Results 1  10
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51
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.
Classification of homomorphisms and dynamical systems
 Trans. Amer. Math. Soc
"... Let A be a unital simple C ∗algebra with tracial rank zero and X be a compact metric space. Suppose that h1, h2: C(X) → A are two unital monomorphisms. We show that h1 and h2 are approximately unitarily equivalent if and only if [h1] = [h2] in KL(C(X), A) and τ ◦ h1(f) = τ ◦ h2(f) for every f ∈ ..."
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Cited by 23 (17 self)
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Let A be a unital simple C ∗algebra with tracial rank zero and X be a compact metric space. Suppose that h1, h2: C(X) → A are two unital monomorphisms. We show that h1 and h2 are approximately unitarily equivalent if and only if [h1] = [h2] in KL(C(X), A) and τ ◦ h1(f) = τ ◦ h2(f) for every f ∈ C(X) and every trace τ of A. Adopting a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let α, β: X → X be two minimal homeomorphisms. Using the above mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a Ktheoretical condition is satisfied. In the case that X is the Cantor set, this notion coincides with strong orbit equivalence of Giordano, Putnam and Skau and the Ktheoretical condition is equivalent to saying that the associate crossed product C ∗algebras are isomorphic. Another application of the above mentioned result is given for C ∗dynamical systems related to a problem of Kishimoto. Let A be a unital simple AHalgebra with no dimension growth and with real rank zero, and let α ∈ Aut(A). We prove that if α r fixes a large subgroup of K0(A) and has the tracial Rokhlin property then A ⋊α Z is again a unital simple AHalgebra with no dimension growth and with real rank zero. 1
Approximate Homotopy of Homomorphisms from C(X) into a Simple C ∗algebra
, 2006
"... Abstract. Let X be a finite CW complex and let h1, h2: C(X) → A be two unital homomorphisms, where A is a unital C ∗algebra. We study the problem when h1 and h2 are approximately homotopy. We present a Ktheoretical necessary and sufficient condition for them to be approximately homotopy under the ..."
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Cited by 23 (17 self)
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Abstract. Let X be a finite CW complex and let h1, h2: C(X) → A be two unital homomorphisms, where A is a unital C ∗algebra. We study the problem when h1 and h2 are approximately homotopy. We present a Ktheoretical necessary and sufficient condition for them to be approximately homotopy under the assumption that A is a unital separable simple C ∗algebra of tracial rank zero, or A is a unital purely infinite simple C ∗algebra. When they are approximately homotopy, we also give a bound for the length of the homotopy. These results are also extended to the case that h1 and h2 are approximately multiplicative contractive completely positive linear maps. Suppose that h: C(X) → A is a monomorphism and u ∈ A is a unitary (with [u] = {0} in K1(A)). We prove that, for any ǫ> 0, and any compact subset F ⊂ C(X), there exists δ> 0 and a finite subset G ⊂ C(X) satisfying the following: if ‖[h(f), u] ‖ < δ and Bott(h, u) = {0}, then there exists a continuous rectifiable path {ut: t ∈ [0, 1]} such that u0 = u, u1 = 1A and ‖[h(g), ut] ‖ < ǫ for all g ∈ F and t ∈ [0,1]. (e 0.1)
Asymptotic unitary equivalence and asymptotically inner automorphisms
, 2008
"... Let C be a unital AHalgebra and let A be a unital separable simple C ∗algebra with tracial rank zero. 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(a)ut = ϕ2(a) for all a ∈ C if and only ..."
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Cited by 17 (9 self)
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Let C be a unital AHalgebra and let A be a unital separable simple C ∗algebra with tracial rank zero. 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(a)ut = ϕ2(a) for all a ∈ C if and only if [ϕ1] = [ϕ2] in KK(C, A), τ ◦ϕ1 = τ ◦ϕ2 for all τ ∈ T(A) and a rotation related map ˜ηϕ1,ϕ2 associated with ϕ1 and ϕ2 is zero. In particular, an automorphism α on a unital separable simple C ∗algebra A in N with tracial rank zero is asymptotically inner if and only if [α] = [idA] in KK(A, A) and the rotation related map ˜ηϕ1,ϕ2 is zero. Let A be a unital AHalgebra (not necessarily simple) and let α ∈ Aut(A) be an automorphism. As an application, we show that the associated crossed product A ⋊α Z can be embedded into a unital simple AFalgebra if and only if A admits a strictly positive αinvariant tracial state.
