August 2013 - Banach - mathdept.okstate.edu

Abstract of a paper by Marek Cuth and Ondrej F.K. Kalenda
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Rich families and elementary submodels" by Marek Cuth and Ondrej F.K. Kalenda. Abstract: We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system an in spaces of density $\aleph_1$. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality indexed by ranges of its projections. Archive classification: math.FA Mathematics Subject Classification: 46B26, 03C30 Submitted from: cuthm5am(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.1818 or http://arXiv.org/abs/1308.1818

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Abstract of a paper by Sun Kwang Kim and Han Ju Lee
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Simultaneously continuous retraction and its application" by Sun Kwang Kim and Han Ju Lee. Abstract: We study the existence of a retraction from the dual space $X^*$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^*}$ which is uniformly continuous in norm topology and continuous in weak-$*$ topology. Such a retraction is called a uniformly simultaneously continuous retraction. It is shown that if $X$ has a normalized unconditional Schauder basis with unconditional basis constant 1 and $X^*$ is uniformly monotone, then a uniformly simultaneously continuous retraction from $X^*$ onto $B_{X^*}$ exists. It is also shown that if $\{X_i\}$ is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity $\delta_i(\eps)$ such that $\inf_i \delta_i(\eps)>0$ and $X= \left[\bigoplus X_i\right]_{c_0}$ or $X=\left[\bigoplus X_i\right]_{\ell_p}$ for $1\le p<\infty$, then a uniformly simultaneously continuous retraction exists from $X^*$ onto $B_{X^*}$. The relation between the existence of a uniformly simultaneously continuous retraction and the Bishsop-Phelps-Bollob\'as property for operators is investigated and it is proved that the existence of a uniformly simultaneously continuous retraction from $X^*$ onto its unit ball implies that a pair $(X, C_0(K))$ has the Bishop-Phelps-Bollob\'as property for every locally compact Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S), C_0(K))$ has the Bishop-Phelps-Bollob\'as property if $C_0(S)$ and $C_0(K)$ are the spaces of all real-valued continuous functions vanishing at infinity on locally compact metric space $S$ and locally compact Hausdorff space $K$ respectively. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, Secondary 46B04, 46B22 Remarks: 15 pages Submitted from: hanjulee(a)dongguk.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.1638 or http://arXiv.org/abs/1308.1638

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Abstract of a paper by Horst Martini, Pier Luigi Papini, and Margarita Spirova
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Complete sets and completion of sets in Banach spaces" by Horst Martini, Pier Luigi Papini, and Margarita Spirova. Abstract: In this paper we study properties of complete sets and of completions of sets in Banach spaces. We consider the family of completions of a given set and its size; we also study in detail the relationships concerning diameters, radii, and centers. The results are illustrated by several examples. Archive classification: math.MG math.FA Mathematics Subject Classification: 46B20, 46B99, 52A05, 52A20, 52A21 Submitted from: margarita.spirova(a)mathematik.tu-chemnitz.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1308.0789 or http://arXiv.org/abs/1308.0789

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Abstract of a paper by Sonia Berrios and Geraldo Botelho
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "A general abstract approach to approximation properties in Banach" by Sonia Berrios and Geraldo Botelho. Abstract: We propose a unifying approach to many approximation properties studied in the literature from the 1930s up to our days. To do so, we say that a Banach space E has the (I,J,{\tau})-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the topology {\tau}, by operators belonging to the operator ideal J. Restricting {\tau} to a class of linear topologies, which we call ideal topologies, this concept recovers many classical/recent approximation properties as particular instances and several important known results are particular cases of more general results that are valid in this abstract framework. Archive classification: math.FA Submitted from: botelho(a)ufu.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1307.8073 or http://arXiv.org/abs/1307.8073

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Abstract of a paper by Charles John Read
by alspach＠math.okstate.edu 20 Aug '13

20 Aug '13

This is an announcement for the paper "Banach spaces with no proximinal subspaces of codimension 2" by Charles John Read. Abstract: The classical theorem of Bishop-Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension of the theorem gives a quantitative description of just how dense the norm-achieving functionals have to be: if (x,f) is in X x X* with ||x||=||f||=1 and |1-f(x)|< h^2/4 then there are (x',f') in X x X* with ||x'||= ||f'||=1, ||x-x'||, ||f-f'||< h and f'(x')=1. This means that there are always "proximinal" hyperplanes H in X (a nonempty subset E of a metric space is said to be "proximinal" if, for x not in E, the distance d(x,E) is always achieved - there is always an e in E with d(x,E)=d(x,e)); for if H= ker f (f in X*) then it is easy to see that H is proximinal if and only if f is norm-achieving. Indeed the set of proximinal hyperplanes H is, in the appropriate sense, dense in the set of all closed hyperplanes H in X. Quite a long time ago [Problem 2.1 in his monograph "The Theory of Best approximation and Functional Analysis" Regional Conference series in Applied Mathematics, SIAM, 1974], Ivan Singer asked if this result generalized to closed subspaces of finite codimension - if every Banach space has a proximinal subspace of codimension 2, for example. In this paper I show that there is a Banach space X such that X has no proximinal subspace of finite codimension n>1. So we have a converse to Bishop-Phelps-Bollobas: a dense set of proximinal hyperplanes can always be found, but proximinal subspaces of larger, finite codimension need not be. Archive classification: math.FA Mathematics Subject Classification: 46B04 (Primary), 46B45, 46B25 (Secondary) Remarks: The paper has been submitted for publication to the Israel Journal of The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1307.7958 or http://arXiv.org/abs/1307.7958

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