Banach January 2015

banach@mathdept.okstate.edu

2 participants
21 discussions

INFORMAL ANALYSIS SEMINAR DEDICATED TO THE WORK OF JOE DIESTEL (March 20-21, 2010)
by Artem Zvavitch 20 Aug '16

20 Aug '16

Dear Friends, On Saturday - Sunday, March 20-21, 2010, (best arrival date Friday, March 19/ Departure Monday, March 22) the Department of Mathematical Sciences at Kent State University will be famous but still very informal. We are happy to announce an: INFORMAL ANALYSIS SEMINAR DEDICATED TO THE WORK OF JOE DIESTEL The following people are among speakers: Peter Casazza, Hans Jarchow, Alexander Koldobsky, Pepe Orihuela, Olek Pelczynski, Thomas Schlumprecht, Lutz Weis. It would be great if you could visit Kent State and participate in the seminar! The conference will have a registration fee of $200 WHICH INCLUDES: pick up/drop off from the airport, THREE DAY STAY in the Microtel hotel at Streetsboro, Saturday/Sunday lunches/dinners to be provided at the department. If you plan to stay fewer than 3 nights or prefer to make your own accommodation arrangements, please reduce your registration fee by $45 per day that you will not use our hotel arrangements. If possible, please send a check for your registration fee, made out to the Department of Mathematical Sciences, Kent State University. The check should be mailed to Virginia Wright, Department of Mathematical Sciences, Kent State University, Kent, OH, 44242. The fee can be also be paid during the registration (check/cash). The conference is supported by NSF; The Department of Mathematical Sciences and Kent State University. Depending on funds availability the fee may be waived for young researchers and people without available funding. Please, write as soon as possible to zvavitch(a)math.kent.edu May we ask you to respond as soon as February 19 (zvavitch(a)math.kent.edu), so that we can gauge the need for housing, lecture room(s), etc. Best Regards, Analysis group at Kent State!

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Abstract of a paper by Ryan Causey
by alspach＠math.okstate.edu 28 Jan '15

28 Jan '15

This is an announcement for the paper "Concerning the Szlenk index" by Ryan Causey. Abstract: We discuss pruning and coloring lemmas on regular families. We discuss several applications of these lemmas to computing the Szlenk index of certain $w^*$ compact subsets of the dual of a separable Banach space. Applications include estimates of the Szlenk index of Minkowski sums, infinite direct sums of separable Banach spaces, constant reduction, and three space properties. We also consider using regular families to construct Banach spaces with prescribed Szlenk index. As a consequence, we give a characterization of which countable ordinals occur as the Szlenk index of a Banach space, prove the optimality of a previous universality result, and compute the Szlenk index of the injective tensor product of separable Banach spaces. Archive classification: math.FA Submitted from: CAUSEYRM(a)mailbox.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.06885 or http://arXiv.org/abs/1501.06885

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Abstract of a paper by S.K. Mercourakis and G. Vassiliadis
by alspach＠math.okstate.edu 28 Jan '15

28 Jan '15

This is an announcement for the paper "Equilateral Sets in Banach Spaces of th form C(K)" by S.K. Mercourakis and G. Vassiliadis. Abstract: We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable. Archive classification: math.FA math.GN Mathematics Subject Classification: Primary 46B20, 46E15, Secondary 46B26, 54D30 Remarks: 17 pages, overlap with arxiv: 1111.2273 v1 Submitted from: smercour(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.06381 or http://arXiv.org/abs/1501.06381

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Abstract of a paper by Hichem Ben-El-Mechaiekh
by alspach＠math.okstate.edu 28 Jan '15

28 Jan '15

This is an announcement for the paper "Intersection Theorems for Closed Convex Sets and Applications" by Hichem Ben-El-Mechaiekh. Abstract: A number of landmark existence theorems of nonlinear functional analysis follow in a simple and direct way from the basic separation of convex closed sets in finite dimension via elementary versions of the Knaster-Kuratowski-Mazurkiewicz principle - which we extend to arbitrary topological vector spaces - and a coincidence property for so-called von Neumann relations. The method avoids the use of deeper results of topological essence such as the Brouwer fixed point theorem or the Sperner's lemma and underlines the crucial role played by convexity. It turns out that the convex KKM principle is equivalent to the Hahn-Banach theorem, the Markov-Kakutani fixed point theorem, and the Sion-von Neumann minimax principle. Archive classification: math.FA Mathematics Subject Classification: Primary: 52A07, 32F32, 32F27, Secondary: 47H04, 47H10, 47N10 Submitted from: hmechaie(a)brocku.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.05813 or http://arXiv.org/abs/1501.05813

