- Banach - mathdept.okstate.edu

Nigel Kalton
by Casazza, Peter 29 Dec '11

29 Dec '11

Dear Friends of Nigel: The Notices of the AMS is having a special section on Nigel Kalton. I need pictures of Nigel lecturing at meetings or with other mathematicians for this issue. If you send pictures, make it clear who the individuals are and where and when the picture was taken and any other pertinent information. I thank you in advance, Pete Casazza casazzap(a)missouri.edu

1 0

Abstract of a paper by H. G. Dales and M. E. Polyakov
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Multi-normed spaces" by H. G. Dales and M. E. Polyakov. Abstract: We modify the very well known theory of normed spaces $(E, \norm)$ within functional analysis by considering a sequence $(\norm_n : n\in\N)$ of norms, where $\norm_n$ is defined on the product space $E^n$ for each $n\in\N$. Our theory is analogous to, but distinct from, an existing theory of `operator spaces'; it is designed to relate to general spaces $L^p$ for $p\in [1,\infty]$, and in particular to $L^1$-spaces, rather than to $L^2$-spaces. After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory that we shall use, we shall present in Chapter 2 our axiomatic definition of a `multi-normed space' $((E^n, \norm_n) : n\in \N)$, where $(E, \norm)$ is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum and maximum multi-norm based on a given space. Multi-norms measure `geometrical features' of normed spaces, in particular by considering their `rate of growth'. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to `multi-topological linear spaces' through `multi-null sequences', and to `multi-bounded' linear operators, which are exactly the `multi-continuous' operators. We define a new Banach space ${\mathcal M}(E,F)$ of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices. We conclude with a theory of `orthogonal decompositions' of a normed space with respect to a multi-norm, and apply this to construct a `multi-dual' space. Archive classification: math.FA Submitted from: matt.daws(a)cantab.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.5148 or http://arXiv.org/abs/1112.5148

1 0

Abstract of a paper by Tomasz Kania and Niels Jakob Laustsen
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,\omega_1])$" by Tomasz Kania and Niels Jakob Laustsen. Abstract: Let $\omega_1$ be the first uncountable ordinal. By a result of Rudin, bounded operators on the Banach space $C([0,\omega_1])$ have a natural representation as $[0,\omega_1]\times 0,\omega_1]$-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on $[0,\omega_1]$ defines a maximal ideal of codimension one in the Banach algebra $\mathscr{B}(C([0,\omega_1]))$ of bounded operators on $C([0,\omega_1])$. We give a coordinate-free characterization of this ideal and deduce from it that $\mathscr{B}(C([0,\omega_1]))$ contains no other maximal ideals. We then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range, and finally we investigate the structure of the lattice of all closed ideals of $\mathscr{B}(C([0,\omega_1]))$. Archive classification: math.FA Mathematics Subject Classification: Primary 47L10, 46H10, Secondary 47L20, 46B26, 47B38 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/1112.4800 or http://arXiv.org/abs/1112.4800

1 0

Abstract of a paper by Deping Ye
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Inequalities for general mixed affine surface areas" by Deping Ye. Abstract: Several general mixed affine surface areas are introduced. We prove some important properties, such as, affine invariance, for these general mixed affine surface areas. We also establish new Alexandrov-Fenchel type inequalities, Santal\'{o}-type inequalities, and affine isoperimetric inequalities for these general mixed affine surface areas. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A20, 53A15 Submitted from: deping.ye(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.5129 or http://arXiv.org/abs/1112.5129

1 0

Abstract of a paper by Serap Oztop and Nico Spronk
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Minimal and maximal $p$-operator space structures" by Serap Oztop and Nico Spronk. Abstract: We show that for $p$-operator spaces, there are natural notions of minimal and maximal structures. This are useful for dealing with tensor products. Archive classification: math.FA math.OA Mathematics Subject Classification: 46L07, 47L25, 46G10 Remarks: 9 pages Submitted from: nspronk(a)math.uwaterloo.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.4884 or http://arXiv.org/abs/1112.4884

