Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Kevin Beanland, Ryan Causey, and Pavlos Motakis
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "Arbitrarily distortable Banach spaces of higher order" by Kevin Beanland, Ryan Causey, and Pavlos Motakis. Abstract: We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank $AD(\cdot)$, introduced by P. Dodos, uses the transfinite Schreier familes and has the property that $AD(X) < \omega_1$ if and only if $X$ is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order $\ell_1$ spreading models. We also compute this rank for for several Banach spaces. In particular, it is shown that class of Banach spaces $\mathfrak{X}^{\omega^\xi}_{0,1}$ , which each admit $\ell_1$ and $c_0$ spreading models hereditarily, and were introduced by S.A. Argyros, the first and third author, satisfy $AD(\mathfrak{X}^{\omega^\xi}_{0,1}) = \omega^\xi + 1$. This answers some questions of Dodos. Archive classification: math.FA Submitted from: CAUSEYRM(a)mailbox.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.5065 or http://arXiv.org/abs/1408.5065

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Abstract of a paper by Dario Trevisan
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "A short proof of Stein's universal multiplier theorem" by Dario Trevisan. Abstract: We give a short proof of Stein's universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods). Archive classification: math.FA Submitted from: dario.trevisan(a)sns.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.4752 or http://arXiv.org/abs/1408.4752

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Abstract of a paper by Yong Jiao, Anming Yang, Lian Wu, and Rui Yi
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "The predual and John-Nirenberg inequalities on generalized BMO martingale spaces" by Yong Jiao, Anming Yang, Lian Wu, and Rui Yi. Abstract: In this paper we introduce the generalized BMO martingale spaces by stopping time sequences, which enable us to characterize the dual spaces of martingale Hardy-Lorentz spaces $H_{p,q}^s$ for $0<p\leq1, 1<q<\infty$. Moreover, by duality we obtain a John-Nirenberg theorem for the generalized BMO martingale spaces when the stochastic basis is regular. We also extend the boundedness of fractional integrals to martingale Hardy-Lorentz spaces. Archive classification: math.FA Mathematics Subject Classification: 60G46, 60G42 Remarks: 23pages Submitted from: jiaoyong(a)csu.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.4641 or http://arXiv.org/abs/1408.4641

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Abstract of a paper by Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava,
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "Ramanujan graphs and the solution of the Kadison-Singer problem" by Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava,. Abstract: We survey the techniques used in our recent resolution of the Kadison-Singer problem and proof of existence of Ramanujan Graphs of every degree: mixed characteristic polynomials and the method of interlacing families of polynomials. To demonstrate the method of interlacing families of polynomials, we give a simple proof of Bourgain and Tzafriri's restricted invertibility principle in the isotropic case. Archive classification: math.SP math.CO math.OA Mathematics Subject Classification: 05C50, 46L05, 26C10 Remarks: A version of this paper will appear in the proceedings of the 2014 The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.4421 or http://arXiv.org/abs/1408.4421

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Abstract of a paper by H. Garth Dales
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "Maximal ideals in commutative Banach algebras" by H. Garth Dales. Abstract: We show that each maximal ideal in a commutative Banach algebra has codimension 1. Archive classification: math.FA math.RA Submitted from: t.kania(a)lancaster.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.3815 or http://arXiv.org/abs/1408.3815

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Abstract of a paper by Eusebio Gardella and Martino Lupini
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "Representations of \'{e}tale groupoids on $L^p$-spaces" by Eusebio Gardella and Martino Lupini. Abstract: For $p\in (1,\infty)$, we study representations of \'{e}tale groupoids on $L^{p}$-spaces. Our main result is a generalization of Renault's disintegration theorem for representations of \'{e}tale groupoids on Hilbert spaces. We establish a correspondence between $L^{p}$-representations of an \'{e}tale groupoid $G$, contractive $L^{p}$-representations of $C_{c}(G)$, and tight regular $L^{p}$-representations of any countable inverse semigroup of open slices of $G$ that is a basis for the topology of $G$. We define analogs $F^{p}(G)$ and $F_{\mathrm{red}}^{p}(G)$ of the full and reduced groupoid C*-algebras using representations on $L^{p}$-spaces. As a consequence of our main result, we deduce that every contractive representation of $F^{p}(G)$ or $F_{\mathrm{red}}^{p}(G)$ is automatically completely contractive. Examples of our construction include the following natural families of Banach algebras: discrete group $L^{p}$-operator algebras, the analogs of Cuntz algebras on $L^{p}$-spaces, and the analogs of AF-algebras on $L^{p} $-spaces. Our results yield new information about these objects: their matricially normed structure is uniquely determined. More generally, groupoid $L^{p}$-operator algebras provide analogs of several families of classical C*-algebras, such as Cuntz-Krieger C*-algebras, tiling C*-algebras, and graph C*-algebras. Archive classification: math.OA Mathematics Subject Classification: 47L10, 22A22 (Primary) 46H05 (Secondary) Remarks: 52 pages Submitted from: mlupini(a)yorku.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.3752 or http://arXiv.org/abs/1408.3752

