Banach October 2005

banach@mathdept.okstate.edu

2 participants
9 discussions

Abstract of a paper by D. Azagra and M. Jimenez-Sevilla
by Dale Alspach 28 Oct '05

28 Oct '05

This is an announcement for the paper "Approximation by smooth functions with no critical points on separable Banach spaces" by D. Azagra and M. Jimenez-Sevilla. Abstract: We characterize the class of separable Banach spaces $X$ such that for every continuous function $f:X\to\mathbb{R}$ and for every continuous function $\varepsilon:X\to\mathbb(0,+\infty)$ there exists a $C^1$ smooth function $g:X\to\mathbb{R}$ for which $|f(x)-g(x)|\leq\varepsilon(x)$ and $g'(x)\neq 0$ for all $x\in X$ (that is, $g$ has no critical points), as those Banach spaces $X$ with separable dual $X^*$. We also state sufficient conditions on a separable Banach space so that the function $g$ can be taken to be of class $C^p$, for $p=1,2,..., +\infty$. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces $\ell_p(\mathbb{N})$ and $L_p(\mathbb{R}^n)$. Some important consequences of the above results are (1) the existence of {\em a non-linear Hahn-Banach theorem} and (2) the smooth approximation of closed sets, on the classes of spaces considered above. Archive classification: Functional Analysis; Differential Geometry Mathematics Subject Classification: 46B20; 46T30; 58E05; 58C25 Remarks: 34 pages The source file(s), critical270905.tex: 127379 bytes, separable2argument.eps: 46690 bytes, separable3argument.eps: 48762 bytes, separable3bargument.eps: 48459 bytes, sinbase2.eps: 47562 bytes, is(are) stored in gzipped form as 0510603.tar.gz with size 90kb. The corresponding postcript file has gzipped size 220kb. Submitted from: daniel_azagra(a)mat.ucm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510603 or http://arXiv.org/abs/math.FA/0510603 or by email in unzipped form by transmitting an empty message with subject line uget 0510603 or in gzipped form by using subject line get 0510603 to: math(a)arXiv.org.

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Abstract of a paper by S. J. Dilworth, E. Odell, Th. Schlumprecht and A. Zsak
by Dale Alspach 28 Oct '05

28 Oct '05

This is an announcement for the paper "Partial Unconditionality" by S. J. Dilworth, E. Odell, Th. Schlumprecht and A. Zsak. Abstract: J. Elton proved that every normalized weakly null sequence in a Banach space admits a subsequence that is nearly unconditional which is a weak form of unconditionality. The notion of near-unconditionality is quantified by a constant $K(\delta)$ depending on a parameter $\delta \in (0,1]$. It is unknown if $\sup_{\delta>0} K(\delta) < \infty$. This problem turns out to be closely related to the question whether every infinite-dimensional Banach space contains a quasi-greedy basic sequence. The notion of a quasi-greedy basic sequence was introduced recently by S. V. Konyagin and V. N. Temlyakov. We present an extension of Elton's result which includes Schreier unconditionality. The proof involves a basic framework which we show can be also employed to prove other partial unconditionality results including that of convex unconditionality due to Argyros, Mercourakis and Tsarpalias. Various constants of partial unconditionality are defined and we investigate the relationships between them. We also explore the combinatorial problem underlying the $\sup_{\delta>0} K(\delta) < \infty$ problem and show that $\sup_{\delta>0} K(\delta) > 5/4$. Archive classification: Functional Analysis Mathematics Subject Classification: 46B15 Remarks: 50 pages The source file(s), partial_unconditionality.tex: 175575 bytes, is(are) stored in gzipped form as 0510609.gz with size 46kb. The corresponding postcript file has gzipped size 196kb. Submitted from: a.zsak(a)dpmms.cam.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510609 or http://arXiv.org/abs/math.FA/0510609 or by email in unzipped form by transmitting an empty message with subject line uget 0510609 or in gzipped form by using subject line get 0510609 to: math(a)arXiv.org.

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Abstract of a paper by Subhash Khot and Assaf Naor
by Dale Alspach 27 Oct '05

27 Oct '05

This is an announcement for the paper "Nonembeddability theorems via Fourier analysis" by Subhash Khot and Assaf Naor. Abstract: Various new nonembeddability results (mainly into $L_1$) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on $\{0,1\}^d$ has $L_1$ distortion $(\log d)^{\frac12-o(1)}$. We also give new lower bounds on the $L_1$ distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 46B20; 68U05 Remarks: With an appendix on quantitative estimates in Bourgain's noise The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510547 or http://arXiv.org/abs/math.FA/0510547 or by email in unzipped form by transmitting an empty message with subject line uget 0510547 or in gzipped form by using subject line get 0510547 to: math(a)arXiv.org.

