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

Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${{KK}}_e(C,A)^{++}$ the set of those $kappa$ for which $kappa(K_0(C)_+setminus{0})subset K_0(A)_+setminus{0}$ and $kappa([1_C])=[1_A]$. Suppose that $kappain {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $phi: Cto A$ such that $[phi]=kappa.$ Suppose that $C$ is a unital AH-algebra and $lambda: mathrm{T}(A)to mathrm{T}_{mathtt{f}}(C)$ is a continuous affine map for which $tau(kappa([p]))=lambda(tau)(p)$ for all projections $p$ in all matrix algebras of $C$ and any $tauin mathrm{T}(A),$ where $mathrm{T}(A)$ is the simplex of tracial states of $A$ and $mathrm{T}_{mathtt{f}}(C)$ is the convex set of faithful tracial states of $C.$ We prove that there is a unital monomorphism $phi: Cto A$ such that $phi$ induces both $kappa$ and $lambda.$ Suppose that $h: Cto A$ is a unital monomorphism and $gamma in mathrm{Hom}(Kone(C), aff(A)).$ We show that there exists a unital monomorphism $phi: Cto A$ such that $[phi]=[h]$ in ${KK}(C,A),$ $taucirc phi=taucirc h$ for all tracial states $tau$ and the associated rotation map can be given by $gamma.$ Applications to classification of simple C*-algebras are also given.