- Banach - mathdept.okstate.edu

Abstract of a paper by Gilles Lancien, Colin Petitjean, Antonín Procházka
by Bentuo Zheng (bzheng) 04 Jun '18

04 Jun '18

This is an announcement for the paper “On the coarse geometry of the James space” by Gilles Lancien<https://arxiv.org/search?searchtype=author&query=Lancien%2C+G>, Colin Petitjean<https://arxiv.org/search?searchtype=author&query=Petitjean%2C+C>, Antonín Procházka<https://arxiv.org/search?searchtype=author&query=Proch%C3%A1zka%2C+A>. Abstract: In this note we prove that Kalton's interlaced graphs do not equi-coarsely embed into the James space J. This allows us to exhibit a coarse invariant for Banach spaces, namely the non equi-coarse embeddability of this family of graphs, which is very close but different from the celebrated property Q of Kalton. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.05171

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Abstract of a paper by R.M. Causey, K. Navoyan
by Bentuo Zheng (bzheng) 04 Jun '18

04 Jun '18

This is an announcement for the paper “Factorization of Asplund operators” by R.M. Causey<https://arxiv.org/search?searchtype=author&query=Causey%2C+R+M>, K. Navoyan<https://arxiv.org/search?searchtype=author&query=Navoyan%2C+K>. Abstract: We give necessary and sufficient conditions for an operator $A:X\to Y$ on a Banach space having a shrinking FDD to factor through a Banach space $Z$ such that the Szlenk index of $Z$ is equal to the Szlenk index of $A$. We also prove that for every ordinal $\xi\in (0, \omega_1)\setminus\{\omega^\eta: \eta<\omega_1\text{\ a limit ordinal}\}$, there exists a Banach space $\mathfrak{G}_\xi$ having a shrinking basis and Szlenk index $\omega^\xi$ such that for any separable Banach space $X$ and any operator $A:X\to Y$ having Szlenk index less than $\omega^\xi$, $A$ factors through a subspace and through a quotient of $\mathfrak{G}_\xi$, and if $X$ has a shrinking FDD, $A$ factors through $\mathfrak{G}_\xi$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.02746

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Abstract of a paper by Fernando Albiac, José L. Ansorena, Pablo M. Berná
by Bentuo Zheng (bzheng) 04 Jun '18

04 Jun '18

This is an announcement for the paper “Asymptotic greediness of the Haar system in the spaces $L_p[0,1], 1<p<\infty$” by Fernando Albiac<https://arxiv.org/search?searchtype=author&query=Albiac%2C+F>, José L. Ansorena<https://arxiv.org/search?searchtype=author&query=Ansorena%2C+J+L>, Pablo M. Berná<https://arxiv.org/search?searchtype=author&query=Bern%C3%A1%2C+P+M>. Abstract: Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant $C_g[\mathcal{H}^{(p)},L_p]$ of the (normalized) Haar system $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ for $1<p<\infty$. We will show that the superdemocracy constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ grows as $p^{\ast}=\max\{p,p/(p-1)\}$ as $p^*$ goes to $\infty$. Thus, since the unconditionality constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ is $p^*-1$, the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and unconditionality) yield that $p^{\ast}\lesssim C_g[\mathcal{H}^{(p)},L_p]\lesssim (p^{\ast})^{2}$. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, $C_g[\mathcal{H}^{(p)},L_p]\approx p^{\ast}$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.01528

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Abstract of a paper by Sergey V. Astashkin
by Bentuo Zheng (bzheng) 04 Jun '18

04 Jun '18

This is an announcement for the paper “Duality problem for disjointly homogeneous rearrangement invariant spaces” by Sergey V. Astashkin<https://arxiv.org/search?searchtype=author&query=Astashkin%2C+S+V>. Abstract: Let $1\leq p<\infty$. A Banach lattice $E$ is said to be disjointly homogeneous (resp. $p$-disjointly homogeneous) if two arbitrary normalized disjoint sequences from $E$ contain equivalent in $E$ subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in $E$ to the unit vector basis of $\ell_p$). Answering a question raised in 2014 by Flores, Hernandez, Spinu, Tradacete, and Troitsky, for each $1< p<\infty$, we construct a reflexive $p$-disjointly homogeneous rearrangement invariant space on $[0,1]$ whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to $\ell_p$, $1\leq p<\infty$, or $c_0$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.00691

