Banach September 2016

banach@mathdept.okstate.edu

1 participants
11 discussions

Announcement of a new book "Open Problems in the Geometry and Analysis of Banach Spaces" by A. J. Guirao, V. Montesinos, and V. Zizler
by Bentuo Zheng (bzheng) 22 Sep '16

22 Sep '16

Dear Colleague: We are glad to announce the appearance of the book "Open Problems in the Geometry and Analysis of Banach Spaces", by A. J. Guirao, V. Montesinos, and V. Zizler, published by Springer. http://www.springer.com/gp/book/9783319335711 All the best, Vicente Montesinos

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Announcement of a new book "Martingales in Banach Spaces" by Gilles Pisier
by Bentuo Zheng (bzheng) 16 Sep '16

16 Sep '16

Dear Colleague, I would like to announce the appearance (at long last !) of my book Martingales in Banach spaces published by Cambridge Univ. Press https://www.cambridge.org/core/books/martingales-in-banach-spaces/A72C7511C… With best wishes (and apologies for this advertisement), gilles

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Abstract of a paper by Omid Zabeti
by Bentuo Zheng (bzheng) 04 Sep '16

04 Sep '16

This is an announcement for the paper “Unbounded absolute weak convergence in Banach lattices” by Omid Zabeti<http://arxiv.org/find/math/1/au:+Zabeti_O/0/1/0/all/0/1>. Abstract: The concepts of unbounded norm convergent nets and unbounded order convergent ones are considered and investigated in several recent papers by Gao, Deng, and et al. In this note, taking idea from these notions, we consider the concept of an unbounded absolute weak convergent (uaw) net in a Banach lattice. A net $(x_\alpha)$ in a Banach lattice E is said to be uaw-convergent to $x\in E$ if for each $u\in E_+$, the net $(|x_\alpha – x|\wedge u)$ converges to zero weakly. We investigate some properties of uaw-convergence and its relationship to other types of unbounded convergent nets. In particular, we characterize order continuous Banach lattices and reflexive Banach lattices in term of uaw-convergence. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.02151

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Abstract of a paper by Mikołaj Krupski, Witold Marciszewski
by Bentuo Zheng (bzheng) 01 Sep '16

01 Sep '16

This is an announcement for the paper “On the weak and pointwise topologies in function spaces II” by Mikołaj Krupski<http://arxiv.org/find/math/1/au:+Krupski_M/0/1/0/all/0/1>, Witold Marciszewski<http://arxiv.org/find/math/1/au:+Marciszewski_W/0/1/0/all/0/1>. Abstract: For a compact space $K$ we denote by $C_w(K) (C_p(K))$ the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we discuss the following basic question which seems to be open: Let K and L be infinite compact spaces. Can it happen that $C_w(K)$ and $C_p(L)$ are homeomorphic? M. Krupski proved that the above problem has a negative answer when $K=L$ and $K$ is finite-dimensional and metrizable. We extend this result to the class of finite-dimensional Valdivia compact spaces $K$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.03883

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Abstract of a paper by M. Kandić, M.A.A. Marabeh, V.G. Troitsky
by Bentuo Zheng (bzheng) 01 Sep '16

01 Sep '16

This is an announcement for the paper “Unbounded Norm Topology in Banach Lattices” by M. Kandić<http://arxiv.org/find/math/1/au:+Kandic_M/0/1/0/all/0/1>, M.A.A. Marabeh<http://arxiv.org/find/math/1/au:+Marabeh_M/0/1/0/all/0/1>, V.G. Troitsky<http://arxiv.org/find/math/1/au:+Troitsky_V/0/1/0/all/0/1>. Abstract: A net $(x_{\alpha})$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $\||x_\alpha-x|\wedge u|\|\rightarrow 0$ for every $u\in X_+$. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff $X$ has a strong unit. Un-topology is metrizable iff $X$ has a quasi-interior point. Suppose that $X$ is order continuous, then un-topology is locally convex iff $X$ is atomic. An order continuous Banach lattice $X$ is a KB-space iff its closed unit ball $B_X$ is un-complete. For a Banach lattice $X$, $B_X$ is un-compact iff $X$ is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak$^*$-convergence.. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.05489

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Abstract of a paper by Lukiel Levy-Moore, Margaret Nichols, Anthony Weston
by Bentuo Zheng (bzheng) 01 Sep '16

