Banach January 2017

banach@mathdept.okstate.edu

2 participants
17 discussions

New Book
by Fred Dashiell 07 Jan '17

07 Jan '17

The following book has recently been released: "Banach Spaces of Continuous Functions as Dual Spaces" by H.G. Dales, F.K. Dashiell Jr., A.T.-M. Lau, and D. Strauss, CMS Books in Mathematics, Springer, 2016. Springer website: http://www.springer.com/us/book/9783319323473 Amazon: https://www.amazon.com/Banach-Spaces-Continuous-Functions-Mathematics/dp/331 9323474/ Thanks Fred Dashiell

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Abstract of a paper by Florent P. Baudier, Ryan Causey, Stephen DIlworth, Denka Kutzarova, Nirina L. Randrianarivony, Thomas Schlumprecht, Sheng Zhang
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “On the geometry of the countably branching diamond graphs” by Florent P. Baudier<https://arxiv.org/find/math/1/au:+Baudier_F/0/1/0/all/0/1>, Ryan Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>, Stephen DIlworth<https://arxiv.org/find/math/1/au:+DIlworth_S/0/1/0/all/0/1>, Denka Kutzarova<https://arxiv.org/find/math/1/au:+Kutzarova_D/0/1/0/all/0/1>, Nirina L. Randrianarivony<https://arxiv.org/find/math/1/au:+Randrianarivony_N/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>, Sheng Zhang<https://arxiv.org/find/math/1/au:+Zhang_S/0/1/0/all/0/1>. Abstract: In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_{\omega k})_{k\in\mathbb{N}$ is investigated. In particular it is shown that for every $\epsilon>0$ and $k\in\mathbb{N}, D_{\omega k}$ embeds bi-Lipschiztly with distortion at most $6(1+\epsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_{\omega k})_{k\in\mathbb{N}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an equivalent asymptotically midpoint uniformly convex norm. Combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces, within the class of separable reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into $L_p$-spaces and to some problems in renorming theory are also discussed. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01984

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Abstract of a paper by Luis García-Lirola, Matías Raja
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Maps with the Radon-Nikodým property” by Luis García-Lirola<https://arxiv.org/find/math/1/au:+Garcia_Lirola_L/0/1/0/all/0/1>, Matías Raja<https://arxiv.org/find/math/1/au:+Raja_M/0/1/0/all/0/1>. Abstract: We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalise the Radon-Nikod\'ym property to a "less linear" frame. We note that a certain part of the theory can be developed in rather great generality. Indeed, we establish that the elements of the dual which are "strongly slicing" for a given uniformly continuous dentable function form a dense $G_{\delta}$ subset of the dual. As a consequence, the space of uniformly continuous dentable maps from a closed convex bounded set to a Banach space is a Banach space. However some interesting applications, as Stegall's variational principle, are no longer true beyond the usual hypotheses, sending us back to the classical case. Moreover, we study the relation between dentable maps and delta-convex maps. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.08585

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Abstract of a paper by Ondřej Kurka
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Tsirelson-like spaces and complexity of classes of Banach spaces” by Ondřej Kurka<https://arxiv.org/find/math/1/au:+Kurka_O/0/1/0/all/0/1>. Abstract: Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of $c_0$ is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the Dunford-Pettis property are $\Pi_2^1$-complete. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.09334

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Abstract of a paper by Fouad Naderi
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Weak* fixed point property for the Reduced Fourier-Stieltjes algebra of a separable locally compact group” by Fouad Naderi<https://arxiv.org/find/math/1/au:+Naderi_F/0/1/0/all/0/1>. Abstract: In this paper we show that if the reduced Fourier-Stieltjes algebra $B_p(G)$ of a separable locally compact group has either weak$^*$ fixed point property or asymptotic center property, then $G$ is compact. These give affirmative answers to open problems raised in [G. Fendler, A. T. Lau, and M. Leinert, {\it Weak$^*$ fixed point property and and asymptotic center for the Fourier-Stieltjes algebra of a locally compact group,} J. Funct. Anal. 264 (1) (2013), 288-302.] Our theorem helps us to provide a negative answer to a question posed by Randrianantoanina. We also show that for a compact scattered topological space $X$, the C$^*$-algebra $C(X)$ has weak fixed point property. This gives a positive answer to a problem posed by Lau and et al. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.08286

