Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Itai Ben Yaacov and C. Ward Henson
by alspach＠math.okstate.edu 17 Dec '12

17 Dec '12

This is an announcement for the paper "Generic orbits and type isolation in the Gurarij space" by Itai Ben Yaacov and C. Ward Henson. Abstract: We study model-theoretic aspects of the separable Gurarij space $\bG$, in particular type isolation and the existence of prime models, without use of formal logic. \begin{enumerate} \item If $E$ is a finite-dimensional Banach space, then the set of isolated types over $E$ is dense, and there exists a prime Gurarij over $E$. This is the unique separable Gurarij space $\bG$ extending $E$ with the unique Hahn-Banach extension property (\emph{property $U$}), and the orbit of $\id\colon E \hookrightarrow \bG$ under the action of $\Aut(\bG)$ is a dense $G_\delta$ in the space of all linear isometric embeddings $E \hookrightarrow \bG$. \item If $E$ is infinite-dimensional then there are no non realised isolated types, and therefore no prime model over $E$ (unless $\bG \cong E$), and all orbits of embeddings $E \hookrightarrow \bG$ are meagre. On the other hand, there are Gurarij spaces extending $E$ with property $U$. \end{enumerate} We also point out that the class of Gurarij space is the class of models of an $\aleph_0$-categorical theory with quantifier elimination, and calculate the density character of the space of types over $E$, answering a question of Avil\'es et al. Archive classification: math.FA math.LO Submitted from: begnac.arxiv(a)free.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.4814 or http://arXiv.org/abs/1211.4814

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Abstract of a paper by Costas Poulios
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "The fixed point property in a Banach space isomorphic to $c_0$" by Costas Poulios. Abstract: We consider a Banach space, which comes naturally from c0 and it appears in the literature, and we prove that this space has the fixed point property for non-expansive mappings. Archive classification: math.FA Submitted from: k-poulios(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.3335 or http://arXiv.org/abs/1211.3335

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Abstract of a paper by Christina Brech and Piotr Koszmider
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "$\ell_\infty$-sums and the Banach space $\ell_\infty/c_0$" by Christina Brech and Piotr Koszmider. Abstract: We prove that the use of the Continuum Hypothesis in some results of Drewnowski and Roberts concerning the Banach space $\ell_\infty/c_0$ cannot be avoided. In particular, we prove that in the $\omega_2$-Cohen model, $\ell_\infty(c_0(\mathfrak{c}))$ does not embed isomorphically into $\ell_\infty/c_0$ where $\mathfrak{c}$ is the cardinality of the continuum. It follows that consistently $\ell_\infty/c_0$ is not isomorphically of the form $\ell_\infty(X)$ for any Banach space $X$. Archive classification: math.FA math.LO 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/1211.3173 or http://arXiv.org/abs/1211.3173

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Abstract of a paper by Pawel Kolwicz, Karol Lesnik and Lech Maligranda
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Pointwise products of some Banach function spaces and factorization" by Pawel Kolwicz, Karol Lesnik and Lech Maligranda. Abstract: The well-known factorization theorem of Lozanovski{\u \i} may be written in the form $L^{1}\equiv E\odot E^{\prime }$, where $\odot $ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize $F$ through $E$, i.e., when $F\equiv E\odot M(E, F) \,$, where $M(E, F) $ is the space of pointwise multipliers from $E$ to $F$. Properties of $M(E, F) $ were investigated in our earlier paper [KLM12] and here we collect and prove some properties of the construction $E\odot F$. The formulas for pointwise product of Calder\'{o}n-Lozanovski{\u \i} $E_{\varphi}$ spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for these spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant Banach function space may be factorized through Marcinkiewicz space. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B20, 46B42, 46A45 Remarks: 43 pages Submitted from: lech.maligranda(a)ltu.se The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.3135 or http://arXiv.org/abs/1211.3135

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Abstract of a paper by Ludek Zajicek
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Gateaux and Hadamard differentiability via directional differentiability" by Ludek Zajicek. Abstract: Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ an arbitrary mapping. Then the following implication holds at each point $x \in X$ except a $\sigma$-directionally porous set:\ If the one-sided Hadamard directional derivative $f'_{H+}(x,u)$ exists in all directions $u$ from a set $S_x \subset X$ whose linear span is dense in $X$, then $f$ is Hadamard differentiable at $x$. This theorem improves and generalizes a recent result of A.D. Ioffe, in which the linear span of $S_x$ equals $X$ and $Y = \R$. An analogous theorem, in which $f$ is pointwise Lipschitz, and which deals with the usual one-sided derivatives and G\^ ateaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which $f$ is supposed to be Lipschitz. Archive classification: math.FA Mathematics Subject Classification: Primary: 46G05, Secondary: 26B05, 49J50 Submitted from: zajicek(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.2604 or http://arXiv.org/abs/1211.2604

