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Register a new userAsymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras
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Authors
Huaxin Lin
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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.
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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 approximately unitarily equivalent if and only if [phi]=[psi] in KL(C,A) taucirc phi=taucirc psi for all tracial states of A and phi^{ddag}=psi^{ddag}, here phi^{ddag} and psi^{ddag} are homomorphisms from $U(C)/CU(C)to U(A)/CU(A) induced by phi and psi, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.
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 $kappa(K_0(C)_+setminus{0})subset K_0(A)_+setminus{0}$ and $kappa([1_C])=[1_A]$. Suppose that $kappain {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $phi: Cto A$ such that $[phi]=kappa.$ Suppose that $C$ is a unital AH-algebra and $lambda: mathrm{T}(A)to mathrm{T}_{mathtt{f}}(C)$ is a continuous affine map for which $tau(kappa([p]))=lambda(tau)(p)$ for all projections $p$ in all matrix algebras of $C$ and any $tauin mathrm{T}(A),$ where $mathrm{T}(A)$ is the simplex of tracial states of $A$ and $mathrm{T}_{mathtt{f}}(C)$ is the convex set of faithful tracial states of $C.$ We prove that there is a unital monomorphism $phi: Cto A$ such that $phi$ induces both $kappa$ and $lambda.$ Suppose that $h: Cto A$ is a unital monomorphism and $gamma in mathrm{Hom}(Kone(C), aff(A)).$ We show that there exists a unital monomorphism $phi: Cto A$ such that $[phi]=[h]$ in ${KK}(C,A),$ $taucirc phi=taucirc h$ for all tracial states $tau$ and the associated rotation map can be given by $gamma.$ Applications to classification of simple C*-algebras are also given.
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 inductive limits of dimension drop circle $C^*$-algebras are also in the class. This unifies some of the previous known classification results for unital simple amenable $C^*$-algebras. We also show that there are many other $C^*$-algebras in the class. We prove that, for any partially ordered, simple weakly unperforated rationally Riesz group $G_0$ with order unit $u,$ any countable abelian group $G_1,$ any metrizable Choquet simplex $S,$ and any surjective affine continuous map $r: Sto S_u(G_0)$ (where $S_u(G_0)$ is the state space of $G_0$) which preserves extremal points, there exists one and only one (up to isomorphism) unital separable simple amenable $C^*$-algebra $A$ in the classifiable class ${cal A}$ such that $$ ((K_0(A), K_0(A)_+, [1_A]), K_1(A), T(A), lambda_A)=((G_0, (G_0)_+, u), G_1,S, r).
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 separable amenable ${cal Z}$-stable $C^*$-algebras. Moreover, it contains all unital simple separable amenable $C^*$-alegbras which satisfy the UCT and have finite rational tracial rank.
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 universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott.
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