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Abstract of a paper by Philip A.H. Brooker
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Banach spaces and operators with non-separable dual” by Philip A.H. Brooker<https://arxiv.org/find/math/1/au:+Brooker_P/0/1/0/all/0/1>. Abstract: Let $W$ and $Z$ be Banach spaces such that $Z$ is separable and let $R: W\rightarrow Z$ be a (continuous, linear) operator. We study consequences of the adjoint operator $R^*$ having non-separable range. From our main technical result we obtain applications to the theory of basic sequences and the existence of universal operators for various classes of operators between Banach spaces. We also obtain an operator-theoretic characterisation of separable Banach spaces with non-separable dual. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.08130

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Abstract of a paper by Philip A.H. Brooker
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Absolutely convex sets of large Szlenk index” by Philip A.H. Brooker<https://arxiv.org/find/math/1/au:+Brooker_P/0/1/0/all/0/1>. Abstract: Let $X$ be a Banach space and $K$ an absolutely convex, weak$^*$-compact subset of $X$. We study consequences of $K$ having a large or undefined Szlenk index, and subsequently derive a number of related results concerning basic sequences and universal operators. We show that if $X$ has a countable Szlenk index then $X$ admits a subspace with a basis and with Szlenk indices comparable to the Szlenk indices of $X$. If X is separable, then $X$ also admits a quotient with these same properties. We also show that for a given ordinal $\xi$ the class of operators whose Szlenk index is not an ordinal less than or equal to $xi$ admits a universal element if and only if $xi<\omega_1$; W.B. Johnson's theorem that the formal identity map from $\ell_1$ to $\ell_{\infty}$ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.08127

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Abstract of a paper by Fernando Muñoz, Eve Oja, Cándido Piñeiro
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited” by Fernando Muñoz<https://arxiv.org/find/math/1/au:+Munoz_F/0/1/0/all/0/1>, Eve Oja<https://arxiv.org/find/math/1/au:+Oja_E/0/1/0/all/0/1>, Cándido Piñeiro<https://arxiv.org/find/math/1/au:+Pineiro_C/0/1/0/all/0/1>. Abstract: Let $X$ and $Y$ be Banach spaces and let $\Omega$ be a compact Hausdorff space. Denote by $\mathcal{C}_p(\Omega, X)$ the space of $p$-continous $X$-valued functions, $1\leq p\leq\infty$. For operators $S\in\mathcal{L}(\mathcal{C}(\Omega), \mathcal{L}(X, Y))$ and $U\in\mathcal{L}(\mathcal{C}_p(\Omega, X), Y)$, we establish integral representation theorems with respect to a vector measure $m:\Sigma\rightarrow\mathcal{L}(X, Y_{**})$, where $\Sigma$ denotes the $\sigma$-algebra of Borel subsets of $\Omega$. The first theorem extends the classical Bartle-Dunford-Schwartz representation theorem. It is used to prove the second theorem, which extends the classical Dinculeanu-Singer representation theorem, also providing to it an alternative simpler proof. For the latter (and the main) result, we build the needed integration theory, relying on a new concept of the $q$-semivariation, $1\leq q\leq\infty$, of a vector measure $m:\Sigma\rightarrow\mathcal{L}(X, Y_{**})$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.07312

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Abstract of a paper by Gonzalo Martínez-Cervantes
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Banach spaces with weak*-sequential dual ball” by Gonzalo Martínez-Cervantes<https://arxiv.org/find/math/1/au:+Martinez_Cervantes_G/0/1/0/all/0/1>. Abstract: A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak$^*$-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak$^*$-sequentially compact dual ball and $Y\subset X$ is a subspace such that $Y$ and $X/Y$ have weak$^*$-sequential dual ball, then X has weak$^*$-sequential dual ball. As an application we obtain that the Johnson-Lindenstrauss space $JL_2$ and $C(K)$ for $K$ scattered compact space of countable height are examples of Banach spaces with weak$^*$-sequential dual ball, answering in this way a question of A. Plichko. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.05948

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Abstract of a paper by Hasan Gül, Süleyman Onal
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Complemented basic sequences in Frechet spaces with finite dimensional decomposition” by Hasan Gül<https://arxiv.org/find/math/1/au:+Gul_H/0/1/0/all/0/1>, Süleyman Onal<https://arxiv.org/find/math/1/au:+Onal_S/0/1/0/all/0/1>. Abstract: Let $E$ be a Frechet-Montel space and $(E_n)_{n\in\mathbb{N}}$ be a finite dimensional unconditional decomposition of $E$ with dim$(E_n)\leq k$ for some fixed $k\in\mathbb{N}$ and for all $n\in\mathbb{N}$. Consider a sequence $(x_n)_{n\in\mathbb{N}}$ formed by taking an element $x_n$ from each $E_n$ for all $n\in\mathbb{N}$. Then $(x_n)_{n\in\mathbb{N}}$ has a subsequence which is complemented in $E$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.05049

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Banach January 2017