Asymptotically 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 no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that ${\varphi }_1,{\varphi }_2:C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t:t\in [0,\infty )\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(c)u_t=ϕ_2(c)\tforal c\in C $$ if and only if $[{\varphi }_1]=[{\varphi }_2]$ in $KK(C,A),$ ${\varphi }_1^{‡}={\varphi }_2^{‡},$ $({\varphi }_1{)}_T=({\varphi }_2{)}_T$ and a rotation related map ${\bar{R}}_{{\varphi }_1,{\varphi }_2}$ associated with ${\varphi }_1$ and ${\varphi }_2$ 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 \CA s which is strictly larger than the class of separable \CA s whose tracial rank are zero or one. The class contains all unital simple ASH-algebras whose state spaces of $K_0$ are the same as the tracial state spaces as well as the simple inductive limits of dimension drop circle algebras. Moreover it contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. 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.

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Cited by in corpus (2)

• Homotopy of unitaries in simple C*-algebras with tracial rank one
• Unitaries in a Simple C*-algebra of Tracial Rank One