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Abstract of a paper by Debmalya Sain, Anubhab Ray, Kallol Paul
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Extreme contractions on finite-dimensional polygonal Banach spaces” by Debmalya Sain<https://arxiv.org/search/math?searchtype=author&query=Sain%2C+D>, Anubhab Ray<https://arxiv.org/search/math?searchtype=author&query=Ray%2C+A>, Kallol Paul<https://arxiv.org/search/math?searchtype=author&query=Paul%2C+K>. Abstract: We explore extreme contractions between finite-dimensional polygonal Banach spaces, from the point of view of attainment of norm of a linear operator. We prove that if $ X $ is an $ n- $dimensional polygonal Banach space and $ Y $ is any Banach space and $ T \in L(X,Y) $ is an extreme contraction, then $ T $ attains norm at $ n $ linearly independent extreme points of $ B_{X}. $ Moreover, if $ T $ attains norm at exactly $ n $ linearly independent extreme points $ x_1, x_2, \ldots, x_n $ of $ B_X $ and does not attain norm at any other extreme point of $ B_X, $ then each $ Tx_i $ is an extreme point of $ B_Y.$ We completely characterize extreme contractions between a finite-dimensional polygonal Banach space and a strictly convex Banach space. We introduce L-P property for a pair of Banach spaces and show that it has natural connections with our present study. We also prove that for any strictly convex Banach space $ X $ and any finite-dimensional polygonal Banach space $ Y, $ the pair $ (X,Y) $ does not have L-P property. Finally, we obtain a characterization of Hilbert spaces among strictly convex Banach spaces in terms of L-P property. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.01881

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Abstract of a paper by Luis C. García-Lirola, Antonín Procházka
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Pełczyński space is isomorphic to the Lipschitz free space over a compact set” by Luis C. García-Lirola<https://arxiv.org/search/math?searchtype=author&query=Garc%C3%ADa-Lirola%2C…>, Antonín Procházka<https://arxiv.org/search/math?searchtype=author&query=Proch%C3%A1zka%2C+A>. Abstract: We prove the result stated in the title. This provides a first example of an infinite-dimensional Banach space whose Lipschitz free space is isomorphic to the free space of a compact set. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.00859

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Abstract of a paper by Antonio Avilés, Gonzalo Martínez-Cervantes
by Bentuo Zheng (bzheng) 30 Sep '18

30 Sep '18

This is an announcement for the paper “Complete metric spaces with property (Z) are length metric spaces” by Antonio Avilés<https://arxiv.org/search/math?searchtype=author&query=Avil%C3%A9s%2C+A>, Gonzalo Martínez-Cervantes<https://arxiv.org/search/math?searchtype=author&query=Mart%C3%ADnez-Cervant…>. Abstract: We prove that every complete metric space with property (Z) is a length space. These answers questions posed by Garc\'{i}a-Lirola, Proch\'{a}zka and Rueda Zoca, and by Becerra Guerrero, L\'{o}pez-P\'{e}rez and Rueda Zoca, related to the structure of Lipschitz-free Banach spaces of metric spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1808.00715

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Banach spaces and optimization: Conference on the occasion of Robert Deville’s 60th birthday
by Tony Prochazka 04 Sep '18

04 Sep '18

Dear colleagues, It is our pleasure to announce *Conference on the occasion of Robert Deville’s 60th birthday: Banach spaces and optimization* which will take place *from June 16 to June 21, 2019* in Métabief, France. Our aim is to bring together experienced and novice researchers interested in Banach space geometry, real analysis, optimization and other fields that Robert’s work has influenced. The conference will feature a series of plenary and short talks on recent advances in these subjects. The following speakers have agreed to give a plenary talk: Daniel Azagra, Aris Daniilidis, Estibalitz Durand-Cartagena, Gilles Godefroy, Antonio Guirao, Petr Hájek, Jesús Jaramillo, Sebastián Lajara, Etienne Matheron, Julian Revalski The registration is open and further information can be found on the website: https://trimestres-lmb.univ-fcomte.fr/Banach-spaces-and-optimization.html Feel free to transmit this announcement to your peers who you suspect might be interested. We are looking forward to meeting you next year in France. Tony Procházka and Matías Raja

