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

By Huaxin Lin, Z. Niu. 2008

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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

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a7e56dc9a82d2f4d20165ecaf3705b855e7c3d12

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Citations in the corpus

Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, methodology): “1]), allowing the application of the classification technique from [98], refined by Lin and Lin-Niu in [54, 55].”
Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture (2009, methodology): “To avoid the task of deriving a succinct and useful form of this theorem for our purpose, we state here a generalization of [28, Theorem 7.1] due to Lin and Niu ([ 11 ]).”
Decomposition rank and Z-stability (2008, background): “1] in connection with [23] and [24] (to remove the remaining K-theory condition of [43]), the class of the corollary satisfies the Elliott conjecture up to Z-stability; but the latter is automatic by Theorem 5.”
Classifying crossed product C*-algebras (2013, methodology): “I am conﬁdent that this method will prove to be as useful for classiﬁcation as the one introduced in [37] (where clas-siﬁcation up to Z -stability was reduced to classiﬁcation up to UHF-stability), see also [17, 19].”
Quasidiagonality of nuclear C*-algebras (2015, background): “4] (which generalises [77]); since Theorem A makes quasidiagonality automatic, the tracial rank hypothesis required in [46] follows from [49, Theorem 6.”
A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group (2014, background): “2] that build on results of Winter [18], and Lin and Niu [12].”