Banach April 2008

banach@mathdept.okstate.edu

1 participants
24 discussions

Abstract of a paper by Dmitry B. Rokhlin
by alspach＠fourier.math.okstate.edu 15 Apr '08

15 Apr '08

This is an announcement for the paper "Kreps-Yan theorem for Banach ideal spaces" by Dmitry B. Rokhlin. Abstract: Let $C$ be a closed convex cone in a Banach ideal space $X$ on a measurable space with a $\sigma$-finite measure. We prove that conditions $C\cap X_+=\{0\}$ and $C\supset -X_+$ imply the existence of a strictly positive continuous functional on $X$, whose restriction to $C$ is non-positive. Archive classification: math.FA Mathematics Subject Classification: 46E30; 46B42 Remarks: 6 pages The source file(s), RokhlinKreps-Yantheoremforbanachidealspaceseng.tex: 18929 bytes, is(are) stored in gzipped form as 0804.2075.gz with size 7kb. The corresponding postcript file has gzipped size 73kb. Submitted from: rokhlin(a)math.rsu.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.2075 or http://arXiv.org/abs/0804.2075 or by email in unzipped form by transmitting an empty message with subject line uget 0804.2075 or in gzipped form by using subject line get 0804.2075 to: math(a)arXiv.org.

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Abstract of a paper by Ariel Blanco and Niels Groenbaek
by alspach＠fourier.math.okstate.edu 14 Apr '08

14 Apr '08

This is an announcement for the paper "Amenability of algebras of approximable operators" by Ariel Blanco and Niels Groenbaek. Abstract: We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space. Archive classification: math.FA Mathematics Subject Classification: 46B20, 47L10 (primary), 16E40 (secondary) Remarks: 20 pages, to appear in Israel Journal of Mathematics The source file(s), OnAmenability2.tex: 82733 bytes, is(are) stored in gzipped form as 0804.1725.gz with size 25kb. The corresponding postcript file has gzipped size 148kb. Submitted from: gronbaek(a)math.ku.dk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.1725 or http://arXiv.org/abs/0804.1725 or by email in unzipped form by transmitting an empty message with subject line uget 0804.1725 or in gzipped form by using subject line get 0804.1725 to: math(a)arXiv.org.

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Abstract of a paper by William Arveson
by alspach＠fourier.math.okstate.edu 10 Apr '08

10 Apr '08

This is an announcement for the paper "Maximal vectors in Hilbert space and quantum entanglement" by William Arveson. Abstract: Given two matrix algebras $M_1$, $M_2$, the natural inclusion of $\mathcal L^1(M_1\otimes M_2)$ in the projective tensor product of Banach spaces $\mathcal L^1(M_1)\hat\otimes \mathcal L^1(M_2)$ is a contraction but not an isometry; and the projective cross norm can be restricted to the convex set $\mathcal S$ of density matrices in $M_1\otimes M_2$to obtain a continuous convex function $E:\mathcal S\to [1,\infty)$. We show that $E$ {\em faithfully measures entanglement} in the sense that a state is entangled if and only if its density matrix $A$ satisfies $E(A)>1$. Moreover, $E(A)$ is maximized at the density matrix $A$ associated with a pure state if and only if the range of $A$ is generated by a maximally entangled unit vector. These concrete results follow from a general analysis of norm-closed subsets $V$ of the unit sphere of a Hilbert space $H$. A {\em maximal vector} (for $V$) is a unit vector $\xi\in H$ whose distance to $V$ is maximum. Maximal vectors generalize the ``maximally entangled" unit vectors of quantum theory. In general, under a mild regularity hypothesis on $V$ we show that there is a {\em norm} on $\mathcal L^1(H)$ whose restriction to the convex set $\mathcal S$ of density operators achieves its minimum precisely on the closed convex hull of the rank one projections associated with vectors in $V$. It achieves its maximum on a rank one projection precisely when its unit vector is a maximal vector. This ``entanglement-measuring norm" is unique, and computation shows it to be the projective cross norm in the above setting of bipartite tensor products $H=H_1\otimes H_2$. Archive classification: math.OA math.FA Mathematics Subject Classification: 46N50,81P68, 94B27 Remarks: 25 pages The source file(s), ent4.tex: 76983 bytes, is(are) stored in gzipped form as 0804.1140.gz with size 21kb. The corresponding postcript file has gzipped size 126kb. Submitted from: arveson(a)math.berkeley.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.1140 or http://arXiv.org/abs/0804.1140 or by email in unzipped form by transmitting an empty message with subject line uget 0804.1140 or in gzipped form by using subject line get 0804.1140 to: math(a)arXiv.org.

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Abstract of a paper by J.M.A.M. van Neerven, M.C. Veraar, and L. Weis
by alspach＠fourier.math.okstate.edu 10 Apr '08

10 Apr '08

This is an announcement for the paper "Stochastic evolution equations in UMD Banach spaces" by J.M.A.M. van Neerven, M.C. Veraar, and L. Weis. Abstract: We discuss existence, uniqueness, and space-time H\"older regularity for solutions of the parabolic stochastic evolution equation \[\left\{\begin{aligned} dU(t) & = (AU(t) + F(t,U(t)))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,\Tend],\\ U(0) & = u_0, \end{aligned} \right. \] where $A$ generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and $W_H$ is a cylindrical Brownian motion with values in a Hilbert space $H$. We prove that if the mappings $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to \mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is $\F_0$-measurable and bounded, then this problem has a unique mild solution, which has trajectories in $C^\l([0,T];\D((-A)^\theta)$ provided $\lambda\ge 0$ and $\theta\ge 0$ satisfy $\l+\theta<\frac12$. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations. Archive classification: math.FA math.PR Mathematics Subject Classification: 47D06; 60H15; 28C20; 46B09 Remarks: Accepted for publication in Journal of Functional Analysis The source file(s), scp_arxiv.tex: 157532 bytes, is(are) stored in gzipped form as 0804.0932.gz with size 44kb. The corresponding postcript file has gzipped size 241kb. Submitted from: mark(a)profsonline.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.0932 or http://arXiv.org/abs/0804.0932 or by email in unzipped form by transmitting an empty message with subject line uget 0804.0932 or in gzipped form by using subject line get 0804.0932 to: math(a)arXiv.org.

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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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Banach April 2008