Banach June 2010

banach@mathdept.okstate.edu

2 participants
22 discussions

Abstract of a paper by Christina Brech and Piotr Koszmider
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "Thin-very tall compact scattered spaces which are hereditarily separable" by Christina Brech and Piotr Koszmider. Abstract: We strengthen the property $\Delta$ of a function $f:[\omega_2]^2\rightarrow [\omega_2]^{\leq \omega}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juh\'asz and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $K$ as above where $K^n$ is hereditarily separable for each $n\in\N$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $C(K)$ is an Asplund space of density $\aleph_2$ which has no Fr\'echet smooth renorming, nor an uncountable biorthogonal system. Archive classification: math.FA math.GN Remarks: accepted to Trans. Amer. Math. Soc. Submitted from: christina.brech(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.3528 or http://arXiv.org/abs/1005.3528

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Abstract of a paper by Christina Brech
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "On the density of Banach {$C(K)$} spaces with the Grothendieck property" by Christina Brech. Abstract: Using the method of forcing we prove that consistently there is a Banach space of continuous functions on a compact Hausdorff space with the Grothendieck property and with density less than the continuum. It follows that the classical result stating that ``no nontrivial complemented subspace of a Grothendieck $C(K)$ space is separable'' cannot be strengthened by replacing ``is separable'' by ``has density less than that of $l_\infty$'', without using an additional set-theoretic assumption. Such a strengthening was proved by Haydon, Levy and Odell, assuming Martin's axiom and the negation of the continuum hypothesis. Moreover, our example shows that certain separation properties of Boolean algebras are quite far from the Grothendieck property. Archive classification: math.FA Citation: Proc. Amer. Math. Soc. 134, No. 12, 3653-3663 (2006) Submitted from: christina.brech(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.3524 or http://arXiv.org/abs/1005.3524

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Abstract of a paper by Tommi Hoynalanmaa
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "Multiresolution analysis for compactly supported interpolating tensor product wavelets" by Tommi Hoynalanmaa. Abstract: We construct a one-dimensional interpolating multiresolution analysis (MRA) of C0(R,K), K = R or K = C, and multidimensional interpolating tensor product MRAs of the function spaces C0(Rn,K) consisting of real or complex valued functions on Rn vanishing at infinity and the function spaces Cu(Rn,K) consisting of bounded and uniformly continuous functions on Rn. The theory of the tensor products of Banach spaces is used. We also generalize the Besov space norm equivalence result from Donoho (1992, Interpolating Wavelet Transforms) for our n-dimensional construction. Archive classification: math.FA Mathematics Subject Classification: 46A32 (Primary), 46B28 (Secondary), 15A69 (Secondary), 46E10 The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.3371 or http://arXiv.org/abs/1005.3371

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Abstract of a paper by Jarno Talponen
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "Directionally Euclidean structures of Banach spaces" by Jarno Talponen. Abstract: We study spaces with directionally asymptotically controlled ellipsoids approximating the unit ball in finite-dimensions. These ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids with a given functional of the dual space. The term asymptotical refers to the fact that we take '$\limsup$' over finite-dimensional subspaces. This leads to some isomorphic and isometric characterizations of Hilbert spaces. An application involving Mazur's rotation problem is given. We also discuss the complexity of the family of ellipsoids as the dimension and geometry vary. Archive classification: math.FA Mathematics Subject Classification: Primary 46C15, Secondary 52A23 Remarks: 10 pages Submitted from: talponen(a)cc.hut.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.2737 or http://arXiv.org/abs/1005.2737

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Abstract of a paper by Daniel Carando and Daniel Galicer
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "The symmetric Radon-Nikod\'ym property for tensor norms" by Daniel Carando and Daniel Galicer. Abstract: We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for finitely generated s-tensor norms $\beta$ of order $n$ and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if $\beta$ is a projective s-tensor norm with the sRN property, then for every Asplund space $E$, the canonical map $\widetilde{\otimes}_{ \beta }^{n,s} E' \rightarrow \Big(\widetilde{\otimes}_{ \beta' }^{n,s} E \Big)'$ is a metric surjection. This can be rephrased as the isometric isomorphism $\mathcal{Q}^{min}(E) = \mathcal{Q}(E)$ for certain polynomial ideal $\Q$. We also relate the sRN property of an s-tensor norm with the Asplund or Radon-Nikod\'{y}m properties of different tensor products. Similar results for full tensor products are also given. As an application, results concerning the ideal of $n$-homogeneous extendible polynomials are obtained, as well as a new proof of the well known isometric isomorphism between nuclear and integral polynomials on Asplund spaces. Archive classification: math.FA Mathematics Subject Classification: 47L22, 46M05, 46B22 Remarks: 17 pages Submitted from: dgalicer(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.2683 or http://arXiv.org/abs/1005.2683

