Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Marek Cuth and Michal Johanis
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their duals” by Marek Cuth and Michal Johanis. Abstract: We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_{\infty}$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of $\ell_{\infty}$ into the dual" and "containing as good copy of $\ell_1$ as possible" in a general Banach space. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.04131

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Abstract of a paper by Tepper L. Gill
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The S-basis and M-basis Problems for Separable Banach Spaces” by Tepper L. Gill. Abstract: This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces $\mcH_1$ and $\mcH_2$, such that $\mcH_1$ is a continuous dense embedding in $\mcB$ and $\mcB$ is a continuous dense embedding in $\mcH_2$. This is the best possible improvement of a theorem due to Mazur (see \cite{BA} and also \cite{PE1}). The second objective is show how $\mcH_2$ allows us to provide a positive answer to the Marcinkiewicz-basis (M-basis) problem. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.03547

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Abstract of a paper by Szymon Draga and Tomasz Kochanek
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The Szlenk power type and tensor products of Banach spaces” by Szymon Draga and Tomasz Kochanek. Abstract: We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most $\omega$. We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.03461

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Abstract of a paper by Beata Randrianantoanina
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “On sign embeddings and narrow operators on $L_2$” by Beata Randrianantoanina. Abstract: The goal of this note is two-fold. First we present a brief overview of "weak" embeddings, with a special emphasis on sign embeddings which were introduced by H. P. Rosenthal in the early 1980s. We also discuss the related notion of narrow operators, which was introduced by A. Plichko and M. Popov in 1990. We give examples of applications of these notions in the geometry of Banach spaces and in other areas of analysis. We also present some open problems. In the second part we prove that Rosenthal's celebrated characterization of narrow operators on $L_1$ is also true for operators on $L_2$. This answers, for $p=2$, a question posed by Plichko and Popov in 1990. For $1<p<2$ the problem remains open. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.02710

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Abstract of a paper by Ryan M. Causey and Stephen J. Dilworth
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Metrical Characterizations of super weakly compact operators” by Ryan M. Causey and Stephen J. Dilworth. Abstract: We characterize super weakly compact operators as those through which binary tree and diamond and Laakso graphs may not be factored with uniform distortion. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.01810

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Abstract of a paper by Sheldon Dantas
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The strong Bishop-Phelps-Bollobás property” by Sheldon Dantas. Abstract: In this paper we introduce the strong Bishop-Phelps-Bollob$\'$as property (sBPBp) for bounded linear operators between two Banach spaces $X$ and $Y$. This property is motivated by a Kim-Lee result which states, under our notation, that a Banach space $X$ is uniformly convex if and only if the pair $(X, N)$ satisfies the sBPBp. Positive results of pairs of Banach spaces $(X, Y)$ satisfying this property are given and concrete pairs of Banach spaces $(X, Y)$ failing it are exhibited. A complete characterization of the sBPBp for the pairs $(\ell_p, \ell_q)$ is also provided. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.01461

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Abstract of a paper by Jose Pedro Moreno and Rolf Schneider
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Multiplication of convex sets in $C(K)$ spaces” by Jose Pedro Moreno and Rolf Schneider. Abstract: Let $C(K)$ denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of $C(K)$, one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. Our main interest is in correlations between properties of the product of closed order intervals in $C(K)$ and properties of the underlying space $K$. When $K$ is finite, the product of two intervals in $C(K)$ is always an interval. Surprisingly, the converse of this result is true for a wide class of compacta. We show that a first-countable space $K$ is finite whenever it has the property that the product of two nonnegative intervals is closed, or the property that the product of an interval with itself is convex. That some assumption on $K$ is needed, can be seen from the fact that, if $K$ is the Stone-Cech compactification of $N$, then the product of two intervals in $C(K)$ with continuous boundary functions is always an interval. For any $K$, it is proved that the product of two positive intervals is always an interval, and that the product of two nonnegative intervals is always convex. Finally, square roots of intervals are investigated, with results of similar type. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.01211

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Abstract of a Paper by Sheldon Dantas, Domingo Garcia, Manuel Maestre and Miguel Martin
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The Bishop-Phelps-Bollobas property for compact operators” by Sheldon Dantas, Domingo Garcia, Manuel Maestre and Miguel Martin. Abstract: We study the Bishop-Phelps-Bollob$\'$as property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X, Y$ be Banach spaces. If $(c_0, Y)$ has the BPBp for compact operators, then so do $(C_0(L), Y)$ for every locally compact Hausdorff topological space $L$ and $(X, Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikod$\'$ym property and $(\ell_1(X), Y)$ has the BPBp for compact operators, then so does $(L_1(\mu, X), Y)$ for every positive measure $\mu$; as a consequence, $(L_1(\mu, X), Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any positive measure $\nu$ and $1<p<\infty$. For $1\leq p<\infty$, if $(X, \ell_p(Y))$ has the BPBp for compact operators, then so does $(X, L_p(\mu, Y))$ for every positive measure $\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X, Y)$ has the BPBp for compact operators, then so do $(X, L_{\infty}(\mu, Y))$ for every $\sigma$-finite positive measure $\mu$ and $(X, C(K, Y))$ for every compact Hausdorff topological space $K$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.00618

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Abstract of a paper by Karim Khanaki
by Bentuo Zheng (bzheng) 29 Mar '16

29 Mar '16

This is an announcement for the paper “Tsirelson’s space has NIP” by Karim Khanaki. Abstract: E. Odell showed that Tsirelson's space is strongly separable. This result implies that Tsirelson's space has the non independence property (NIP). The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1603.08134

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Abstract of a paper by A. Perez and M Raja
by Bentuo Zheng (bzheng) 29 Mar '16

29 Mar '16

This is an announcement for the paper “A Bourgain-like property of Banach spaces with no copies of $c_0$” by A. Perez and M Raja. Abstract: We give a characterization of the existence of copies of $c_0$ in Banach spaces in terms of indexes. As an application, we deduce new proofs of James Distortion theorem and Bessaga-Pelczynski theorem about weakly unconditionally Cauchy series. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1603.08706

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