Almost multiplicative morphisms and almost commuting matrices
 J. Operator Theory
, 1998
"... Abstract. We prove that a contractive positive linear map which is approximately multiplicative and approximately injective from C(X) into certain unital simple C∗algebras of real rank zero and stable rank one is close to a homomorphism (with finite dimensional range) if a necessary Ktheoretical ..."
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Cited by 13 (9 self)
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Abstract. We prove that a contractive positive linear map which is approximately multiplicative and approximately injective from C(X) into certain unital simple C∗algebras of real rank zero and stable rank one is close to a homomorphism (with finite dimensional range) if a necessary Ktheoretical obstruction vanishes and dimension of X is no more than two. We also show that the above is false it the dimension of X is greater than 2, in general.
The corona factorization property and approximate unitary equivalence
 HOUSTON JOURNAL OF MATHEMATICS
, 2006
"... We study Rørdam’s group, KL(A,B), and a corona factorization condition. Our key technical result is a lemma showing that approximate unitary equivalence preserves the purely large property of Elliott and Kucerovsky [10]. Using this, we characterize KL(A,B) as a group of purely large extensions under ..."
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Cited by 13 (3 self)
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We study Rørdam’s group, KL(A,B), and a corona factorization condition. Our key technical result is a lemma showing that approximate unitary equivalence preserves the purely large property of Elliott and Kucerovsky [10]. Using this, we characterize KL(A,B) as a group of purely large extensions under approximate unitary equivalence, generalizing a theorem of Kasparov’s. Then we prove the following: Let B be a stable and separable C∗algebra. Then the following are equivalent (for absorption of weakly nuclear extensions): i) The corona algebra of B has a certain quasiinvertibility property, which we here call the corona factorization property. ii) Rørdam’s group KL1nuc(A,B) is isomorphic to the group of full essential extensions of A by B. iii) Every strongly full and positive element of the corona algebra of B is properly infinite. iv) Every normfull extension of B is absorbing, with respect to approximate unitary equivalence. v) Every normfull extension of B is absorbing, with respect to ordinary unitary equivalence. vi) Every normfull extension of B is absorbing, with respect to weak equivalence. vii) Every normfull trivial extension of B is absorbing, with respect to unitary equivalence. viii) A Ktheoretical uniqueness result for maps into M(B)/B. We show that if X is the infinite Cartesian product of spheres, then C(X)⊗K does not have the corona factorization property. We apply our technical lemma to study quasidiagonality and weak quasidiagonality of extensions.
The corona factorization property
, 2005
"... Abstract. The corona factorization property is a property with connections to extension theory, Ktheory and the structure of C ∗algebras. This paper is a short survey of the subject, together with some new results and open questions. 1. ..."
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Cited by 13 (1 self)
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Abstract. The corona factorization property is a property with connections to extension theory, Ktheory and the structure of C ∗algebras. This paper is a short survey of the subject, together with some new results and open questions. 1.
Approximate unitary equivalence in simple C∗algebras of tracial rank one
 Trans. Amer. Math. Soc
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Some remarks on the universal coefficient theorem in KKtheory
 In Operator algebras and mathematical physics (Constanţa
, 2003
"... Abstract. If a nuclear separable C*algebra A can be approximated by C*subalgebras satisfying the UCT, thenA satisfies the UCT. It is also shown that the validity of the UCT for all separable nuclear C*algebras is equivalent to a certain finite dimensional approximation property. 1. ..."
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Cited by 11 (3 self)
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Abstract. If a nuclear separable C*algebra A can be approximated by C*subalgebras satisfying the UCT, thenA satisfies the UCT. It is also shown that the validity of the UCT for all separable nuclear C*algebras is equivalent to a certain finite dimensional approximation property. 1.