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Huaxin Lin - One of the best experts on this subject based on the ideXlab platform.
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Non-amenable simple C*-algebras with Tracial approximation
arXiv: Operator Algebras, 2021Co-Authors: Huaxin LinAbstract:We construct two types of unital separable simple $C^*$-alebras $A_z^{C_1}$ and $A_z^{C_2},$ one is exact but not amenable, and the other is non-exact. Both have the same Elliott invariant as the Jiang-Su algebra, namely, $A_z^{C_i}$ has a unique Tracial State, $$(K_0(A_z^{C_i}), K_0(A_z^{C_i})_+, [1_{A_z^{C_i}} ])=(\mathbb Z, \mathbb Z_+,1)$$ and $K_{1}(A_z^{C_i})=\{0\}$ ($i=1,2$). We show that $A_z^{C_i}$ ($i=1,2$) is essentially Tracially in the class of separable ${\cal Z}$-stable $C^*$-alebras of nuclear dimension 1. $A_z^{C_i}$ has stable rank one, strict comparison for positive elements and no 2-quasitrace other than the unique Tracial State. We also produce models of unital separable simple non-exact $C^*$-alebras which are essentially Tracially in the class of simple separable nuclear ${\cal Z}$-stable $C^*$-alebras and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential Tracial approximation.
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Homomorphisms from AH-algebras
Journal of Topology and Analysis, 2017Co-Authors: Huaxin LinAbstract:Let C be a general unital AH-algebra and let A be a unital simple C∗-algebra with Tracial rank at most one. Suppose that φ,ψ : C → A are two unital monomorphisms. We show that φ and ψ are approximately unitarily equivalent if and only if [φ] = [ψ]in KL(C,A), φ♯ = ψ♯and(1) φρ = ψρ, where φ♯ and ψ♯ are continuous affine maps from Tracial State space T(A) of A to faithful Tracial State space Tf(C) of C induced by φ and ψ, respectively, and φρ and ψρ are induced homomorphisms s from K1(C) into Aff(T(A))/ρA(K0(A))¯, where Aff(T(A)) is the space of all real affine continuous functions on T(A) and ρA(K0(A))¯ is the closure of the image of K0(A) in the affine space Aff(T(A)). In particular, the above holds for C = C(X), the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements κ ∈ KLe(C,A)++, an affine map γ : T(C) → Tf(C) and a homomorphisms α : K1(C) →Aff(T(A))/ρA(K0(A))¯, there exists a unital monomor...
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Homomorphisms from AH-algebras
Journal of Topology and Analysis, 2017Co-Authors: Huaxin LinAbstract:Let [Formula: see text] be a general unital AH-algebra and let [Formula: see text] be a unital simple [Formula: see text]-algebra with Tracial rank at most one. Suppose that [Formula: see text] are two unital monomorphisms. We show that [Formula: see text] and [Formula: see text] are approximately unitarily equivalent if and only if [Formula: see text] [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are continuous affine maps from Tracial State space [Formula: see text] of [Formula: see text] to faithful Tracial State space [Formula: see text] of [Formula: see text] induced by [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] and [Formula: see text] are induced homomorphisms s from [Formula: see text] into [Formula: see text], where [Formula: see text] is the space of all real affine continuous functions on [Formula: see text] and [Formula: see text] is the closure of the image of [Formula: see text] in the affine space [Formula: see text]. In particular, the above holds for [Formula: see text], the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements [Formula: see text], an affine map [Formula: see text] and a homomorphisms [Formula: see text], there exists a unital monomorphism [Formula: see text] such that [Formula: see text] and [Formula: see text].
