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Marius Junge - One of the best experts on this subject based on the ideXlab platform.
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Rosenthal Operator Spaces
Studia Mathematica, 2020Co-Authors: Marius Junge, Niels Jørgen Nielsen, Timur OikhbergAbstract:In 1969 Lindenstrauss and Rosenthal showed that if a Banach Space is isomorphic to a complemented subSpace of an Lp-Space, then it is either an Lp-Space or isomorphic to a Hilbert Space. This is the motivation of this paper where we study non- Hilbertian complemented Operator subSpaces of non-commutative Lp-Spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of Operator Spaces for every 2 < p < 1 which can be considered as Operator Space analogues of the Rosenthal sequence Spaces from Banach Space theory, constructed in 1970. Under the usual conditions on the defining sequence we prove that most of these Spaces are Operator Lp-Spaces, not completely isomorphic to previously known such Spaces. However, it turns out that some column and row versions of our Spaces are not Operator Lp-Spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.
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capacity bounds via Operator Space methods
Journal of Mathematical Physics, 2018Co-Authors: Marius Junge, Nicholas LaracuenteAbstract:We prove that for generalized dephasing channels, the coherent information and reverse coherent information coincides. It also implies an alternative approach for the strong super-additivity and strong converse of generalized dephasing channels using the Operator Space technique. Our argument is based on an improved Renyi relative entropy estimate via analyzing the channel’s Stinespring Space. We also apply this estimate to new examples of quantum channels arising from quantum group co-representation and Kitave’s quantum computation model. In particular, we find concrete examples of non-degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity.We prove that for generalized dephasing channels, the coherent information and reverse coherent information coincides. It also implies an alternative approach for the strong super-additivity and strong converse of generalized dephasing channels using the Operator Space technique. Our argument is based on an improved Renyi relative entropy estimate via analyzing the channel’s Stinespring Space. We also apply this estimate to new examples of quantum channels arising from quantum group co-representation and Kitave’s quantum computation model. In particular, we find concrete examples of non-degradable channels that our estimates are tight and give a formula of nontrivial quantum capacity.
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unbounded violations of bipartite bell inequalities via Operator Space theory
Communications in Mathematical Physics, 2010Co-Authors: Marius Junge, Carlos Palazuelos, David Perezgarcia, Ignacio Villanueva, Michael M WolfAbstract:In this work we show that bipartite quantum states with local Hilbert Space dimension n can violate a Bell inequality by a factor of order \({{\rm \Omega} \left(\frac{\sqrt{n}}{\log^2n} \right)}\) when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and Operator Space theory and, in particular, the very recent noncommutative Lp embedding theory.
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Operator Space theory a natural framework for bell inequalities
Physical Review Letters, 2010Co-Authors: Marius Junge, Carlos Palazuelos, David Perezgarcia, Ignacio Villanueva, Michael M WolfAbstract:In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local Hilbert Space dimension n can violate a Bell inequality by a factor of order $\frac{\sqrt{n}}{\log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert Space dimension estimates and communication complexity are given.
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mixed norm inequalities and Operator Space lp embedding theory
2010Co-Authors: Marius Junge, Javier ParcetAbstract:The authors prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function Spaces which appear in quantum probability as square functions, conditioned square functions and maximal functions.
Zhongjin Ruan - One of the best experts on this subject based on the ideXlab platform.
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Cb-frames for Operator Spaces
Journal of Functional Analysis, 2016Co-Authors: Zhongjin RuanAbstract:Abstract In this paper, we introduce the concept of cb-frames for Operator Spaces. We show that there is a concrete cb-frame for the reduced free group C ⁎ -algebra C r ⁎ ( F 2 ) , which is derived from the infinite convex decomposition of the biorthogonal system ( λ s , δ s ) s ∈ F 2 . We show that, in general, a separable Operator Space X has a cb-frame if and only if it has the completely bounded approximation property if and only if it is completely isomorphic to a completely complemented subSpace of an Operator Space with a cb-basis. Therefore, a discrete group Γ is weakly amenable if and only if the reduced group C⁎-algebra C r ⁎ ( Γ ) has a cb-frame. Finally, we show that, in contrast to Banach Space case, there exists a separable Operator Space, which cannot be completely isomorphic to a subSpace of an Operator Space with a cb-basis.
