Banach August 2014

banach@mathdept.okstate.edu

1 participants
23 discussions

Abstract of a paper by M. G. Cabrera-Padilla, J. A. Chavez-Dominguez, A. Jimenez-Vargas, and Moises Villegas-Vallecillos
by alspach＠math.okstate.edu 22 Aug '14

22 Aug '14

This is an announcement for the paper "Lipschitz tensor product" by M. G. Cabrera-Padilla, J. A. Chavez-Dominguez, A. Jimenez-Vargas, and Moises Villegas-Vallecillos. Abstract: Inspired by ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes E of a pointed metric space and a Banach space E as a certain linear subspace of the algebraic dual of Lipo(X,E^*). We prove that <Lipo(X,E^*),X\boxtimes E> forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the linear space of all finite-rank continuous linear operators from (X^#,T) into E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise convergence of X^#. The concept of Lipschitz tensor product of elements of X^# and E^* yields the space X^#\boxast E^* as a certain linear subspace of the algebraic dual of X\boxtimes E. To ensure the good behavior of a norm on X\boxtimes E with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz cross-norm on X\boxtimes E is defined. We show that the Lipschitz injective norm epsilon, the Lipschitz projective norm pi and the Lipschitz p-nuclear norm d_p (1<=p<=infty) are uniform dualizable Lipschitz cross-norms on X\boxtimes E. In fact, epsilon is the least dualizable Lipschitz cross-norm and pi is the greatest Lipschitz cross-norm on X\boxtimes E. Moreover, dualizable Lipschitz cross-norms alpha on X\boxtimes E are characterized by satisfying the relation epsilon<=alpha<=pi. In addition, the Lipschitz injective (projective) norm on X\boxtimes E can be identified with the injective (respectively, projective) tensor norm on the Banach-space tensor product between the Lipschitz-free space over X and E. In terms of the space X^#\boxast E^*, we describe the spaces of Lipschitz compact (finite-rank, approximable) operators from X to E^$. Archive classification: math.FA Mathematics Subject Classification: 26A16, 46B28, 46E15, 47L20 Remarks: 31 pages Submitted from: ajimenez(a)ual.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.1874 or http://arXiv.org/abs/1408.1874

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Abstract of a paper by William B. Johnson, Amir Bahman Nasseri, Gideon Schechtman and Tomasz Tkocz
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Injective Tauberian operators on $L_1$ and operators with dense range on $\ell_\infty$" by William B. Johnson, Amir Bahman Nasseri, Gideon Schechtman and Tomasz Tkocz. Abstract: There exist injective Tauberian operators on $L_1(0,1)$ that have dense, non closed range. This gives injective, non surjective operators on $\ell_\infty$ that have dense range. Consequently, there are two quasi-complementary, non complementary subspaces of $\ell_\infty$ that are isometric to $\ell_\infty$. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B08, 47A53 Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.1443 or http://arXiv.org/abs/1408.1443

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Abstract of a paper by J. Lopez-Abad and P. Tradacete
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Bases of random unconditional convergence in Banach spaces" by J. Lopez-Abad and P. Tradacete. Abstract: We study random unconditional convergence for a basis in a Banach space. The connections between this notion and classical unconditionality are explored. In particular, we analyze duality relations, reflexivity, uniqueness of these bases and existence of unconditional subsequences. Archive classification: math.FA Mathematics Subject Classification: 46B09, 46B15 Submitted from: ptradace(a)math.uc3m.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.0478 or http://arXiv.org/abs/1408.0478

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Abstract of a paper by Kevin Beanland, Daniel Freeman and Pavlos Motakis
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "The stabilized set of $p$'s in Krivine's theorem can be disconnected" by Kevin Beanland, Daniel Freeman and Pavlos Motakis. Abstract: For any closed subset $F$ of $[1,\infty]$ which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space $X$ with a 1-unconditional basis is constructed so that in each block subspace $Y$ of $X$, $\ell_p$ is finitely block represented in $Y$ if and only if $p \in F$. In particular, this solves the question as to whether the stabilized Krivine set for a Banach space had to be connected. We also prove that for every infinite dimensional subspace $Y$ of $X$ there is a dense subset $G$ of $F$ such that the spreading models admitted by $Y$ are exactly the $\ell_p$ for $p\in G$. Archive classification: math.FA Mathematics Subject Classification: 46B03, 46B06, 46B07, 46B25, 46B45 Remarks: 25 pages Submitted from: pmotakis(a)central.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.0265 or http://arXiv.org/abs/1408.0265

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Abstract of a paper by Antonin Prochazka and Luis Sanchez-Gonzalez
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Low distortion embeddings between $C(K)$ spaces" by Antonin Prochazka and Luis Sanchez-Gonzalez. Abstract: We show that, for each ordinal $\alpha<\omega_1$, the space $C([0,\omega^\alpha])$ does not embed into $C(K)$ with distortion strictly less than $2$ unless $K^{(\alpha)}\neq \emptyset$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B85 Remarks: 11 pages Submitted from: antonin.prochazka(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1408.0211 or http://arXiv.org/abs/1408.0211

