Banach

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Abstract of a paper by Piotr Koszmider
by alspach＠math.okstate.edu 06 Apr '15

06 Apr '15

This is an announcement for the paper "Uncountable equilateral sets in Banach spaces of the form $C(K)$" by Piotr Koszmider. Abstract: The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin's axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form $C(K)$ has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form $C(K)$ which has no uncountable equilateral set (or equivalently no uncountable $(1+\varepsilon)$-separated set in the unit sphere for any $\varepsilon>0$) making another consistent combinatorial assumption. The compact $K$ is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than $C(K)$ which has no uncountable equilateral set. It follows from the results of S. Mercourakis, G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples which have no uncountable equilateral sets in any equivalent renorming. It follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in in which it has an uncountable equilateral set. 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/1503.06356 or http://arXiv.org/abs/1503.06356

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Abstract of a paper by Martin Rmoutil
by alspach＠math.okstate.edu 23 Mar '15

23 Mar '15

This is an announcement for the paper "Norm-attaining functionals and proximinal subspaces" by Martin Rmoutil. Abstract: G. Godefroy asked whether, on any Banach space, the set of norm-attaining functionals contains a 2-dimensional linear subspace. We prove that a recent construction due to C.J. Read provides an example of a space which does not have this property. This is done through a study of the relation between the following two sentences where X is a Banach space and Y is a closed subspace of finite codimension in X: (A) Y is proximinal in X. (B) The annihilator of Y consists of norm-attaining functionals. We prove that these are equivalent if X is the Read's space. Moreover, we prove that any non-reflexive Banach space X with any given closed subspace Y of finite codimension at least 2 admits an equivalent norm such that (B) is true and (A) is false. Archive classification: math.FA Mathematics Subject Classification: 46B10, 46B20, 46B03 Submitted from: martin(a)rmoutil.eu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.06112 or http://arXiv.org/abs/1503.06112

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Abstract of a paper by Henry Towsner
by alspach＠math.okstate.edu 23 Mar '15

23 Mar '15

This is an announcement for the paper "An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space" by Henry Towsner. Abstract: The proof that the James space is not locally unconditional appears to be non-constructive, since it makes use of an ultraproduct construction. Using proof mining, we extract a constructive proof and obtain a lower bound on the growth of the local unconditionality constants. Archive classification: math.LO math.FA Mathematics Subject Classification: 46B15 Submitted from: htowsner(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.04745 or http://arXiv.org/abs/1503.04745

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Abstract of a paper by A.Vershik
by alspach＠math.okstate.edu 23 Mar '15

23 Mar '15

This is an announcement for the paper "Equipped graded graphs, projective limits of simplices, and their boundaries" by A.Vershik. Abstract: In this paper, we develop a theory of equipped graded graphs (or Bratteli diagrams) and an alternative theory of projective limits of finite-dimensional simplices. An equipment is an additional structure on the graph, namely, a system of ``cotransition'' probabilities on the set of its paths. The main problem is to describe all probability measures on the path space of a graph with given cotransition probabilities; it goes back to the problem, posed by E.~B.~Dynkin in the 1960s, of describing exit and entrance boundaries for Markov chains. The most important example is the problem of describing all central measures, to which one can reduce the problems of describing states on AF-algebras or characters on locally finite groups. We suggest an unification of the whole theory, an interpretation of the notions of Martin, Choquet, and Dynkin boundaries in terms of equipped graded graphs and in terms of the theory of projective limits of simplices. In the last section, we study the new notion of ``standardness'' of projective limits of simplices and of equipped Bratteli diagrams, as well as the notion of ``lacunarization.'' Archive classification: math.FA Mathematics Subject Classification: 37L40, 60J20 Remarks: 21 pp.Ref. 12 Submitted from: avershik(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.04447 or http://arXiv.org/abs/1503.04447

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Abstract of a paper by Kallol Paul, Debmalya Sain and Puja Ghosh
by alspach＠math.okstate.edu 23 Mar '15

23 Mar '15

This is an announcement for the paper "Smoothness of bounded linear operators" by Kallol Paul, Debmalya Sain and Puja Ghosh. Abstract: We prove that for a bounded linear operator $T$ on a Hilbert space $\mathbb{H},$ $T \bot_B A \Leftrightarrow \langle Tx, Ax \rangle = 0 $ for some $x \in S_{\mathbb{H}}, \|Tx\| = \|T\| $ iff the norm attaining set $M_T = \{ x \in S_{\mathbb{H}} : \|Tx\| = \|T\|\} $ is a unit sphere of some finite dimensional subspace $H_0$ of $\mathbb{H}$ i.e., $M_T = S_{H_0} $ and $\|T\|_{{H_0}^{\bot}} < \|T\|.$ We also prove that if $T$ is a bounded linear operator on a Banach space $\mathbb{X}$ with the norm attaining set $M_T = D \cup(-D)$ ( $D$ is a non-empty compact connected subset of $S_{\mathbb{X}}$) and $\sup_{y \in C} \|Ty\| < \|T\|$ for all closed subsets $C$ of $S_{\mathbb{X}}$ with $d(M_T,C) > 0,$ then $T \bot_B A \Leftrightarrow Tx \bot_B Ax $ for some $x \in M_T.$ Using these results we characterize smoothness of compact operators on normed linear spaces and smoothness of bounded linear operators on Hilbert as well as Banach spaces. This is for the first time that a characterization of smoothness of bounded linear operators on a normed linear space has been obtained. We prove that $T \in B(\mathbb{X}, \mathbb{Y})$ (where $\mathbb{X}$ is a real Banach space and $\mathbb{Y}$ is a real normed linear space) is smooth iff $T$ attains its norm at unique (upto muliplication by scalar) vector $ x \in S_{\mathbb{X}},$ $Tx$ is a smooth point of $\mathbb{Y} $ and $\sup_{y \in C} \|Ty\| < \|T\|$ for all closed subsets $C$ of $S_{\mathbb{X}}$ with $d(\pm x,C) > 0.$ Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B50 Remarks: 13 pages Submitted from: kalloldada(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.03683 or http://arXiv.org/abs/1503.03683