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Abstract of a paper by Tomasz Kania
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "A reflexive space whose algebra of operators is not a Grothendieck" by Tomasz Kania. Abstract: By a result of Johnson, the Banach space $F=(\bigoplus_{n=1}^\infty \ell_1^n)_{\ell_\infty}$ contains a complemented copy of $\ell_1$. We identify $F$ with a complemented subspace of the space of (bounded, linear) operators on the reflexive space $(\bigoplus_{n=1}^\infty \ell_1^n)_{\ell_p}$ ($p\in (1,\infty))$, thus giving a negative answer to the problem posed in the monograph of Diestel and Uhl which asks whether the space of operators on a reflexive Banach space is Grothendieck. Archive classification: math.FA math.OA Mathematics Subject Classification: 46B25, 47L10 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/1211.2867 or http://arXiv.org/abs/1211.2867

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Abstract of a paper by William B. Johnson and Sofia Ortega Castillo
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "The cluster value problem in spaces of continuous functions" by William B. Johnson and Sofia Ortega Castillo. Abstract: We study the cluster value problem for certain Banach algebras of holomorphic functions defined on the unit ball of a complex Banach space X. The main results are for spaces of the form X = C(K). Archive classification: math.FA math.CV Mathematics Subject Classification: Several complex variables and analytic spaces, Functional Submitted from: ortega(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.2339 or http://arXiv.org/abs/1211.2339

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Abstract of a paper by David Alonso-Gutierrez, Nikos Dafnis, Maria A. Hernandez Cifre, and Joscha Prochno
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "On mean outer radii of random polytopes" by David Alonso-Gutierrez, Nikos Dafnis, Maria A. Hernandez Cifre, and Joscha Prochno. Abstract: In this paper we introduce a new sequence of quantities for random polytopes. Let $K_N=\conv\{X_1,\dots,X_N\}$ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body $K$ of $\R^n$. We prove that the so-called $k$-th mean outer radius $\widetilde R_k(K_N)$ has order $\max\{\sqrt{k},\sqrt{\log N}\}L_K$ with high probability if $n^2\leq N\leq e^{\sqrt{n}}$. We also show that this is also the right order of the expected value of $\widetilde R_k(K_N)$ in the full range $n\leq N\leq e^{\sqrt{n}}$. Archive classification: math.FA Mathematics Subject Classification: Primary 52A22, Secondary 52A23, 05D40 Remarks: 14 pages Submitted from: prochno(a)math.uni-kiel.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.2336 or http://arXiv.org/abs/1211.2336

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Abstract of a paper by Joanna Garbulinska
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "Isometric uniqueness of a complementably universal Banach space for Schauder decompositions" by Joanna Garbulinska. Abstract: We present an isometric version of the complementably universal Banach space $\mathbb{P}$ with a Schauder decomposition. The space $\mathbb{P}$ is isomorphic to Pe\l czy\'nski's space with a universal basis as well as to Kadec' complementably universal space with the bounded approximation property. Archive classification: math.FA Mathematics Subject Classification: Primary: 46B04. Secondary:46M15, 46M40 Submitted from: jgarbulinska(a)ujk.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1211.2211 or http://arXiv.org/abs/1211.2211

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Abstract of a paper by Pawel Wolff
by alspach＠math.okstate.edu 15 Nov '12

15 Nov '12

This is an announcement for the paper "On randomness reduction in the Johnson-Lindenstrauss lemma" by Pawel Wolff. Abstract: A refinement of so-called fast Johnson-Lindenstrauss transform, due to Ailon and Chazelle (2006), and Matou\v{s}ek (2008), is proposed. While it preserves the time efficiency and simplicity of implementation of the original construction, it reduces randomness used to generate the random transformation. In the analysis of the construction two auxiliary results are established which might be of independent interest: a Bernstein-type inequality for a sum of = a random sample from a family of independent random variables and a normal approximation result for such a sum. Archive classification: math.PR math.FA Mathematics Subject Classification: 60E15, 46B85 Submitted from: pwolff(a)mimuw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1202.5500 or http://arXiv.org/abs/1202.5500

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