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 φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(c)ut = φ2(c) for all c ∈ C if and only if [φ1] = [φ2] in KK(C,A), φ ‡ 1 = φ 2 , (φ1)T = (φ2)T and a rotation related map Rφ1,φ2 associated with φ1 and φ2 is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class A of unital separable simple amenable C-algebras which is strictly larger than the class of separable C-algebras whose tracial rank are zero or one. Tensor products of two C-algebras in A are again in A. Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras whose state spaces of K0 is the same as the tracial state spaces as well as some unital simple ASH-algebras whose K0-group is Z and tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are Z-stable are isomorphic to ones with no dimension growth.