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

• Autores: Huaxin Lin, Zhuang Niu
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 219, Nº 5, 2008, págs. 1729-1769
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.07.011
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• Resumen

  • Let 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 ? in KK(C,A) for which ?(K0(C)+{0})K0(A)+{0} and ?([1C])=[1A]. Suppose that ?KKe(C,A)++. We show that there is a unital monomorphism :C?A such that []=?. Suppose that C is a unital AH-algebra and is a continuous affine map for which t(?([p]))=?(t)(p) for all projections p in all matrix algebras of C and any tT(A), where T(A) is the simplex of tracial states of A and is the convex set of faithful tracial states of C. We prove that there is a unital monomorphism :C?A such that induces both ? and ?.

    Suppose that h:C?A is a unital monomorphism and ?Hom(K1(C),Aff(A)). We show that there exists a unital monomorphism :C?A such that []=[h] in KK(C,A), t?=t?h for all tracial states t and the associated rotation map can be given by ?. Denote by KKT(C,A)++ the set of compatible pairs (?,?), where ?KLe(C,A)++ and ? is a continuous affine map from T(A) to . 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 , where 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 -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 -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