Article

Inventiones mathematicae

, Volume 183, Issue 2, pp 385-450

First online:

Asymptotic unitary equivalence and classification of simple amenable C -algebras

  • Huaxin LinAffiliated withDepartment of Mathematics, East China Normal University Email author 

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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:CA 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.