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Abstract of a paper by Ulrich Kohlenbach and Laurentiu Leustean
by Dale Alspach 27 Oct '05

27 Oct '05

This is an announcement for the paper "The approximate fixed point property in product spaces" by Ulrich Kohlenbach and Laurentiu Leustean. Abstract: In this paper we generalize to unbounded convex subsets C of hyperbolic spaces results obtained by W.A. Kirk and R. Espinola on approximate fixed points of nonexpansive mappings in product spaces $(C\times M)_\infty$, where M is a metric space and C is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families $(C_u)_{u\in M}$ of unbounded convex subsets of a hyperbolic space. The key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to Borwein, Reich and Shafrir, obtained by the authors in a previous paper using techniques from mathematical logic. Inspired by that, we introduce in the last section the notion of uniform approximate fixed point property for sets C and classes of self-mappings of C. The paper ends with an open problem. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 47H10, 47H09 (Primary) 03F10 (Secondary) Remarks: 15 pages The source file(s), AFPP.tex: 38299 bytes, is(are) stored in gzipped form as 0510563.gz with size 10kb. The corresponding postcript file has gzipped size 62kb. Submitted from: leustean(a)mathematik.tu-darmstadt.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510563 or http://arXiv.org/abs/math.FA/0510563 or by email in unzipped form by transmitting an empty message with subject line uget 0510563 or in gzipped form by using subject line get 0510563 to: math(a)arXiv.org.

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Abstract of a paper by Jordi Lopez Abad and Antonis Manoussakis
by Dale Alspach 20 Oct '05

20 Oct '05

This is an announcement for the paper "A classification of Tsirelson type spaces" by Jordi Lopez Abad and Antonis Manoussakis. Abstract: We give a complete classification of mixed Tsirelson spaces T[( F\_i, theta\_i)\_{i=1}^r ] for finitely many pairs of given compact and hereditary families F\_i of finite sets of integers and 0<theta\_i<1 in terms of the Cantor-Bendixson indexes of the families F\_i, and theta\_i (0< i < r+1). We prove that there are unique countable ordinal alpha and 0<theta<1 such that every block sequence of T[( F\_i, theta\_i)\_{i=1}^r ] has a subsequence equivalent to a subsequence of the natural basis of the T( S\_{omega^alpha},theta ). Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: MSC 46b20, 05d10 The source file(s), lop-man.tex: 131413 bytes, is(are) stored in gzipped form as 0510410.gz with size 37kb. The corresponding postcript file has gzipped size 150kb. Submitted from: abad(a)logique.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510410 or http://arXiv.org/abs/math.FA/0510410 or by email in unzipped form by transmitting an empty message with subject line uget 0510410 or in gzipped form by using subject line get 0510410 to: math(a)arXiv.org.

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Abstract of a paper by Jordi Lopez Abad and S. Todorcevic
by Dale Alspach 20 Oct '05

20 Oct '05

This is an announcement for the paper "Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces" by Jordi Lopez Abad and S. Todorcevic. Abstract: We provide a somewhat general framework for studying weakly null sequences in Banach spaces using Ramsey theory of families of finite subsets of integers Archive classification: Functional Analysis; Logic Mathematics Subject Classification: MSC: 05d10, 46b20 The source file(s), lop-tod.tex: 145237 bytes, is(are) stored in gzipped form as 0510407.gz with size 41kb. The corresponding postcript file has gzipped size 136kb. Submitted from: abad(a)logique.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510407 or http://arXiv.org/abs/math.FA/0510407 or by email in unzipped form by transmitting an empty message with subject line uget 0510407 or in gzipped form by using subject line get 0510407 to: math(a)arXiv.org.

