Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Miroslav Bacak and Petr Hajek
by alspach＠fourier.math.okstate.edu 07 Apr '08

07 Apr '08

This is an announcement for the paper "Mazur intersection property for Asplund spaces" by Miroslav Bacak and Petr Hajek. Abstract: The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character $\om$ has a renorming with the Mazur intersection property. Combined with the previous result of Jim\' enez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP renormability of Asplund spaces of density $\om$ is undecidable in ZFC. Archive classification: math.FA Mathematics Subject Classification: 46B03 Remarks: 6 pages The source file(s), bacak-hajek.tex: 25023 bytes, is(are) stored in gzipped form as 0804.0583.gz with size 9kb. The corresponding postcript file has gzipped size 70kb. Submitted from: bacak(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.0583 or http://arXiv.org/abs/0804.0583 or by email in unzipped form by transmitting an empty message with subject line uget 0804.0583 or in gzipped form by using subject line get 0804.0583 to: math(a)arXiv.org.

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Abstract of a paper by Emanuel Milman
by alspach＠fourier.math.okstate.edu 07 Apr '08

07 Apr '08

This is an announcement for the paper "On the role of convexity in functional and isoperimetric inequalities" by Emanuel Milman. Abstract: This is a continuation of our previous work http://arxiv.org/abs/0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz-Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe--Cattiaux--Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, $q$-log-Sobolev inequalities ($q \in [1,2]$) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry--Ledoux and Bobkov--Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the $CD(0,\infty)$ curvature-dimension condition of Bakry--\'Emery. Archive classification: math.FA math.PR Mathematics Subject Classification: 32F32, 26D10, 46E35, 31C15 Remarks: 42 pages The source file(s), Dingir120.eps: 7755 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.0453 or http://arXiv.org/abs/0804.0453 or by email in unzipped form by transmitting an empty message with subject line uget 0804.0453 or in gzipped form by using subject line get 0804.0453 to: math(a)arXiv.org.

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Abstract of a paper by Z. Brzezniak, J. M. A. M. van Neerven, M. C. Veraar and L. Weis
by alspach＠fourier.math.okstate.edu 07 Apr '08

07 Apr '08

This is an announcement for the paper "Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation" by Z. Brzezniak, J. M. A. M. van Neerven, M. C. Veraar and L. Weis. Abstract: Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation. Archive classification: math.PR math.FA Mathematics Subject Classification: 60H15; 28C20; 35R60; 46B09; 60B11 Remarks: Accepted for publication in Journal of Differential Equations The source file(s), zakai_01_04-2008_arxiv.tex: 83664 bytes, is(are) stored in gzipped form as 0804.0302.gz with size 25kb. The corresponding postcript file has gzipped size 148kb. Submitted from: mark(a)profsonline.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.0302 or http://arXiv.org/abs/0804.0302 or by email in unzipped form by transmitting an empty message with subject line uget 0804.0302 or in gzipped form by using subject line get 0804.0302 to: math(a)arXiv.org.

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Abstract of a paper by Marianne Morillon
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Uniform Eberlein spaces and the finite axiom of choice" by Marianne Morillon. Abstract: We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ({\em resp.} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of $I$, ({\em resp.} the countable axiom of choice $\ACD$) implies that $F$ is compact. This enhances previous results where $\ACD$ ({\em resp.} the axiom of Dependent Choices $\DC$) was required. Moreover, if $I$ is linearly orderable (for example $I=\IR$), the closed unit ball of $\ell^2(I)$ is weakly compact (in $\ZF$). Archive classification: math.FA math.GN math.LO Mathematics Subject Classification: 03E25 , 54B10, 54D30, 46B26 The source file(s), icone-ermit.eps: 24310 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.0154 or http://arXiv.org/abs/0804.0154 or by email in unzipped form by transmitting an empty message with subject line uget 0804.0154 or in gzipped form by using subject line get 0804.0154 to: math(a)arXiv.org.