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Abstract of a paper by Morten Nielsen
by alspach＠math.okstate.edu 28 Jan '15

28 Jan '15

This is an announcement for the paper "On Schauder Bases Properties of Multiply Generated Gabor Systems" by Morten Nielsen. Abstract: Let $A$ be a finite subset of $L^2(\mathbb{R})$ and $p,q\in\mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R})$ of the Gabor system $$G(1,p/q,A)=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z}, g\in A\},$$ with a specific ordering on $\mathbb{Z}\times \mathbb{Z}\times A$. The characterization is given in terms of a Muckenhoupt matrix $A_2$ condition on an associated Zibulski-Zeevi type matrix. Archive classification: math.FA Mathematics Subject Classification: 42C15, 46B15, 42C40 Remarks: 14 pages Submitted from: mnielsen(a)math.aau.dk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.05794 or http://arXiv.org/abs/1501.05794

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Abstract of a paper by Yanni Chen
by alspach＠math.okstate.edu 28 Jan '15

28 Jan '15

This is an announcement for the paper "A General Beurling-Helson-Lowdenslager Theorem on the Disk" by Yanni Chen. Abstract: We give a simple proof of the Beurling-Helson-Lowdenslager invariant subspace theorem for a very general class of norms on $L^{\infty}\left( \mathbb{T}% \right) . Archive classification: math.FA Submitted from: yanni.chen(a)unh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.05718 or http://arXiv.org/abs/1501.05718

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Abstract of a paper by Assaf Naor and Gideon Schechtman
by alspach＠math.okstate.edu 28 Jan '15

28 Jan '15

This is an announcement for the paper "Pythagorean powers of hypercubes" by Assaf Naor and Gideon Schechtman. Abstract: For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the vector space $\mathbb{F}_2^n$, where $\mathbb{F}_2$ is the field of size two. Endow $\mathbb{F}_2^n$ with the Hamming metric, i.e., with the metric induced by the $\ell_1^n$ norm when one identifies $\mathbb{F}_2^n$ with $\{0,1\}^n\subseteq \mathbb{R}^n$. Denote by $\ell_2^n(\mathbb{F}_2^n)$ the $n$-fold Pythagorean product of $\mathbb{F}_2^n$, i.e., the space of all $x=(x_1,\ldots,x_n)\in \prod_{j=1}^n \mathbb{F}_2^n$, equipped with the metric $$ \forall\, x,y\in \prod_{j=1}^n \mathbb{F}_2^n,\qquad d_{\ell_2^n(\mathbb{F}_2^n)}(x,y)= \sqrt{ \|x_1-y_1\|_1^2+\ldots+\|x_n-y_n\|_1^2}. $$ It is shown here that the bi-Lipschitz distortion of any embedding of $\ell_2^n(\mathbb{F}_2^n)$ into $L_1$ is at least a constant multiple of $\sqrt{n}$. This is achieved through the following new bi-Lipschitz invariant, which is a metric version of (a slight variant of) a linear inequality of Kwapie{\'n} and Sch\"utt (1989). Letting $\{e_{jk}\}_{j,k\in \{1,\ldots,n\}}$ denote the standard basis of the space of all $n$ by $n$ matrices $M_n(\mathbb{F}_2)$, say that a metric space $(X,d_X)$ is a KS space if there exists $C=C(X)>0$ such that for every $n\in 2\mathbb{N}$, every mapping $f:M_n(\mathbb{F}_2)\to X$ satisfies \begin{equation*}\label{eq:metric KS abstract} \frac{1}{n}\sum_{j=1}^n\mathbb{E}\left[d_X\Big(f\Big(x+\sum_{k=1}^ne_{jk}\Big),f(x)\Big)\right]\le C \mathbb{E}\left[d_X\Big(f\Big(x+\sum_{j=1}^ne_{jk_j}\Big),f(x)\Big)\right], \end{equation*} where the expectations above are with respect to $x\in M_n(\mathbb{F}_2)$ and $k=(k_1,\ldots,k_n)\in \{1,\ldots,n\}^n$ chosen uniformly at random. It is shown here that $L_1$ is a KS space (with $C= 2e^2/(e^2-1)$, which is best possible), implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems. Archive classification: math.FA math.MG Submitted from: naor(a)math.princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.05213 or http://arXiv.org/abs/1501.05213