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Abstract of a paper by Amit Maji and P. D. Srivastava
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "On some geometric properties of generalized Musielak-Orlicz sequence space and corresponding operator ideals" by Amit Maji and P. D. Srivastava. Abstract: Let $\bold{\Phi}=(\phi_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{\bold {\Phi}}^{A}(X)$ is introduced. It is shown that the space is complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{\bold{\Phi}}^{A}(X)$ is a $\sigma$- Dedikind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of $s$-type $l_{\bold{\Phi}}^{A}$ and operator ideals under certain conditions on the matrix $A$ are discussed. Archive classification: math.FA Mathematics Subject Classification: 46A45, 47B06, 47L20 Remarks: 18 pages Submitted from: amaji(a)maths.iitkgp.ernet.in The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.3528 or http://arXiv.org/abs/1408.3528

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Abstract of a paper by Thomas Schlumprecht
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "On Zippin's embedding theorem of Banach spaces into Banach spaces with bases" by Thomas Schlumprecht. Abstract: We present a new proof of Zippin's Embedding Theorem, that every separable reflexive Banach space embeds into one with shrinking and boundedly complete basis, and every Banach space with a separable dual embeds into one with a shrinking basis. This new proof leads to improved versions of other embedding results. Archive classification: math.FA Mathematics Subject Classification: 46B03 Submitted from: schlump(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.3311 or http://arXiv.org/abs/1408.3311

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Abstract of a paper by Enrique A. Sanchez-Perez
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "The product duality formula in Banach space theory" by Enrique A. Sanchez-Perez. Abstract: In this paper we analyze a definition of product of Banach spaces that is naturally associated by duality with an abstract notion of space of multiplication operators. This dual relation allows to understand several constructions coming from different fields of the functional analysis, that can be seen as instances of the abstract one when a particular product is considered. Some relevant examples and applications are shown. Archive classification: math.FA Mathematics Subject Classification: 46A32, 46E30, 47A30, 46B10 Remarks: 14 pages Submitted from: easancpe(a)mat.upv.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.2147 or http://arXiv.org/abs/1408.2147

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Abstract of a paper by M. G. Cabrera-Padilla, J. A. Chavez-Dominguez, A. Jimenez-Vargas, and Moises Villegas-Vallecillos
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "Lipschitz tensor product" by M. G. Cabrera-Padilla, J. A. Chavez-Dominguez, A. Jimenez-Vargas, and Moises Villegas-Vallecillos. Abstract: Inspired by ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes E of a pointed metric space and a Banach space E as a certain linear subspace of the algebraic dual of Lipo(X,E^*). We prove that <Lipo(X,E^*),X\boxtimes E> forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the linear space of all finite-rank continuous linear operators from (X^#,T) into E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise convergence of X^#. The concept of Lipschitz tensor product of elements of X^# and E^* yields the space X^#\boxast E^* as a certain linear subspace of the algebraic dual of X\boxtimes E. To ensure the good behavior of a norm on X\boxtimes E with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz cross-norm on X\boxtimes E is defined. We show that the Lipschitz injective norm epsilon, the Lipschitz projective norm pi and the Lipschitz p-nuclear norm d_p (1<=p<=infty) are uniform dualizable Lipschitz cross-norms on X\boxtimes E. In fact, epsilon is the least dualizable Lipschitz cross-norm and pi is the greatest Lipschitz cross-norm on X\boxtimes E. Moreover, dualizable Lipschitz cross-norms alpha on X\boxtimes E are characterized by satisfying the relation epsilon<=alpha<=pi. In addition, the Lipschitz injective (projective) norm on X\boxtimes E can be identified with the injective (respectively, projective) tensor norm on the Banach-space tensor product between the Lipschitz-free space over X and E. In terms of the space X^#\boxast E^*, we describe the spaces of Lipschitz compact (finite-rank, approximable) operators from X to E^$. Archive classification: math.FA Mathematics Subject Classification: 26A16, 46B28, 46E15, 47L20 Remarks: 31 pages Submitted from: ajimenez(a)ual.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.1874 or http://arXiv.org/abs/1408.1874

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