01 Sep '16

This is an announcement for the paper “Comparing the generalized roundness of metric spaces” by Lukiel Levy-Moore<http://arxiv.org/find/math/1/au:+Levy_Moore_L/0/1/0/all/0/1>, Margaret Nichols<http://arxiv.org/find/math/1/au:+Nichols_M/0/1/0/all/0/1>, Anthony Weston<http://arxiv.org/find/math/1/au:+Weston_A/0/1/0/all/0/1>. Abstract: Motivated by the local theory of Banach spaces we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalized roundness of metric spaces. We illustrate this technique in two different ways by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if $\mathcal{U}$ is any ultrafilter and $X$ is any Banach space, then the second dual $X^{**}$ and the ultrapower $(X)_{\mathcal{U}}$ have the same generalized roundness as $X$, and (2) no Banach space of positive generalized roundness is uniformly homeomorphic to $c_0$ or $\ell_p, 2<p<\infty$. Our technique also leads to the identification of new classes of metric trees of generalized roundness one. In particular, we give the first examples of metric trees of generalized roundness one that have finite diameter. These results on metric trees provide a natural sequel to a paper of Caffarelli, Doust and Weston. In addition, we show that metric trees of generalized roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.03699

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Abstract of a paper by Marek Cuth
by Bentuo Zheng (bzheng) 01 Sep '16

01 Sep '16

This is an announcement for the paper “Separable determination in Banach space” by Marek Cuth<http://arxiv.org/find/math/1/au:+Cuth_M/0/1/0/all/0/1>. Abstract: We study a relation between three different formulations of theorems on separable determination - one using the concept of rich families, second via the concept of suitable models and third, a new one, suggested in this paper, using the notion of $\omega$-monotone mappings. In particular, we show that in Banach spaces all those formulations are in a sense equivalent and we give a positive answer to two questions of O. Kalenda and the author. Our results enable us to obtain new statements concerning separable determination of $sigma$-porosity (and of similar notions) in the language of rich families; thus, not using any terminology from logic or set theory. Moreover, we prove that in Asplund spaces, generalized lushness is separably determined. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.03685

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Abstract of a paper by Ryan M Causey
by Bentuo Zheng (bzheng) 01 Sep '16

01 Sep '16

This is an announcement for the paper “A note on the relationship between the Szlenk and w∗-dentability indices of arbitrary w∗-compact sets” by Ryan M Causey<http://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: We prove the optimal estimate between the Szlenk and $w^*$-dentability indices of an arbitrary $w^*$-compact subset of the dual of a Banach space. For a given $w^*$-compact, convex subset $K$ of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of $K$. We give applications to the $w^*$-dentability index of a Banach space and of an operator. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.03668

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Abstract of a paper by Ryan M Causey
by Bentuo Zheng (bzheng) 01 Sep '16

01 Sep '16

This is an announcement for the paper “Power type ξ-asymptotically uniformly smooth norms” by Ryan M Causey<http://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: We extend a precise renorming result of Godefroy, Kalton, and Lancien regarding asymptotically uniformly smooth norms of separable Banach spaces with Szlenk index $\omega$. For every ordinal $\xi$, we characterize the operators, and therefore the Banach spaces, which admit a $\xi$ -asymptotically uniformly smooth norm with power type modulus and compute for those operators the best possible exponent in terms of the values of $S_{z\xi}(\cdot, \epsilon)$. We also introduce the $\xi$-Szlenk power type and investigate ideal and factorization properties of classes associated with the $\xi$-Szlenk power type. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.03666

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Abstract of a paper by Debmalya Sain
by Bentuo Zheng (bzheng) 01 Sep '16

01 Sep '16

This is an announcement for the paper “On the norm attainment set of a bounded linear operator” by Debmalya Sain<http://arxiv.org/find/math/1/au:+Sain_D/0/1/0/all/0/1>. Abstract: In this paper we explore the properties of a bounded linear operator defined on a Banach space, in light of operator norm attainment. Using Birkhoff-James orthogonality techniques, we give a necessary condition for a bounded linear operator attaining norm at a particular point of the unit sphere. We prove a number of corollaries to establish the importance of our study. As part of our exploration, we also obtain a characterization of smooth Banach spaces in terms of operator norm attainment and Birkhoff-James orthogonality. Restricting our attention to $\ell_p^2 (p\in N, p\geq 2)$ spaces, we obtain an upper bound for the number of points at which any linear operator, which is not a scalar multiple of an isometry, may attain norm. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.00755

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Banach September 2016