May 2017 - Banach - mathdept.okstate.edu

Workshop at Texas A&M
by Bill Johnson 04 May '17

04 May '17

Workshop in Analysis and Probability Department of Mathematics Texas A&M University Summer 2017 The Summer 2017 Workshop in Analysis and Probability at Texas A&M University will be in session from July 5 to August 4. All activities will take place in the Blocker Building. The homepage of the Workshop can be found at http://www.math.tamu.edu/~kerr/workshop [www.math.tamu.edu] The Summer Informal Regional Functional Analysis Seminar (SUMIRFAS) will be held July 21-23. Its homepage is located at http://www.math.tamu.edu/~kerr/workshop/sumirfas2017 [www.math.tamu.edu] June 5-9 there will be a Concentration Week, "Ergodic Theory and Operator Algebras", organized by Lewis Bowen and David Kerr. The goal of the meeting is to explore current trends in abstract ergodic theory and its interactions with the structure theory of von Neumann algebras and C*-algebras. Topics include entropy, measured equivalence relations, amenability, soficity, rigidity, random processes on networks, invariant random subgroups, and C*-simplicity. The homepage of the Concentration Week is located at http://www.math.tamu.edu/~kerr/etoa2017 [www.math.tamu.edu] July 17-21 there will be a Concentration Week, "Probabilistic and Algebraic Methods in Quantum Information Theory", organized by Michael Brannan and Benoit Collins. The past decade has seen a spectacular development of powerful new mathematical tools in quantum information theory, including random matrix theory, free probability theory, representation theory, tensor categories, quantum groups, non-commutative harmonic analysis, operator spaces, and the theory of non-local games. The aim of this concentration week is to bring together both leading experts and young researchers in these fields to further explore these emerging connections. It is intended to be multidisciplinary, with the hope of fostering communications between researchers with different backgrounds and interests related to quantum information theory. The homepage of the Concentration Week is located at https://sites.google.com/site/probabalgebramethodsinquantum [sites.google.com] The Workshop is supported in part by grants from the National Science Foundation (NSF). Minorities, women, graduate students, and young researchers are especially encouraged to attend. For logistical support, including requests for support, please contact Cara Starmer <cara(a)math.tamu.edu>. For more information on the Workshop itself, please contact William Johnson <johnson(a)math.tamu.edu>, David Kerr <kerr(a)math.tamu.edu>, or Gilles Pisier <pisier(a)math.tamu.edu>. For information about the Concentration Week "Ergodic Theory and Operator Algebras" please contact Lewis Bowen <lpbowen(a)math.utexas.edu> or David Kerr <kerr(a)math.tamu.edu>. For information about the Concentration Week "Probabilistic and Algebraic Methods in Quantum Information Theory", please contact Michael Brannan <mbrannan(a)math.tamu.edu> or Benoit Collins <collins(a)math.kyoto-u.ac.jp>.

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Abstract of a paper by Vladimir Kadets, Mariia Soloviova
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “Quantitative version of the Bishop-Phelps-Bollobás theorem for operators with values in a space with the property $\beta$” by Vladimir Kadets<https://arxiv.org/find/math/1/au:+Kadets_V/0/1/0/all/0/1>, Mariia Soloviova<https://arxiv.org/find/math/1/au:+Soloviova_M/0/1/0/all/0/1>. Abstract: The Bishop-Phelps-Bollob\'as property for operators deals with simultaneous approximation of an operator $T$ and a vector $x$ at which $T: X\rightarrow Y$ nearly attains its norm by an operator $F$ and a vector $z$, respectively, such that $F$ attains its norm at $z$. We study the possible estimates from above and from below for parameters that measure the rate of approximation in the Bishop-Phelps-Bollob\'as property for operators for the case of $Y$ having the property $\beta$ of Lindenstrauss. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.07095

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Abstract of a paper by Olesia Zavarzina
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “Non-expansive bijections between unit balls of Banach spaces” by Olesia Zavarzina<https://arxiv.org/find/math/1/au:+Zavarzina_O/0/1/0/all/0/1>. Abstract: It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M\rightarrow B_M$ is an isometry. We extend these results to non-expansive bijections $F: B_E\rightarrow B_M$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictly convex, or the space $\ell_1$ then every non-expansive bijection $F: B_E\rightarrow B_M$is an isometry. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.06961

