Banach

banach@mathdept.okstate.edu

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2nd Announcement of BWB 2018
by Christina Brech 16 Dec '17

16 Dec '17

2nd Announcement of BWB 2018 Second Brazilian Workshop in Geometry of Banach Spaces August 13-17, 2018 Maresias, Sao Paulo State, Brazil. (Satellite Conference of the ICM 2018) We would like to remind you that we are organizing the Second Brazilian Workshop in Geometry of Banach Spaces, as a satellite conference of the ICM 2018. We remind you that the *deadline for abstract submissions to the ICM (Rio de Janeiro) is January 5* and further details can be found at www.icm2018.org. The BWB will take place at the Beach Hotel Maresias, on the coast of Sao Paulo State, in Maresias, in the week August 13-17, 2018. The scientific program will focus on the theory of geometry of Banach spaces, with emphasis on the following directions: large scale geometry of Banach spaces; nonlinear theory; homological theory and set theory. Registration will start early 2018 and additional scientific, practical and financial information can be found on website www.ime.usp.br/~ banach/bwb2018/. Plenary speakers: S. A. Argyros (Nat. Tech. U. Athens) G. Godefroy (Paris 6) S. Grivaux* (U. Picardie Jules Verne) R. Haydon* (U. Oxford) W. B. Johnson (Texas A&M) J. Lopez-Abad (U. Paris 7) A. Naor* (U. Princeton) D. Pellegrino (UFPB) G. Pisier* (Paris 6 & Texas A&M) B. Randrianantoanina (Miami U.) C. Rosendal (U. Illinois Chicago) N. Weaver (Washington U.) (* to be confirmed) Scientific committee J. M. F. Castillo (U. Extremadura) R. Deville (U. Bordeaux) V. Ferenczi (U. Sao Paulo) M. Gonzalez (U. Cantabria) V. Pestov (U. Ottawa & UFSC) G. Pisier (U. Paris 6 & Texas A&M) D. Preiss (U. Warwick) B. Randrianantoanina (Miami U.) We are looking forward to meeting you next year in Brazil, C. Brech, L. Candido, W. Cuellar, V. Ferenczi and P. Kaufmann

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Abstract of a paper by
by Bentuo Zheng (bzheng) 03 Dec '17

03 Dec '17

This is an announcement for the paper “Valuations on Banach lattices” by Pedro Tradacete<https://arxiv.org/find/math/1/au:+Tradacete_P/0/1/0/all/0/1>, Ignacio Villanueva<https://arxiv.org/find/math/1/au:+Villanueva_I/0/1/0/all/0/1>. Abstract: We provide a general framework for the study of valuations on Banach lattices. This complements and expands several recent works about valuations on function spaces, including $L_p(\mu)$, Orlicz spaces and spaces $C(K)$ of continuous functions on a compact Hausdorff space. In particular, we study decomposition properties, boundedness and integral representation of continuous valuations. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1711.11558

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Abstract of a paper by Petr Hájek, Tomasz Kania, Tommaso Russo
by Bentuo Zheng (bzheng) 03 Dec '17

03 Dec '17

This is an announcement for the paper “Symmetrically separated sequences in the unit sphere of a Banach space” by Petr Hájek<https://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1>, Tomasz Kania<https://arxiv.org/find/math/1/au:+Kania_T/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 the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\|>1$ for distinct elements $x, y\in A$, thereby answering a question of J. M. F. Castillo. In the case where $X$ contains an unconditional basic sequence, the set $A$ may be chosen in a way that $\|x\pm y\|>1+\epsilon$ for some $\epsilon>0$ and distinct $x, y\in A$. Under additional structural properties of $X$, such as non-trivial cotype, we obtain quantitative estimates for the said $\epsilon$. Certain renorming results are also presented. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1711.05149

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Abstract of a paper by Leandro Antunes, Valentin Ferenczi, Sophie Grivaux, Christian Rosendal
by Bentuo Zheng (bzheng) 03 Dec '17

03 Dec '17

This is an announcement for the paper “Light groups of isomorphisms of Banach spaces and invariant LUR renormings” by Leandro Antunes<https://arxiv.org/find/math/1/au:+Antunes_L/0/1/0/all/0/1>, Valentin Ferenczi<https://arxiv.org/find/math/1/au:+Ferenczi_V/0/1/0/all/0/1>, Sophie Grivaux<https://arxiv.org/find/math/1/au:+Grivaux_S/0/1/0/all/0/1>, Christian Rosendal<https://arxiv.org/find/math/1/au:+Rosendal_C/0/1/0/all/0/1>. Abstract: Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1711.03482

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Abstract of a paper by Ondřej Kurka
by Bentuo Zheng (bzheng) 03 Dec '17