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Abstract of a paper by Marius Junge and Christian Le Merdy
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Dilations and rigid factorisations on noncommutative L^p-spaces" by Marius Junge and Christian Le Merdy. Abstract: We study some factorisation and dilation properties of completely positive maps on noncommutative L^p-spaces. We show that Akcoglu's dilation theorem for positive contractions on classical (=commutative) L^p-spaces has no reasonable analog in the noncommutative setting. Our study relies on non symmetric analogs of Pisier's operator space valued noncommutative L^p-spaces that we investigate in the first part of the paper. Archive classification: math.FA math.OA Mathematics Subject Classification: 46L07, 46L51, 48B28 Remarks: To be published in Journal of Functional Analysis The source file(s), JLRevised.tex: 91495 bytes, is(are) stored in gzipped form as 0803.4410.gz with size 26kb. The corresponding postcript file has gzipped size 178kb. Submitted from: clemerdy(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.4410 or http://arXiv.org/abs/0803.4410 or by email in unzipped form by transmitting an empty message with subject line uget 0803.4410 or in gzipped form by using subject line get 0803.4410 to: math(a)arXiv.org.

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Abstract of a paper by Christian Le Merdy, Eric Ricard, and Jean Roydor
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Completely 1-complemented subspaces of Schatten spaces" by Christian Le Merdy, Eric Ricard, and Jean Roydor. Abstract: We consider the Schatten spaces S^p in the framework of operator space theory and for any $1\leq p\not=2<\infty$, we characterize the completely 1-complemented subspaces of S^p. They turn out to be the direct sums of spaces of the form S^p(H,K), where H,K are Hilbert spaces. This result is related to some previous work of Arazy-Friedman giving a description of all 1-complemented subspaces of S^p in terms of the Cartan factors of types 1-4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L^p-spaces. Also we show that for any $n\geq 2$, there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2n which is not completely isometric, and we investigate L^p-versions of such isomorphisms. Archive classification: math.FA math.OA Mathematics Subject Classification: 46L07; 46L89; 17C65 Remarks: To be pubished in the Transactions of the American Mathematical The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.4408 or http://arXiv.org/abs/0803.4408 or by email in unzipped form by transmitting an empty message with subject line uget 0803.4408 or in gzipped form by using subject line get 0803.4408 to: math(a)arXiv.org.

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Abstract of a paper by Christian Le Merdy and Fedor Sukochev
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Rademacher averages on noncommutative symmetric spaces" by Christian Le Merdy and Fedor Sukochev. Abstract: Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let $(\varepsilon_k)_k$ be a Rademacher sequence, on some probability space $\Omega$. For finite sequences $(x_k)_k of E(M), we consider the Rademacher averages $\sum_k \varepsilon_k\otimes x_k$ as elements of the noncommutative function space $E(L^\infty(\Omega)\otimes M)$ and study estimates for their norms $\Vert \sum_k \varepsilon_k \otimes x_k\Vert_E$ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, the latter norm is equivalent to the infimum of $\Vert (\sum y_k^*y_k)^{\frac{1}{2}}\Vert + \Vert (\sum z_k z_k^*)^{\frac{1}{2}}\Vert$ over all $y_k,z_k$ in E(M) such that $x_k=y_k+z_k$ for any k. Dual estimates are given when E is 2-convex and has a non trivial upper Boyd index. We also study Rademacher averages for doubly indexed families of E(M). Archive classification: math.FA math.OA Mathematics Subject Classification: 46L52; 46M35; 47L05 The source file(s), KHTot.tex: 72248 bytes, is(are) stored in gzipped form as 0803.4404.gz with size 20kb. The corresponding postcript file has gzipped size 152kb. Submitted from: clemerdy(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.4404 or http://arXiv.org/abs/0803.4404 or by email in unzipped form by transmitting an empty message with subject line uget 0803.4404 or in gzipped form by using subject line get 0803.4404 to: math(a)arXiv.org.

