Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras

Let A and C be two unital simple C*-algebras with tracial rank zero. Suppose that C is amenable and satisfies the Universal Coefficient Theorem. Denote by KKe(C,A) ++ the set of those κ for which κ(K0(C)+ \ {0}) ⊂ K0(A)+ \ {0} and κ([1C ]) = [1A]. Suppose that κ ∈ KKe(C,A) . We show that there is a unital monomorphism φ : C → A such that [φ] = κ. Suppose that C is a unital AH-algebra and λ : T(A) → Tf(C) is a continuous affine map for which τ (κ([p])) = λ(τ )(p) for all projections p in all matrix algebras of C and any τ ∈ T(A), where T(A) is the simplex of tracial states of A and Tf(C) is the convex set of faithful tracial states of C. We prove that there is a unital monomorphism φ : C → A such that φ induces both κ and λ. Suppose that h : C → A is a unital monomorphism and γ ∈ Hom(K1(C),Aff(A)). We show that there exists a unital monomorphism φ : C → A such that [φ] = [h] in KK(C,A), τ ◦ φ = τ ◦ h for all tracial states τ and the associated rotation map can be given by γ. Denote by KKT (C,A) the set of compatible pairs (κ, λ), where κ ∈ KLe(C,A) ++ and λ is a continuous affine map from T(A) to Tf(C). Together with a result of asymptotic unitary equivalence in [14], this provides a bijection from the asymptotic unitary equivalence classes of unital monomorphisms from C to A to (KKT (C,A),Hom(K1(C),Aff(T(A)))/ < R0 >), where < R0 > is a subgroup related to vanishing rotation maps. As an application, combining with a result of W. Winter ([23]), we show that two unital amenable simple Z-stable C*-algebras are isomorphic if they have the same Elliott invariant and the tensor products of these C*-algebras with any UHF-algebras have tracial rank zero. In particular, if A and B are two unital separable simple Z-stable C*-algebras which are inductive limits of C*-algebras of type I with unique tracial states, then they are isomorphic if and only if they have isomorphic Elliott invariant.