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Homomorphisms from AH-algebras
arXiv: Operator Algebras, 2011Co-Authors: Huaxin LinAbstract:Let $C$ be a general unital AH-algebra and let $A$ be a unital simple $C^*$-algebra with Tracial rank at most one. Suppose that $\phi, \psi: C\to A$ are two unital monomorphisms. We show that $\phi$ and $\psi$ are approximately unitarily equivalent if and only if \beq[\phi]&=&[\psi] {\rm in} KL(C,A), \phi_{\sharp}&=&\psi_{\sharp}\tand \phi^{\dag}&=&\psi^{\dag}, \eneq where $\phi_{\sharp}$ and $\psi_{\sharp}$ are continuous affine maps from Tracial State space $T(A)$ of $A$ to faithful Tracial State space $T_{\rm f}(C)$ of $C$ induced by $\phi$ and $\psi,$ respectively, and $\phi^{\ddag}$ and $\psi^{\ddag}$ are induced homomorphisms from $K_1(C)$ into $\Aff(T(A))/\bar{\rho_A(K_0(A))},$ where $\Aff(T(A))$ is the space of all real affine continuous functions on $T(A)$ and $\bar{\rho_A(K_0(A))}$ is the closure of the image of $K_0(A)$ in the affine space $\Aff(T(A)).$ In particular, the above holds for $C=C(X),$ the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements $\kappa\in KL_e(C,A)^{++},$ an affine map $\gamma: T(C)\to T_{\rm f}(C)$ and a \hm $\af: K_1(C)\to \Aff(T(A))/\bar{\rho_A(K_0(A))},$ there exists a unital monomorphism $\phi: C\to A$ such that $[h]=\kappa,$ $h_{\sharp}=\gamma$ and $\phi^{\dag}=\af.$
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Asymptotic unitary equivalence and classification of simple amenable C ∗ -algebras
Inventiones mathematicae, 2010Co-Authors: Huaxin LinAbstract:Let C and A be two unital separable amenable simple C ∗-algebras with Tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ϕ 1,ϕ 2:C→A are two unital monomorphisms. We show that there is a continuous path of unitaries {u t :t∈[0,∞)} of A such that $$\lim_{t\to\infty}u_t^*\varphi_1(c)u_t=\varphi_2(c)\quad\mbox{for all }c\in C$$ if and only if [ϕ 1]=[ϕ 2] in $KK(C,A),\varphi_{1}^{\ddag}=\varphi_{2}^{\ddag},(\varphi_{1})_{T}=(\varphi _{2})_{T}$ and a rotation related map $\overline{R}_{\varphi_{1},\varphi_{2}}$ associated with ϕ 1 and ϕ 2 is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${\mathcal{A}}$ of unital separable simple amenable C ∗-algebras which is strictly larger than the class of separable C ∗-algebras with Tracial rank zero or one. Tensor products of two C ∗-algebras in ${\mathcal{A}}$ are again in ${\mathcal{A}}$ . Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras for which the State space of K 0 is the same as the Tracial State space and also some unital simple ASH-algebras whose K 0-group is ℤ and whose Tracial State spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are ${\mathcal{Z}}$ -stable are isomorphic to ones with no dimension growth.
Radu Balan - One of the best experts on this subject based on the ideXlab platform.
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INDEPENDENCE, AND SPECTRAL PROPERTIES OF THE ALGEBRA OF TIME-FREQUENCY SHIFT OPERATORS
2015Co-Authors: Radu BalanAbstract:Abstract. In this paper we analyze the Banach *-algebra of time-frequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful Tracial State on this algebra which proves the algebra contains no compact operators. As a corollary we obtain a special case of the Heil-Ramanathan-Topiwala conjecture regarding linear independence of finitely many time-frequency shifts of one L2 function. We also estimate the coefficient decay of the inverse of finite linear combinations of time-frequency shifts. 1
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the noncommutative wiener lemma linear independence and spectral properties of the algebra of time frequency shift operators
Transactions of the American Mathematical Society, 2008Co-Authors: Radu BalanAbstract:In this paper we analyze the Banach *-algebra of time-frequency shifts with absolutely summable coecients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful Tracial State on this algebra which proves the algebra contains no compact operators. As a corollary we obtain a special case of the Heil-Ramanathan-Topiwala conjecture regarding linear independence of nitely many time-frequency shifts of one L 2 function. We also estimate the coecient decay of the inverse of nite linear combinations of time-frequency shifts.
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a noncommutative wiener lemma and a faithful Tracial State on banach algebras of time frequency shift operators
arXiv: Functional Analysis, 2008Co-Authors: Radu BalanAbstract:In this paper we analyze the Banach *-algebra of time-frequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wienerlemma. We also construct a faithful Tracial State on this algebra which implies the algebra contains no compact operators. As a corollary we obtain a special case of the Heil-Ramanathan-Topiwala conjecture regarding linear independence of finitely many time-frequency shifts of one L 2 function.