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Abstract of a paper by Saak S. Gabriyelyan, Sidney A. Morris
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “A topological group observation on the Banach--Mazur separable quotient problem” by Saak S. Gabriyelyan<https://arxiv.org/search?searchtype=author&query=Gabriyelyan%2C+S+S>, Sidney A. Morris<https://arxiv.org/search?searchtype=author&query=Morris%2C+S+A>. Abstract: The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fr\'{e}chet space has an infinite-dimensional separable quotient space, namely $R^{\omega}$. It is proved in this paper that every infinite-dimensional Fr\'{e}chet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace which is an infinite-dimensional Fr\'{e}chet space, has an infinite-dimensional (in the topological sense) separable metrizable quotient group, namely $T^{\omega}$, where $T$ denotes the compact unit circle group. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.02652

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Abstract of a paper by Michiya Mori, Narutaka Ozawa
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Mankiewicz's theorem and the Mazur--Ulam property for $C^*$-algebras” by Michiya Mori<https://arxiv.org/search?searchtype=author&query=Mori%2C+M>, Narutaka Ozawa<https://arxiv.org/search?searchtype=author&query=Ozawa%2C+N>. Abstract: We prove that every unital $C^*$-algebra $A$, possibly except for the $2$ by $2$ matrix algebra, has the Mazur--Ulam property. Namely, every surjective isometry from the unit sphere $S_A$ of $A$ onto the unit sphere $S_Y$ of another normed space $Y$ extends to a real linear map. This extends the result of A. M. Peralta and F. J. Fernandez-Polo who have proved the same under the additional assumption that both $A$ and $Y$ are von Neumann algebras. In the course of the proof, we strengthen Mankiewicz's theorem and prove that every surjective isometry from a closed unit ball with enough extreme points onto an arbitrary convex subset of a normed space is necessarily affine. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.10674

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Abstract of a paper by Antonio Avilés, Gonzalo Martínez-Cervantes, José Rodríguez
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Weak$^*$-sequential properties of Johnson-Lindenstrauss spaces” by Antonio Avilés<https://arxiv.org/search?searchtype=author&query=Avil%C3%A9s%2C+A>, Gonzalo Martínez-Cervantes<https://arxiv.org/search?searchtype=author&query=Mart%C3%ADnez-Cervantes%2C…>, José Rodríguez<https://arxiv.org/search?searchtype=author&query=Rodr%C3%ADguez%2C+J>. Abstract: A Banach space $X$ is said to have Efremov's property $(\epsilon)$ if every element of the weak$^*$-closure of a convex bounded set $C\subset X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of $\mathbb{N}$ for which the corresponding Johnson-Lindenstrauss spaces enjoy (resp. fail) property $(\epsilon)$. This is related to a gap in [A. Plichko, Three sequential properties of dual Banach spaces in the weak$^*$ topology, Topology Appl. 190 (2015), 93--98] and allows to answer (consistently) questions of Plichko and Yost. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.10350

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Abstract of a paper by Vladimir P. Fonf, Sebastian Lajara, Stanimir Troyanski, Clemente Zanco
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper Norming subspaces of Banach spaces” by Vladimir P. Fonf<https://arxiv.org/search?searchtype=author&query=Fonf%2C+V+P>, Sebastian Lajara<https://arxiv.org/search?searchtype=author&query=Lajara%2C+S>, Stanimir Troyanski<https://arxiv.org/search?searchtype=author&query=Troyanski%2C+S>, Clemente Zanco<https://arxiv.org/search?searchtype=author&query=Zanco%2C+C>. Abstract: We show that, if $X$ is a closed subspace of a Banach space $E$ and $Z$ is a closed subspace of $E^*$ such that $Z$ is norming for $X$ and $X$ is total over $Z$ (as well as $X$ is norming for $Z$ and $Z$ is total over $X$), then $X$ and the pre-annihilator of $Z$ are complemented in $E$ whenever $Z$ is $w^*$-closed or $X$ is reflexive. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.09968

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Abstract of a paper by Karim Khanaki
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Remarks on Banach spaces determined by their finite dimensional subspaces” by Karim Khanaki<https://arxiv.org/search?searchtype=author&query=Khanaki%2C+K>. Abstract: A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without model theory) of the fact that every finitely determined space $X$ (isometrically) contains every (separable) space $Y$ which is finitely representable in $X$. We also point out how a similar argument proves the Krivine-Maurey theorem on stable Banach spaces, and give the model theoretic interpretations of some results. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.08446

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Abstract of a paper by Pablo M. Berná
by Bentuo Zheng (bzheng) 05 May '18

05 May '18

This is an announcement for the paper “Equivalence between almost-greedy bases and semi-greedy bases” by Pablo M. Berná<https://arxiv.org/search?searchtype=author&query=Bern%C3%A1%2C+P+M>. Abstract: In this paper we show that almost-greedy bases are equivalent to semi-greedy bases in the context of Schauder bases in Banach spaces. Moreover, using this result, we answer a question asked in [3]. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.05730

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