1 0

Abstract of a paper by C. Hugo Jimenez, Marton Naszodi, and Rafael Villa
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Push forward measures and concentration phenomena" by C. Hugo Jimenez, Marton Naszodi, and Rafael Villa. Abstract: In this note we study how a concentration phenomenon can be transmitted from one measure $\mu$ to a push-forward measure $\nu$. In the first part, we push forward $\mu$ by $\pi:supp(\mu)\rightarrow \Ren$, where $\pi x=\frac{x}{\norm{x}_L}\norm{x}_K$, and obtain a concentration inequality in terms of the medians of the given norms (with respect to $\mu$) and the Banach-Mazur distance between them. This approach is finer than simply bounding the concentration of the push forward measure in terms of the Banach-Mazur distance between $K$ and $L$. As a corollary we show that any normed probability space with good concentration is far from any high dimensional subspace of the cube. In the second part, two measures $\mu$ and $\nu$ are given, both related to the norm $\norm{\cdot}_L$, obtaining a concentration inequality in which it is involved the Banach-Mazur distance between $K$ and $L$ and the Lipschitz constant of the map that pushes forward $\mu$ into $\nu$. As an application, we obtain a concentration inequality for the cross polytope with respect to the normalized Lebesgue measure and the $\ell_1$ norm. Archive classification: math.FA Mathematics Subject Classification: 46B06, 46b07, 46B09, 52A20 Remarks: 12 pages Submitted from: carloshugo(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.4765 or http://arXiv.org/abs/1112.4765

1 0

Abstract of a paper by David Alonso-Gutierrez, C. Hugo Jimenez, and Rafael Villa
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Brunn-Minkowski and Zhang inequalities for convolution bodies" by David Alonso-Gutierrez, C. Hugo Jimenez, and Rafael Villa. Abstract: A quantitative version of Minkowski sum, extending the definition of $\theta$-convolution of convex bodies, is studied to obtain extensions of the Brunn-Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given. Archive classification: math.FA Mathematics Subject Classification: 52A40 (Primary), 52A20, 52A23 (Secondary) Remarks: 16 pages Submitted from: carloshugo(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.4757 or http://arXiv.org/abs/1112.4757

1 0

Abstract of a paper by Antonio Aviles and Piotr Koszmider
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "A continuous image of a Radon-Nikod\'{y}m compact which is not Radon-Nikod\'{y}m" by Antonio Aviles and Piotr Koszmider. Abstract: We construct a continuous image of a Radon-Nikod\'{y}m compact space which is not Radon-Nikod\'{y}m compact, solving the problem posed in the 80ties by Isaac Namioka. Archive classification: math.FA math.GN math.LO Submitted from: avileslo(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.4152 or http://arXiv.org/abs/1112.4152

1 0

Abstract of a paper by Fedor Nazarov, Dmitry Ryabogin and Artem Zvavitch
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Non-uniqueness of convex bodies with prescribed volumes of sections and projections" by Fedor Nazarov, Dmitry Ryabogin and Artem Zvavitch. Abstract: We show that if $d\ge 4$ is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions. Archive classification: math.CA math.FA math.MG Mathematics Subject Classification: Primary: 52A20, 52A40, secondary: 52A38 Submitted from: zvavitch(a)math.kent.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1112.3976 or http://arXiv.org/abs/1112.3976

1 0

Abstract of a paper by Manor Mendel and Assaf Naor
by alspach＠math.okstate.edu 29 Dec '11

29 Dec '11

This is an announcement for the paper "Ultrametric skeletons" by Manor Mendel and Assaf Naor. Abstract: We prove that for every $\epsilon\in (0,1)$ there exists $C_\epsilon\in (0,\infty)$ with the following property. If $(X,d)$ is a compact metric space and $\mu$ is a Borel probability measure on $X$ then there exists a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\epsilon)$, and a probability measure $\nu$ supported on $S$ satisfying $\nu\left(B_d(x,r)\right)\le \left(\mu(B_d(x,C_\epsilon r)\right)^{1-\epsilon}$ for all $x\in X$ and $r\in (0,\infty)$. The dependence of the distortion on $\epsilon$ is sharp. We discuss an extension of this statement to multiple measures, as well as how it implies Talagrand's majorizing measures theorem. Archive classification: math.MG 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/1112.3416 or http://arXiv.org/abs/1112.3416

1 0