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Abstract of a paper by Eva Pernecka
by alspach＠math.okstate.edu 23 Mar '15

23 Mar '15

This is an announcement for the paper "On uniformly differentiable mappings" by Eva Pernecka. Abstract: We are concerned with the rigidity of $\ell_\infty$ and $\ell_\infty^n$ with respect to uniformly differentiable mappings. Our main result is a non-linear analogy of the classical result on the rigidity of $\ell_\infty$ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on the non-complementability of $c_0$ in $\ell_\infty$ due to Phillips. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46T20 Submitted from: pernecka(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.03536 or http://arXiv.org/abs/1503.03536

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Abstract of a paper by Daniel Carando, Veronica Dimant, Santiago Muro, and Damian Pinasco
by alspach＠math.okstate.edu 23 Mar '15

23 Mar '15

This is an announcement for the paper "An integral formula for multiple summing norms of operators" by Daniel Carando, Veronica Dimant, Santiago Muro, and Damian Pinasco. Abstract: We prove that the multiple summing norm of multilinear operators defined on some $n$-dimensional real or complex vector spaces with the $p$-norm may be written as an integral with respect to stables measures. As an application we show inclusion and coincidence results for multiple summing mappings. We also present some contraction properties and compute or estimate the limit orders of this class of operators. Archive classification: math.FA Mathematics Subject Classification: 15A69, 15A60, 47B10, 47H60, 46G25 Remarks: 19 pages Submitted from: smuro(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.01638 or http://arXiv.org/abs/1503.01638

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Abstract of a paper by Danie Carando, Andreas Defant, and Pablo Sevilla-Peris
by alspach＠math.okstate.edu 23 Mar '15

23 Mar '15

This is an announcement for the paper "Some polynomial versions of cotype and applications" by Danie Carando, Andreas Defant, and Pablo Sevilla-Peris. Abstract: We introduce non-linear versions of the classical cotype of Banach spaces. We show that spaces with l.u.st and cotype, and that spaces having Fourier cotype enjoy our non-linear cotype. We apply these concepts to get results on convergence of vector-valued power series in infinite many variables and on $\ell_{1}$-multipliers of vector-valued Dirichlet series. Finally we introduce cotype with respect to indexing sets, an idea that includes our previous definitions. Archive classification: math.FA Submitted from: psevilla(a)mat.upv.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.00850 or http://arXiv.org/abs/1503.00850

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Abstract of a paper by Esteban Andruchow, Eduardo Chiumiento and Maria Eugenia Di Iorio y Lucero
by alspach＠math.okstate.edu 23 Mar '15

23 Mar '15

This is an announcement for the paper "Proper subspaces and compatibility" by Esteban Andruchow, Eduardo Chiumiento and Maria Eugenia Di Iorio y Lucero. Abstract: Let $\mathcal{E}$ be a Banach space contained in a Hilbert space $\mathcal{L}$. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambicki\v{\i}, we say that a bounded operator on $\mathcal{E}$ is a proper operator if it admits an adjoint with respect to the inner product of $\mathcal{L}$. By a proper subspace $\mathcal{S}$ we mean a closed subspace of $\mathcal{E}$ which is the range of a proper projection. If there exists a proper projection which is also self-adjoint with respect to the inner product of $\mathcal{L}$, then $\mathcal{S}$ belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition to ensure that a proper subspace is compatible. Each proper subspace $\mathcal{S}$ has a supplement $\mathcal{T}$ which is also a proper subspace. We give a characterization of the compatibility of both subspaces $\mathcal{S}$ and $\mathcal{T}$. Several examples are provided that illustrate different situations between proper and compatible subspaces. Archive classification: math.FA Remarks: 18 pages Submitted from: eduardo(a)mate.unlp.edu.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.00596 or http://arXiv.org/abs/1503.00596

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Relations Between Banach Space Theory and Geometric Measure Theory workshop, 08 - 12 June 2015, University of Warwick, UK
by Olga Maleva 19 Mar '15

19 Mar '15

2nd ANNOUNCEMENT OF THE WORKSHOP Relations Between Banach Space Theory and Geometric Measure Theory 08 - 12 June 2015 University of Warwick United Kingdom Confirmed plenary speakers include: Jesus M F Castillo (Universidad de Extremadura) Gilles Godefroy (Université Paris VI) William B Johnson (Texas A&M University) Assaf Naor (Princeton University) Mikhail Ostrovskii (St. John’s University) Gideon Schechtman (Weizmann Institute) Thomas Schlumprecht (Texas A&M University) The homepage of the workshop is: http://tinyurl.com/BanachGMT To register please follow the links on the homepage of the workshop. NEW: List of currently registered participants is available on the website of the workshop For further information on the workshop please contact the organisers: * David Preiss <d dot preiss at warwick dot ac dot uk> * Olga Maleva <o dot maleva at bham dot ac dot uk> We expect to be able to cover some expenses for a number of participants. Please read more information on the homepage about the funding. Please register your attendance at the workshop by 15 April 2015. The Workshop is supported by a European Research Council grant.

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