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Abstract of a paper by Assaf Naor and Scott Sheffield
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "Absolutely minimal Lipschitz extension of tree-valued mappings" by Assaf Naor and Scott Sheffield. Abstract: We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called {\bf Politics} --- a generalization of a game called {\bf Tug of War} that has been used in~\cite{PSSW09} to study real-valued AMLEs. Archive classification: math.MG math.AP math.FA math.PR Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.2535 or http://arXiv.org/abs/1005.2535

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Abstract of a paper by Peter G. Casazza, Matt Fickus, Dustin Mixon and Janet C. Tremain
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "Concrete constructions of non-pavable projections" by Peter G. Casazza, Matt Fickus, Dustin Mixon and Janet C. Tremain. Abstract: It is known that the paving conjecture fails for $2$-paving projections with constant diagonal $1/2$. But the proofs of this fact are existence proofs. We will give concrete examples of these projections and projections with constant diagonal $1/r$ which are not $r$-pavable in a very strong sense. Archive classification: math.FA Mathematics Subject Classification: 42C15, 46C05, 46C07 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/1005.2164 or http://arXiv.org/abs/1005.2164

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Abstract of a paper by Pascal Lefevre, Daniel Li, Herve Queffelec, and Luis Rodriguez-Piazza
by Dale Alspach 03 Jun '10

03 Jun '10

This is an announcement for the paper "The canonical injection of the Hardy-Orlicz space $H^\Psi$ into the Bergman-Orlicz space ${\mathfrak B}^\Psi$" by Pascal Lefevre, Daniel Li, Herve Queffelec, and Luis Rodriguez-Piazza. Abstract: We study the canonical injection from the Hardy-Orlicz space $H^\Psi$ into the Bergman-Orlicz space ${\mathfrak B}^\Psi$. Archive classification: math.FA Remarks: 21 pages Submitted from: daniel.li(a)euler.univ-artois.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1996 or http://arXiv.org/abs/1005.1996

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Abstract of a paper by Jordi Marzo and Kristian Seip
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "$L^\infty$ to $L^p$ constants for Riesz projections" by Jordi Marzo and Kristian Seip. Abstract: The norm of the Riesz projection from $L^\infty(\T^n)$ to $L^p(\T^n)$ is considered. It is shown that for $n=1$, the norm equals $1$ if and only if $p\le 4$ and that the norm behaves asymptotically as $p/(\pi e)$ when $p\to \infty$. The critical exponent $p_n$ is the supremum of those $p$ for which the norm equals $1$. It is proved that $2+2/(2^n-1)\le p_n <4$ for $n>1$; it is unknown whether the critical exponent for $n=\infty$ exceeds $2$. Archive classification: math.FA math.CV Mathematics Subject Classification: 41A44, 42B05, 46E30 Submitted from: seip(a)math.ntnu.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1842 or http://arXiv.org/abs/1005.1842

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Abstract of a paper by Alan D. Sokal
by alspach＠fourier.math.okstate.edu 03 Jun '10

03 Jun '10

This is an announcement for the paper "A really simple elementary proof of the uniform boundedness theorem" by Alan D. Sokal. Abstract: I give a proof of the uniform boundedness theorem that is elementary (i.e. does not use any version of the Baire category theorem) and also extremely simple. Archive classification: math.FA Mathematics Subject Classification: 46B99 (Primary), 46B20, 46B28 (Secondary) Remarks: LaTex2e, 5 pages Submitted from: sokal(a)nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1585 or http://arXiv.org/abs/1005.1585

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Banach June 2010