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Abstract of a paper by Th. Schlumprecht and N. Sivakumar
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian" by Th. Schlumprecht and N. Sivakumar. Abstract: Let $\lambda$ be a positive number, and let $(x_j:j\in\mathbb Z)\subset\mathbb R$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\{\mathbb R\ni t\mapsto e^{ix_jt}:j\in\mathbb Z\}$ is a Riesz basis ({\it i.e.,\/} unconditionalbasis) for $L_2[-\pi,\pi]$. Given a function $f\in L_2(\mathbb R)$ whose Fourier transform is zero almost everywhere outside the interval $[-\pi,\pi]$, there is a unique square-summable sequence $(a_j:j\in\mathbb Z)$, depending on $\lambda$ and $f$, such that the function$$I_\lambda(f)(x):=\sum_{j\in\mathbb Z}a_je^{-\lambda(x-x_j)^2}, \qquad x\in\mathbb R, $$ is continuous and square integrable on $(-\infty,\infty)$, and satisfies the interpolatory conditions $I_\lambda (f)(x_j)=f(x_j)$, $j\in\mathbb Z$. It is shown that $I_\lambda(f)$ converges to $f$ in $L_2(\mathbb R)$, and also uniformly on $\mathbb R$, as $\lambda\to0^+$. A multidimensional version of this result is also obtained. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on $\ell_p(\mathbb Z)$ for every $p\in[1,\infty]$. Archive classification: math.CA math.FA Mathematics Subject Classification: 41A05 46E15 The source file(s), scsi1_5.tex: 93892 bytes, is(are) stored in gzipped form as 0803.4344.gz with size 27kb. The corresponding postcript file has gzipped size 165kb. 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/0803.4344 or http://arXiv.org/abs/0803.4344 or by email in unzipped form by transmitting an empty message with subject line uget 0803.4344 or in gzipped form by using subject line get 0803.4344 to: math(a)arXiv.org.

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Abstract of a paper by Anton R. Schep
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Products and factors of Banach function spaces" by Anton R. Schep. Abstract: Given two Banach function spaces we study the pointwise product space E.F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E . M(E, F)=F, where M(E,F) denotes the space of multiplication operators from E into F. Archive classification: math.FA Mathematics Subject Classification: 46E30; 47B38 Remarks: 16 pages The source file(s), product-bfs.bbl: 4503 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.4336 or http://arXiv.org/abs/0803.4336 or by email in unzipped form by transmitting an empty message with subject line uget 0803.4336 or in gzipped form by using subject line get 0803.4336 to: math(a)arXiv.org.

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Abstract of a paper by Yauhen Radyna, Yakov Radyno, and Anna Sidorik
by alspach＠fourier.math.okstate.edu 02 Apr '08

02 Apr '08

This is an announcement for the paper "Characterizing Hilbert spaces using Fourier transform over the field of p-adic numbers" by Yauhen Radyna, Yakov Radyno, and Anna Sidorik. Abstract: We characterize Hilbert spaces in the class of all Banach spaces using Fourier transform of vector-valued functions over the field $Q_p$ of $p$-adic numbers. Precisely, Banach space $X$ is isomorphic to a Hilbert one if and only if Fourier transform $F: L_2(Q_p,X)\to L_2(Q_p,X)$ in space of functions, which are square-integrable in Bochner sense and take value in $X$, is a bounded operator. Archive classification: math.FA Mathematics Subject Classification: 46C15, 43A25 Citation: Yauhen Radyna, Yakov Radyno, Anna Sidorik, Characterizing Hilbert The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.3646 or http://arXiv.org/abs/0803.3646 or by email in unzipped form by transmitting an empty message with subject line uget 0803.3646 or in gzipped form by using subject line get 0803.3646 to: math(a)arXiv.org.

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