Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Gilles Pisier
by alspach＠math.okstate.edu 04 Feb '11

04 Feb '11

This is an announcement for the paper "Grothendieck's Theorem, past and present" by Gilles Pisier. Abstract: Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as ``the fundamental theorem in the metric theory of tensor products''. That is now commonly referred to as ``Grothendieck's theorem'' (GT in short), or sometimes as ``Grothendieck's inequality''. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in $C^*$-algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of ``operator spaces'' or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields:\ in connection with Bell's inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by ``semidefinite programming' and hence solved in polynomial time. In this expository paper, we present a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author's 1986 CBMS notes. Archive classification: math.FA math-ph math.MP math.OA Mathematics Subject Classification: 46B28, 46B07 Submitted from: pisier(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.4195 or http://arXiv.org/abs/1101.4195

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Abstract of a paper by Daniel Carando, Silvia Lassalle, and Pablo Schmidberg
by alspach＠math.okstate.edu 28 Jan '11

28 Jan '11

This is an announcement for the paper "The reconstruction formula for Banach frames and duality" by Daniel Carando, Silvia Lassalle, and Pablo Schmidberg. Abstract: We study conditions on a Banach frame that ensures the validity of a reconstruction formula. In particular, we show that any Banach frames for (a subspace of) $L_p$ or $L_{p,q}$ ($1\le p < \infty$) with respect to a solid sequence space always satisfies an unconditional reconstruction formula. The existence of reconstruction formulae allows us to prove some James-type results for atomic decompositions: an unconditional atomic decomposition (or unconditional Schauder frame) for $X$ is shrinking (respectively, boundedly complete) if and only if $X$ does not contain an isomorphic copy of $\ell_1$ (respectively, $c_0$). Archive classification: math.FA math.CA Mathematics Subject Classification: 41A65, 42C15, 46B10, 46B15 Remarks: 16 pages Submitted from: slassall(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.2430 or http://arXiv.org/abs/1101.2430

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Abstract of a paper by Ngai-Ching Wong
by alspach＠math.okstate.edu 28 Jan '11

28 Jan '11

This is an announcement for the paper "Operator ideals arising from generating sequences" by Ngai-Ching Wong. Abstract: In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated. Archive classification: math.FA Mathematics Subject Classification: 47L20, 47B10 46A11, 46A17 Remarks: 17 pages, for the Proceedings of International Conference on Algebra 2010, World Scientific. (The International Conference on Algebra in honor of the 70th birthday of Professor Shum Kar Ping was held by Universitas Gadjah Mada (UGM)in Yogyakarta, Indonesia on October 7-10, 2010.) Submitted from: wong(a)math.nsysu.edu.tw The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.2085 or http://arXiv.org/abs/1101.2085

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Abstract of a paper by Daniel Pellegrino and Joilson Ribeiro
by alspach＠math.okstate.edu 28 Jan '11

28 Jan '11

This is an announcement for the paper "On multi-ideals and polynomial ideals of Banach spaces" by Daniel Pellegrino and Joilson Ribeiro. Abstract: The notion of coherent sequences of polynomial ideals and the notion of compatibility of a polynomial ideal with a given operator ideal were recently introduced by D. Carando, V. Dimant and S. Muro. These concepts play an important role in the theory of polynomial ideals, since they offer some properties that polynomial ideals must satisfy in order to keep the spirit of a given operator ideal and also maintain some coherence between the different levels of $n$-homogeneity. However, it seems to exist no reason to omit the multi-ideals from these cycle of ideas. In the present paper we revisit these notions; more precisely, we propose that these concepts are considered for a pair $(\mathcal{P}_{k},\mathcal{M}_{k})_{k=1}^{\infty}$, where $(\mathcal{P}% _{k})_{k=1}^{\infty}$ is a polynomial ideal and $(\mathcal{M}_{k}% )_{k=1}^{\infty}$ is a multi-ideal. The construction of our approach is inspired by the important special case of absolutely summing operators. Archive classification: math.FA Remarks: 16 pages Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.1992 or http://arXiv.org/abs/1101.1992

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Abstract of a paper by Adriano Thiago L. Bernardino and Daniel Pellegrino
by alspach＠math.okstate.edu 28 Jan '11

28 Jan '11

This is an announcement for the paper "Some remarks on absolutely summing multilinear operators" by Adriano Thiago L. Bernardino and Daniel Pellegrino. Abstract: This short note has a twofold purpose: (i) to answer a question from a recent paper of D. Popa on multilinear variants of Pietsch's composition theorem for absolutely summing operators. More precisely, we show that there is a natural (and very simple) perfect extension of Pietsch's composition theorem to the multilinear setting; (ii) to investigate extensions of some results of the aforementioned paper for particular situations, by exploring cotype properties of the spaces involved. Archive classification: math.FA Remarks: 7 pages Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.2119 or http://arXiv.org/abs/1101.2119

