Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Martino Lupini
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Fraisse limits in functional analysis" by Martino Lupini. Abstract: We provide a unified approach to many Fra\"{\i}ss\'{e} limits in functional analysis, including the Gurarij space, the Poulsen simplex, and their noncommutative analogs. We recover in this general framework many classical results about the Gurarij space and the Poulsen simplex, and at the same time obtain their noncommutative generalizations. Particularly, we construct noncommutative analogs of universal operators in the sense of Rota. Archive classification: math.FA math.LO math.OA Mathematics Subject Classification: 46L07, 46A55 (Primary) 46L89, 03C30, 03C98 (Secondary) Remarks: 28 pages Submitted from: lupini(a)caltech.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.05188 or http://arXiv.org/abs/1510.05188

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Abstract of a paper by D.I. Florentin, V. D. Milman, and A. Segal
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Identifying Set Inclusion by Projective Positions and Mixed Volumes" by D.I. Florentin, V. D. Milman, and A. Segal. Abstract: We study a few approaches to identify inclusion (up to a shift) between two convex bodies in ${\mathbb R}^n$. To this goal we use mixed volumes and fractional linear maps. We prove that inclusion may be identified by comparing volume or surface area of all projective positions of the sets. We prove similar results for Minkowski sums of the sets. Archive classification: math.FA math.MG Mathematics Subject Classification: 52A05, 52A20, 52A38, 52A39, 51N15, 46B20 Citation: Identifying Set Inclusion by Projective Positions and Mixed The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.03844 or http://arXiv.org/abs/1510.03844

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Abstract of a paper by Martin Dolezal, Martin Rmoutil, Benjamin Vejnar, and Vaclav Vlasak
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Haar meager sets revisited" by Martin Dolezal, Martin Rmoutil, Benjamin Vejnar, and Vaclav Vlasak. Abstract: In the present article we investigate Darji's notion of Haar meager sets from several directions. We consider alternative definitions and show that some of them are equivalent to the original one, while others fail to produce interesting notions. We define Haar meager sets in nonabelian Polish groups and show that many results, including the facts that Haar meager sets are meager and form a $\sigma$-ideal, are valid in the more general setting as well. The article provides various examples distinguishing Haar meager sets from Haar null sets, including decomposition theorems for some subclasses of Polish groups. As a corollary we obtain, for example, that $\mathbb Z^\omega$, $\mathbb R^\omega$ or any Banach space can be decomposed into a Haar meager set and a Haar null set. We also establish the stability of non-Haar meagerness under Cartesian product. Archive classification: math.GN math.FA Remarks: 19 pages Submitted from: dolezal(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.01613 or http://arXiv.org/abs/1510.01613

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Abstract of a paper by Francisco J. Garcia-Pacheco, Alejandro Miralles, and Daniele Puglisi
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Dual maps and the Dunford-Pettis property" by Francisco J. Garcia-Pacheco, Alejandro Miralles, and Daniele Puglisi. Abstract: We characterize the points of $\left\|\cdot\right\|$-$w^*$ continuity of dual maps, turning out to be the smooth points. We prove that a Banach space has the Schur property if and only if it has the Dunford-Pettis property and there exists a dual map that is sequentially $w$-$w$ continuous at $0$. As consequence, we show the existence of smooth Banach spaces on which the dual map is not $w$-$w$ continuous at $0$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B10 Remarks: 6 pages Submitted from: mirallea(a)uji.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.01531 or http://arXiv.org/abs/1510.01531

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Abstract of a paper by Joel Blot and Philippe Cieutat
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Completeness of Sums of Subspace of Bounded Functions and Applications" by Joel Blot and Philippe Cieutat. Abstract: We give a new proof of a characterization of the closeness of the range of a continuous linear operator and of the closeness of the sum of two closed vector subspaces of a Banach space. Then we state sufficient conditions for the closeness of the sum of two closed subspaces of the Banach space of bounded functions and apply this result on various pseudo almost periodic spaces and pseudo almost automorphic spaces. Archive classification: math.FA Submitted from: blot(a)univ-paris1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.01160 or http://arXiv.org/abs/1510.01160

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Abstract of a paper by Guillaume Aubrun and Stanislaw Szarek
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Dvoretzky's theorem and the complexity of entanglement detection" by Guillaume Aubrun and Stanislaw Szarek. Abstract: The well-known Horodecki criterion asserts that a state $\rho$ on $\mathbb{C}^d \otimes \mathbb{C}^d$ is entangled if and only if there exists a positive map $\Phi : \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(\Phi \otimes \mathsf{I})(\rho)$ is not positive semi-definite. We show that that the number of such maps needed to detect all the robustly entangled states (i.e., states $\rho$ which remain entangled even in the presence of substantial randomizing noise) exceeds $\exp(c d^3 / \log d)$. The proof is based on a study of the approximability of the set of states (resp. of separable states) by polytopes with few vertices or with few faces, and ultimately relies on the Dvoretzky--Milman theorem about the dimension of almost spherical sections of convex bodies. The result can be interpreted as a geometrical manifestation of the complexity of entanglement detection. Archive classification: quant-ph math.FA Mathematics Subject Classification: 81P40, 46B07 Submitted from: aubrun(a)math.univ-lyon1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.00578 or http://arXiv.org/abs/1510.00578

