The Experts below are selected from a list of 246 Experts worldwide ranked by ideXlab platform
Uffe Haagerup - One of the best experts on this subject based on the ideXlab platform.
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Ultraproducts of von neumann algebras
Journal of Functional Analysis, 2014Co-Authors: Hiroshi Ando, Uffe HaagerupAbstract:Abstract We study several notions of Ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the Ultraproduct M ω introduced by Ocneanu is a corner of the Ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the Ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu Ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu Ultraproduct of type III factors, which are not present in the tracial Ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω .
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Ultraproducts of von neumann algebras
arXiv: Operator Algebras, 2012Co-Authors: Hiroshi Ando, Uffe HaagerupAbstract:We study several notions of Ultraproducts of von Neumann algebras from a unifying viewpoint. In particular, we show that for a sigma-finite von Neumann algebra $M$, the Ultraproduct $M^{\omega}$ introduced by Ocneanu is a corner of the Ultraproduct $\prod^{\omega}M$ introduced by Groh and Raynaud. Using this connection, we show that the Ultraproduct action of the modular automorphism group of a normal faithful state $\varphi$ of $M$ on the Ocneanu Ultraproduct is the modular automorphism group of the ultrapower state ($\sigma_t^{\varphi^{\omega}}=(\sigma_t^{\varphi})^{\omega}$). Applying these results, we obtain several phenomena of the Ocneanu Ultraproduct of type III factors, which are not present in the tracial Ultraproducts. For instance, it turns out that the ultrapower $M^{\omega}$ of a Type III$_0$ factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of $M$ in $M^{\omega}$ and Connes' asymptotic centralizer algebra $M_{\omega}$.
Hiroshi Ando - One of the best experts on this subject based on the ideXlab platform.
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Ultraproducts of von neumann algebras
Journal of Functional Analysis, 2014Co-Authors: Hiroshi Ando, Uffe HaagerupAbstract:Abstract We study several notions of Ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the Ultraproduct M ω introduced by Ocneanu is a corner of the Ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the Ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu Ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu Ultraproduct of type III factors, which are not present in the tracial Ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω .
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Ultraproducts of von neumann algebras
arXiv: Operator Algebras, 2012Co-Authors: Hiroshi Ando, Uffe HaagerupAbstract:We study several notions of Ultraproducts of von Neumann algebras from a unifying viewpoint. In particular, we show that for a sigma-finite von Neumann algebra $M$, the Ultraproduct $M^{\omega}$ introduced by Ocneanu is a corner of the Ultraproduct $\prod^{\omega}M$ introduced by Groh and Raynaud. Using this connection, we show that the Ultraproduct action of the modular automorphism group of a normal faithful state $\varphi$ of $M$ on the Ocneanu Ultraproduct is the modular automorphism group of the ultrapower state ($\sigma_t^{\varphi^{\omega}}=(\sigma_t^{\varphi})^{\omega}$). Applying these results, we obtain several phenomena of the Ocneanu Ultraproduct of type III factors, which are not present in the tracial Ultraproducts. For instance, it turns out that the ultrapower $M^{\omega}$ of a Type III$_0$ factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of $M$ in $M^{\omega}$ and Connes' asymptotic centralizer algebra $M_{\omega}$.
Teresa Piovesan - One of the best experts on this subject based on the ideXlab platform.
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on the closure of the completely positive semidefinite cone and linear approximations to quantum colorings
Other publications TiSEM, 2017Co-Authors: Sabine Burgdorf, Monique Laurent, Teresa PiovesanAbstract:We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+$, consisting of all the $n \times n$ symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set $\mathcal Q$ of bipartite quantum correlations, as projection of an affine section of it. We have two main results concerning the structure of the completely positive semidefinite cone, namely about its interior and about its closure. On the one hand we construct a hierarchy of polyhedral cones covering the interior of $\mathcal{CS}_+$, which we use for computing some variants of the quantum chromatic number by way of a linear program. On the other hand we give an explicit description of the closure of the completely positive semidefinite cone by showing that it consists of all matrices admitting a Gram representation in the tracial Ultraproduct of matrix algebras.
