Banach

banach@mathdept.okstate.edu

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Abstract of a paper by S. J. Dilworth, Denka Kutzarova, G. Lancien, and N. L. Randrianarivony Title: Asymptotic geometry of Banach spaces and uniform quotient maps
by alspach＠math.okstate.edu 20 Sep '12

20 Sep '12

This is an announcement for the paper "Asymptotic geometry of Banach spaces and uniform quotient maps" by S. J. Dilworth, Denka Kutzarova, G. Lancien, and N. L. Randrianarivony. Abstract: Recently, Lima and Randrianarivony pointed out the role of the property $(\beta)$ of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of $(\beta)$ of the domain space. We also provide conditions under which this comparison can be improved. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B80 (Primary), 46B20 (Secondary) Submitted from: nrandria(a)slu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.0501 or http://arXiv.org/abs/1209.0501

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Abstract of a paper by Anatolij Plichko
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "On uniform continuity of convex bodies with respect to measures in Banach spaces" by Anatolij Plichko. Abstract: Let $\mu$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $\mu$-continuous if $\mu(\partial U)=0$. In the paper the $\mu$-continuity and uniform $\mu$-continuity of convex bodies in $X$, especially of balls and half-spaces, is considered. The $\mu$-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F.~Tops{\o}e is given. Archive classification: math.FA Submitted from: aplichko(a)pk.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.6407 or http://arXiv.org/abs/1208.6407

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Abstract of a paper by Daniel Nunez-Alarcon
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "A note on the polynomial Bohnenblust-Hille inequality" by Daniel Nunez-Alarcon. Abstract: Recently, in paper published in the Annals of Mathematics, it was shown that the Bohnenblust-Hille inequality for (complex) homogeneous polynomials is hypercontractive. However, and to the best of our knowledge, there is no result providing (nontrivial) lower bounds for the optimal constants for n-homogeneous polynomials (n > 2). In this short note we provide lower bounds for these famous constants. Archive classification: math.FA Submitted from: danielnunezal(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.6238 or http://arXiv.org/abs/1208.6238

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Abstract of a paper by Jian Ding, James R. Lee, and Yuval Peres
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Markov type and threshold embeddings" by Jian Ding, James R. Lee, and Yuval Peres. Abstract: For two metric spaces $X$ and $Y$, say that $X$ {\em threshold-embeds into $Y$} if there exist a number $K > 0$ and a family of Lipschitz maps $\{\varphi_{\tau} : X \to Y : \tau > 0 \}$ such that for every $x,y \in X$, $$ d_X(x,y) \geq \tau \implies d_Y(\varphi_{\tau}(x),\varphi_{\tau}(y)) \geq \|\varphi_{\tau}\|_{\Lip} \tau/K\,, $$ where $\|\varphi_{\tau}\|_{\Lip}$ denotes the Lipschitz constant of $\varphi_{\tau}$. We show that if a metric space $X$ threshold-embeds into a Hilbert space, then $X$ has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space $X$ threshold-embeds into a $p$-uniformly smooth Banach space, then $X$ has Markov type $p$. The preceding result, together with Kwapien's theorem, is used to show that if a Banach space threshold-embeds into a Hilbert space then it is linearly isomorphic to a Hilbert space. This suggests some non-linear analogs of Kwapien's theorem. For instance, a subset $X \subseteq L_1$ threshold-embeds into Hilbert space if and only if $X$ has Markov type 2. Archive classification: math.MG math.FA math.PR Submitted from: jrl(a)cs.washington.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.6088 or http://arXiv.org/abs/1208.6088

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Abstract of a paper by Piotr Koszmider and Saharon Shelah
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Independent families in Boolean algebras with some separation" by Piotr Koszmider and Saharon Shelah. Abstract: We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed. Archive classification: math.LO math.FA math.GN Submitted from: piotr.math(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.0177 or http://arXiv.org/abs/1209.0177

