Banach October 2006

banach@mathdept.okstate.edu

2 participants
18 discussions

Abstract of a paper by Hun Hee Lee
by Dale Alspach 10 Oct '06

10 Oct '06

This is an announcement for the paper "Unconditionality with respect to orthonormal systems in noncommutative $L_2$ spaces" by Hun Hee Lee. Abstract: Orthonormal systems in commutative $L_2$ spaces can be used to classify Banach spaces. When the system is complete and satisfies certain norm condition the unconditionality with respect to the system characterizes Hilbert spaces. As a noncommutative analogue we introduce the notion of unconditionality of operator spaces with respect to orthonormal systems in noncommutative $L_2$ spaces and show that the unconditionality characterizes operator Hilbert spaces when the system is complete and satisfy certain norm condition. The proof of the main result heavily depends on free probabilistic tools such as contraction principle for $*$-free Haar unitaries, comparision of averages with respect to $*$-free Haar unitaries and $*$-free circular elements and $K$-covexity, type 2 and cotype 2 with respect to $*$-free circular elements. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: 47L25; 46L53 Remarks: 18 pages The source file(s), Unc-NoncomONS.tex: 56149 bytes, is(are) stored in gzipped form as 0610245.gz with size 15kb. The corresponding postcript file has gzipped size 92kb. 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/math.FA/0610245 or http://arXiv.org/abs/math.FA/0610245 or by email in unzipped form by transmitting an empty message with subject line uget 0610245 or in gzipped form by using subject line get 0610245 to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi, Alain Louveau, and Christian Rosendal
by Dale Alspach 10 Oct '06

10 Oct '06

This is an announcement for the paper "The complexity of classifying separable Banach spaces up to isomorphism" by Valentin Ferenczi, Alain Louveau, and Christian Rosendal. Abstract: It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 46B03; 03E15 The source file(s), ComplexityIsomorphism14.tex: 82408 bytes, is(are) stored in gzipped form as 0610289.gz with size 25kb. The corresponding postcript file has gzipped size 101kb. Submitted from: rosendal(a)math.uiuc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0610289 or http://arXiv.org/abs/math.FA/0610289 or by email in unzipped form by transmitting an empty message with subject line uget 0610289 or in gzipped form by using subject line get 0610289 to: math(a)arXiv.org.

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Abstract of a paper by Fran\c coise Lust-Piquard and Quanhua Xu
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators" by Fran\c coise Lust-Piquard and Quanhua Xu. Abstract: We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0,\;\8)$. Let $E(\M)$ be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from $E(\M)$ into a Hilbert space $\mathcal H$ corresponds a positive norm one functional $f\in E_{(2)}(\M)^*$ such that $$\forall\; x\in E(\M)\quad \|T(x)\|^2\le K^2\,\|T\|^2 f(x^*x+xx^*),$$ where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for $E(\M)$ when $E$ is either 2-concave or 2-convex and $q$-concave for some $q<\8$. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one. Archive classification: Functional Analysis; Operator Algebras Mathematics Subject Classification: Primary 46L52; Secondary 46L50; 47A63 Remarks: 14 pages. To appear in J. Funct. Anal The source file(s), petitgro.tex: 50432 bytes, is(are) stored in gzipped form as 0609356.gz with size 16kb. The corresponding postcript file has gzipped size 74kb. Submitted from: qx(a)math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0609356 or http://arXiv.org/abs/math.FA/0609356 or by email in unzipped form by transmitting an empty message with subject line uget 0609356 or in gzipped form by using subject line get 0609356 to: math(a)arXiv.org.

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Abstract of a paper by M.D. Voisei
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "The sum and chain rules for maximal monotone operators" by M.D. Voisei. Abstract: This paper is primarily concerned with the problem of maximality for the sum $A+B$ and composition $L^{*}ML$ in non-reflexive Banach space settings under qualifications constraints involving the domains of $A,B,M$. Here $X$, $Y$ are Banach spaces with duals $X^{*}$, $Y^{*}$, $A,B:X\rightrightarrows X^{*}$, $M:Y\rightrightarrows Y^{*}$ are multi-valued maximal monotone operators, and $L:X\rightarrow Y$ is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results on the topic are presented. Archive classification: Functional Analysis Mathematics Subject Classification: 47H05, 46N10 Remarks: 17 pages, submitted to Set-Valued Analysis The source file(s), tscr.tex: 42800 bytes, is(are) stored in gzipped form as 0609296.gz with size 12kb. The corresponding postcript file has gzipped size 60kb. Submitted from: mvoisei(a)utpa.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0609296 or http://arXiv.org/abs/math.FA/0609296 or by email in unzipped form by transmitting an empty message with subject line uget 0609296 or in gzipped form by using subject line get 0609296 to: math(a)arXiv.org.

