Banach

banach@mathdept.okstate.edu

2549 discussions

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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Abstract of a paper by Jan Rozendaal, Fedor Sukochev and Anna Tomskova
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Operator Lipschitz functions on Banach spaces" by Jan Rozendaal, Fedor Sukochev and Anna Tomskova. Abstract: Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates of the form \begin{align*} \|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq \textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)} \end{align*} for a large class of functions $f$, where $A\in\mathcal{L}(X)$, $B\in \mathcal{L}(Y)$ are scalar type operators and $S\in \mathcal{L}(X,Y)$. In particular, we establish this estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\ell_{p}$ and $Y=\ell_{q}$, for $p<q$ and $p=q=1$, and for $X=Y=\mathrm{c}_{0}$. We also obtain results for $p\geq q$. We study the estimate above in the setting of Banach ideals in $\mathcal{L}(X,Y)$. The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix. Archive classification: math.FA math.OA Mathematics Subject Classification: Primary 47A55, 47A56, secondary 47B47 Remarks: 30 pages Submitted from: janrozendaalmath(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.03267 or http://arXiv.org/abs/1501.03267

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Abstract of a paper by Grigory Ivanov
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "Convex hull deviation and contractibility" by Grigory Ivanov. Abstract: We study the Hausdorff distance between a set and its convex hull. Let $X$ be a Banach space, define the CHD-module of space $X$ as the supremum of this distance for all subset of the unit ball in $X$. In the case of finite dimensional Banach spaces we obtain the exact upper bound of the CHD-module depending on the dimension of the space. We give an upper bound for the CHD-module in $L_p$ spaces. We prove that CHD-module is not greater than the maximum of the Lipschitz constants of metric projection operator onto hyperplanes. This implies that for a Hilbert space CHD-module equals 1. We prove criterion of the Hilbert space and study the contractibility of proximally smooth sets in uniformly convex and uniformly smooth Banach spaces. Archive classification: math.FA Submitted from: grigory.ivanov(a)phystech.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1501.02596 or http://arXiv.org/abs/1501.02596

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Abstract of a paper by Leandro Candido and Piotr Koszmider
by alspach＠math.okstate.edu 16 Jan '15

16 Jan '15

This is an announcement for the paper "On complemented copies of $c_0(\omega_1)$ in $C(K^n)$ spaces" by Leandro Candido and Piotr Koszmider. Abstract: Given a compact Hausdorff space $K$ we consider the Banach space of real continuous functions $C(K^n)$ or equivalently the $n$-fold injective tensor product $\hat\bigotimes_{\varepsilon}C(K)$ or the Banach space of vector valued continuous functions $C(K, C(K, C(K ..., C(K)...)$. We address the question of the existence of complemented copies of $c_0(\omega_1)$ in $\hat\bigotimes_{\varepsilon}C(K)$ under the hypothesis that $C(K)$ contains an isomorphic copy of $c_0(\omega_1)$. This is related to the results of E. Saab and P. Saab that $X\hat\otimes_\varepsilon Y$ contains a complemented copy of $c_0$, if one of the infinite dimensional Banach spaces $X$ or $Y$ contains a copy of $c_0$ and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if $C(K)$ has density $\omega_1$ and contains a copy of $c_0(\omega_1)$, then $C(K\times K)$ contains a complemented copy $c_0(\omega_1)$. The main result is that under the assumption of $\clubsuit$ for every $n\in N$ there is a compact Hausdorff space $K_n$ of weight $\omega_1$ such that $C(K)$ is Lindel\"of in the weak topology, $C(K_n)$ contains a copy of $c_0(\omega_1)$, $C(K_n^n)$ does not contain a complemented copy of $c_0(\omega_1)$ while $C(K_n^{n+1})$ does contain a complemented copy of $c_0(\omega_1)$. This shows that additional set-theoretic assumptions in Galego and Hagler's nonseparable version of Cembrano and Freniche's theorem are necessary as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc. Archive classification: math.FA math.GN math.LO 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/1501.01785 or http://arXiv.org/abs/1501.01785

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