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Abstract of a paper by Marek Cuth
by alspach＠math.okstate.edu 28 Jan '11

28 Jan '11

This is an announcement for the paper "Separable reduction theorems by the method of elementary submodels" by Marek Cuth. Abstract: We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to a special separable subspace, dependent only on the given set (function). We are interested in properties of sets ``to be dense, nowhere dense, meager, residual or porous'' and in properties of functions ``to be continuous, semicontinuous or Fr\'echet differentiable''. Our method of creating separable subspaces enables us to combine our results, so we easily get separable reductions of function properties such as ``be continuous on a dense subset'', ``be Fr\'echet differentiable on a residual subset'', etc. Finally, we show some applications of presented separable reduction theorems and demonstrate that some results of Zajicek, Lindenstrauss and Preiss hold in nonseparable setting as well. Archive classification: math.FA Mathematics Subject Classification: 46B26, 03C30 Remarks: 27 pages Submitted from: cuthm5am(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1101.1627 or http://arXiv.org/abs/1101.1627

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2011 Workshop at A&M
by Bill Johnson 28 Jan '11

28 Jan '11

Workshop in Analysis and Probability Department of Mathematics Texas A&M University Summer 2011 The Summer 2011 Workshop in Analysis and Probability at Texas A&M University will be in session from July 5 until August 5. For information about the Workshop, consult the Workshop Home Page, whose URL is http://www.math.tamu.edu/conferences/linanalysis/ The Informal Regional Functional Analysis Seminar (SUMIRFAS) will be held July 29 - 31. Steve Dilworth, Daniel Freeman, Denka Kuzarova, Edward Odell (co-chair), and Thomas Schlumprecht (co-chair) are organizing a Concentration Week on "Greedy Algorithms in Banach spaces and Compressed Sensing" for the week of July 18-22. When encoding or reconstructing a vector using an iterative algorithm, a natural approach is to take the best or biggest approximation at each iteration. Such techniques are referred to as greedy algorithms. The theory of compressed sensing is concerned with encoding and reconstructing vectors which are sparsely represented with respect to a given basis. Kevin Ford will present a series of talks on deterministic construction of matrices with the restrictive isometry property. There will be a second series of talks devoted to greedy algorithms and bases. The home page for this Concentration Week is at http://www.math.utexas.edu/users/freeman/greedy11/index.html Florent Baudier (chair), Bill Johnson, Piotr Nowak, and Bunyamin Sari are organizing a Concentration Week on "Non-Linear Geometry of Banach Spaces, Geometric Group Theory, and Differentiability" for the week of August 1-5. The program will include an introductory course by Mark Sapir on coarse embeddings and their applications to geometric group theory, and a series of lectures by Gilles Godefroy on the recent work of the late Nigel Kalton on the coarse classification of Banach spaces. The home page for this Concentration Week is at http://www.math.tamu.edu/~pnowak/index/cw.html The Workshop is supported in part by grants from the National Science Foundation (NSF). Minorities, women, graduate students, and young researchers are especially encouraged to attend. For logistical support, including requests for support, please contact Cara Barton <cara(a)math.tamu.edu>. For more information on the Workshop itself, please contact William Johnson <johnson(a)math.tamu.edu>, David Larson <larson(a)math.tamu.edu>, Gilles Pisier <pisier(a)math.tamu.edu>, or Joel Zinn <jzinn(a)math.tamu.edu>. For information about the Concentration Week "Greedy Algorithms in Banach spaces and Compressed Sensing", contact Thomas Schlumprecht <schlump(a)math.tamu.edu> or Ted Odell <odell(a)mail.ma.utexas.edu>. For information about the Concentration Week "Non-Linear Geometry of Banach Spaces, Geometric Group Theory, and Differentiability", contact Florent Baudier <florent(a)math.tamu.edu>.

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Second announcement of IVM4 in honor of J.Diestel
by IVM4 Murcia 2011 05 Jan '11

