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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 $phi_1, phi_2: Cto A$ are two unital monomorphisms. We show that there is a continuous path of unitaries ${u_t: tin [0, infty)}$ of $A$ such that $$ lim_{ttoinfty}u_t^*phi_1(c)u_t=phi_2(c)tforal cin C $$ if and only if $[phi_1]=[phi_2]$ in $KK(C,A),$ $phi_1^{ddag}=phi_2^{ddag},$ $(phi_1)_T=(phi_2)_T$ and a rotation related map $bar{R}_{phi_1,phi_2}$ 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 ${cal A}$ 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 ${cal Z}$-stable are isomorphic to ones with no dimension growth.
Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $phi, psi: Cto A$ are two unital monomorphisms. With some restriction on $C,$ we show that $phi$ and $psi$ are approxi
Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${{KK}}_e(C,A)^{++}$ the set of those $kappa$ for which $kappa(K_0(C)_+setminus{0})subset
We study the range of a classifiable class ${cal A}$ of unital separable simple amenable $C^*$-algebras which satisfy the Universal Coefficient Theorem. The class ${cal A}$ contains all unital simple AH-algebras. We show that all unital simple induct
We present a classification theorem for a class of unital simple separable amenable ${cal Z}$-stable $C^*$-algebras by the Elliott invariant. This class of simple $C^*$-algebras exhausts all possible Elliott invariant for unital stably finite simple
We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $Aotimes Q$ has generalized tracial rank at most one, where $Q$ is the