Banach March 2011

banach@mathdept.okstate.edu

2 participants
20 discussions

Abstract of a paper by Vladimir Kadets, Miguel Martin, Javier Meri, and Dirk Werner
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces" by Vladimir Kadets, Miguel Martin, Javier Meri, and Dirk Werner. Abstract: We show that for spaces with 1-unconditional bases lushness, the alternative Daugavet property and numerical index~1 are equivalent. In the class of rearrangement invariant (r.i.)\ sequence spaces the only examples of spaces with these properties are $c_0$, $\ell_1$ and $\ell_\infty$. The only lush r.i.\ separable function space on $[0,1]$ is $L_1[0,1]$; the same space is the only r.i.\ separable function space on $[0,1]$ with the Daugavet property over the reals. Archive classification: math.FA Mathematics Subject Classification: Primary 46B04. Secondary 46E30 Submitted from: werner(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.1282 or http://arXiv.org/abs/1103.1282

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Auerbach bases and minimal volume sufficient enlargements" by Mikhail I. Ostrovskii. Abstract: Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement which is a parallelepiped, some spaces have ``exotic'' minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having ``exotic'' minimal-volume sufficient enlargements in terms of Auerbach bases. Archive classification: math.FA Mathematics Subject Classification: 46B07 (primary), 52A21, 46B15 (secondary) Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.0997 or http://arXiv.org/abs/1103.0997

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Embeddability of locally finite metric spaces into Banach spaces is finitely determined" by Mikhail I. Ostrovskii. Abstract: The main purpose of the paper is to prove the following results: Let $A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz embedding into $X$. Let $A$ be a locally finite metric space whose finite subsets admit uniformly coarse embeddings into a Banach space $X$. Then $A$ admits a coarse embedding into $X$. These results generalize previously known results of the same type due to Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author (2006, 2009). One of the main steps in the proof is: each locally finite subset of an ultraproduct $X^\mathcal{U}$ admits a bilipschitz embedding into $X$. We explain how this result can be used to prove analogues of the main results for other classes of embeddings. Archive classification: math.FA Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.0748 or http://arXiv.org/abs/1103.0748

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Abstract of a paper by Radoslaw Adamczak, Rafal Latala, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Geometry of log-concave ensembles of random matrices and approximate reconstruction" by Radoslaw Adamczak, Rafal Latala, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann. Abstract: We study the Restricted Isometry Property of a random matrix $\Gamma$ with independent isotropic log-concave rows. To this end, we introduce a parameter $\Gamma_{k,m}$ that controls uniformly the operator norm of sub-matrices with $k$ rows and $m$ columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of projections of an isotropic log-concave vector. Archive classification: math.PR math.FA math.MG Mathematics Subject Classification: Primary 52A23, 46B06, 46B09, 60E15 Secondary 15B52, 94B75 Submitted from: radamcz(a)mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.0401 or http://arXiv.org/abs/1103.0401

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Abstract of a paper by Pericles D Pavlakos and Minos Petrakis
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "On the structure of non dentable subsets of C({\omega}^{\omega}^k)" by Pericles D Pavlakos and Minos Petrakis. Abstract: It is shown that there is no K closed convex bounded non-dentable subset of C({\omega}^{\omega} ^k) such that on the subsets of K the PCP and the RNP are equivalent properties. Then applying Schachermayer-Rosenthal theorem, we conclude that every non-dentable K contains non-dentable subset L so that on L the weak topology coincides with the norm one. It follows from known results that the RNP and the KMP are equivalent properties on the subsets of C({\omega}^{\omega} ^k). Archive classification: math.FA Remarks: 18 pages,accepted in Studia Mathematica Submitted from: minos(a)science.tuc.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.0366 or http://arXiv.org/abs/1103.0366

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Abstract of a paper by Marek Kosiek and Krzysztof Rudol
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "A disintegration theorem" by Marek Kosiek and Krzysztof Rudol. Abstract: A new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption. The main tool is a result on fibers in the spectrum of algebra of essentially bounded functions established recently by the first-named author. Archive classification: math.FA Mathematics Subject Classification: Primary: 28A50, Secondary: 46J10 Remarks: 3 pages Submitted from: Marek.Kosiek(a)im.uj.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.0255 or http://arXiv.org/abs/1103.0255

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Abstract of a paper by Antonio Aviles, Grzegorz Plebanek, and Jose Rodriguez
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Measurability in C(2^k) and Kunen cardinals" by Antonio Aviles, Grzegorz Plebanek, and Jose Rodriguez. Abstract: A cardinal k is called a Kunen cardinal if the sigma-algebra on k x k generated by all products AxB, coincides with the power set of k x k. For any cardinal k, let C(2^k) be the Banach space of all continuous real-valued functions on the Cantor cube 2^k. We prove that k is a Kunen cardinal if and only if the Baire sigma-algebra on C(2^k) for the pointwise convergence topology coincides with the Borel sigma-algebra on C(2^k) for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given. Archive classification: math.FA Mathematics Subject Classification: 28A05, 28B05 Submitted from: avileslo(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.0247 or http://arXiv.org/abs/1103.0247

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Abstract of a paper by Gideon Schechtman
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Approximate Gaussian isoperimetry for k sets" by Gideon Schechtman. Abstract: Given $2\le k\le n$, the minimal $(n-1)$-dimensional Gaussian measure of the union of the boundaries of $k$ disjoint sets of equal Gaussian measure in $\R^n$ whose union is $\R^n$ is of order $\sqrt{\log k}$. A similar results holds also for partitions of the sphere $S^{n-1}$ into $k$ sets of equal Haar measure. Archive classification: math.PR math.FA Mathematics Subject Classification: 60E15, 52A40 Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1102.4102 or http://arXiv.org/abs/1102.4102

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Abstract of a paper by Ran Levy and Gideon Schechtman
by alspach＠math.okstate.edu 21 Mar '11

21 Mar '11

This is an announcement for the paper "Stabilizing isomorphisms from $\ell_p(\ell_2)$ into $L_p[0,1]$" by Ran Levy and Gideon Schechtman. Abstract: Let $1<p\not=2<\infty$, $\epsilon>0$ and let $T:\ell_p(\ell_2)\overset{into}{\rightarrow}L_p[0,1]$ be an isomorphism. Then there is a subspace $Y\subset \ell_p(\ell_2)$ $(1+\epsilon)$-isomorphic to $\ell_p(\ell_2)$ such that: $T_{|Y}$ is an $(1+\epsilon)$-isomorphism and $T\left(Y\right)$ is $K_p$-complemented in $L_p[0,1]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+\epsilon)\gamma_p$ if $p>2$ and $K_p\le (1+\epsilon)\gamma_{p/(p-1)}$ if $1<p<2$, where $\gamma_r$ is the $L_r$ norm of a standard Gaussian variable. Archive classification: math.FA Mathematics Subject Classification: 46E30 Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.0047 or http://arXiv.org/abs/1103.0047

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M. Kadec
by Vladimir Kadets 07 Mar '11

07 Mar '11

I regret to inform that my father Professor Mikhail Kadets (M. Kadec) died on Monday, March 7. He was 87 years old. Vladimir Kadets.

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Banach March 2011