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44
Lifting KKelements, asymptotical unitary equivalence and classification of simple C ∗ algebras
, 2008
"... 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 κ for which κ(K0(C)+ \ {0}) ⊂ K0(A)+ \ {0} and κ([1C]) = [1A]. Suppose that κ ∈ KKe(C, A) ++. We show that there ..."
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Cited by 32 (8 self)
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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 κ 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 AHalgebra 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 of asymptotic unitary equivalence in [14], 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 with a result of W. Winter ([23]), we show that two unital amenable simple Zstable C*algebras are isomorphic if they have the same Elliott invariant and the tensor products of these C*algebras with any UHFalgebras have tracial rank zero. In particular, if A and B are two unital separable simple Zstable C*algebras which are inductive limits of C*algebras of type I with unique tracial states, then they are isomorphic if and only if they have isomorphic Elliott invariant. 1
Asymptotically unitary equivalence and classification of simple amenable C*algebras
, 2009
"... 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 ϕ1, ϕ2: C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut: t ∈ [0, ∞)} of A such ..."
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Cited by 28 (13 self)
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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 ϕ1, ϕ2: C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut: t ∈ [0, ∞)} of A such that lim t→ ∞ u∗tϕ1(c)ut = ϕ2(c) for all c ∈ C if and only if [ϕ1] = [ϕ2] in KK(C, A), ϕ ‡ 1 = ϕ ‡ 2, (ϕ1)T = (ϕ2)T and a rotation related map Rϕ1,ϕ2 associated with ϕ1 and ϕ2 is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class A of unital separable simple amenable C∗algebras which is strictly larger than the class of separable C∗algebras whose tracial rank are zero or one. Tensor products of two C∗algebras in A are again in A. Moreover, this class is closed under inductive limits and contains all unital simple ASHalgebras whose state spaces of K0 is the same as the tracial state spaces as well as some unital simple ASHalgebras whose K0group is Z and tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AHalgebras which are Zstable are isomorphic to ones with no dimension growth.
The Rohlin property for automorphisms of the JiangSu algebra
, 2009
"... For projectionless C∗algebras absorbing the JiangSu algebra tensorially, we study a kind of the Rohlin property for autmorphisms. We show that the crossed products obtained by automorphisms with this Rohlin property also absorb the JiangSu algebra tensorially under a mild technical condition on t ..."
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Cited by 19 (4 self)
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For projectionless C∗algebras absorbing the JiangSu algebra tensorially, we study a kind of the Rohlin property for autmorphisms. We show that the crossed products obtained by automorphisms with this Rohlin property also absorb the JiangSu algebra tensorially under a mild technical condition on the C∗algebras. In particular, for the JiangSu algebra we show the uniqueness up to outer conjugacy of the automorphism with this Rohlin property.
Trivialization of C(X)algebras with strongly selfabsorbing fibres
 Bull. Soc. Math. France
"... Abstract. Suppose A is a separable unital C(X)algebra each fibre of which is isomorphic to the same strongly selfabsorbing and K1injective C ∗algebra D. We show that A and C(X) ⊗ D are isomorphic as C(X)algebras provided the compact Hausdorff space X is finitedimensional. This statement is kn ..."
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Cited by 18 (10 self)
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Abstract. Suppose A is a separable unital C(X)algebra each fibre of which is isomorphic to the same strongly selfabsorbing and K1injective C ∗algebra D. We show that A and C(X) ⊗ D are isomorphic as C(X)algebras provided the compact Hausdorff space X is finitedimensional. This statement is known not to extend to the infinitedimensional case. A unital and separable C ∗algebra D ̸ = C is strongly selfabsorbing if there is an isomorphism D ∼ = → D ⊗ D which is approximately unitarily equivalent to the inclusion map D → D⊗D, d ↦ → d⊗1D, cf. [10]. Strongly selfabsorbing C ∗algebras
MINIMAL DYNAMICS AND KTHEORETIC RIGIDITY: ELLIOTT’S CONJECTURE
, 903
"... Abstract. Let X be an infinite, compact, metrizable space of finite covering dimension and α: X → X a minimal homeomorphism. We prove that the crossed product C(X) ⋊α Z absorbs the JiangSu algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to is ..."
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Cited by 15 (6 self)
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Abstract. Let X be an infinite, compact, metrizable space of finite covering dimension and α: X → X a minimal homeomorphism. We prove that the crossed product C(X) ⋊α Z absorbs the JiangSu algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to isomorphism by their graded ordered Ktheory under the necessary condition that their projections separate traces. This result applies, in particular, to those crossed products arising from uniquely ergodic homeomorphisms. From its earliest days the theory of operator algebras has been entwined with dynamics, and some of the most important developments in the subject revolve around this interaction. The groupmeasure space construction of Murray and von Neumann provided the first examples of nontypeI factors; recently we have
Localizing the Elliott Conjecture at Strongly Selfabsorbing C ∗algebras, II —–An Appendix
, 709
"... This note provides some technical support to the proof of a result of W. Winter which shows that two unital separable simple amenable Zabsorbing C ∗algebras with locally finite decomposition property satisfying the UCT whose projections separate the traces are isomorphic if their Ktheory is finit ..."
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Cited by 13 (10 self)
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This note provides some technical support to the proof of a result of W. Winter which shows that two unital separable simple amenable Zabsorbing C ∗algebras with locally finite decomposition property satisfying the UCT whose projections separate the traces are isomorphic if their Ktheory is finitely generated and their Elliott invariants are the same. 1
Approximate unitary equivalence in simple C∗algebras of tracial rank one
 Trans. Amer. Math. Soc
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The range of a class of classifiable separable simple amenable
 C*algebras, Adv. Math
"... We study the range of a classifiable class A of unital separable simple amenable C ∗algebras which satisfy the Universal Coefficient Theorem. The class A contains all unital simple AHalgebras. We show that all unital simple inductive limits of dimension drop circle C ∗algebras are also in the cla ..."
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Cited by 9 (5 self)
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We study the range of a classifiable class A of unital separable simple amenable C ∗algebras which satisfy the Universal Coefficient Theorem. The class A contains all unital simple AHalgebras. 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 G0 with order unit u, any countable abelian group G1, any metrizable Choquet simplex S, and any surjective affine continuous map r: S → Su(G0) (where Su(G0) is the state space of G0) which preserves extremal points, there exists one and only one (up to isomorphism) unital separable simple amenable C ∗algebra A in the classifiable class A such that