Banach December 2011

banach@mathdept.okstate.edu

2 participants
34 discussions

Abstract of a paper by Manor Mendel and Assaf Naor
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Ultrametric skeletons" by Manor Mendel and Assaf Naor. Abstract: We prove that for every $\epsilon\in (0,1)$ there exists $C_\epsilon\in (0,\infty)$ with the following property. If $(X,d)$ is a compact metric space and $\mu$ is a Borel probability measure on $X$ then there exists a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\epsilon)$, and a probability measure $\nu$ supported on $S$ satisfying $\nu\left(B_d(x,r)\right)\le \left(\mu(B_d(x,C_\epsilon r)\right)^{1-\epsilon}$ for all $x\in X$ and $r\in (0,\infty)$. The dependence of the distortion on $\epsilon$ is sharp. We discuss an extension of this statement to multiple measures, as well as how it implies Talagrand's majorizing measures theorem. Archive classification: math.MG math.FA math.PR Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.3416 or http://arXiv.org/abs/1112.3416

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Abstract of a paper by Marek Cuth and Martin Rmoutil
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Sigma-porosity is separably determined" by Marek Cuth and Martin Rmoutil. Abstract: We prove a separable reduction theorem for sigma-porosity of Suslin sets. In particular, if A is a Suslin subset in a Banach space X, then each separable subspace of X can be enlarged to a separable subspace V such that A is sigma-porous in X if and only if the intersection of A and V is sigma-porous in V. Such a result is proved for several types of sigma-porosity. The proof is done using the method of elementary submodels, hence the results can be combined with other separable reduction theorems. As an application we extend a theorem of L.Zajicek on differentiability of Lipschitz functions on separable Asplund spaces to the nonseparable setting. Archive classification: math.FA Mathematics Subject Classification: 28A05, 54E35, 58C20 Submitted from: cuthm5am(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.3813 or http://arXiv.org/abs/1112.3813

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Test-space characterizations of some classes of Banach spaces" by Mikhail I. Ostrovskii. Abstract: Let $\mathcal{P}$ be a class of Banach spaces and let $T=\{T_\alpha\}_{\alpha\in A}$ be a set of metric spaces. We say that $T$ is a set of {\it test-spaces} for $\mathcal{P}$ if the following two conditions are equivalent: (1) $X\notin\mathcal{P}$; (2) The spaces $\{T_\alpha\}_{\alpha\in A}$ admit uniformly bilipschitz embeddings into $X$. The first part of the paper is devoted to a simplification of the proof of the following test-space characterization obtained in M.I. Ostrovskii [Different forms of metric characterizations of classes of Banach spaces, Houston J. Math., to appear]: For each sequence $\{X_m\}_{m=1}^\infty$ of finite-dimensional Banach spaces there is a sequence $\{H_n\}_{n=1}^\infty$ of finite connected unweighted graphs with maximum degree $3$ such that the following conditions on a Banach space $Y$ are equivalent: (A) $Y$ admits uniformly isomorphic embeddings of $\{X_m\}_{m=1}^\infty$; (B) $Y$ admits uniformly bilipschitz embeddings of $\{H_n\}_{n=1}^\infty$. The second part of the paper is devoted to the case when $\{X_m\}_{m=1}^\infty$ is an increasing sequence of spaces. It is shown that in this case the class of spaces given by (A) can be characterized using one test-space, which can be chosen to be an infinite graph with maximum degree 3. Archive classification: math.FA math.CO math.MG Mathematics Subject Classification: Primary: 46B07, Secondary: 05C12, 46B85, 54E35 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.3086 or http://arXiv.org/abs/1112.3086

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Abstract of a paper by Steven Heilman, Aukosh Jagannath, and Assaf Naor
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Solution of the propeller conjecture in $\R^3$" by Steven Heilman, Aukosh Jagannath, and Assaf Naor. Abstract: It is shown that every measurable partition $\{A_1,\ldots, A_k\}$ of $\R^3$ satisfies \begin{equation}\label{eq:abs} \sum_{i=1}^k\left\|\int_{A_i} xe^{-\frac12\|x\|_2^2}dx\right\|_2^2\le 9\pi^2. \end{equation} Let $\{P_1,P_2,P_3\}$ be the partition of $\R^2$ into $120^\circ$ sectors centered at the origin. The bound~\eqref{eq:abs} is sharp, with equality holding if $A_i=P_i\times \R$ for $i\in \{1,2,3\}$ and $A_i=\emptyset$ for $i\in \{4,\ldots,k\}$ (up to measure zero corrections, orthogonal transformations and renumbering of the sets $\{A_1,\ldots,A_k\}$). This settles positively the $3$-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of~\eqref{eq:abs} reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of~\eqref{eq:abs} is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with $4\times 4$ centered and spherical hypothesis matrix equals $\frac{2\pi}{3}$. Archive classification: cs.DS math.FA math.MG Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.2993 or http://arXiv.org/abs/1112.2993