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Abstract of a paper by Ivan S. Yaroslavtsev
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Burkholder-Davis-Gundy inequalities in UMD Banach spaces” by Ivan S. Yaroslavtsev<https://arxiv.org/search/math?searchtype=author&query=Yaroslavtsev%2C+I+S>. Abstract: In this paper we prove Burkholder-Davis-Gundy inequalities for a general martingale $M$ with values in a UMD Banach space $X$. Assuming that $M_0=0$, we show that the following two-sided inequality holds for all $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{{$\star$}} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E \gamma([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} Here $ \gamma([\![M]\!]_t) $ is the $L^2$-norm of the unique Gaussian measure on $X$ having $[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle, \langle M,y^*\rangle]_t$ as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of \eqref{eq:main} was proved for UMD Banach functions spaces $X$. We show that for continuous martingales, \eqref{eq:main} holds for all $0<p<\infty$, and that for purely discontinuous martingales the right-hand side of \eqref{eq:main} can be expressed more explicitly in terms of the jumps of $M$. For martingales with independent increments, \eqref{eq:main} is shown to hold more generally in reflexive Banach spaces $X$ with finite cotype. In the converse direction, we show that the validity of \eqref{eq:main} for arbitrary martingales implies the UMD property for $X$. As an application we prove various It\^o isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide It\^o isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.05573

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Abstract of a paper by Cleon S. Barroso
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “On the fixed point property in Banach spaces isomorphic to $c_0$” by Cleon S. Barroso<https://arxiv.org/search/math?searchtype=author&query=Barroso%2C+C+S>. Abstract: We prove that every Banach space containing a subspace isomorphic to $\co$ fails the fixed point property. The proof is based on an amalgamation approach involving a suitable combination of known results and techniques, including James's distortion theorem, Ramsey's combinatorial theorem, Brunel-Sucheston spreading model techniques and Dowling, Lennard and Turett's fixed point methodology employed in their characterization of weak compactness in $\co$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.11614

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Abstract of a paper by Eve Oja, Tauri Viil, Dirk Werner
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Totally smooth renormings” by Eve Oja<https://arxiv.org/search/math?searchtype=author&query=Oja%2C+E>, Tauri Viil<https://arxiv.org/search/math?searchtype=author&query=Viil%2C+T>, Dirk Werner<https://arxiv.org/search/math?searchtype=author&query=Werner%2C+D>. Abstract: We study the problem of totally smooth renormings of Banach spaces and provide such renormings for spaces which are weakly compactly generated. We also consider renormings for $(a,B,c)$-ideals. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.07335

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Abstract of a paper by Mikhail I. Ostrovskii, Beata Randrianantoanina
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “A characterization of superreflexivity through embeddings of lamplighter groups” by Mikhail I. Ostrovskii<https://arxiv.org/search/math?searchtype=author&query=Ostrovskii%2C+M+I>, Beata Randrianantoanina<https://arxiv.org/search/math?searchtype=author&query=Randrianantoanina%2C+B>. Abstract: We prove that finite lamplighter groups $\{\mathbb{Z}_2\wr\mathbb{Z}_n\}_{n\ge 2}$ with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space, and therefore form a set of test-spaces for superreflexivity. Our proof is inspired by the well known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover $\mathbb{Z}_2\wr\mathbb{Z}_n$ by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings. https://arxiv.org/abs/1807.06692

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Abstract of a paper by William B. Johnson, Tomasz Kania
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Embedding Banach spaces into the space of bounded functions with countable support” by William B. Johnson<https://arxiv.org/search/math?searchtype=author&query=Johnson%2C+W+B>, Tomasz Kania<https://arxiv.org/search/math?searchtype=author&query=Kania%2C+T>. Abstract: We prove that a WLD subspace of the space $\ell_\infty^c(\Gamma)$ consisting of all bounded, countably supported functions on a set $\Gamma$ embeds isomorphically into $\ell_\infty$ if and only if it does not contain isometric copies of $c_0(\omega_1)$. Moreover, a subspace of $\ell_\infty^c(\omega_1)$ is constructed that has an unconditional basis, does not embed into $\ell_\infty$, and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of $c_0(\omega_1)$). The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.05239

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Abstract of a paper by Stephen J. Dilworth, Denka Kutzarova, Mikhail I. Ostrovskii
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Lipschitz free spaces on finite metric spaces” by Stephen J. Dilworth<https://arxiv.org/search/math?searchtype=author&query=Dilworth%2C+S+J>, Denka Kutzarova<https://arxiv.org/search/math?searchtype=author&query=Kutzarova%2C+D>, Mikhail I. Ostrovskii<https://arxiv.org/search/math?searchtype=author&query=Ostrovskii%2C+M+I>. Abstract: Main results of the paper: (1) For any finite metric space $M$ the Lipschitz free space on $M$ contains a large well-complemented subspace which is close to $\ell_1^n$. (2) Lipschitz free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of graphs which are not necessarily graph automorphisms; (b) In the case of such recursive families of graphs as Laakso graphs we use the well-known approach of Gr\"unbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.03814

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