Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Tony K. Nogueira and Daniel Pellegrino
by alspach＠math.okstate.edu 18 May '15

18 May '15

This is an announcement for the paper "On the size of certain subsets of invariant Banach sequence spaces" by Tony K. Nogueira and Daniel Pellegrino. Abstract: In this note we improve some recent results of G. Botelho and V. F\'{a}varo on invariant Banach sequence spaces. Our main result shows that, under very weak assumptions, more general versions of some subsets of invariant sequence spaces investigated by G. Botelho and V. F\'{a}varo in 2014 contain, up to the null vector, a closed infinite-dimensional subspace . Archive classification: math.FA Submitted from: pellegrino(a)pq.cnpq.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.08238 or http://arXiv.org/abs/1504.08238

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Abstract of a paper by Spiros A. Argyros, Ioannis Gasparis and Pavlos Motakis
by alspach＠math.okstate.edu 18 May '15

18 May '15

This is an announcement for the paper "On the structure of separable $\mathcal{L}_\infty$-spaces" by Spiros A. Argyros, Ioannis Gasparis and Pavlos Motakis. Abstract: Based on a construction method introduced by J. Bourgain and F. Delbaen, we give a general definition of a Bourgain-Delbaen space and prove that every infinite dimensional separable $\mathcal{L}_\infty$-space is isomorphic to such a space. Furthermore, we provide an example of a $\mathcal{L}_\infty$ and asymptotic $c_0$ space not containing $c_0$. Archive classification: math.FA Mathematics Subject Classification: 46B03, 46B06, 46B07 Remarks: 15 pages Submitted from: pmotakis(a)central.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.08223 or http://arXiv.org/abs/1504.08223

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Abstract of a paper by Liang Hong
by alspach＠math.okstate.edu 18 May '15

18 May '15

This is an announcement for the paper "On the relationship between order bounded operators, topologically bounded operators and topologically continuous operators" by Liang Hong. Abstract: The relationship between order bounded operators and order continuous operators has been investigated by several authors. The purpose of this paper is to study the relationship between order bounded operators, topologically bounded operators and topologically continuous operators. We give conditions for (i) the space of topologically continuous operators to be an ideal of the space of order bounded operators; this result generalizes the Nakano-Roberts Theorem; (ii) the space of topologically continuous operators to be a band of the space of order bounded operators; (iii) the space of order bounded operators to coincide with the space of topologically bounded operators; (iv) the space of order bounded operators to coincide with the space of topologically continuous operators. In addition, a set of counterexamples are given for illustration purpose; these counterexamples are interesting in their own rights and contribute to the literature. Archive classification: math.FA Mathematics Subject Classification: Primary 47B60, 47B65, Secondary 46A40, 06B30, 06F30 Submitted from: hong(a)rmu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.08016 or http://arXiv.org/abs/1504.08016

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Abstract of a paper by Triloki Nath
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "Differentiability of Distance Function and The Proximinal Condition implying Convexity" by Triloki Nath. Abstract: A necessary and sufficient condition for the differentiability of the distance function generated by an almost proximinal closed set has been given for normed linear spaces with locally uniformly convex and differentiable norm. We prove that the proximinal condition of Giles [6] is true for almost sun. In such spaces if the proximinal condition is satisfied and the distance function is uniformly differentiable on a dense set then it turn in the differentiability on all off the set (generating the distance function). The proximinal condition ensures about the convexity of almost sun in some spaces under a differentiability condition of the distance function. A necessary and sufficient condition is derived for the convexity of Chebyshev sets in Banach spaces with rotund dual. Archive classification: math.FA Mathematics Subject Classification: 41A65, 46B20 Remarks: 9 pages Submitted from: tnath(a)dhsgsu.ac.in The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.07292 or http://arXiv.org/abs/1504.07292

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Abstract of a paper by Gilles Lancien, Antonin Prochazka, and Matias Raja
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "Szlenk indices of convex hulls" by Gilles Lancien, Antonin Prochazka, and Matias Raja. Abstract: We study the general measures of non compactness defined on subsets of a dual Banach space, their associated derivations and their $\omega$-iterates. We introduce the notion of convexifiable measure of non compactness and investigate the properties of its associated fragment and slice derivations. We apply our results to the Kuratowski measure of non compactness and to the study of the Szlenk index of a Banach space. As a consequence, we obtain, for any countable ordinal $\alpha$, a characterization of the Banach spaces with Szlenk index bounded by $\omega^{\alpha+1}$ in terms of the existence an equivalent renorming. This extends a result by Knaust, Odell and Schlumprecht on Banach spaces with Szlenk index equal to $\omega$. Archive classification: math.FA Mathematics Subject Classification: 46B20 Submitted from: gilles.lancien(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.06997 or http://arXiv.org/abs/1504.06997