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Abstract of a paper by Ondrej F.K. Kalenda and Jiri Spurny
by alspach＠math.okstate.edu 28 Jan '15

28 Jan '15

This is an announcement for the paper "Preserving affine Baire classes by perfect affine maps" by Ondrej F.K. Kalenda and Jiri Spurny. Abstract: Let $\varphi\colon X\to Y$ be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on $Y$ in the space of real-valued affine continuous functions on $X$ is complemented. We show that if $F$ is a topological vector space, then $f\colon Y\to F$ is of affine Baire class $\alpha$ whenever the composition $f\circ\varphi$ is of affine Baire class $\alpha$. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings. Archive classification: math.FA Mathematics Subject Classification: 46A55, 26A21, 54H05 Remarks: 10 pages Submitted from: kalenda(a)karlin.mff.cuni.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.05118 or http://arXiv.org/abs/1501.05118

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Abstract of a paper by Isaac Goldbring and Martino Lupini
by alspach＠math.okstate.edu 28 Jan '15

28 Jan '15

This is an announcement for the paper "Model-theoretic aspects of the Gurarij operator space" by Isaac Goldbring and Martino Lupini. Abstract: We show that the theory of the Gurarij operator space is the model-completion of the theory of operator spaces, it has a unique separable $1$-exact model, continuum many separable models, and no prime model. We also establish the corresponding facts for the Gurarij operator system. The proofs involve establishing that the theories of the Fra\"iss\'{e} limits of the classes of finite-dimensional $M_q$-spaces and $M_q$-systems are separably categorical and have quantifier-elimination. We conclude the paper by showing that no existentially closed operator system can be completely order isomorphic to a C$^*$ algebra. Archive classification: math.LO math.FA math.OA Remarks: 21 pages Submitted from: isaac(a)math.uic.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.04332 or http://arXiv.org/abs/1501.04332

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Abstract of a paper by Piotr Koszmider and Cristobal Rodriguez-Porras
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "On automorphisms of the Banach space $\ell_\infty/c_0$" by Piotr Koszmider and Cristobal Rodriguez-Porras. Abstract: We investigate Banach space automorphisms $T:\ell_\infty/c_0\rightarrow\ell_\infty/c_0 $ focusing on the possibility of representing their fragments of the form $$T_{B,A}:\ell_\infty(A)/c_0(A)\rightarrow \ell_\infty(B)/c_0(B)$$ for $A, B\subseteq N$ infinite by means of linear operators from $\ell_\infty(A)$ into $\ell_\infty(B)$, infinite $A\times B$-matrices, continuous maps from $B^*=\beta B\setminus B$ into $A^*$, or bijections from $B$ to $A$. This leads to the analysis of general linear operators on $\ell_\infty/c_0$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for other give examples showing that it is impossible. In particular, we show that there are automorphisms of $\ell_\infty/c_0$ which cannot be lifted to operators on $\ell_\infty$ and assuming OCA+MA we show that every automorphism of $\ell_\infty/c_0$ with no fountains or with no funnels is locally, i.e., for some infinite $A, B\subseteq N$ as above, induced by a bijection from $B$ to $A$. This additional set-theoretic assumption is necessary as we show that the continuum hypothesis implies the existence of counterexamples of diverse flavours. However, many basic problems, some of which are listed in the last section, remain open. Archive classification: math.FA 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/1501.03466 or http://arXiv.org/abs/1501.03466

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Banach January 2015