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Abstract of a paper by Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, Artem Zvavitch
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “On the monotonicity of Minkowski sums towards convexity” by Matthieu Fradelizi<https://arxiv.org/find/math/1/au:+Fradelizi_M/0/1/0/all/0/1>, Mokshay Madiman<https://arxiv.org/find/math/1/au:+Madiman_M/0/1/0/all/0/1>, Arnaud Marsiglietti<https://arxiv.org/find/math/1/au:+Marsiglietti_A/0/1/0/all/0/1>, Artem Zvavitch<https://arxiv.org/find/math/1/au:+Zvavitch_A/0/1/0/all/0/1>. Abstract: Let us define for a compact set $A\subset\mathbb{R}_n$ the sequence $$ A(k)=\{\frac{a_1+\cdots+a_k}{k}: a_1, \cdots, a_k\in A\}=\frac{1}{k}(A+\cdots+ A). $$ By a theorem of Shapley, Folkman and Starr (1969), $A(k)$ approaches the convex hull of $A$ in Hausdorff distance as $k$ goes to $\infty$. Bobkov, Madiman and Wang (2011) conjectured that Vol$_n(A(k))$ is non-decreasing in $k$, where Vol$_n$ denotes the $n$-dimensional Lebesgue measure, or in other words, that when one has convergence in the Shapley-Folkman-Starr theorem in terms of a volume deficit, then this convergence is actually monotone. We prove that this conjecture holds true in dimension $1$ but fails in dimension $n\geq 12$. We also discuss some related inequalities for the volume of the Minkowski sum of compact sets, showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex. Then we consider whether one can have monotonicity in the Shapley-Folkman-Starr theorem when measured using alternate measures of non-convexity, including the Hausdorff distance, effective standard deviation or inner radius, and a non-convexity index of Schneider. For these other measures, we present several positive results, including a strong monotonicity of Schneider's index in general dimension, and eventual monotonicity of the Hausdorff distance and effective standard deviation. Along the way, we clarify the interrelationships between these various notions of non-convexity, demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.05486

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Abstract of a paper by Bruno de Mendonça Braga, Andrew Swift
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “Coarse embeddings into superstable spaces” by Bruno de Mendonça Braga<https://arxiv.org/find/math/1/au:+Braga_B/0/1/0/all/0/1>, Andrew Swift<https://arxiv.org/find/math/1/au:+Swift_A/0/1/0/all/0/1>. Abstract: Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of $\ell_p$, for some $p\in[1, \infty)$. In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then $X$ must contain an isomorphic copy of $\ell_p$, for some $p\in[1, \infty)$. In these notes, we show that if a Banach space coarsely embeds into a superstable Banach space, then $X$ has a spreading model isomorphic to $\ell_p$, for some $p\in[1, \infty)$. In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any superstable Banach space. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.04468

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Abstract of a paper by Gilles Pisier
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “A note on Sidon sets in bounded orthonormal systems” by Gilles Pisier<https://arxiv.org/find/math/1/au:+Pisier_G/0/1/0/all/0/1>. Abstract: We give a simple example of an $n$-tuple of orthonormal elements in $L_2$ (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant but that are Sidon only with constant $\approx\sqrt{n}$. This is optimal. The first example of this kind was given by Bourgain and Lewko, but with constant $\approx\sqrt{\log n}$. We deduce from our example that there are two $n$-tuples each Sidon with constant $1$, lying in orthogonal linear subspaces and such that their union is Sidon only with constant $\approx\sqrt{n}$. This is again asymptotically optimal. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.02969

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Abstract of a paper by Bernardo Cascales, Antonio J. Guirao, Vladimir Kadets, Mariia Soloviova
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “$\Gamma$-flatness and Bishop-Phelps-Bollobás type theorems for operators” by Bernardo Cascales<https://arxiv.org/find/math/1/au:+Cascales_B/0/1/0/all/0/1>, Antonio J. Guirao<https://arxiv.org/find/math/1/au:+Guirao_A/0/1/0/all/0/1>, Vladimir Kadets<https://arxiv.org/find/math/1/au:+Kadets_V/0/1/0/all/0/1>, Mariia Soloviova<https://arxiv.org/find/math/1/au:+Soloviova_M/0/1/0/all/0/1>. Abstract: The Bishop-Phelps-Bollob\'{a}s property deals with simultaneous approximation of an operator $T$ and a vector x at which $T$ nearly attains its norm by an operator $T_0$ and a vector x0, respectively, such that $T_0$ attains its norm at x0. In this note we extend the already known results about {the} Bishop-Phelps-Bollob\'{a}s property for Asplund operators to a wider class of Banach spaces and to a wider class of operators. Instead of proving a BPB-type theorem for each space separately we isolate two main notions: $\Gamma$-flat operators and Banach spaces with ACK$_{\rho}$ structure. In particular, we prove a general BPB-type theorem for $\Gamma$-flat operators acting to a space with ACK$_{\rho}$ structure and show that uniform algebras and spaces with the property $\beta$ have ACK$_{\rho}$ structure. We also study the stability of the ACK$_{\rho}$ structure under some natural Banach space theory operations. As a consequence, we discover many new examples of spaces $Y$ such that the Bishop-Phelps-Bollob\'{a}s property for Asplund operators is valid for all pairs of the form $(X, Y)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.01768

