Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Mikael de la Salle
by Dale Alspach 06 Jul '07

06 Jul '07

This is an announcement for the paper "Equimeasurabily and isometries in noncommutative Lp-spaces" by Mikael de la Salle. Abstract: We prove some noncommutative analogues of a theorem by Rudin and Plotkin about equimeasurability and isometries in L_p-spaces. Let 0<p<\infty, p not an even integer. The main result of this paper states that in the category of unital subspaces of noncommutative probability Lp-spaces, the unital completely isometric maps come from *-isomorphisms of the underlying von Neumann algebras. Unfortunately we are only able to treat the case of bounded operators. Archive classification: math.OA math.FA Mathematics Subject Classification: 46L53; 46L51; 47L05 Remarks: 11 pages The source file(s), article_arxiv.bbl: 2056 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.0427 or http://arXiv.org/abs/0707.0427 or by email in unzipped form by transmitting an empty message with subject line uget 0707.0427 or in gzipped form by using subject line get 0707.0427 to: math(a)arXiv.org.

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Abstract of a paper by Marius Junge and Hun Hee Lee
by Dale Alspach 06 Jul '07

06 Jul '07

This is an announcement for the paper "A Maurey type result for operator spaces" by Marius Junge and Hun Hee Lee. Abstract: The little Grothendieck theorem for Banach spaces says that every bounded linear operator between $C(K)$ and $\ell_2$ is 2-summing. However, it is shown in \cite{J05} that the operator space analogue fails. Not every cb-map $v : \K \to OH$ is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem : Every cb-map $v : \K \to OH$ is $(q,cb)$-summing for any $q>2$ and hence admits a factorization $\|v(x)\| \leq c(q) \|v\|_{cb} \|axb\|_q$ with $a,b$ in the unit ball of the Schatten class $S_{2q}$. Archive classification: math.FA math.OA Mathematics Subject Classification: 47L25; 46B07 Remarks: 29 pages The source file(s), MaureyTypeResultOS.tex: 99707 bytes, is(are) stored in gzipped form as 0707.0152.gz with size 25kb. The corresponding postcript file has gzipped size 184kb. Submitted from: lee.hunhee(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.0152 or http://arXiv.org/abs/0707.0152 or by email in unzipped form by transmitting an empty message with subject line uget 0707.0152 or in gzipped form by using subject line get 0707.0152 to: math(a)arXiv.org.

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Abstract of a paper by Hun Hee Lee
by Dale Alspach 06 Jul '07

06 Jul '07

This is an announcement for the paper "Tsirelson like operator spaces" by Hun Hee Lee. Abstract: We construct nontrivial examples of weak-$C_p$ ($1\leq p \leq \infty$) operator spaces with the local operator space structure very close to $C_p = [R, C]_{\frac{1}{p}}$. These examples are non-homogeneous Hilbertian operator spaces, and their constructions are similar to that of 2-convexified Tsirelson's space by W. B. Johnson. Archive classification: math.FA math.OA Mathematics Subject Classification: 47L25; 46B07 Remarks: 19 pages The source file(s), TsirelsonLikeOS.tex: 54208 bytes, is(are) stored in gzipped form as 0707.0147.gz with size 13kb. The corresponding postcript file has gzipped size 113kb. Submitted from: lee.hunhee(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.0147 or http://arXiv.org/abs/0707.0147 or by email in unzipped form by transmitting an empty message with subject line uget 0707.0147 or in gzipped form by using subject line get 0707.0147 to: math(a)arXiv.org.

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Abstract of a paper by Shin-ichi Ohta
by Dale Alspach 06 Jul '07

06 Jul '07

This is an announcement for the paper "Markov type of Alexandrov spaces of nonnegative curvature" by Shin-ichi Ohta. Abstract: We prove that Alexandrov spaces $X$ of nonnegative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of $X$ into a 2-uniformly convex Banach space is extended as a Lipschitz continuous map on the entire space $X$. Archive classification: math.MG math.FA Mathematics Subject Classification: 46B20, 53C21, 60J10 Remarks: 16 pages The source file(s), type+.tex: 40468 bytes, is(are) stored in gzipped form as 0707.0102.gz with size 11kb. The corresponding postcript file has gzipped size 103kb. Submitted from: sohta(a)math.kyoto-u.ac.jp The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.0102 or http://arXiv.org/abs/0707.0102 or by email in unzipped form by transmitting an empty message with subject line uget 0707.0102 or in gzipped form by using subject line get 0707.0102 to: math(a)arXiv.org.

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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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