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Abstract of a paper by Matthew Neal and Bernard Russo
by Dale Alspach 20 Oct '05

20 Oct '05

This is an announcement for the paper "Classification of contractively complemented Hilbertian operator spaces" by Matthew Neal and Bernard Russo. Abstract: We construct some separable infinite dimensional homogeneous Hilbertian operator spaces which generalize the row and column spaces R and C. We show that separable infinite-dimensional Hilbertian JC*-triples are completely isometric to an element of the set of (infinite) intersections of these spaces. This set includes the operator spaces R, C, their intersection, and the space spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that these new spaces are completely isometric to the space of creation (resp. annihilation) operators on the anti-symmetric tensors of the Hilbert space. Together with the finite-dimensional case studied in our previous paper, this gives a full operator space classification of all rank-one JC*-triples in terms of creation and annihilation operator spaces. We use the above to show that all contractive projections on a C*-algebra A with infinite dimensional Hilbertian range are ``expansions'' (which we define precisely) of normal contractive projections from the second dual of A onto a Hilbertian space which is completely isometric to one of the four spaces mentioned above. This generalizes the well known result, first proved for B(H) by Robertson, that all Hilbertian operator spaces that are completely contractively complemented in a C*-algebra are completely isometric to R or C. We also compute various completely bounded Banach-Mazur distances between these spaces. Archive classification: Operator Algebras; Functional Analysis Mathematics Subject Classification: 46L07 Remarks: 24 pages, submitted The source file(s), fock0827.tex: 87070 bytes, is(are) stored in gzipped form as 0510323.gz with size 27kb. The corresponding postcript file has gzipped size 117kb. Submitted from: brusso(a)math.uci.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.OA/0510323 or http://arXiv.org/abs/math.OA/0510323 or by email in unzipped form by transmitting an empty message with subject line uget 0510323 or in gzipped form by using subject line get 0510323 to: math(a)arXiv.org.

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Conference on Banach spaces and their Applications in Analysis
by Dale Alspach 06 Oct '05

06 Oct '05

Second Announcement for the conference ``Banach spaces and their applications in Analysis'', in honor of Nigel Kalton's 60th birthday, to be held May 22-27, 2006 at Miami University in Oxford, Ohio. We would like to announce that the website for the conference has been updated and now includes information about registration fees and deadlines for submitting abstracts and payments. At the conference we will emphasize the following themes: - -- Nonlinear theory (Lipschitz classifications of Banach and metric spaces and related topics), - -- Isomorphic theory of Banach spaces including connections with combinatorics and set theory, - -- Algebraic and homological methods in Banach spaces, - -- Approximation theory and algorithms in Banach spaces (greedy algorithms, interpolation etc.), - -- Functional calculus and applications to Partial Differential Equations. The following people have agreed to be principal speakers at the conference: Yuri Brudnyi (Technion - Israel Institute of Technology, Israel) Jesus M. F. Castillo (University of Extremadura, Spain) Marianna Csornyei (University College, London, UK) Stephen Dilworth (University of South Carolina) Gilles Godefroy (UniversitÃ¨ Paris VI, France) William B. Johnson (Texas A&M University) Joram Lindenstrauss (Hebrew University, Israel) Assaf Naor (Microsoft Research) Edward Odell (University of Texas) Aleksander Pelczynski (Polish Academy of Sciences, Poland) David Preiss (University College, London, UK) Gideon Schechtman (Weizmann Institute of Science, Israel) Thomas Schlumprecht (Texas A&M University) Vladimir Temlyakov (University of South Carolina) Nicole Tomczak-Jaegermann (University of Alberta, Canada) Roman Vershynin (University of California - Davis) Lutz Weis (Universitat Karlsruhe, Germany) Przemyslaw Wojtaszczyk (Warsaw University, Poland) More information about the conference, its location, registration fees, deadlines, accommodations and a printable poster are available at the website: http://www.users.muohio.edu/randrib/bsaa2006.html Please direct any questions to either of the organizers at randrib(a)muohio.edu or randrin(a)muohio.edu Sincerely yours, Beata Randrianantoanina Narcisse Randrianantoanina.

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Abstract of a paper by Peter G. Casazza, Matt Fickus, Janet C. Tremain, and Eric Weber
by Dale Alspach 04 Oct '05

04 Oct '05

This is an announcement for the paper "The Kadison-Singer problem in mathematics and engineering" by Peter G. Casazza, Matt Fickus, Janet C. Tremain, and Eric Weber. Abstract: We will show that the famous, intractible 1959 Kadison-Singer problem in $C^{*}$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer. Archive classification: Functional Analysis Mathematics Subject Classification: 42C15; 46B03; 46C05; 47A05; 46L05; 46L10 The source file(s), KSSubmit.tex: 163819 bytes, is(are) stored in gzipped form as 0510024.gz with size 46kb. The corresponding postcript file has gzipped size 199kb. Submitted from: pete(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510024 or http://arXiv.org/abs/math.FA/0510024 or by email in unzipped form by transmitting an empty message with subject line uget 0510024 or in gzipped form by using subject line get 0510024 to: math(a)arXiv.org.

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Banach October 2005