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Abstract of a paper by Frank Oertel and Mark Owen
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "On utility-based super-replication prices of contingent claims with unbounded payoffs" by Frank Oertel and Mark Owen. Abstract: Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at $-\infty$ we prove that the utility-based super-replication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite {\it loss-entropy}. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, the former set is the closure of the latter under a suitable weak topology. Central to our proof is the representation of a cone $C_U$ of utility-based super-replicable contingent claims as the polar cone to the set of finite loss-entropy pricing measures. The cone $C_U$ is defined as the closure, under a relevant weak topology, of the cone of all (sufficiently integrable) contingent claims that can be dominated by a zero-financed terminal wealth. We investigate also the natural dual of this result and show that the polar cone to $C_U$ is generated by those separating measures with finite loss-entropy. The full two-sided polarity we achieve between measures and contingent claims yields an economic justification for the use of the cone $C_U$, and an open question. Archive classification: Probability; Functional Analysis; Optimization and Control Mathematics Subject Classification: 1B16, 46N10, 60G44 The source file(s), 051102reversed.tex: 29375 bytes, is(are) stored in gzipped form as 0609403.gz with size 10kb. The corresponding postcript file has gzipped size 53kb. Submitted from: f.oertel(a)ucc.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.PR/0609403 or http://arXiv.org/abs/math.PR/0609403 or by email in unzipped form by transmitting an empty message with subject line uget 0609403 or in gzipped form by using subject line get 0609403 to: math(a)arXiv.org.

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Abstract of a paper by Frank Oertel and Mark Owen
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "Utility-based super-replication prices of unbounded contingent claims and duality of cones" by Frank Oertel and Mark Owen. Abstract: Consider a financial market in which an agent trades with utility-induced restrictions on wealth. We prove that the utility-based super-replication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite entropy. Central to our proof is the representation of a cone $C_\V$ of utility-based super-replicable contingent claims as the polar cone of the set of finite entropy separating measures. $C_\V$ is shown to be the closure, under a relevant weak topology, of the cone of all (sufficiently integrable) contingent claims that can be dominated by a zero-financed terminal wealth. As our approach shows, those terminal wealths need {\it not} necessarily stem from {\it admissible} trading strategies only. We investigate also the natural dual of this result, and show that the polar cone of $C_\V$ is the cone generated by separating measures with {\it finite loss-entropy}. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, we prove that the former set is the closure of the latter under a suitable weak topology. Finally, we show how our framework can be applied to another field of mathematical economics and how it sheds a different light on Farkas' Lemma and its infinite dimensional version there. Archive classification: Probability; Functional Analysis; Optimization and Control Mathematics Subject Classification: 1B16, 46N10, 60G44 The source file(s), Final_Version_17_01_06_submitted.tex: 56116 bytes, is(are) stored in gzipped form as 0609402.gz with size 17kb. The corresponding postcript file has gzipped size 82kb. Submitted from: f.oertel(a)ucc.ie The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.PR/0609402 or http://arXiv.org/abs/math.PR/0609402 or by email in unzipped form by transmitting an empty message with subject line uget 0609402 or in gzipped form by using subject line get 0609402 to: math(a)arXiv.org.

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Abstract of a paper by Matthieu Fradelizi and Mathieu Meyer
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "Some functional forms of Blaschke-Santal\'o inequality" by Matthieu Fradelizi and Mathieu Meyer. Abstract: We establish new functional versions of the Blaschke-Santal\'o inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of K.~Ball and we give a simple proof of the case of equality. As a corollary, we get some inequalities for $\log$-concave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman, with its equality case. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 52A40 Remarks: 19 pages, to appear in Mathematische Zeitschrift The source file(s), Blaschke-Santalo-final.tex: 48038 bytes, is(are) stored in gzipped form as 0609553.gz with size 15kb. The corresponding postcript file has gzipped size 71kb. Submitted from: matthieu.fradelizi(a)univ-mlv.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0609553 or http://arXiv.org/abs/math.FA/0609553 or by email in unzipped form by transmitting an empty message with subject line uget 0609553 or in gzipped form by using subject line get 0609553 to: math(a)arXiv.org.

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Abstract of a paper by Konrad J. Swanepoel
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "A problem of Kusner on equilateral sets" by Konrad J. Swanepoel. Abstract: R. B. Kusner [R. Guy, Amer. Math. Monthly 90 (1983), 196--199] asked whether a set of vectors in a d-dimensional real vector space such that the l-p distance between any pair is 1, has cardinality at most d+1. We show that this is true for p=4 and any d >= 1, and false for all 1<p<2 with d sufficiently large, depending on p. More generally we show that the maximum cardinality is at most $(2\lceil p/4\rceil-1)d+1$ if p is an even integer, and at least $(1+\epsilon_p)d$ if 1<p<2, where $\epsilon_p>0$ depends on p. Archive classification: Metric Geometry; Functional Analysis Mathematics Subject Classification: 52C10 (Primary) 52A21, 46B20 (Secondary) Citation: Archiv der Mathematik (Basel) 83 (2004), no. 2, 164--170 Remarks: 6 pages. Small correction to Proposition 2 The source file(s), kusner-corrected.tex: 19322 bytes, is(are) stored in gzipped form as 0309317.gz with size 7kb. The corresponding postcript file has gzipped size 43kb. Submitted from: swanekj(a)unisa.ac.za The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0309317 or http://arXiv.org/abs/math.MG/0309317 or by email in unzipped form by transmitting an empty message with subject line uget 0309317 or in gzipped form by using subject line get 0309317 to: math(a)arXiv.org.

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Banach October 2006