Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Raf Cluckers and Daniel J. Miller
by alspach＠math.okstate.edu 28 Sep '12

28 Sep '12

This is an announcement for the paper "Lebesgue classes and preparation of real constructible functions" by Raf Cluckers and Daniel J. Miller. Abstract: We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any $q > 0$ and constructible functions $f$ and $\mu$ on $E\times\RR^n$, we prove a theorem describing the structure of the set of all $(x,p)$ in $E \times (0,\infty]$ for which $y \mapsto f(x,y)$ is in $L^p(|\mu|_{x}^{q})$, where $|\mu|_{x}^{q}$ is the positive measure on $\RR^n$ whose Radon-Nikodym derivative with respect to the Lebesgue measure is $y\mapsto |\mu(x,y)|^q$. We also prove a closely related preparation theorem for $f$ and $\mu$. These results relate analysis (the study of $L^p$-spaces) with geometry (the study of zero loci). Archive classification: math.AG math.FA math.LO Mathematics Subject Classification: 46E30, 32B20, 14P15 (Primary) 42B35, 03C64 (Secondary) Submitted from: dmille10(a)emporia.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.3439 or http://arXiv.org/abs/1209.3439

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Abstract of a paper by Miguel Martin and Yoshimichi Ueda
by alspach＠math.okstate.edu 28 Sep '12

28 Sep '12

This is an announcement for the paper "On the geometry of von Neumann algebra preduals" by Miguel Martin and Yoshimichi Ueda. Abstract: Let $M$ be a von Neumann algebra and let $M_\star$ be its (unique) predual. We study when for every $\varphi\in M_\star$ there exists $\psi\in M_\star$ solving the equation $\|\varphi \pm \psi\|=\|\varphi\|=\|\psi\|$. This is the case when $M$ does not contain type I nor type III$_1$ factors as direct summands and it is false at least for the unique hyperfinite type III$_1$ factor. An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras. Archive classification: math.OA Remarks: 9 pages Submitted from: ueda(a)math.kyushu-u.ac.jp The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.3391 or http://arXiv.org/abs/1209.3391

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Abstract of a paper by Antonio Aviles and Piotr Koszmider
by alspach＠math.okstate.edu 28 Sep '12

28 Sep '12

This is an announcement for the paper "A Banach space in which every injective operator is surjective" by Antonio Aviles and Piotr Koszmider. Abstract: We construct an infinite dimensional Banach space of continuous functions C(K) such that every one-to-one operator on C(K) is onto. Archive classification: 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/1209.3042 or http://arXiv.org/abs/1209.3042

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Abstract of a paper by S. V. Kislyakov, D. V. Maksimov, and D. M. Stolyarov
by alspach＠math.okstate.edu 20 Sep '12

20 Sep '12

This is an announcement for the paper "Differential expressions with mixed homogeneity and spaces of smooth functions they generate" by S. V. Kislyakov, D. V. Maksimov, and D. M. Stolyarov. Abstract: Let $\{T_1,\dots,T_l\}$ be a collection of differential operators with constant coefficients on the torus $\mathbb{T}^n$. Consider the Banach space $X$ of functions $f$ on the torus for which all functions $T_j f$, $j=1,\dots,l$, are continuous. Extending the previous work of the first two authors, we analyse the embeddability of $X$ into some space $C(K)$ as a complemented subspace. We prove the following. Fix some pattern of mixed homogeneity and extract the senior homogeneous parts (relative to the pattern chosen) $\{\tau_1,\dots,\tau_l\}$ from the initial operators $\{T_1,\dots,T_l\}$. Let $N$ be the dimension of the linear span of $\{\tau_1,\dots,\tau_l\}$. If $N\geqslant 2$, then $X$ is not isomorphic to a complemented subspace of $C(K)$ for any compact space $K$. The main ingredient of the proof of this fact is a new Sobolev-type embedding theorem. Archive classification: math.FA math.CA Remarks: 37 pages Submitted from: dms239(a)mail.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.2078 or http://arXiv.org/abs/1209.2078

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Abstract of a paper by Gilles Pisier
by alspach＠math.okstate.edu 20 Sep '12

20 Sep '12

This is an announcement for the paper "Quantum expanders and geometry of operator spaces" by Gilles Pisier. Abstract: We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the ``local theory" of operator spaces. This allows us to provide sharp estimates for the growth of the multiplicity of $M_N$-spaces needed to represent (up to a constant $C>1$) the $M_N$-version of the $n$-dimensional operator Hilbert space $OH_n$ as a direct sum of copies of $M_N$. We show that, when $C$ is close to 1, this multiplicity grows as $\exp{\beta n N^2}$ for some constant $\beta>0$. The main idea is to identify quantum expanders with ``smooth" points on the matricial analogue of the unit sphere. This generalizes to operator spaces a classical geometric result on $n$-dimensional Hilbert space (corresponding to $N=1$). Our work strongly suggests to further study a certain class of operator spaces that we call matricially subGaussian. In a second part, we introduce and study a generalization of the notion of exact operator space that we call subexponential. Using Random Matrices we show that the factorization results of Grothendieck type that are known in the exact case all extend to the subexponential case, and we exhibit (a continuum of distinct) examples of non-exact subexponential operator spaces. We also show that $OH$, $R+C$ and $\max(\ell_2)$ (or any other maximal operator space) are not subexponential. Archive classification: math.OA math-ph math.FA math.MP Submitted from: pisier(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.2059 or http://arXiv.org/abs/1209.2059