05 Jan '11

This is the second circular regarding the meeting ''Integration, Vector Measures and Related Topics IV'' Dedicated to Joe Diestel, that will be held in Murcia, March 2 - March 5, 2011. We kindly invite you all to participate in the meeting. Those who would like to participate and didn't register yet please do. Our webpage at http://www.um.es/beca/Murcia2011/index.php is fully functional and contains all the information regarding the meeting, including: 1.- Plenary speakers. 2.- Detailed program (yet without lecture titles). 3.- Committees. 4.- Inscription. 5.- Participants: we are already 75 people. 6.- Abstracts. 7.- Hotels and travel. 8.- How to contact us. Each one of the above items contains up-to-date information regarding the meeting. We stress in particular for you to read the information about Hotels and Travel where you can find the explanations of how to book the hotel conference, prices, etc. as well as useful information of how to reach Murcia downtown or the hotel conference by yourself: the organizers will fleet a bus departing from Murcia downtown on the 1st of March, arriving to the hotel conference on time to check-in and have dinner: more details will be posted eventually. Upcoming information will be posted regularly at the web page: please, STAY WIRED. IMPORTANT INFORMATION FOR THIS SECOND ANNOUNCEMENT 1.- To cover the few expenses for which we cannot use our grants funded with public money, we need to charge 50 EUROS per person participating in the meeting, to be paid upon arrival to the meeting. 2.- As of now we can offer 8 GRANTS to cover the stay and meals at the conference hotel. These grants are intended mainly for young researchers without economical support. Those willing to apply for the grant should send ASAP a message to Bernardo Cascales at ''beca(a)um.es''. 3.- DEADLINE FOR ABSTRACT SUBMISSION: We plan to have 42 short lectures in two parallel sessions. We already have a good number of abstracts that have been sent to us, some of which have been already accepted. Some others are still being scrutinized by the Scientific Committee. The 31st OF JANUARY 2011 IS THE DEADLINE for abstracts submission. Please do send us yours before the deadline. 4.- The announced mini-course will have the title ''NEW TRENDS IN VECTOR MEASURES AND INTEGRATION''. The Scientific Committee has invited three young researchers to present a theme of their liking with special stress in recent research and open problems within the theme of the conference. These three researchers are: Qingying Bu (University of Mississippi, USA), Jose Calabuig (Universidad Politecnica de Valencia, Spain) and Jose Rodriguez (Universidad de Murcia, Spain). We believe that this all for the moment. Those of you needing some further explanations please do contact us at the e-mail address set up for the meeting ''banach(a)um.es'', or alternatively send a message to any of the local organizers. We are all looking forward to seeing you in Murcia, at the conference by the beach. We wish you all a splendid winter vacation time and a Happy New 2011. The organizers. -- Bernardo Cascales. Departamento de Matematicas. Universidad de Murcia. Office Phone: +34868884174 Web page (Personal): http://webs.um.es/beca Web (Research Group): http://www.um.es/beca On behalf of the organizers. Integration, Vector Measures and Related Topics IV. Murcia. March 2-5. 2011 http://www.um.es/beca/Murcia2011/index.php Eliminar

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Abstract of a paper by Mark W. Meckes
by alspach＠math.okstate.edu 30 Dec '10

30 Dec '10

This is an announcement for the paper "Positive definite metric spaces" by Mark W. Meckes. Abstract: Magnitude is a numerical invariant of finite metric spaces, recently introduced by T.\ Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended to infinite metric spaces in several a priori distinct ways. This paper develops the theory of a class of metric spaces, positive definite metric spaces, for which magnitude is more tractable than in general. In particular, it is shown that all the proposed definitions of magnitude coincide for compact positive definite metric spaces. Some additional results are proved about the behavior of magnitude as a function of such spaces, and a number of examples of positive definite metric spaces are found, including all subsets of $L_p$ for $p\le 2$ and Euclidean spheres equipped with the geodesic distance. Finally, some facts about the magnitude of compact subsets of $\ell_p^n$ for $p \le 2$ are proved, generalizing results of Leinster for $p=1,2$, using properties of these spaces which are somewhat stronger than positive definiteness. Archive classification: math.MG math.FA math.GN Submitted from: mark.meckes(a)case.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1012.5863 or http://arXiv.org/abs/1012.5863

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Abstract of a paper by Orest Bucicovschi and Jiri Lebl
by alspach＠math.okstate.edu 30 Dec '10

30 Dec '10

This is an announcement for the paper "On the continuity and regularity of convex extensions" by Orest Bucicovschi and Jiri Lebl. Abstract: We study continuity and regularity of convex extensions of functions from a compact set $C$ to its convex hull $K$. We show that if $C$ contains the relative boundary of $K$, and $f$ is a continuous convex function on $C$, then $f$ extends to a continuous convex function on $K$ using the standard convex roof construction. In fact, a necessary and sufficient condition for $f$ to extend from any set to a continuous convex function on the convex hull is that $f$ extends to a continuous convex function on the relative boundary of the convex hull. We give examples showing that the hypotheses in the results are necessary. In particular, if $C$ does not contain the entire relative boundary of $K$, then there may not exist any continuous convex extension of $f$. Finally, when $\partial K$ and $f$ are $C^1$ we give a necessary and sufficient condition for the convex roof construction to be $C^1$ on all of $K$. We also discuss an application of the convex roof construction in quantum computation. Archive classification: math.FA Mathematics Subject Classification: 52A41, 81P68 Remarks: 12 pages, 2 figures Submitted from: jlebl(a)math.ucsd.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1012.5796 or http://arXiv.org/abs/1012.5796

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