Banach November 2014

banach@mathdept.okstate.edu

2 participants
26 discussions

Abstract of a paper by Gerard Buskes and Chris Schwanke
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "Functional completions of archimedean vector lattices" by Gerard Buskes and Chris Schwanke. Abstract: We study completions of Archimedean vector lattices relative to any nonempty set of positively-homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper. Archive classification: math.FA Mathematics Subject Classification: 06F20, 46A40 Submitted from: mmbuskes(a)olemiss.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5878 or http://arXiv.org/abs/1410.5878

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Abstract of a paper by Pandelis Dodos, Vassilis Kanellopoulos and Thodoris Karageorgos
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "Szemer\'{e}di's regularity lemma via martingales" by Pandelis Dodos, Vassilis Kanellopoulos and Thodoris Karageorgos. Abstract: We prove a variant of the abstract probabilistic version of Szemer\'{e}di's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences. Archive classification: math.CO math.FA math.PR Remarks: 24 pages, no figures Submitted from: pdodos(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5966 or http://arXiv.org/abs/1410.5966

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Abstract of a paper by Pandelis Dodos, Vassilis Kanellopoulos and Konstantinos Tyros
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "A concentration inequality for product spaces" by Pandelis Dodos, Vassilis Kanellopoulos and Konstantinos Tyros. Abstract: We prove a concentration inequality which asserts that, under some mild regularity conditions, every random variable defined on the product of sufficiently many probability spaces exhibits pseudorandom behavior. Archive classification: math.PR math.CO math.FA Remarks: 11 pages, no figures Submitted from: pdodos(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5965 or http://arXiv.org/abs/1410.5965

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Abstract of a paper by Silvia Lassalle and Pablo Turco
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "The weak bounded approximation property for $\mathcal A$" by Silvia Lassalle and Pablo Turco. Abstract: Fixed a Banach operator ideal $\mathcal A$, we introduce and investigate the weak bounded approximation property for $\mathcal A$, which is strictly weaker than the bounded approximation property for $\mathcal A$ of Lima, Lima and Oja (2010). We relate the weak BAP for $\mathcal A$ with approximation properties given by tensor norms and show that the metric approximation property of order $p$ of Saphar is the weak BAP for the ideal of $p'$-summing operators, $1<p<\infty$, $\frac 1p + \frac 1{p'}=1$. Under this framework, we address the question of approximation properties passing from $X'$ to $X$ or from $X''$ to $X'$. Archive classification: math.FA Mathematics Subject Classification: 47B10, 46A32, 46B28 Remarks: 15 Pages Submitted from: paturco(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5670 or http://arXiv.org/abs/1410.5670

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Abstract of a paper by Mario Chica, Vladimir Kadets, Miguel Martin, Javier Meri, and Soloviova
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "Two refinements of the Bishop-Phelps-Bollob\'as modulus" by Mario Chica, Vladimir Kadets, Miguel Martin, Javier Meri, and Soloviova. Abstract: Extending the celebrated result by Bishop and Phelps that the set of norm attaining functionals is always dense in the topological dual of a Banach space, Bollob\'as proved the nowadays known as the Bishop-Phelps-Bollob\'as theorem, which allows to approximate at the same time a functional and a vector in which it almost attains the norm. Very recently, two Bishop-Phelps-Bollob\'as moduli of a Banach space have been introduced [J. Math. Anal. Appl. 412 (2014), 697--719] to measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in this space. In this paper we present two refinements of the results of that paper. On the one hand, we get a sharp general estimation of the Bishop-Phelps-Bollob\'as modulus as a function of the norms of the point and the functional, and we also calculate it in some examples, including Hilbert spaces. On the other hand, we relate the modulus of uniform non-squareness with the Bishop-Phelps-Bollob\'as modulus obtaining, in particular, a simpler and quantitative proof of the fact that a uniformly non-square Banach space cannot have the maximum value of the Bishop-Phelps-Bollob\'as modulus. Archive classification: math.FA Mathematics Subject Classification: 46B04 Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5570 or http://arXiv.org/abs/1410.5570

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Abstract of a paper by Jesus M. F. Castillo, Valentin Ferenczi and Manuel Gonzalez
by alspach＠math.okstate.edu 12 Nov '14

12 Nov '14

This is an announcement for the paper "Singular twisted sums generated by complex interpolation" by Jesus M. F. Castillo, Valentin Ferenczi and Manuel Gonzalez. Abstract: We present new methods to obtain singular twisted sums $X\oplus_\Omega X$ (i.e., exact sequences $0\to X\to X\oplus_\Omega X \to X\to 0$ in which the quotient map is strictly singular), in which $X$ is the interpolation space arising from a complex interpolation scheme and $\Omega$ is the induced centralizer. Although our methods are quite general, in our applications we are mainly concerned with the choice of $X$ as either a Hilbert space, or Ferenczi's uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces, including the only known example so far: the Kalton-Peck space $Z_2$. In the second case we obtain the first example of an H.I. twisted sum of an H.I. space. We then use Rochberg's description of iterated twisted sums to show that there is a sequence $\mathcal F_n$ of H.I. spaces so that $\mathcal F_{m+n}$ is a singular twisted sum of $\mathcal F_m$ and $\mathcal F_n$, while for $l>n$ the direct sum $\mathcal F_n \oplus \mathcal F_{l+m}$ is a nontrivial twisted sum of $\mathcal F_l$ and $\mathcal F_{m+n}$. We also introduce and study the notion of disjoint singular twisted sum of K\"othe function spaces and construct several examples involving reflexive $p$-convex K\"othe function spaces, which include the function version of the Kalton-Peck space $Z_2$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B70, 46M18 Submitted from: castillo(a)unex.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1410.5505 or http://arXiv.org/abs/1410.5505

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Banach November 2014