Banach August 2007

banach@mathdept.okstate.edu

1 participants
11 discussions

Abstract of a paper by Yun Sung Choi, Han Ju Lee, and Hyun Gwi Song
by Dale Alspach 31 Aug '07

31 Aug '07

This is an announcement for the paper "Bishop's theorem and differentiability of a subspace of $C_b(K)$" by Yun Sung Choi, Han Ju Lee, and Hyun Gwi Song. Abstract: Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the G\^{a}teaux and Fr\'echet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_\delta$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ and a metrizable compact space $K$. In the case that $X$ is a complex Banach space with the Radon-Nikod\'ym property, we show that the set of all strong peak functions in $A_b(B_X)=\{ f\in C_b(B_X) : f|_{B_X^\circ} \mbox{ is holomorphic}\}$ is dense. As an application, we show that the smallest closed norming subset of $A_b(B_X)$ is the closure of the set of all strong peak points for $A_b(B_X)$. This implies that the norm of $A_b(B_X)$ is G\^{a}teaux differentiable on a dense subset of $A_b(B_X)$, even though the norm is nowhere Fr\'echet differentiable when $X$ is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary. Archive classification: math.FA Mathematics Subject Classification: 46B04; 46G20; 46G25; 46B22 The source file(s), bishop-070130.tex: 87264 bytes, is(are) stored in gzipped form as 0708.4069.gz with size 25kb. The corresponding postcript file has gzipped size 157kb. 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/0708.4069 or http://arXiv.org/abs/0708.4069 or by email in unzipped form by transmitting an empty message with subject line uget 0708.4069 or in gzipped form by using subject line get 0708.4069 to: math(a)arXiv.org.

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Abstract of a paper by Yun Sung Choi Kwang Hee Han Han Ju Lee
by Dale Alspach 31 Aug '07

31 Aug '07

This is an announcement for the paper "Boundaries for algebras of holomorphic functions on Banach spaces" by Yun Sung Choi Kwang Hee Han Han Ju Lee. Abstract: We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\lambda_{\varphi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\lambda_{\varphi, w}$ if $\varphi$ satisfies the $\delta_2$-condition. Archive classification: math.FA Mathematics Subject Classification: 46E50; 46B20; 46B45 The source file(s), shilovboundary-final-corrected.tex: 39013 bytes, is(are) stored in gzipped form as 0708.4068.gz with size 12kb. The corresponding postcript file has gzipped size 102kb. 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/0708.4068 or http://arXiv.org/abs/0708.4068 or by email in unzipped form by transmitting an empty message with subject line uget 0708.4068 or in gzipped form by using subject line get 0708.4068 to: math(a)arXiv.org.

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Abstract of a paper by N.J. Kalton and G. Lancien
by Dale Alspach 31 Aug '07

31 Aug '07

This is an announcement for the paper "Best constants for Lipschitz embeddings of metric spaces into $c_0$" by N.J. Kalton and G. Lancien. Abstract: We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into $c_0$ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical $\ell_p-$spaces into $c_0$ and give other applications. We prove that if a Banach space embeds almost isometrically into $c_0$, then it embeds linearly almost isometrically into $c_0$. We also study Lipschitz embeddings into $c_0^+$. Archive classification: math.FA Mathematics Subject Classification: 46B20; 46T99 Remarks: 22 pages The source file(s), kaltonlancienarxiv.tex: 58313 bytes, is(are) stored in gzipped form as 0708.3924.gz with size 16kb. The corresponding postcript file has gzipped size 122kb. Submitted from: gilles.lancien(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0708.3924 or http://arXiv.org/abs/0708.3924 or by email in unzipped form by transmitting an empty message with subject line uget 0708.3924 or in gzipped form by using subject line get 0708.3924 to: math(a)arXiv.org.