Yasuhiko Sato - One of the best experts on this subject based on the ideXlab platform.
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Actions of amenable groups and crossed products of Z-absorbing C*-algebras
arXiv: Operator Algebras, 2016Co-Authors: Yasuhiko SatoAbstract:We study actions of countable discrete amenable groups on unital separable simple nuclear Z-absorbing C*-algebras. Under a certain assumption on Tracial States, which is automatically satisfied in the case of a unique Tracial State, the crossed product is shown to absorb the Jiang-Su algebra Z tensorially.
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Decomposition rank of UHF-absorbing C*-algebras
Duke Mathematical Journal, 2014Co-Authors: Hiroki Matui, Yasuhiko SatoAbstract:Let A be a unital separable simple C*-algebra with a unique Tracial State. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal UHF-algebra has decomposition rank at most one. Then it is proved that A is nuclear, quasidiagonal and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF-algebra has Tracial rank zero. Applying this characterization, we obtain a counter-example to the Powers-Sakai conjecture.
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trace spaces of simple nuclear c algebras with finite dimensional extreme boundary
arXiv: Operator Algebras, 2012Co-Authors: Yasuhiko SatoAbstract:Let A be a unital separable simple infinite-dimensional nuclear C*-algebra with at least one Tracial State. We prove that if the trace space of A has compact finite-dimensional extreme boundary then there exist unital embeddings of matrix algebras into a certain central sequence algebra of A which is determined by the uniform topology on the trace space. As an application, it is shown that if furthermore A has strict comparison then A absorbs the Jiang-Su algebra tensorially.
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Certain aperiodic automorphisms of unital simple projectionless C*-algebras
International Journal of Mathematics, 2009Co-Authors: Yasuhiko SatoAbstract:Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K1(A) ≅ G and a unique Tracial State, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut(A)/WInn(A) are conjugate, where WInn(A) means the subgroup of Aut(A) consisting of automorphisms which are inner in the Tracial representation. In the second part of this paper, we consider a class of unital simple C*-algebras with a unique Tracial State which contains the class of unital simple A𝕋-algebras of real rank zero with a unique Tracial State. This class is closed under inductive limits and crossed products by actions of ℤ with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism ᾶ of A with the Rohlin property such that ᾶ and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.
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Certain aperiodic automorphisms of unital simple projectionless C*-algebras
arXiv: Operator Algebras, 2008Co-Authors: Yasuhiko SatoAbstract:Let $G$ be an inductive limit of finite cyclic groups and let $A$ be a unital simple projectionless C*-algebra with $K_1(A) \cong G$ and with a unique Tracial State, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in $\Aut(A)/\WInn(A)$ are conjugate, where $\WInn(A)$ means the subgroup of $\Aut(A)$ consisting of automorphisms which are inner in the Tracial representation. In the second part of this paper, we consider a class of unital simple C*-algebras with a unique Tracial State which contains the class of unital simple AT-algebras of real rank zero with a unique Tracial State. This class is closed under inductive limits and under crossed products by actions of $\Z$ with the Rohlin property. Let $A$ be a TAF-algebra in this class. We show that for any automorphism $\alpha$ of $A$ there exists an automorphism $\widetilde{\alpha}$ of $A$ with the Rohlin property such that $\widetilde{\alpha}$ and $\alpha$ are asymptotically unitarily equivalent. In its proof we use an aperiodic automorphism of the Jiang-Su algebra.
Sylvia Pulmannová - One of the best experts on this subject based on the ideXlab platform.
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S-map and Tracial States
Information Sciences, 2009Co-Authors: Olga Nánásiová, Sylvia PulmannováAbstract:The notion of s-map introduced in quantum logics to define conditional expectations for non-compatible events is studied in projection lattices of von Neumann algebras. It is shown that every Tracial State gives rise to an s-map, and conversely, every s-map defines a Tracial State.
Ken Miyake - One of the best experts on this subject based on the ideXlab platform.
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The Extreme Points of the Tracial State Space of Noncommutative 3-Spheres
Letters in Mathematical Physics, 2009Co-Authors: Ken MiyakeAbstract:We determine the extreme points of the Tracial State space of the noncommutative 3-sphere \({C^{\rm alg}(S_{\theta}^{3})}\) when the deformation parameter θ is an irrational number.
