Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Asuman Guven Aksoy and Grzegorz Lewicki
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Limit theorems for numerical index" by Asuman Guven Aksoy and Grzegorz Lewicki. Abstract: We improve upon on a limit theorem for numerical index for large classes of Banach spaces including vector valued $\ell_p$-spaces and $\ell_p$-sums of Banach spaces where\\ $1\leq p \leq \infty$. We first prove $ n_1( X) = \displaystyle \lim_m n_1( X_m)$ for a modified numerical index $n_1(\, .\, )$. Later, we establish if a norm on $X$ satisfies the local characterization condition, then $n(X) = \displaystyle\lim_m n(X_m).$ We also present an example of a Banach space where the local characterization condition is satisfied. Archive classification: math.FA math.OA Submitted from: aaksoy(a)cmc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.4822 or http://arXiv.org/abs/1106.4822

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Abstract of a paper by Antonio Falco and Anthony Nouy
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces" by Antonio Falco and Anthony Nouy. Abstract: Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called Proper Generalized Decompositions methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated Proper Generalized Decompositions for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space. Archive classification: math.NA math.FA math.OC Mathematics Subject Classification: 65K10, 49M29 Submitted from: afalco(a)uch.ceu.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.4424 or http://arXiv.org/abs/1106.4424

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Abstract of a paper by B. F. Svaiter
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Gauge functions for convex cones" by B. F. Svaiter. Abstract: We analyze a class of sublinear functionals which characterize the interior and the exterior of a convex cone in a normed linear space. Archive classification: math.FA math.OC Mathematics Subject Classification: 46B99, 46N10 Submitted from: benar(a)impa.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.3342 or http://arXiv.org/abs/1106.3342

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Abstract of a paper by Justin Tatch Moore
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Amenability and Ramsey Theory" by Justin Tatch Moore. Abstract: The purpose of this article is to connect the notion of the amenability of a discrete group with a new form of structural Ramsey theory. The Ramsey theoretic reformulation of amenability constitutes a considerable weakening of the F\o lner criterion. As a bi-product, it will be shown that in any non amenable group G, there is a subset E of G such that no finitely additive probability measure on G measures all translates of E equally. Archive classification: math.GR math.CO math.FA math.LO Mathematics Subject Classification: 05D10, 05C55, 20F38, 20F65, 43A07 Remarks: 13 pages. Comments welcome Submitted from: justin(a)math.cornell.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.3127 or http://arXiv.org/abs/1106.3127

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Abstract of a paper by Nikhil Srivastava and Roman Vershynin
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Covariance estimation for distributions with 2+ε moments" by Nikhil Srivastava and Roman Vershynin. Abstract: We study the minimal sample size N=N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, and with an arbitrary fixed accuracy. We establish the optimal bound N = O(n) for every distribution whose k-dimensional marginals have uniformly bounded 2+\epsilon moments outside the sphere of radius O(\sqrt{k}). In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan-Lovasz-Simonovits problem, which was recently solved by Adamczak, Litvak, Pajor and Tomczak-Jaegermann. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption -- uniformly bounded 2+\epsilon moments of one-dimensional marginals. Our argument proceeds by randomizing the spectral sparsification technique of Batson, Spielman and Srivastava. The spectral edges of the sample covariance matrix are controlled via the Stieltjes transform evaluated at carefully chosen random points. Archive classification: math.PR math.ST Submitted from: nikhils(a)math.ias.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.2775 or http://arXiv.org/abs/1106.2775

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Abstract of a paper by Piotr Koszmider
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "On large indecomposable Banach spaces" by Piotr Koszmider. Abstract: Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an indecomposable Banach space of density two to continuum. The space exists consistently, is of the form C(K) and it has few operators in the sense that any bounded linear operator T on C(K) satisfies T(f)=gf+S(f) for every f in C(K), where g is in C(K) and S is weakly compact (strictly singular). Archive classification: math.FA math.GN math.LO Submitted from: piotr.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.2916 or http://arXiv.org/abs/1106.2916

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Abstract of a paper by Piotr Koszmider
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "A C(K) Banach space which does not have the Schroeder-Bernstein property" by Piotr Koszmider. Abstract: We construct a totally disconnected compact Hausdorff space N which has clopen subsets M included in L included in N such that N is homeomorphic to M and hence C(N) is isometric as a Banach space to C(M) but C(N) is not isomorphic to C(L). This gives two nonisomorphic Banach spaces of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces of the form C(K). N is obtained as a particular compactification of the pairwise disjoint union of a sequence of Ks for which C(K)s have few operators. Archive classification: math.FA math.GN Submitted from: piotr.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.2917 or http://arXiv.org/abs/1106.2917

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Abstract of a paper by Ranjana Jain and Ajay Kumar
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Operator space projective tensor product: Embedding into second dual and ideal structure" by Ranjana Jain and Ajay Kumar. Abstract: We prove that for operator spaces $V$ and $W$, the operator space $V^{**}\otimes_h W^{**}$ can be completely isometrically embedded into $(V\otimes_h W)^{**}$, $\otimes_h$ being the Haagerup tensor product. It is also shown that, for exact operator spaces $V$ and $W$, a jointly completely bounded bilinear form on $V\times W$ can be extended uniquely to a separately $w^*$-continuous jointly completely bounded bilinear form on $ V^{**}\times W^{**}$. This paves the way to obtain a canonical embedding of $V^{**}\widehat{\otimes} W^{**}$ into $(V\widehat{\otimes} W)^{**}$ with a continuous inverse, where $\widehat{\otimes}$ is the operator space projective tensor product. Further, for $C^*$-algebras $A$ and $B$, we study the (closed) ideal structure of $A\widehat{\otimes}B$, which, in particular, determines the lattice of closed ideals of $B(H)\widehat{\otimes} B(H)$ completely. Archive classification: math.FA Mathematics Subject Classification: 46L06, 46L07, 47L25 Remarks: 13 pages Submitted from: rjain.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.2644 or http://arXiv.org/abs/1106.2644

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Abstract of a paper by Guillaume Aubrun, Stanislaw J. Szarek and Deping Ye
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "Entanglement thresholds for random induced states" by Guillaume Aubrun, Stanislaw J. Szarek and Deping Ye. Abstract: For a random quantum state on $H=C^d \otimes C^d$ obtained by partial tracing a random pure state on $H \otimes C^s$, we consider the whether it is typically separable or typically entangled. We show that a threshold occurs when the environment dimension $s$ is of order roughly $d^3$. More precisely, when $s \leq cd^3$, such a random state is entangled with very large probability, while when $s \geq Cd^3 \log^2 d$, it is separable with very large probability (here $C,c>0$ are appropriate effectively computable universal constants). Our proofs rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces. Our methods work also for multipartite systems and for "unbalanced" systems such as $C^{d} \otimes C^{d'}$, $d \neq d' $. Archive classification: quant-ph math.FA math.PR Report Number: Mittag-Leffler-2010fall Remarks: 29 pages Submitted from: szarek(a)cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.2264 or http://arXiv.org/abs/1106.2264

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Abstract of a paper by Taras Banakh and Ivan Hetman
by alspach＠math.okstate.edu 27 Jun '11

27 Jun '11

This is an announcement for the paper "A hidden characterization of polyhedral convex sets" by Taras Banakh and Ivan Hetman. Abstract: We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not = \emptyset$ for any distinct points $x,y\in A$. Archive classification: math.FA math.CO Mathematics Subject Classification: 46A55, 52B05, 52A07, 52A37 Remarks: 8 pages Submitted from: tbanakh(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1106.2227 or http://arXiv.org/abs/1106.2227

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