(A)). In particular, the above holds for C=C(X), the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements κ∈ KLₑ(C,A)⁺⁺, an affine map γ: T(C)→ T f(C) and a: K₁(C)→ (T(A))/ρₐ(K₀(A)), there exists a unital monomorphism ϕ: C→ A such that [h]=κ, h=γ and ϕ†=.