Banach June 2007

banach@mathdept.okstate.edu

1 participants
15 discussions

Abstract of a paper by Han Ju Lee
by Dale Alspach 28 Jun '07

28 Jun '07

This is an announcement for the paper "Randomized series and geometry of Banach spaces" by Han Ju Lee. Abstract: We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a K\"othe function space $E$ is upper locally uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is midpoint locally uniformly convex. Archive classification: math.FA Mathematics Subject Classification: 46B20;46B07;46B09 The source file(s), randomized-series2007-01-29.tex: 33940 bytes, is(are) stored in gzipped form as 0706.3740.gz with size 10kb. The corresponding postcript file has gzipped size 96kb. Submitted from: hahnju(a)postech.ac.kr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.3740 or http://arXiv.org/abs/0706.3740 or by email in unzipped form by transmitting an empty message with subject line uget 0706.3740 or in gzipped form by using subject line get 0706.3740 to: math(a)arXiv.org.

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Abstract of a paper by Jeff Cheeger and Bruce Kleiner
by Dale Alspach 28 Jun '07

28 Jun '07

This is an announcement for the paper "Characterizations of the Radon-Nikodym Property in terms of inverse limits" by Jeff Cheeger and Bruce Kleiner. Abstract: We show that a separable Banach space has the Radon-Nikodym Property if and only if it is isomorphic to the limit of an inverse system, V_1<--- V_2<---...<--- V_k<---..., where the V_i's are finite dimensional Banach spaces, and the bonding maps V_{k-1}<--- V_k are quotient maps. We also show that the inverse system can be chosen to be a good finite dimensional approximation (GFDA), a notion introduced our earlier paper "On the differentiability of Lipschtz maps from metric measure spaces into Banach spaces". As a corollary, it follows that the differentiation and bi-Lipschitz non-embedding theorems in that paper, which were proved for maps into GFDA targets, are optimal in the sense that they hold for targets with the Radon-Nikodym Property. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B22;46G05 The source file(s), gfda.bbl: 1902 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.3389 or http://arXiv.org/abs/0706.3389 or by email in unzipped form by transmitting an empty message with subject line uget 0706.3389 or in gzipped form by using subject line get 0706.3389 to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi and Eloi Medina Galego
by Dale Alspach 28 Jun '07

28 Jun '07

This is an announcement for the paper "Countable groups of isometries on Banach spaces" by Valentin Ferenczi and Eloi Medina Galego. Abstract: A group $G$ is representable in a Banach space $X$ if $G$ is isomorphic to the group of isometries on $X$ in some equivalent norm. We prove that a countable group $G$ is representable in a separable real Banach space $X$ in several general cases, including when $G=\{-1,1\} \times H$, $H$ finite and $\dim X \geq |H|$, or when $G$ contains a normal subgroup with two elements and $X$ is of the form $c_0(Y)$ or $\ell_p(Y)$, $1 \leq p <+\infty$. This is a consequence of a result inspired by methods of S. Bellenot and stating that under rather general conditions on a separable real Banach space $X$ and a countable bounded group $G$ of isomorphisms on $X$ containing $-Id$, there exists an equivalent norm on $X$ for which $G$ is equal to the group of isometries on $X$. We also extend methods of K. Jarosz to prove that any complex Banach space of dimension at least $2$ may be renormed to admit only trivial real isometries, and that any real Banach space which is a cartesian square may be renormed to admit only trivial and conjugation real isometries. It follows that every real space of dimension at least $4$ and with a complex structure up to isomorphism may be renormed to admit exactly two complex structures up to isometry, and that every real cartesian square may be renormed to admit a unique complex structure up to isometry. Archive classification: math.FA Mathematics Subject Classification: 46B03; 46B04 Remarks: 43 pages The source file(s), ferenczigalego_isometries.tex: 104441 bytes, is(are) stored in gzipped form as 0706.3861.gz with size 29kb. The corresponding postcript file has gzipped size 137kb. Submitted from: ferenczi(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.3861 or http://arXiv.org/abs/0706.3861 or by email in unzipped form by transmitting an empty message with subject line uget 0706.3861 or in gzipped form by using subject line get 0706.3861 to: math(a)arXiv.org.