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Abstract of a paper by Gilles Pisier
by Dale Alspach 28 Aug '07

28 Aug '07

This is an announcement for the paper "A remark on hypercontractive semigroups and operator ideals" by Gilles Pisier. Abstract: In this note, we answer a question raised by Johnson and Schechtman \cite{JS}, about the hypercontractive semigroup on $\{-1,1\}^{\NN}$. More generally, we prove the folllowing theorem. Let $1<p<2$. Let $(T(t))_{t>0}$ be a holomorphic semigroup on $L_p$ (relative to a probability space). Assume the following mild form of hypercontractivity: for some large enough number $s>0$, $T(s)$ is bounded from $L_p$ to $L_2$. Then for any $t>0$, $T(t)$ is in the norm closure in $B(L_p)$ (denoted by $\overline{\Gamma_2}$) of the subset (denoted by ${\Gamma_2}$) formed by the operators mapping $L_p$ to $L_2$ (a fortiori these operators factor through a Hilbert space). Archive classification: math.FA Mathematics Subject Classification: 47D06 The source file(s), hyper.tex: 11355 bytes, is(are) stored in gzipped form as 0708.3423.gz with size 5kb. The corresponding postcript file has gzipped size 50kb. Submitted from: gip(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0708.3423 or http://arXiv.org/abs/0708.3423 or by email in unzipped form by transmitting an empty message with subject line uget 0708.3423 or in gzipped form by using subject line get 0708.3423 to: math(a)arXiv.org.

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Abstract of a paper by Denny H. Leung and Wee-Kee Tang
by Dale Alspach 24 Aug '07

24 Aug '07

This is an announcement for the paper "Semilattice structures of spreading models" by Denny H. Leung and Wee-Kee Tang. Abstract: Given a Banach space X, denote by SP_{w}(X) the set of equivalence classes of spreading models of X generated by normalized weakly null sequences in X. It is known that SP_{w}(X) is a semilattice, i.e., it is a partially ordered set in which every pair of elements has a least upper bound. We show that every countable semilattice that does not contain an infinite increasing sequence is order isomorphic to SP_{w}(X) for some separable Banach space X. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B15 The source file(s), LeungTangSemiLatticeStructureSpdMod.tex: 37531 bytes, is(are) stored in gzipped form as 0708.3126.gz with size 11kb. The corresponding postcript file has gzipped size 92kb. Submitted from: weekee.tang(a)nie.edu.sg The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0708.3126 or http://arXiv.org/abs/0708.3126 or by email in unzipped form by transmitting an empty message with subject line uget 0708.3126 or in gzipped form by using subject line get 0708.3126 to: math(a)arXiv.org.

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Abstract of a paper by Horst Martini, Konrad J Swanepoel and Gunter Weiss
by Dale Alspach 24 Aug '07

24 Aug '07

This is an announcement for the paper "The geometry of Minkowski spaces --- a survey. Part I" by Horst Martini, Konrad J Swanepoel and Gunter Weiss. Abstract: We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A21 (Primary), 46B07, 46B20 (Secondary) Citation: Expositiones Mathematicae 19 (2001) 97-142 Remarks: 56 pages, 28 figures The source file(s), fig10.eps: 54544 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0708.2900 or http://arXiv.org/abs/0708.2900 or by email in unzipped form by transmitting an empty message with subject line uget 0708.2900 or in gzipped form by using subject line get 0708.2900 to: math(a)arXiv.org.

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Abstract of a paper by Ronen Eldan and Boaz Klartag
by Dale Alspach 21 Aug '07

21 Aug '07

This is an announcement for the paper "Pointwise estimates for marginals of convex bodies" by Ronen Eldan and Boaz Klartag. Abstract: We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the probability density of the projection of X onto E. We show that the ratio between this probability density and the standard gaussian density in E is very close to 1 in large parts of E. Here c > 0 is a universal constant. This complements a recent result by the second named author, where the total-variation metric between the densities was considered. Archive classification: math.MG math.FA Remarks: 17 pages The source file(s), pointwise.tex: 43054 bytes, is(are) stored in gzipped form as 0708.2513.gz with size 13kb. The corresponding postcript file has gzipped size 100kb. Submitted from: bklartag(a)princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0708.2513 or http://arXiv.org/abs/0708.2513 or by email in unzipped form by transmitting an empty message with subject line uget 0708.2513 or in gzipped form by using subject line get 0708.2513 to: math(a)arXiv.org.

