Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Florent Baudier, Gilles Lancien, Thomas Schlumprecht
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Not every infinite dimensional Banach space coarsely contains Hilbert space” by Florent Baudier<https://arxiv.org/find/math/1/au:+Baudier_F/0/1/0/all/0/1>, Gilles Lancien<https://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>. Abstract: In this article a new concentration inequality is proven for Lipschitz maps on the infinite Hamming graphs and taking values in Tsirelson's original space. This concentration inequality is then used to disprove the conjecture that the separable infinite dimensional Hilbert space coarsely embeds into every infinite dimensional Banach space. Some positive embeddability results are proven for the infinite Hamming graphs and the countably branching trees using the theory of spreading models. A purely metric characterization of finite dimensionality is also obtained, as well as a rigidity result pertaining to the spreading model set for Banach spaces coarsely embeddable into Tsirelson's original space. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.06797

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Abstract of a paper by
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Power type asymptotically uniformly smooth and asymptotically uniformly flat norms” by Ryan M Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: We provide a short characterization of $p$-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for $1<p<\infty$, the class of $p$-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class a Banach ideal. We also prove that the class of asymptotically uniformly flattenable operators can be endowed with an ideal norm making this class a Banach ideal. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.05484

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Abstract of a paper by
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper 1-Greedy renormings of Garling sequence spaces” by Fernado Albiac<https://arxiv.org/find/math/1/au:+Albiac_F/0/1/0/all/0/1>, José L. Ansorena<https://arxiv.org/find/math/1/au:+Ansorena_J/0/1/0/all/0/1>, Ben Wallis<https://arxiv.org/find/math/1/au:+Wallis_B/0/1/0/all/0/1>. Abstract: Garling sequence spaces admit a renorming with respect to which their standard unit vector basis is 1-greedy. We also discuss some additional properties of these Banach spaces related to uniform convexity and superreflexivity. In particular, our approach to the study of the superreflexivity of Garling sequence space provides an example of how essentially non-linear tools from greedy approximation can be used to shed light into the linear structure of the spaces. The paper may be downloaded from t

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Abstract of a paper by Trond A. Abrahamsen, Petr Hájek, Olav Nygaard, Stanimir Troyanski
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper Strongly extreme points and approximation properties” by Trond A. Abrahamsen<https://arxiv.org/find/math/1/au:+Abrahamsen_T/0/1/0/all/0/1>, Petr Hájek<https://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Olav Nygaard<https://arxiv.org/find/math/1/au:+Nygaard_O/0/1/0/all/0/1>, Stanimir Troyanski<https://arxiv.org/find/math/1/au:+Troyanski_S/0/1/0/all/0/1>. Abstract: We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but none of these points are denting. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.02625

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Abstract of a paper by Luis García-Lirola, Matías Raja
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “On strong asymptotic uniform smoothness and convexity” by Luis García-Lirola<https://arxiv.org/find/math/1/au:+Garcia_Lirola_L/0/1/0/all/0/1>, Matías Raja<https://arxiv.org/find/math/1/au:+Raja_M/0/1/0/all/0/1>. Abstract: We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to uniformly smooth spaces admitting a monotone FDD, extending a result by Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov. Our techniques also provide a characterisation of Orlicz functions $M, N$ such that the space of compact operators $\mathcal{K}(h_M, h_N)$ is asymptotically uniformly smooth. Finally we show that $\mathcal{K}(X, Y)$ is not strictly convex whenever $X$ and $Y$ are at least two-dimensional, which extends a result by Dilworth and Kutzarova. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.02020

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Abstract of a paper by Jinlu Li
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “On the Normality, Regularity and Chain-completeness of Partially Ordered Banach Spaces and Applications” by Jinlu Li<https://arxiv.org/find/math/1/au:+Li_J/0/1/0/all/0/1>. Abstract: In this paper, we study the connections between the normality, regularity, full regularity, and chain-complete property in partially ordered Banach spaces. Then, by applying these properties, we prove some fixed point theorems on partially ordered Banach spaces. As applications of these fixed point theorems, we prove the existence of solutions of some integral equations, such as Hammerstein integral equations, in Banach spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.01580

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Abstract of a paper by Petr Hájek, Tommaso Russo
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Some remarks on smooth renormings of Banach spaces” by Petr Hájek<https://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Tommaso Russo<https://arxiv.org/find/math/1/au:+Russo_T/0/1/0/all/0/1>. Abstract: We prove that in every separable Banach space X with a Schauder basis and a $C_k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C_k$-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler.. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.01384