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Cb-frames for Operator Spaces
arXiv: Operator Algebras, 2016Co-Authors: Zhongjin RuanAbstract:In this paper, we introduce the concept of cb-frames for Operator Spaces. We show that there is a concrete cb-frame for the reduced free group C*-algebra $C_r^*(F_2)$, which is derived from the infinite convex decomposition of the biorthogonal system $(\lambda_s, \delta_s)_{s \in F_2}$. We show that, in general, a separable Operator Space X has a cb-frame if and only if it has the completely bounded approximation property if and only if it is completely isomorphic to a completely complemented subSpace of an Operator Space with a cb-basis. Therefore, a discrete group $\Gamma$ is weakly amenable if and only if the reduced group C*-algebra $C^*_r(\Gamma)$ has a cb-frame. Finally, we show that, in contrast to Banach Space case, there exists a separable Operator Space, which can not be completely isomorphic to a subSpace of an Operator Space with a cb-basis.
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On p-approximation properties for p-Operator Spaces
Journal of Functional Analysis, 2010Co-Authors: Guimei An, Zhongjin RuanAbstract:Abstract This paper has a two-fold purpose. Let 1 p ∞ . We first introduce the p -Operator Space injective tensor product and study various properties related to this tensor product, including the p -Operator Space approximation property, for p -Operator Spaces on L p -Spaces. We then apply these properties to the study of the pseudofunction algebra PF p ( G ) , the pseudomeasure algebra PM p ( G ) , and the Figa–Talamanca–Herz algebra A p ( G ) of a locally compact group G . We show that if G is a discrete group, then most of approximation properties for the reduced group C ∗ -algebra C λ ∗ ( G ) , the group von Neumann algebra VN ( G ) , and the Fourier algebra A ( G ) (related to amenability, weak amenability, and approximation property of G ) have the natural p -analogues for PF p ( G ) , PM p ( G ) , and A p ( G ) , respectively. The p -completely bounded multiplier algebra M cb A p ( G ) plays an important role in this work.
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Operator Space tensor products and hopf convolution algebras
Journal of Operator Theory, 2003Co-Authors: Edward G Effros, Zhongjin RuanAbstract:It is shown how one may use Operator Space tensor product to define Hopf algebraic operations on the preduals of Hopf von Neumann algebras. A careful discussion of the extended Haagerup tensor product is presented which includes a useful technique for handling computations with products of infinite matrices.
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On the Local Lifting Property for Operator Spaces
Journal of Functional Analysis, 1999Co-Authors: Zhongjin RuanAbstract:We study the local lifting property for Operator Spaces. This is a natural non-commutative analogue of the Banach Space local lifting property, but is very different from the local lifting property studied in C*-algebra theory. We show that an Operator Space has the λ-local lifting property if and only if it is an LΓ1, λ Space. These Operator Space are λ-completely isomorphic to the Operator subSpaces of the Operator preduals of von Neumann algebras, and thus λ-locally reflexive. Moreover, we show that an Operator Space V has the λ-local lifting property if and only if its Operator Space dual V* is λ-injective.
Gilles Pisier - One of the best experts on this subject based on the ideXlab platform.
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Random matrices and subexponential Operator Spaces
Israel Journal of Mathematics, 2020Co-Authors: Gilles PisierAbstract:We introduce and study a generalization of the notion of exact Operator Space that we call subexponential. Using Random Matrices we show that the factorization results of Grothendieck type that are known in the exact case all extend to the subexponential case, but we exhibit (a continuum of distinct) examples of non-exact subexponential Operator Spaces, as well as a C*-algebra that is subexponential with constant 1 but not exact. We also show that OH, R + C and max(l2) (or any other maximal Operator Space) are not subexponential.