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Abstract of a paper by Tim Austin, Assaf Naor, and Yuval Peres
by Dale Alspach 21 Jun '07

21 Jun '07

This is an announcement for the paper "The wreath product of Z with Z has Hilbert compression exponent 2/3" by Tim Austin, Assaf Naor, and Yuval Peres. Abstract: We consider the wreath product $\Z\bwr \Z $, and prove that any Lipschitz function $f:\Z\bwr \Z \to L_2$ satisfies $$\liminf_{d_{\Z\bwr\Z}(x,y)\to \infty}\frac{\|f(x)-f(y)\|_2}{d_{\Z\bwr\Z}(x,y)^{2/3}}<\infty. $$ On the other hand, as as shown by Tessera in~\cite{Tess06}, there exists a Lipschitz function $g:\Z\bwr \Z \to L_2$ and a real $c>0$ such that $\|f(x)-f(y)\|_2 \ge c\,d_{\Z\bwr\Z}(x,y)^{2/3}$ for all $x,y \in \Z\bwr\Z$. Thus the Hilbert compression exponent of $\Z\bwr \Z$ is exactly $\frac23$, answering a question posed by Arzhantseva, Guba and Sapir~\cite{AGS06} and by Tessara~\cite{Tess06}. Our proof is based on an application of K. Ball's notion of Markov type. Archive classification: math.MG math.FA The source file(s), ZwreathZ.bbl: 3412 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.1943 or http://arXiv.org/abs/0706.1943 or by email in unzipped form by transmitting an empty message with subject line uget 0706.1943 or in gzipped form by using subject line get 0706.1943 to: math(a)arXiv.org.

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Abstract of a paper by Omer Friedland and Sasha Sodin
by Dale Alspach 21 Jun '07

21 Jun '07

This is an announcement for the paper "An extension of a Bourgain--Lindenstrauss--Milman inequality" by Omer Friedland and Sasha Sodin. Abstract: Let || . || be a norm on R^n. Averaging || (\eps_1 x_1, \cdots, \eps_n x_n) || over all the 2^n choices of \eps = (\eps_1, \cdots, \eps_n) in \{ -1, +1 \}^n, we obtain an expression ||| . ||| which is an unconditional norm on R^n. Bourgain, Lindenstrauss and Milman showed that, for a certain (large) constant \eta > 1, one may average over (\eta n) (random) choices of \eps and obtain a norm that is isomorphic to ||| . |||. We show that this is the case for any \eta > 1. Archive classification: math.FA math.PR The source file(s), kkh_18.6.tex: 12943 bytes, is(are) stored in gzipped form as 0706.2638.gz with size 5kb. The corresponding postcript file has gzipped size 63kb. Submitted from: sodinale(a)post.tau.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.2638 or http://arXiv.org/abs/0706.2638 or by email in unzipped form by transmitting an empty message with subject line uget 0706.2638 or in gzipped form by using subject line get 0706.2638 to: math(a)arXiv.org.

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Abstract of a paper by Chang-Pao Chen, Hao-Wei Huang, and Chun-Yen Shen
by Dale Alspach 21 Jun '07

21 Jun '07

This is an announcement for the paper "Characterization of the matrix whose norm is determined by its action on decreasing sequences" by Chang-Pao Chen, Hao-Wei Huang, and Chun-Yen Shen. Abstract: Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. Archive classification: math.FA Mathematics Subject Classification: 15A60, 40G05, 47A30, 47B37 The source file(s), HWHshenfinal.tex: 34262 bytes, is(are) stored in gzipped form as 0706.1098.gz with size 11kb. The corresponding postcript file has gzipped size 96kb. Submitted from: shenc(a)indiana.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.1098 or http://arXiv.org/abs/0706.1098 or by email in unzipped form by transmitting an empty message with subject line uget 0706.1098 or in gzipped form by using subject line get 0706.1098 to: math(a)arXiv.org.

