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

作者：Lin, Huaxin^*

来源：Inventiones Mathematicae, 2011, 183(2): 385-450.

DOI：10.1007/s00222-010-0280-9

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 phi(1), phi(2) : C -> A are two unital monomorphisms. We show that there is a continuous path of unitaries {u(t) : t is an element of [0, infinity)} of A such that
lim(t ->infinity) u*(t)phi(1)(c)u(t) = phi(2)(c) for all c is an element of C
if and only if [phi(1)] = [phi(2)] in KK(C, A), phi(double dagger)(1) = phi(double dagger)(2), (phi(1))(T) = (phi(2))(T) and a rotation related map (R) over bar (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 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 A are again in 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 (sic)-stable are isomorphic to ones with no dimension growth.

• 出版日期2011-2
• 单位华东师范大学