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Abstract of a paper by P.A.H. Brooker and G. Lancien
by alspach＠math.okstate.edu 20 Sep '12

20 Sep '12

This is an announcement for the paper "Three-space property for asymptotically uniformly smooth renormings" by P.A.H. Brooker and G. Lancien. Abstract: We prove that if $Y$ is a closed subspace of a Banach space $X$ such that $Y$ and $X/Y$ admit an equivalent asymptotically uniformly smooth norm, then $X$ also admits an equivalent asymptotically uniformly smooth norm. The proof is based on the use of the Szlenk index and yields a few other applications to renorming theory. Archive classification: math.FA Submitted from: gilles.lancien(a)univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.1567 or http://arXiv.org/abs/1209.1567

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Abstract of a paper by S. J. Dilworth, S. Gogyan, and Denka Kutzarova
by alspach＠math.okstate.edu 20 Sep '12

20 Sep '12

This is an announcement for the paper "On the convergence of a weak greedy algorithm for the multivariate Haar basis" by S. J. Dilworth, S. Gogyan, and Denka Kutzarova. Abstract: We define a family of weak thresholding greedy algorithms for the multivariate Haar basis for $L_1[0,1]^d$ ($d \ge 1$). We prove convergence and uniform boundedness of the weak greedy approximants for all $f \in L_1[0,1]^d$. Archive classification: math.FA math.CA Mathematics Subject Classification: Primary: 41A65. Secondary: 42A10, 46B20 Remarks: 25 pages Submitted from: dilworth(a)math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.1378 or http://arXiv.org/abs/1209.1378

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Abstract of a paper by Stanislav Shkarin
by alspach＠math.okstate.edu 20 Sep '12

20 Sep '12

This is an announcement for the paper "On the spectrum of frequently hypercyclic operators" by Stanislav Shkarin. Abstract: A bounded linear operator $T$ on a Banach space $X$ is called frequently hypercyclic if there exists $x\in X$ such that the lower density of the set $\{n\in\N:T^nx\in U\}$ is positive for any non-empty open subset $U$ of $X$. Bayart and Grivaux have raised a question whether there is a frequently hypercyclic operator on any separable infinite dimensional Banach space. We prove that the spectrum of a frequently hypercyclic operator has no isolated points. It follows that there are no frequently hypercyclic operators on all complex and on some real hereditarily indecomposable Banach spaces, which provides a negative answer to the above question. Archive classification: math.FA math.DS Mathematics Subject Classification: 47A16, 37A25 Citation: Proc. AMS 137 (2009), 123-134 Submitted from: s.shkarin(a)qub.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.1221 or http://arXiv.org/abs/1209.1221

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Abstract of a paper by Stanislav Shkarin
by alspach＠math.okstate.edu 20 Sep '12

20 Sep '12

This is an announcement for the paper "Norm attaining operators and pseudospectrum" by Stanislav Shkarin. Abstract: It is shown that if $1<p<\infty$ and $X$ is a subspace or a quotient of an $\ell_p$-direct sum of finite dimensional Banach spaces, then for any compact operator $T$ on $X$ such that $\|I+T\|>1$, the operator $I+T$ attains its norm. A reflexive Banach space $X$ and a bounded rank one operator $T$ on $X$ are constructed such that $\|I+T\|>1$ and $I+T$ does not attain its norm. Archive classification: math.FA Mathematics Subject Classification: 47A30, 47A10 Citation: Integral Equations and Operator Theory 64 (2009), 115-136 Submitted from: s.shkarin(a)qub.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.1218 or http://arXiv.org/abs/1209.1218

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Abstract of a paper by Gilles Pisier
by alspach＠math.okstate.edu 20 Sep '12

20 Sep '12

This is an announcement for the paper "Martingale inequalities and operator space structures on $L_p$" by Gilles Pisier. Abstract: We describe a new operator space structure on $L_p$ when $p$ is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder's martingale inequalities have a very natural form:\ the span of the Rademacher functions is completely isomorphic to the operator Hilbert space $OH$, and the square function of a martingale difference sequence $d_n$ is $\Sigma \ d_n\otimes \bar d_n$. Various inequalities from harmonic analysis are also considered in the same operator valued framework. Moreover, the new operator space structure also makes sense for non commutative $L_p$-spaces with analogous results. Archive classification: math.OA math.FA math.PR Submitted from: pisier(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1209.1071 or http://arXiv.org/abs/1209.1071

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