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Random Matrices and Subexponential Operator Spaces
arXiv: Operator Algebras, 2012Co-Authors: Gilles PisierAbstract:We introduce and study a generalization of the notion of exact Operator Space that we call subexponential. Using Random Matrices we show that the factorization results of Grothendieck type that are known in the exact case all extend to the subexponential case, but we exhibit (a continuum of distinct) examples of non-exact subexponential Operator Spaces, as well as a $C^*$-algebra that is subexponential with constant 1 but not exact. We also show that $OH$, $R+C$ and $\max(\ell_2)$ (or any other maximal Operator Space) are not subexponential.
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Grothendieck’s theorem for Operator Spaces
Inventiones Mathematicae, 2002Co-Authors: Gilles Pisier, Dimitri ShlyakhtenkoAbstract:We prove several versions of Grothendieck’s Theorem for completely bounded linear maps T:E→F *, when E and F are Operator Spaces. We prove that if E, F are C *-algebras, of which at least one is exact, then every completely bounded T:E→F * can be factorized through the direct sum of the row and column Hilbert Operator Spaces. Equivalently T can be decomposed as T=T r +T c where T r (resp. T c ) factors completely boundedly through a row (resp. column) Hilbert Operator Space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Blecher on the factorization of completely bounded bilinear forms on C *-algebras. Moreover, our result holds more generally for any pair E, F of “exact” Operator Spaces. This yields a characterization of the completely bounded maps from a C *-algebra (or from an exact Operator Space) to the Operator Hilbert Space OH. As a corollary we prove that, up to a complete isomorphism, the row and column Hilbert Operator Spaces and their direct sums are the only Operator Spaces E such that both E and its dual E * are exact. We also characterize the Schur multipliers which are completely bounded from the Space of compact Operators to the trace class.
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The “maximal” tensor product of Operator Spaces
Proceedings of the Edinburgh Mathematical Society, 1999Co-Authors: Timur Oikhberg, Gilles PisierAbstract:In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two Operator Spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional Operator Space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert Spaces). Moreover, we show that if an Operator Space E is such that, for any Operator Space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert Space.
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The ``maximal" tensor product of Operator Spaces
arXiv: Functional Analysis, 1997Co-Authors: Timur Oikhberg, Gilles PisierAbstract:In analogy with the maximal tensor product of $C^*$-algebras, we define the ``maximal" tensor product $E_1\otimes_\mu E_2$ of two Operator Spaces $E_1$ and $E_2$ and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: \ $E_1\otimes_h E_2 + E_2\otimes_h E_1$. Let $E$ be an $n$-dimensional Operator Space. As an application, we show that the equality $E^* \otimes_\mu E=E^* \otimes_{\rm min} E$ holds isometrically iff $E = R_n$ or $E=C_n$ (the row or column $n$-dimensional Hilbert Spaces). Moreover, we show that if an Operator Space $E$ is such that, for any Operator Space $F$, we have $F\otimes_{\min} E=F\otimes_{\mu} E$ isomorphically, then $E$ is completely isomorphic to either a row or a column Hilbert Space.
David P Blecher - One of the best experts on this subject based on the ideXlab platform.
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Involutive Operator algebras
Positivity, 2020Co-Authors: David P Blecher, Zhenhua WangAbstract:Examples of Operator algebras with involution include the Operator $$*$$ ∗ -algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix algebras, (complexifications) of real Operator algebras, and an Operator algebraic version of the complex symmetric Operators studied by Garcia, Putinar, Wogen, Zhu, and others. We investigate the general theory of involutive Operator algebras, and give many applications, such as a characterization of the symmetric Operator algebras introduced in the early days of Operator Space theory.