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Abstract of a paper by Pavlos Motakis, Daniele Puglisi and Despoina Zisimopoulou
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "A hierarchy of separable commutative Calkin algebras" by Pavlos Motakis, Daniele Puglisi and Despoina Zisimopoulou. Abstract: For specific well founded countably branching trees $\mathcal{T}$ we construct $\mathcal{L}_\infty$ spaces $X_{\mathcal{T}}$. For each such tree $\mathcal{T}$ the Calkin algebra of $X_{\mathcal{T}}$ strongly resembles $C(\mathcal{T})$, the algebra of continuous functions defined on $\mathcal{T}$ and in the case in which $\mathcal{T}$ has finite height, those two algebras are homomorphic. We conclude that for every countable compact metric space $K$ with finite Cantor-Bendixson index there exists a $\mathcal{L}_\infty$ space whose Calkin algebra is isomorphic, as a Banach algebra, to $C(K)$. Archive classification: math.FA Mathematics Subject Classification: 46B03, 46B25, 46B28 Remarks: 28 pages Submitted from: pmotakis(a)central.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.8073 or http://arXiv.org/abs/1407.8073

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Abstract of a paper by Yanni Chen
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Lebesgue and Hardy spaces for symmetric norms I" by Yanni Chen. Abstract: In this paper, we define and study a class $\mathcal{R}_{c}$ of norms on $L^{\infty}\left( \mathbb{T}\right) $, called $continuous\ rotationally\ symmetric \ norms$, which properly contains the class $\left \{ \left \Vert \cdot \right \Vert _{p}:1\leq p<\infty \right \} .$ For $\alpha \in \mathcal{R}% _{c}$ we define $L^{\alpha}\left( \mathbb{T}\right) $ and the Hardy space $H^{\alpha}\left( \mathbb{T}\right) $, and we extend many of the classical results, including the dominated convergence theorem, convolution theorems, dual spaces, Beurling-type invariant spaces, inner-outer factorizations, characterizing the multipliers and the closed densely-defined operators commuting with multiplication by $z$. We also prove a duality theorem for a version of $L^{\alpha}$ in the setting of von Neumann algebras. Archive classification: math.OA Submitted from: yet2(a)wildcats.unh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.7920 or http://arXiv.org/abs/1407.7920

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Abstract of a paper by Sun Kwang Kim and Han Ju Lee
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "The Bishop-Phelps-Bollob\'as property for operators from $\mathcal{C}(K)$ to uniformly convex spaces" by Sun Kwang Kim and Han Ju Lee. Abstract: We show that the pair $(C(K),X)$ has the Bishop-Phelps-Bolloba\'as property for operators if $K$ is a compact Hausdorff space and $X$ is a uniformly convex space. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, Secondary 46B04, 46B22 Citation: To apprear in J. Math. Anal. Appl. 2014 Remarks: 7 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/1407.7872 or http://arXiv.org/abs/1407.7872

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Abstract of a paper by Yun Sung Choi, Sun Kwang Kim, Han Ju Lee and Miguel Martin
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "On Banach spaces with the approximate hyperplane series property" by Yun Sung Choi, Sun Kwang Kim, Han Ju Lee and Miguel Martin. Abstract: We present a sufficient condition for a Banach space to have the approximate hyperplane series property (AHSP) which actually covers all known examples. We use this property to get a stability result to vector-valued spaces of integrable functions. On the other hand, the study of a possible Bishop-Phelps-Bollob\'{a}s version of a classical result of V. Zizler leads to a new characterization of the AHSP for dual spaces in terms of $w^*$-continuous operators and other related results. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, Secondary 46B04, 46B22 Remarks: 12 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/1407.7848 or http://arXiv.org/abs/1407.7848

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Abstract of a paper by A. Jimenez-Vargas, J.M. Sepulcre, and Moises Villegas-Vallecillos
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Biduality and density in Lipschitz function spaces" by A. Jimenez-Vargas, J.M. Sepulcre, and Moises Villegas-Vallecillos. Abstract: For pointed compact metric spaces $(X,d)$, we address the biduality problem as to when the space of Lipschitz functions $\mathrm{Lip}_0(X,d)$ is isometrically isomorphic to the bidual of the space of little Lipschitz functions $\mathrm{lip}_0(X,d)$, and show that this is the case whenever the closed unit ball of $\mathrm{lip}_0(X,d)$ is dense in the closed unit ball of $\mathrm{Lip}_0(X,d)$ with respect to the topology of pointwise convergence. Then we apply our density criterion to prove in an alternate way the real version of a classical result which asserts that $\mathrm{Lip}_0(X,d^\alpha)$ is isometrically isomorphic to $\mathrm{lip}_0(X,d^\alpha)^{**}$ for any $\alpha$ in $(0,1)$. Archive classification: math.FA Mathematics Subject Classification: 46E10, 46E15, 46J10 Remarks: 7 pages Submitted from: ajimenez(a)ual.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.7599 or http://arXiv.org/abs/1407.7599

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Banach August 2014