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on the closure of the completely positive semidefinite cone and linear approximations to quantum colorings
Conference on Theory of Quantum Computation Communication and Cryptography, 2015Co-Authors: Sabine Burgdorf, Monique Laurent, Teresa PiovesanAbstract:We investigate structural properties of the completely positive semidefinite cone CS^n_+, consisting of all the n x n symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set Q of bipartite quantum correlations, as projection of an affine section of it. We have two main results concerning the structure of the completely positive semidefinite cone, namely about its interior and about its closure. On the one hand we construct a hierarchy of polyhedral cones which covers the interior of CS^n_+, which we use for computing some variants of the quantum chromatic number by way of a linear program. On the other hand we give an explicit description of the closure of the completely positive semidefinite cone, by showing that it consists of all matrices admitting a Gram representation in the tracial Ultraproduct of matrix algebras.
Yoshimichi Ueda - One of the best experts on this subject based on the ideXlab platform.
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on the geometry of von neumann algebra preduals
Positivity, 2014Co-Authors: Miguel Martin, Yoshimichi UedaAbstract:Let \(M\) be a von Neumann algebra and let \(M_\star \) be its (unique) predual. We study when for every \(\varphi \in M_\star \) there exists \(\psi \in M_\star \) solving the equation \(\Vert \varphi \pm \psi \Vert =\Vert \varphi \Vert =\Vert \psi \Vert \). This is the case when \(M\) does not contain type I nor type III\(_1\) factors as direct summands and it is false at least for the unique hyperfinite type III\(_1\) factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of \(M_\star \) of length \(4\). An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the Ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such Ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.
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on the geometry of von neumann algebra preduals
arXiv: Operator Algebras, 2012Co-Authors: Miguel Martin, Yoshimichi UedaAbstract:Let $M$ be a von Neumann algebra and let $M_\star$ be its (unique) predual. We study when for every $\varphi\in M_\star$ there exists $\psi\in M_\star$ solving the equation $\|\varphi \pm \psi\|=\|\varphi\|=\|\psi\|$. This is the case when $M$ does not contain type I nor type III$_1$ factors as direct summands and it is false at least for the unique hyperfinite type III$_1$ factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of $M_\star$ of length 4. An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the Ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such Ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.
Sabine Burgdorf - One of the best experts on this subject based on the ideXlab platform.
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on the closure of the completely positive semidefinite cone and linear approximations to quantum colorings
Other publications TiSEM, 2017Co-Authors: Sabine Burgdorf, Monique Laurent, Teresa PiovesanAbstract:We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+$, consisting of all the $n \times n$ symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set $\mathcal Q$ of bipartite quantum correlations, as projection of an affine section of it. We have two main results concerning the structure of the completely positive semidefinite cone, namely about its interior and about its closure. On the one hand we construct a hierarchy of polyhedral cones covering the interior of $\mathcal{CS}_+$, which we use for computing some variants of the quantum chromatic number by way of a linear program. On the other hand we give an explicit description of the closure of the completely positive semidefinite cone by showing that it consists of all matrices admitting a Gram representation in the tracial Ultraproduct of matrix algebras.
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on the closure of the completely positive semidefinite cone and linear approximations to quantum colorings
Conference on Theory of Quantum Computation Communication and Cryptography, 2015Co-Authors: Sabine Burgdorf, Monique Laurent, Teresa PiovesanAbstract:We investigate structural properties of the completely positive semidefinite cone CS^n_+, consisting of all the n x n symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set Q of bipartite quantum correlations, as projection of an affine section of it. We have two main results concerning the structure of the completely positive semidefinite cone, namely about its interior and about its closure. On the one hand we construct a hierarchy of polyhedral cones which covers the interior of CS^n_+, which we use for computing some variants of the quantum chromatic number by way of a linear program. On the other hand we give an explicit description of the closure of the completely positive semidefinite cone, by showing that it consists of all matrices admitting a Gram representation in the tracial Ultraproduct of matrix algebras.
