Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras

Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that $phi_1, phi_2: Cto A$ are two unital monomorphisms. We show that there is a continuous path of unitaries ${u_t: tin [0, infty)}$ of $A$ such that $$ lim_{ttoinfty}u_t^*phi_1(c)u_t=phi_2(c)tforal cin C $$ if and only if $[phi_1]=[phi_2]$ in $KK(C,A),$ $phi_1^{ddag}=phi_2^{ddag},$ $(phi_1)_T=(phi_2)_T$ and a rotation related map $bar{R}_{phi_1,phi_2}$ associated with $phi_1$ and $phi_2$ is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${cal A}$ of unital separable simple amenable CA s which is strictly larger than the class of separable CA s whose tracial rank are zero or one. The class contains all unital simple ASH-algebras whose state spaces of $K_0$ are the same as the tracial state spaces as well as the simple inductive limits of dimension drop circle algebras. Moreover it contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. One consequence of the main result is that all unital simple AH-algebras which are ${cal Z}$-stable are isomorphic to ones with no dimension growth.