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Abstract of a paper by Gilles Lancien, Matias Raja
by Bentuo Zheng (bzheng) 01 Jun '17

01 Jun '17

This is an announcement for the paper “Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces” by Gilles Lancien<https://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1>, Matias Raja<https://arxiv.org/find/math/1/au:+Raja_M/0/1/0/all/0/1>. Abstract: In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we show for instance, that for $1\leq q<p$, a $q$-asymptotically uniformly convex Banach space does not coarse Lipschitz embed into a $p$-asymptotically uniformly smooth quasi-reflexive Banach space. This extends a recent result of B.M. Braga. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.00577

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Workshop at Texas A&M
by Bill Johnson 04 May '17

04 May '17

Workshop in Analysis and Probability Department of Mathematics Texas A&M University Summer 2017 The Summer 2017 Workshop in Analysis and Probability at Texas A&M University will be in session from July 5 to August 4. All activities will take place in the Blocker Building. The homepage of the Workshop can be found at http://www.math.tamu.edu/~kerr/workshop [www.math.tamu.edu] The Summer Informal Regional Functional Analysis Seminar (SUMIRFAS) will be held July 21-23. Its homepage is located at http://www.math.tamu.edu/~kerr/workshop/sumirfas2017 [www.math.tamu.edu] June 5-9 there will be a Concentration Week, "Ergodic Theory and Operator Algebras", organized by Lewis Bowen and David Kerr. The goal of the meeting is to explore current trends in abstract ergodic theory and its interactions with the structure theory of von Neumann algebras and C*-algebras. Topics include entropy, measured equivalence relations, amenability, soficity, rigidity, random processes on networks, invariant random subgroups, and C*-simplicity. The homepage of the Concentration Week is located at http://www.math.tamu.edu/~kerr/etoa2017 [www.math.tamu.edu] July 17-21 there will be a Concentration Week, "Probabilistic and Algebraic Methods in Quantum Information Theory", organized by Michael Brannan and Benoit Collins. The past decade has seen a spectacular development of powerful new mathematical tools in quantum information theory, including random matrix theory, free probability theory, representation theory, tensor categories, quantum groups, non-commutative harmonic analysis, operator spaces, and the theory of non-local games. The aim of this concentration week is to bring together both leading experts and young researchers in these fields to further explore these emerging connections. It is intended to be multidisciplinary, with the hope of fostering communications between researchers with different backgrounds and interests related to quantum information theory. The homepage of the Concentration Week is located at https://sites.google.com/site/probabalgebramethodsinquantum [sites.google.com] The Workshop is supported in part by grants from the National Science Foundation (NSF). Minorities, women, graduate students, and young researchers are especially encouraged to attend. For logistical support, including requests for support, please contact Cara Starmer <cara(a)math.tamu.edu>. For more information on the Workshop itself, please contact William Johnson <johnson(a)math.tamu.edu>, David Kerr <kerr(a)math.tamu.edu>, or Gilles Pisier <pisier(a)math.tamu.edu>. For information about the Concentration Week "Ergodic Theory and Operator Algebras" please contact Lewis Bowen <lpbowen(a)math.utexas.edu> or David Kerr <kerr(a)math.tamu.edu>. For information about the Concentration Week "Probabilistic and Algebraic Methods in Quantum Information Theory", please contact Michael Brannan <mbrannan(a)math.tamu.edu> or Benoit Collins <collins(a)math.kyoto-u.ac.jp>.

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Abstract of a paper by Vladimir Kadets, Mariia Soloviova
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “Quantitative version of the Bishop-Phelps-Bollobás theorem for operators with values in a space with the property $\beta$” by Vladimir Kadets<https://arxiv.org/find/math/1/au:+Kadets_V/0/1/0/all/0/1>, Mariia Soloviova<https://arxiv.org/find/math/1/au:+Soloviova_M/0/1/0/all/0/1>. Abstract: The Bishop-Phelps-Bollob\'as property for operators deals with simultaneous approximation of an operator $T$ and a vector $x$ at which $T: X\rightarrow Y$ nearly attains its norm by an operator $F$ and a vector $z$, respectively, such that $F$ attains its norm at $z$. We study the possible estimates from above and from below for parameters that measure the rate of approximation in the Bishop-Phelps-Bollob\'as property for operators for the case of $Y$ having the property $\beta$ of Lindenstrauss. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.07095

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