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Metric characterizations of isometries and of unital Operator Spaces and systems
Proceedings of the American Mathematical Society, 2011Co-Authors: David P Blecher, Matthew NealAbstract:We give some new characterizations of unitaries, isometries, unital Operator Spaces, unital function Spaces, Operator systems, C * -algebras, and related objects. These characterizations only employ the vector Space and Operator Space structure (not mentioning products, involutions, or any kind of function on the Space).
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Metric characterizations of isometries and of unital Operator Spaces and systems
arXiv: Operator Algebras, 2008Co-Authors: David P Blecher, Matthew NealAbstract:We give some new characterizations of unitaries, isometries, unital Operator Spaces, unital function Spaces, Operator systems, C*-algebras, and related objects. These characterizations only employ the vector Space and Operator Space structure.
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the shilov boundary of an Operator Space and the characterization theorems
Journal of Functional Analysis, 2001Co-Authors: David P BlecherAbstract:Abstract We study Operator Spaces, Operator algebras, and Operator modules from the point of view of the noncommutative Shilov boundary. In this attempt to utilize some noncommutative Choquet theory, we find that Hilbert C *-modules and their properties, which we studied earlier in the Operator Space framework, replace certain topological tools. We introduce certain multiplier Operator algebras and C *-algebras of an Operator Space, which generalize the algebras of adjointable Operators on a C *-module and the imprimitivity C *-algebra. It also generalizes a classical Banach Space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify and strengthen several theorems characterizing Operator algebras and modules. We also include some general notes on the commutative case of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about function modules.
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Multipliers of Operator Spaces, and the injective envelope
arXiv: Operator Algebras, 1999Co-Authors: David P Blecher, Vern I. PaulsenAbstract:We study the injective envelope I(X) of an Operator Space X, showing amongst other things that it is a self-dual C$^*-$module. We describe the diagonal corners of the injective envelope of the canonical Operator system associated with X. We prove that if X is an Operator $A-B$-bimodule, then A and B can be represented completely contractively as subalgebras of these corners. Thus, the Operator algebras that can act on X are determined by these corners of I(X) and consequently bimodules actions on X extend naturally to actions on I(X). These results give another characterization of the multiplier algebra of an Operator Space, which was introduced by the first author, and a short proof of a recent characterization of Operator modules, and a related result. As another application, we extend Wittstock's module map extension theorem, by showing that an Operator $A-B$-bimodule is injective as an Operator $A-B$-bimodule if and only if it is injective as an Operator Space.
D. Popescu - One of the best experts on this subject based on the ideXlab platform.
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A Variable Length Tentacle Manipulator Control System
Proceedings of the 2005 IEEE International Conference on Robotics and Automation, 2005Co-Authors: M. Ivanescu, N. Popescu, D. PopescuAbstract:The control problem of a class of tentacle arm, with variable length, that can achieve any position and orientation in 3D Space and can increase the length in order to get a better control in the constraint Operator Space is presented. First, the dynamic model of the system is inferred. In order to avoid the difficulties generated by the complexity of the nonlinear integral - differential model, the control problem is based on the artificial potential method. Then, the method is used for constrained motion in an environment with obstacles. Numerical simulations for spatial and planar tentacle models are presented in order to illustrate the efficiency of the method.
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A variable length hyperredundant arm control system
IEEE International Conference Mechatronics and Automation 2005, 2005Co-Authors: M. Ivanescu, N. Popescu, D. PopescuAbstract:The control problem of a class of hyperredundant arm, with variable length, that can achieve any position and orientation in 3D Space and can increase the length in order to get a better control in the constraint Operator Space is presented. First, the dynamic model of the system is inferred. In order to avoid the difficulties generated by the complexity of the nonlinear integral-differential model, the control problem is based on the artificial potential method. Then, the method is used for constrained motion in an environment with obstacles. Numerical simulations for spatial and planar models are presented in order to illustrate the efficiency of the method.