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Abstract of a paper by , L. Maligranda
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “$L_p+L_{\infty}$ and $L_p\cap L_{\infty}$ are not isomorphic for all $1\leq p<\infty, p\neq 2$” by S.V. Astashkin<https://arxiv.org/find/math/1/au:+Astashkin_S/0/1/0/all/0/1>, L. Maligranda<https://arxiv.org/find/math/1/au:+Maligranda_L/0/1/0/all/0/1>. Abstract: Isomorphic classification of symmetric spaces is an important problem related to the study of symmetric structures in arbitrary Banach spaces. This research was initiated in the seminal work of Johnson, Maurey, Schechtman and Tzafriri (JMST, 1979). Somewhat later it was extended by Kalton to lattice structures (1993). In particular, in JMST (see also Lindenstrauss-Tzafriri book [1979, Section 2.f]) it was shown that the space $L_p\cap L_{\infty}$ for $2\leq p<\infty$ (resp. $L_p+L_{\infty}$ for $1<p\leq 2$) is isomorphic to $L_p$. A detailed investigation of various properties of separable sums and intersections of $L_p$-spaces (i.e., with $p<\infty$) was undertaken by Dilworth in the papers from 1988 and 1990. In contrast to that, we focus here on the problem if the nonseparable spaces $L_p+L_{\infty}$ and $L_p\cap L_{\infty}$, $1\leq p<\infty$, are isomorphic or not. We prove that these spaces are not isomorphic if $1\leq p<\infty, p\neq 2$. It comes as a consequence of the fact that the space $L_p\cap L_{\infty}$, $1\leq p<\infty, p\neq 2$, does not contain a complemented subspace isomorphic to $L_p$. In particular, as a subproduct, we show that $L_p\cap L_{\infty}$ contains a complemented subspace isomorphic to $\ell_2$ if and only if $p=2$. The problem if $L_p+L_{\infty}$ and $L_p\cap L_{\infty}$ are isomorphic or not remains open. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.01717

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Abstract of a paper by Daniel Núñez-Alarcón, Diana M. Serrano-Rodríguez
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “The Khintchine Inequality is equivalent to the Mixed $(\ell_{\frac{p}{p-1}}, \ell_2)$-Littlewood Inequality” by Daniel Núñez-Alarcón<https://arxiv.org/find/math/1/au:+Nunez_Alarcon_D/0/1/0/all/0/1>, Diana M. Serrano-Rodríguez<https://arxiv.org/find/math/1/au:+Serrano_Rodriguez_D/0/1/0/all/0/1>. Abstract: n this paper we prove that the Khintchine Inequality is equivalent to the mixed $(\ell_{\frac{p}{p-1}}, \ell_2)$-Littlewood inequality. Moreover, we obtain the optimal constants of the Multiple Khintchine inequality. As application, we obtain the optimal constants of the multilinear mixed $(\ell_{\frac{p}{p-1}}, \ell_2)$-Littlewood inequality, completing the estimates in \cite{racsam}.. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.01029

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Abstract of a paper by Vladimir Kadets, Gines Lopez, Miguel Martin
by Bentuo Zheng (bzheng) 30 Apr '17

30 Apr '17

This is an announcement for the paper “Some geometric properties of Read's space” by Vladimir Kadets<https://arxiv.org/find/math/1/au:+Kadets_V/0/1/0/all/0/1>, Gines Lopez<https://arxiv.org/find/math/1/au:+Lopez_G/0/1/0/all/0/1>, Miguel Martin<https://arxiv.org/find/math/1/au:+Martin_M/0/1/0/all/0/1>. Abstract: We study geometric properties of the Banach space $\mathcal{R}$ constructed recently by C.\ Read (arXiv 1307.7958<https://arxiv.org/abs/1307.7958>) which does not contain proximinal subspaces of finite codimension greater than or equal to two. Concretely, we show that the bidual of $\mathcal{R}$ is strictly convex and that $\mathcal{R}$ is weakly locally uniformly rotund (but it is not locally uniformly rotund). Apart of the own interest of the results, they provide a simplification of the proof by M.\ Rmoutil (J.\ Funct.\ Anal.\ 272 (2017), 918--928) that the set of norm-attaining functionals over $\mathcal{R}$ does not contain any linear subspace of dimension greater than or equal to two. Besides, this provides positive answer to the questions of whether the dual of $\mathcal{R}$ is smooth and that whether $\mathcal{R}$ is weakly locally uniformly rotund (Rmoutil, J.\ Funct.\ Anal.\ 272 (2017), 918--928). Finally, we present a renorming of Read's space which is smooth, whose dual is smooth, and which does not contain proximinal subspaces of finite codimension greater than or equal to two and such that its set of norm-attaining functionals does not contain any linear subspace of dimension greater than of equal to two. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.00791

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