Asymptotic unitary equivalence and classification of simple amenable C ∗-algebras

Abstract

Let C and A be two unital separable amenable simple C ∗-algebras with tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ϕ 1,ϕ 2:C→A are two unital monomorphisms. We show that there is a continuous path of unitaries {u t :t∈[0,∞)} of A such that $$\lim_{t\to\infty}u_t^*\varphi_1(c)u_t=\varphi_2(c)\quad\mbox{for all }c\in C$$ if and only if [ϕ 1]=[ϕ 2] in $KK(C,A),\varphi_{1}^{\ddag}=\varphi_{2}^{\ddag},(\varphi_{1})_{T}=(\varphi _{2})_{T}$ and a rotation related map $\overline{R}_{\varphi_{1},\varphi_{2}}$ associated with ϕ 1 and ϕ 2 is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${\mathcal{A}}$ of unital separable simple amenable C ∗-algebras which is strictly larger than the class of separable C ∗-algebras with tracial rank zero or one. Tensor products of two C ∗-algebras in ${\mathcal{A}}$ are again in ${\mathcal{A}}$ . Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras for which the state space of K 0 is the same as the tracial state space and also some unital simple ASH-algebras whose K 0-group is ℤ and whose tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are ${\mathcal{Z}}$ -stable are isomorphic to ones with no dimension growth.