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Abstract of a paper by Tuomas Hytonen, Sean Li, and Assaf Naor
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Quantitative affine approximation for UMD targets" by Tuomas Hytonen, Sean Li, and Assaf Naor. Abstract: It is shown here that if $(Y,\|\cdot\|_Y)$ is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists $c=c(Y)\in (0,\infty)$ with the following property. For every $n\in \mathbb{N}$ and $\varepsilon\in (0,1/2]$, if $(X,\|\cdot\|_X)$ is an $n$-dimensional normed space with unit ball $B_X$ and $f:B_X\to Y$ is a $1$-Lipschitz function then there exists an affine mapping $\Lambda:X\to Y$ and a sub-ball $B^*=y+\rho B_X\subseteq B_X$ of radius $\rho\ge \exp(-(1/\varepsilon)^{cn})$ such that $\|f(x)-\Lambda(x)\|_Y\le \varepsilon \rho$ for all $x\in B^*$. This estimate on the macroscopic scale of affine approximability of vector-valued Lipschitz functions is an asymptotic improvement (as $n\to \infty$) over the best previously known bound even when $X$ is $\mathbb{R}^n$ equipped with the Euclidean norm and $Y$ is a Hilbert space. Archive classification: math.FA math.MG Submitted from: naor(a)math.princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.00276 or http://arXiv.org/abs/1510.00276

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Abstract of a paper by Xiao Chun Fang and Marat Pliev
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Narrow Orthogonally Additive Operators on Lattice-Normed Spaces" by Xiao Chun Fang and Marat Pliev. Abstract: The aim of this article is to extend results of M.~Popov and second named author about orthogonally additive narrow operators on vector lattices. The main object of our investigations are an orthogonally additive narrow operators between lattice-normed spaces. We prove that every $C$-compact laterally-to-norm continuous orthogonally additive operator from a Banach-Kantorovich space $V$ to a Banach lattice $Y$ is narrow. We also show that every dominated Uryson operator from Banach-Kantorovich space over an atomless Dedekind complete vector lattice $E$ to a sequence Banach lattice $\ell_p(\Gamma)$ or $c_0(\Gamma)$ is narrow. Finally, we prove that if an orthogonally additive dominated operator $T$ from lattice-normed space $(V,E)$ to Banach-Kantorovich space $(W,F)$ is order narrow then the order narrow is its exact dominant $\ls T\rs$. Archive classification: math.FA Mathematics Subject Classification: 46B99. 47B99 Remarks: 16 pages Submitted from: martin.weber(a)tu-dresden.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.09189 or http://arXiv.org/abs/1509.09189

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Abstract of a paper by Alexandr Andoni, Assaf Naor, and Ofer Neiman
by alspach＠math.okstate.edu 22 Oct '15

22 Oct '15

This is an announcement for the paper "Snowflake universality of Wasserstein spaces" by Alexandr Andoni, Assaf Naor, and Ofer Neiman. Abstract: For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for every $\varepsilon>0$, every $\theta\in (0,1/p]$ and every finite metric space $(X,d_X)$, the metric space $(X,d_{X}^{\theta})$ embeds into $\mathscr{P}_p(\mathbb{R}^3)$ with distortion at most $1+\varepsilon$. We show that this is sharp when $p\in (1,2]$ in the sense that the exponent $1/p$ cannot be replaced by any larger number. In fact, for arbitrarily large $n\in \mathbb{N}$ there exists an $n$-point metric space $(X_n,d_n)$ such that for every $\alpha\in (1/p,1]$ any embedding of the metric space $(X_n,d_n^\alpha)$ into $\mathscr{P}_p(\mathbb{R}^3)$ incurs distortion that is at least a constant multiple of $(\log n)^{\alpha-1/p}$. These statements establish that there exists an Alexandrov space of nonnegative curvature, namely $\mathscr{P}_{\! 2}(\mathbb{R}^3)$, with respect to which there does not exist a sequence of bounded degree expander graphs. It also follows that $\mathscr{P}_{\! 2}(\mathbb{R}^3)$ does not admit a uniform, coarse, or quasisymmetric embedding into any Banach space of nontrivial type. Links to several longstanding open questions in metric geometry are discussed, including the characterization of subsets of Alexandrov spaces, existence of expanders, the universality problem for $\mathscr{P}_{\! 2}(\mathbb{R}^k)$, and the metric cotype dichotomy problem. Archive classification: math.MG math.FA Submitted from: naor(a)math.princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.08677 or http://arXiv.org/abs/1509.08677

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Analytic and Probabilistic Techniques in Modern Convex Geometry, Nov 7-9, 2015
by Pivovarov, Peter 30 Sep '15

30 Sep '15

Dear Colleagues: The Mathematics Department at the University of Missouri-Columbia is pleased to host a conference on Analytic and Probabilistic Techniques in Modern Convex Geometry, dedicated to Alexander Koldobsky on the occassion of his 60th birthday, November 7-9, 2015. We aim to bring together experienced and early-stage researchers to discuss the latest developments on slicing inequalities for convex sets, geometry of high-dimensional measures, affine isoperimetric inequalities and non-asymptotic random matrix theory. Information is available at http://www.bengal.missouri.edu/~pivovarovp/APTMCG/index.html Funding is still available to cover the local and travel expenses of a limited number of participants. Graduate students, postdoctoral researchers, and members of underrepresented groups are particularly encouraged to apply for support. Please register online or contact Peter Pivovarov at pivovarovp(a)missouri.edu. A poster session will be held for researchers to display their work. Graduate students are particularly encouraged to submit a poster. Yours sincerely, Peter Pivovarov on behalf of the organizers: Grigoris Paouris Peter Pivovarov Mark Rudelson Artem Zvavitch

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