On the classification of simple amenable C∗C*-algebras with finite decomposition rank, II

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 $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott.Comment: submitted. Some minor upda

Lifting KK-elements, asymptotic unitary equivalence and classification of simple C∗-algebras

AbstractLet A and C be two unital simple C∗-algebras with tracial rank zero. Suppose that C is amenable and satisfies the Universal Coefficient Theorem. Denote by KKe(C,A)++ the set of those κ in KK(C,A) for which κ(K0(C)+∖{0})⊂K0(A)+∖{0} and κ([1C])=[1A]. Suppose that κ∈KKe(C,A)++. We show that there is a unital monomorphism ϕ:C→A such that [ϕ]=κ. Suppose that C is a unital AH-algebra and λ:T(A)→Tf(C) is a continuous affine map for which τ(κ([p]))=λ(τ)(p) for all projections p in all matrix algebras of C and any τ∈T(A), where T(A) is the simplex of tracial states of A and Tf(C) is the convex set of faithful tracial states of C. We prove that there is a unital monomorphism ϕ:C→A such that ϕ induces both κ and λ.Suppose that h:C→A is a unital monomorphism and γ∈Hom(K1(C),Aff(A)). We show that there exists a unital monomorphism ϕ:C→A such that [ϕ]=[h] in KK(C,A), τ○ϕ=τ○h for all tracial states τ and the associated rotation map can be given by γ. Denote by KKT(C,A)++ the set of compatible pairs (κ,λ), where κ∈KLe(C,A)++ and λ is a continuous affine map from T(A) to Tf(C). Together with a result on asymptotic unitary equivalence in [H. Lin, Asymptotic unitary equivalence and asymptotically inner automorphisms, arXiv:math/0703610, 2007], this provides a bijection from the asymptotic unitary equivalence classes of unital monomorphisms from C to A to (KKT(C,A)++,Hom(K1(C),Aff(T(A)))/R0), where R0 is a subgroup related to vanishing rotation maps.As an application, combining these results with a result of W. Winter [W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C∗-algebras, arXiv:0708.0283v3, 2007], we show that two unital amenable simple Z-stable C∗-algebras are isomorphic if they have the same Elliott invariant and the tensor products of these C∗-algebras with any UHF-algebra have tracial rank zero. In particular, if A and B are two unital separable simple Z-stable C∗-algebras with unique tracial states which are inductive limits of C∗-algebras of type I, then they are isomorphic if and only if they have isomorphic Elliott invariants