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Abstract of a paper by Stanislaw Prus and Andrzej Wisnicki
by Dale Alspach 21 Jun '07

21 Jun '07

This is an announcement for the paper "On the fixed point property in direct sums of Banach spaces with strictly monotone norms" by Stanislaw Prus and Andrzej Wisnicki. Abstract: It is shown that if a Banach space X has the super fixed point property for nonexpansive mappings or admits a 1-unconditional basis and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. Archive classification: math.FA Mathematics Subject Classification: 47H09; 46B20 Remarks: 12 pages The source file(s), direct_p.tex: 35126 bytes, is(are) stored in gzipped form as 0706.0915.gz with size 10kb. The corresponding postcript file has gzipped size 86kb. Submitted from: awisnic(a)golem.umcs.lublin.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.0915 or http://arXiv.org/abs/0706.0915 or by email in unzipped form by transmitting an empty message with subject line uget 0706.0915 or in gzipped form by using subject line get 0706.0915 to: math(a)arXiv.org.

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Abstract of a paper by Edward Odell, Thomas Schlumprecht and Andras Zsak
by Dale Alspach 06 Jun '07

06 Jun '07

This is an announcement for the paper "A new infinite game in Banach spaces with applications" by Edward Odell, Thomas Schlumprecht and Andras Zsak. Abstract: We consider the following two-player game played on a separable, infinite-dimensional Banach space X. Player S chooses a positive integer k_1 and a finite-codimensional subspace X_1 of X. Then player P chooses x_1 in the unit sphere of X_1. Moves alternate thusly, forever. We study this game in the following setting. Certain normalized, 1-unconditional sequences (u_i) and (v_i) are fixed so that S has a winning strategy to force P to select x_i's so that if the moves are (k_1,X_1,x_1,k_2,X_2,x_2,...), then (x_i) is dominated by (u_{k_i}) and/or (x_i) dominates (v_{k_i}). In particular, we show that for suitable (u_i) and (v_i) if X is reflexive and S can win both of the games above, then X embeds into a reflexive space Z with an FDD which also satisfies analogous block upper (u_i) and lower (v_i) estimates. Certain universal space consequences ensue. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 30 pages, uses mypreamble.tex The source file(s), mypreamble.tex: 7670 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.0651 or http://arXiv.org/abs/0706.0651 or by email in unzipped form by transmitting an empty message with subject line uget 0706.0651 or in gzipped form by using subject line get 0706.0651 to: math(a)arXiv.org.

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Abstract of a paper by P. Holicky, O. Kalenda, L. Vesely, and L. Zajicek
by Dale Alspach 06 Jun '07

06 Jun '07

This is an announcement for the paper "Quotients of continuous convex functions on nonreflexive Banach spaces" by P. Holicky, O. Kalenda, L. Vesely, and L. Zajicek. Abstract: On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction gives also a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals. Archive classification: math.FA Mathematics Subject Classification: 46B10; 46B03 Remarks: 5 pages The source file(s), 06HKVZscisly.tex: 19081 bytes, is(are) stored in gzipped form as 0706.0633.gz with size 7kb. The corresponding postcript file has gzipped size 71kb. Submitted from: zajicek(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.0633 or http://arXiv.org/abs/0706.0633 or by email in unzipped form by transmitting an empty message with subject line uget 0706.0633 or in gzipped form by using subject line get 0706.0633 to: math(a)arXiv.org.

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Abstract of a paper by James R. Lee, Assaf Naor, and Yuval Peres
by Dale Alspach 06 Jun '07

06 Jun '07

This is an announcement for the paper "Trees and Markov convexity" by James R. Lee, Assaf Naor, and Yuval Peres. Abstract: We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors. Archive classification: math.MG math.FA The source file(s), TreeMarkov-GAFA.tex: 228845 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0706.0545 or http://arXiv.org/abs/0706.0545 or by email in unzipped form by transmitting an empty message with subject line uget 0706.0545 or in gzipped form by using subject line get 0706.0545 to: math(a)arXiv.org.

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Banach June 2007