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Abstract of a paper by Martin Bohata, Jan Hamhalter and Ondrej F.K. Kalenda
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "On Markushevich bases in preduals of von Neumann algebras" by Martin Bohata, Jan Hamhalter and Ondrej F.K. Kalenda. Abstract: We prove that the predual of any von Neumann algebra is $1$-Plichko, i.e., it has a countably $1$-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U.~Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the self-adjoint part of the predual is $1$-Plichko as well. Archive classification: math.FA math.OA Mathematics Subject Classification: 46B26, 46L10 Remarks: 13 pages Submitted from: kalenda(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.06981 or http://arXiv.org/abs/1504.06981

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Abstract of a paper by Ondrej Kurka
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "Amalgamations of classes of Banach spaces with a monotone basis" by Ondrej Kurka. Abstract: It was proved by Argyros and Dodos that, for many classes $ C $ of separable Banach spaces which share some property $ P $, there exists an isomorphically universal space that satisfies $ P $ as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that $ C $ consists of spaces with a monotone Schauder basis. For example, we prove that if $ C $ is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every $ X \in C $ is reflexive and has a monotone Schauder basis, then there exists a separable reflexive Banach space that is isometrically universal for $ C $. Archive classification: math.FA Mathematics Subject Classification: 46B04, 54H05 (Primary) 46B15, 46B20, 46B70 (Secondary) Submitted from: kurka.ondrej(a)seznam.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.06862 or http://arXiv.org/abs/1504.06862

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Abstract of a paper by Michal Doucha
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "An example of a non-commutative uniform Banach group" by Michal Doucha. Abstract: We construct a non-commutative uniform Banach group which has the free group of countably many generators as a dense subgroup. Archive classification: math.FA math.GN math.GR Submitted from: m.doucha(a)post.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.05841 or http://arXiv.org/abs/1504.05841

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Abstract of a paper by Jean Bourgain and Mark Lewko
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "Sidonicity and variants of Kaczmarz's problem" by Jean Bourgain and Mark Lewko. Abstract: We prove that a uniformly bounded system of orthonormal functions satisfying the $\psi_2$ condition: (1) must contain a Sidon subsystem of proportional size, (2) must satisfy the Rademacher-Sidon property, and (3) must have its 5-fold tensor satisfy the Sidon property. On the other hand, we construct a uniformly bounded orthonormal system that satisfies the $\psi_2$ condition but which is not Sidon. These problems are variants of Kaczmarz's Scottish book problem (problem 130) which, in its original formulation, was answered negatively by Rudin. A corollary of our argument is a new, elementary proof of Pisier's theorem that a set of characters satisfying the $\psi_2$ condition is Sidon. Archive classification: math.CA math.FA math.PR Remarks: 22 pages, no figures Submitted from: mlewko(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.05290 or http://arXiv.org/abs/1504.05290

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Abstract of a paper by S. Gabriyelyan, J. Kakol, and G. Plebanek
by alspach＠math.okstate.edu 29 Apr '15

29 Apr '15

This is an announcement for the paper "The Ascoli property for function spaces and the weak topology of Banach and Fr\'echet spaces" by S. Gabriyelyan, J. Kakol, and G. Plebanek. Abstract: Following [3] we say that a Tychonoff space $X$ is an Ascoli space if every compact subset $\mathcal{K}$ of $C_k(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_\mathbb{R}$-space, hence any $k$-space, is Ascoli. Let $X$ be a metrizable space. We prove that the space $C_{k}(X)$ is Ascoli iff $C_{k}(X)$ is a $k_\mathbb{R}$-space iff $X$ is locally compact. Moreover, $C_{k}(X)$ endowed with the weak topology is Ascoli iff $X$ is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of $\ell_1$, we show that the following assertions are equivalent for a Banach space $E$: (i) $E$ does not contain isomorphic copy of $\ell_1$, (ii) every real-valued sequentially continuous map on the unit ball $B_{w}$ with the weak topology is continuous, (iii) $B_{w}$ is a $k_\mathbb{R}$-space, (iv) $B_{w}$ is an Ascoli space. We prove also that a Fr\'{e}chet lcs $F$ does not contain isomorphic copy of $\ell_1$ iff each closed and convex bounded subset of $F$ is Ascoli in the weak topology. However we show that a Banach space $E$ in the weak topology is Ascoli iff $E$ is finite-dimensional. We supplement the last result by showing that a Fr\'{e}chet lcs $F$ which is a quojection is Ascoli in the weak topology iff either $F$ is finite dimensional or $F$ is isomorphic to the product $\mathbb{K}^{\mathbb{N}}$, where $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$. Archive classification: math.FA math.GN Mathematics Subject Classification: 46A04, 46B03, 54C30 Submitted from: saak(a)bgu.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1504.04202 or http://arXiv.org/abs/1504.04202

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