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Abstract of a paper by Assaf Naor and Yuval Peres
by Dale Alspach 15 Aug '07

15 Aug '07

This is an announcement for the paper "Embeddings of discrete groups and the speed of random walks" by Assaf Naor and Yuval Peres. Abstract: For a finitely generated group G and a banach space X let \alpha^*_X(G) (respectively \alpha^#_X(G)) be the supremum over all \alpha\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such that for all x,y\in G we have \|f(x)-f(y)\|\ge c\cdot d_G(x,y)^\alpha. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is \alpha^*(G)=\alpha^*_{L_2}(G) (respectively \alpha^#(G)= \alpha_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then \alpha^#_X(G)\le \frac{1}{p\beta^*(G)}. Here \beta^*(G) is the largest \beta\ge 0 for which there exists a set of generators S of G and c>0 such that for all t\in \N we have \E\big[d_G(W_t,e)\big]\ge ct^\beta, where \{W_t\}_{t=0}^\infty is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X=L_p, generalizes a theorem of Guentner and Kaminker and answers a question posed by Tessera. We also show that if \alpha^*(G)\ge 1/2 then \alpha^*(G\bwr \Z)\ge \frac{2\alpha^*(G)}{2\alpha^*(G)+1}. This improves the previous bound due to Stalder and ValetteWe deduce that if we write \Z_{(1)}= \Z and \Z_{(k+1)}\coloneqq \Z_{(k)}\bwr \Z then \alpha^*(\Z_{(k)})=\frac{1}{2-2^{1-k}}, and use this result to answer a question posed by Tessera in on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C_2\bwr C_n embed into L_1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres. Finally, we use these results to show that edge Markov type need not imply Enflo type. Archive classification: math.MG math.FA math.GR Remarks: 23 pages The source file(s), , is(are) stored in gzipped form as with size . The corresponding postcript file has gzipped size . 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/0708.0853 or http://arXiv.org/abs/0708.0853 or by email in unzipped form by transmitting an empty message with subject line uget 0708.0853 or in gzipped form by using subject line get 0708.0853 to: math(a)arXiv.org.

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Abstract of a paper by William B. Johnson and Gideon Schechtman
by Dale Alspach 06 Aug '07

06 Aug '07

This is an announcement for the paper "Multiplication operators on $L(L_p)$ and $\ell_p$-strictly singular operators" by William B. Johnson and Gideon Schechtman. Abstract: A classification of weakly compact multiplication operators on $L(L_p)$, $1<p<\infty$, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of $\ell_p$-strictly singular operators, and we also investigate the structure of general $\ell_p$-strictly singular operators on $L_p$. The main result is that if an operator $T$ on $L_p$, $1<p<2$, is $\ell_p$-strictly singular and $T_{|X}$ is an isomorphism for some subspace $X$ of $L_p$, then $X$ embeds into $L_r$ for all $r<2$, but $X$ need not be isomorphic to a Hilbert space. It is also shown that if $T $ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p <2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976. Archive classification: math.FA Mathematics Subject Classification: 46B20; 46E30 The source file(s), JSElemOpAug3.07.tex: 53364 bytes, is(are) stored in gzipped form as 0708.0560.gz with size 17kb. The corresponding postcript file has gzipped size 120kb. 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/0708.0560 or http://arXiv.org/abs/0708.0560 or by email in unzipped form by transmitting an empty message with subject line uget 0708.0560 or in gzipped form by using subject line get 0708.0560 to: math(a)arXiv.org.

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Abstract of a paper by Han Ju Lee
by Dale Alspach 06 Aug '07

06 Aug '07

This is an announcement for the paper "Notes on the geometry of space of polynomials" by Han Ju Lee. Abstract: We show that the symmetric injective tensor product space $\hat{\otimes}_{n,s,\varepsilon}E$ is not complex strictly convex if $E$ is a complex Banach space of $\dim E \ge 2$ and if $n\ge 2$ holds. It is also reproved that $\ell_\infty$ is finitely represented in $\hat{\otimes}_{n,s,\varepsilon}E$ if $E$ is infinite dimensional and if $n\ge 2$ holds, which was proved in the other way by Dineen. Archive classification: math.FA Mathematics Subject Classification: 46B20 The source file(s), Notes-on-geometry-polynomials-revised.tex: 12189 bytes, is(are) stored in gzipped form as 0708.0331.gz with size 4kb. The corresponding postcript file has gzipped size 53kb. 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/0708.0331 or http://arXiv.org/abs/0708.0331 or by email in unzipped form by transmitting an empty message with subject line uget 0708.0331 or in gzipped form by using subject line get 0708.0331 to: math(a)arXiv.org.

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Banach August 2007