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Abstract of a paper by Jesus Ferrer, Piotr Koszmider, and Wieslaw Kubis
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Almost disjoint families of countable sets and separable properties" by Jesus Ferrer, Piotr Koszmider, and Wieslaw Kubis. Abstract: We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindel\"of in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithic and an example of a space $C(K_{\mathcal A})$ with controlled and continuous SCP which has neither a projectional skeleton nor a projectional resolution of the identity. Finally, we describe the structure of almost disjoint families of cardinality $\omega_1$ which induce monolithic spaces of the form $K_{ \mathcal A}$: They can be obtained from countably many ladder systems and pairwise disjoint families applying simple operations. Archive classification: math.FA Mathematics Subject Classification: Primary: 46E15, 03E75. Secondary: 46B20, 46B26 Remarks: 21 pages Submitted from: kubis(a)math.cas.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.0199 or http://arXiv.org/abs/1209.0199

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Abstract of a paper by Alexey I. Popov and Adi Tcaciuc
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Every operator has almost-invariant subspaces" by Alexey I. Popov and Adi Tcaciuc. Abstract: We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we show that the same is true for operators which have non-eigenvalues in the boundary of their spectrum. In the Hilbert space, our methods produce perturbations that are also small in norm, improving on an old result of Brown and Pearcy. Archive classification: math.FA Mathematics Subject Classification: 47A15 (Primary) 47A55 (Secondary) Remarks: 11 pages Submitted from: atcaciuc(a)ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.5831 or http://arXiv.org/abs/1208.5831

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Abstract of a paper by Claudia Correa and Daniel V. Tausk
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "A note on the continuous self-maps of the ladder system space" by Claudia Correa and Daniel V. Tausk. Abstract: We give a partial characterization of the continuous self-maps of the ladder system space K_S. Our results show that K_S is highly nonrigid. We also discuss reasonable notions of "few operators" for spaces C(K) with scattered K and we show that C(K_S) does not have few operators for such notions. Archive classification: math.FA math.GN Mathematics Subject Classification: 54G12, 46E15 Remarks: 5 pages Submitted from: tausk(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.5454 or http://arXiv.org/abs/1208.5454

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Abstract of a paper by H. G. Dales, Tomasz Kania, Tomasz Kochanek, Piotr Koszmider, and Niels Jakob Laustsen
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Maximal left ideals of the Banach algebra of bounded operators on a Banach space" by H. G. Dales, Tomasz Kania, Tomasz Kochanek, Piotr Koszmider, and Niels Jakob Laustsen. Abstract: We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach space $E$: (I) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (II) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form $\{ T\in\mathscr{B}(E) : Tx = 0\}$ (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of $\mathscr{B}(E)$ contains $\mathscr{F}(E)$; (iii) the answer to Question (II) is positive for many, but not all, Banach spaces. Archive classification: math.FA math.OA Mathematics Subject Classification: Primary 47L10, 46H10, Secondary 47L20 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/1208.4762 or http://arXiv.org/abs/1208.4762

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Abstract of a paper by Tomasz Kochanek
by alspach＠math.okstate.edu 11 Sep '12

11 Sep '12

This is an announcement for the paper "Stability of vector measures and twisted sums of Banach spaces" by Tomasz Kochanek. Abstract: A Banach space $X$ is said to have the $\mathsf{SVM}$ (stability of vector measures) property if there exists a~constant $v<\infty$ such that for any algebra of sets $\mathcal F$, and any function $\nu\colon\mathcal F\to X$ satisfying $$\|\nu(A\cup B)-\nu(A)-\nu(B)\|\leq 1\quad\mbox{for disjoint }A,B\in\mathcal F,$$there is a~vector measure $\mu\colon\mathcal F\to X$ with $\|\nu(A)-\mu(A)\|\leq v$ for all $A\in\mathcal F$. If this condition is valid when restricted to set algebras $\mathcal F$ of cardinality less than some fixed cardinal number $\kappa$, then we say that $X$ has the $\kappa$-$\mathsf{SVM}$ property. The least cardinal $\kappa$ for which $X$ does not have the $\kappa$-$\mathsf{SVM}$ property (if it exists) is called the $\mathsf{SVM}$ character of $X$. We apply the machinery of twisted sums and quasi-linear maps to characterise these properties and to determine $\mathsf{SVM}$ characters for many classical Banach spaces. We also discuss connections between the $\kappa$-$\mathsf{SVM}$ property, $\kappa$-injectivity and the `three-space' problem. Archive classification: math.FA Mathematics Subject Classification: Primary 28B05, 46G10, 46B25, Secondary 46B03 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/1208.4755 or http://arXiv.org/abs/1208.4755

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