03 Dec '17

This is an announcement for the paper “The isomorphism class of c0 is not Borel” by Ondřej Kurka<https://arxiv.org/find/math/1/au:+Kurka_O/0/1/0/all/0/1>. Abstract: We show that the class of all Banach spaces which are isomorphic to $c_0$ is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. The proof employs a recent Bourgain-Delbaen construction by Argyros, Gasparis and Motakis. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1711.03328

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Abstract of a paper by Tomasz Kobos
by Bentuo Zheng (bzheng) 03 Dec '17

03 Dec '17

This is an announcement for the paper “Extremal Banach-Mazur distance between a symmetric convex body and an arbitrary convex body on the plane” by Tomasz Kobos<https://arxiv.org/find/math/1/au:+Kobos_T/0/1/0/all/0/1>. Abstract: We prove that if $K, L\subset\mathbb{R}^2$ are convex bodies such that $L$ is symmetric and the Banach-Mazur distance between $K$ and $L$ is equal to $2$, then $K$ is the triangle. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1711.01787

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Abstract of a paper by Vladimir Kadets, Olesia Zavarzina
by Bentuo Zheng (bzheng) 03 Dec '17

03 Dec '17

This is an announcement for the paper “Non-expansive bijections to the unit ball of $\ell_1$-sum of strictly convex Banach spaces” by Vladimir Kadets<https://arxiv.org/find/math/1/au:+Kadets_V/0/1/0/all/0/1>, Olesia Zavarzina<https://arxiv.org/find/math/1/au:+Zavarzina_O/0/1/0/all/0/1>. Abstract: Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i, i\in I$ of strictly convex Banach spaces every non-expansive bijection from the unit ball of $X$ to the unit ball of sum of $Z_i$ by $\ell_1$ is an isometry. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1711.00262

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Abstract of a paper by David Alonso-Gutierrez, Joscha Prochno, Christoph Thaele
by Bentuo Zheng (bzheng) 02 Nov '17

02 Nov '17

This is an announcement for the paper “Gaussian fluctuations for high-dimensional random projections of $\ell_p^n$-balls” by David Alonso-Gutierrez<https://arxiv.org/find/math/1/au:+Alonso_Gutierrez_D/0/1/0/all/0/1>, Joscha Prochno<https://arxiv.org/find/math/1/au:+Prochno_J/0/1/0/all/0/1>, Christoph Thaele<https://arxiv.org/find/math/1/au:+Thaele_C/0/1/0/all/0/1>. Abstract: In this paper, we study high-dimensional random projections of $\ell_p^n$-balls. More precisely, for any n∈ℕ let En be a random subspace of dimension $k_n\in\{1,…,n\}$ and $X_n$ be a random point in the unit ball of $\ell_p^n$. Our work provides a description of the Gaussian fluctuations of the Euclidean norm $\|P_{E_n}X_n\|_2$ of random orthogonal projections of $X_n$ onto $E_n$. In particular, under the condition that $k_n\rightarrow\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_n\rightarrow\infty$ fast enough, we provide a Berry-Esseen bound on the rate of convergence in the central limit theorem. At the end we provide a discussion of the large deviations counterpart to our central limit theorem. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.10130

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Abstract of a paper by Ian Doust, Shaymaa Al-shakarchi
by Bentuo Zheng (bzheng) 02 Nov '17

02 Nov '17

This is an announcement for the paper “Isomorphisms of $AC(\sigma)$ spaces for countable sets” by Ian Doust<https://arxiv.org/find/math/1/au:+Doust_I/0/1/0/all/0/1>, Shaymaa Al-shakarchi<https://arxiv.org/find/math/1/au:+Al_shakarchi_S/0/1/0/all/0/1>. Abstract: It is known that the classical Banach--Stone theorem does not extend to the class of $AC(\sigma)$ spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if $\sigma$ is restricted to the set of compact polygons, then all the corresponding $AC(\sigma)$ spaces are isomorphic. In this paper we examine the case where $\sigma$ is the spectrum of a compact operator, and show that in this case one can obtain an infinite family of homeomorphic sets for which the corresponding function spaces are not isomorphic. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.09073

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Abstract of a paper by Bruno de Mendonça Braga
by Bentuo Zheng (bzheng) 02 Nov '17

02 Nov '17

This is an announcement for the paper “Nonlinear weakly sequentially continuous embeddings between Banach spaces” by Bruno de Mendonça Braga<https://arxiv.org/find/math/1/au:+Braga_B/0/1/0/all/0/1>. Abstract: In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies $$\|e_1+…+e_k\|_{\bar{\delta_Y}\leq\|e_1+…+e_k\|_S$$, where $\bar{\delta_Y}$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.07852

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