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Abstract of a paper by Guixiang Hong and Zhi Yin
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Wavelet approach to operator-valued Hardy spaces" by Guixiang Hong and Zhi Yin. Abstract: This paper is devoted to the study of operator-valued Hardy spaces via wavelet method. This approach is parallel to that in noncommutative martingale case. We show that our Hardy spaces defined by wavelet coincide with those introduced by Tao Mei via the usual Lusin and Littlewood-Paley square functions. As a consequence, we give an explicit complete unconditional basis of the Hardy space H1(R) when H1(R) is equipped with an appropriate operator space structure. Archive classification: math.FA math.CA Submitted from: ghong(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.2912 or http://arXiv.org/abs/1112.2912

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Abstract of a paper by Andrea Colesanti and Ilaria Fragala
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "The area measure of log-concave functions and related inequalities" by Andrea Colesanti and Ilaria Fragala. Abstract: On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such first variation, which lead to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution. Archive classification: math.FA math.MG Remarks: 36 pages Submitted from: colesant(a)math.unifi.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.2555 or http://arXiv.org/abs/1112.2555

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Abstract of a paper by Valentin Ferenczi and Thomas Schlumprecht
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Subsequential minimality in Gowers and Maurey spaces" by Valentin Ferenczi and Thomas Schlumprecht. Abstract: We define block sequences $(x_n)$ in every block subspace of a variant of the space of Gowers and Maurey so that the map $x_{2n-1}\mapsto x_{2n} $ extends to an isomorphism. This implies the existence of a subsequentially minimal HI space, which solves a question in \cite{FR}. Archive classification: math.FA Mathematics Subject Classification: 46B03, 03E15 Submitted from: schlump(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.2411 or http://arXiv.org/abs/1112.2411

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Abstract of a paper by Tal Orenshtein and Boaz Tsaban
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Pointwise convergence of partial functions: The Gerlits-Nagy Problem" by Tal Orenshtein and Boaz Tsaban. Abstract: For a set X of real numbers, let B(X) denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product R^X. Assume that for each countable subset A of B(X), each f in the closure of A is in the closure of $A$ under pointwise limits of sequences of partial functions. We show that in this case, B(X) is countably Frechet-Urysohn, that is, each point in the closure of a countable set is a limit of a sequence of elements of that set. This solves a problem of Arnold Miller. The continuous version of this problem is equivalent to a notorious open problem of Gerlits and Nagy. Answering a question of Salvador Hernandez, we show that the same result holds for the space of all Baire class 1 functions on X. We conjecture that the answer to the continuous version of this problem is negative, but we identify a nontrivial class of sets X of real numbers, for which we can provide a positive solution to this problem. The proofs establish new local-to-global correspondences, and use methods of infinite-combinatorial topology, including a new fusion result of Francis Jordan. Archive classification: math.GN math.CA math.CO math.FA math.LO Remarks: Submitted for publication Submitted from: tsaban(a)math.biu.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.2373 or http://arXiv.org/abs/1112.2373

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Kalton memorial website
by Dale Alspach 28 Dec '11

28 Dec '11

Dear Colleagues, The Memorial website in honor of Nigel Kalton is now active: http://kaltonmemorial.missouri.edu/ The website is not entirely finished yet, in the end we expect nearly all (perhaps, all) of his publications freely available as pdf files. We welcome additional contributions such as, photos, stories, reminiscences, etc. In particular, we hope to receive more contributions describing various aspects of Nigel's work. Please send all material to Fritz Gesztesy Department of Mathematics University of Missouri Columbia, MO 65211 USA E-mail: gesztesyf(a)missouri.edu Best regards, Fritz Gesztesy

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Abstract of a paper by Marcel de Jeu and Marten Wortel
by alspach＠math.okstate.edu 20 Dec '11

20 Dec '11

This is an announcement for the paper "Compact groups of positive operators on Banach lattices" by Marcel de Jeu and Marten Wortel. Abstract: In this paper we study groups of positive operators on Banach lattices. If a certain factorization property, for which we are not aware of counterexamples, holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we again have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered. Archive classification: math.FA math.RT Mathematics Subject Classification: Primary 22D12, Secondary 22C05, 46B42 Remarks: 21 pages Submitted from: marten.wortel(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.1611 or http://arXiv.org/abs/1112.1611

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Banach December 2011