Inventiones mathematicae

, Volume 183, Issue 2, pp 385–450

Asymptotic unitary equivalence and classification of simple amenable C-algebras

Article

DOI: 10.1007/s00222-010-0280-9

Cite this article as:
Lin, H. Invent. math. (2011) 183: 385. doi:10.1007/s00222-010-0280-9

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 ϕ1,ϕ2:CA are two unital monomorphisms. We show that there is a continuous path of unitaries {ut: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 [ϕ1]=[ϕ2] 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 ϕ1 and ϕ2 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 K0 is the same as the tracial state space and also some unital simple ASH-algebras whose K0-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.

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal UniversityShanghaiChina