- Banach - mathdept.okstate.edu

Abstract of a paper by Denny H. Leung
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Ideals of operators on $(\oplus \ell^\infty(n))_{\ell^1}$" by Denny H. Leung. Abstract: The unique maximal ideal in the Banach algebra $L(E)$, $E = (\oplus \ell^\infty(n))_{\ell^1}$, is identified. The proof relies on techniques developed by Laustsen, Loy and Read and a dichotomy result for operators mapping into $L^1$ due to Laustsen, Odell, Schlumprecht and Zs\'{a}k. Archive classification: math.FA Mathematics Subject Classification: 46L10, 46H10 Submitted from: matlhh(a)nus.edu.sg The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.7352 or http://arXiv.org/abs/1310.7352

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Abstract of a paper by Denny H. Leung and Lei Li
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Order isomorphisms on function spaces" by Denny H. Leung and Lei Li. Abstract: The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure, respectively, of the space $C(X)$. In this paper, it is shown that for rather general subspaces $A(X)$ and $A(Y)$ of $C(X)$ and $C(Y)$ respectively, any linear bijection $T: A(X) \to A(Y)$ such that $f \geq 0$ if and only if $Tf \geq 0$ gives rise to a homeomorphism $h: X \to Y$ with which $T$ can be represented as a weighted composition operator. The three classical results mentioned above can be derived as corollaries. Generalizations to noncompact spaces and other function spaces such as spaces of uniformly continuous functions, Lipschitz functions and differentiable functions are presented. Archive classification: math.FA Mathematics Subject Classification: 46E15 Submitted from: matlhh(a)nus.edu.sg The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.7351 or http://arXiv.org/abs/1310.7351

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Abstract of a paper by Andrei Dorogovtsev and Mikhail Popov
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Basis entropy in Banach spaces" by Andrei Dorogovtsev and Mikhail Popov. Abstract: We introduce and study two notions of entropy in a Banach space X with a normalized Schauder basis . The geometric entropy E(A) of a subset A of X is defined to be the infimum of radii of compact bricks containing A. We obtain several compactness characterizations for bricks (Theorem 3.7) useful for main results. We also obtain sufficient conditions on a set in a Hilbert space to have finite unconditional entropy. For Banach spaces without a Schauder basis we offer another entropy, called the Auerbach entropy. Finally, we pose some open problems. Archive classification: math.FA Mathematics Subject Classification: 46B50, 46B15, 60H07 Remarks: 22 pages Submitted from: adoro(a)imath.kiev.ua The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.7248 or http://arXiv.org/abs/1310.7248

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Abstract of a paper by M A Sofi
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Around finite-dimensionality in functional analysis" by M A Sofi. Abstract: As objects of study in functional analysis, Hilbert spaces stand out as special objects of study as do nuclear spaces in view of a rich geometrical structure they possess as Banach and Frechet spaces, respectively. On the other hand, there is the class of Banach spaces including certain function spaces and sequence spaces which are distinguished by a poor geometrical structure and are subsumed under the class of so-called Hilbert-Schmidt spaces. It turns out that these three classes of spaces are mutually disjoint in the sense that they intersect precisely in finite dimensional spaces. However, it is remarkable that despite this mutually exclusive character, there is an underlying commonality of approach to these disparate classes of objects in that they crop up in certain situations involving a single phenomenon-the phenomenon of finite dimensionality-which, by definition, is a generic term for those properties of Banach spaces which hold good in finite dimensional spaces but fail in infinite dimension. Archive classification: math.FA Mathematics Subject Classification: 46A11, 46C15 Citation: RACSAM 2013 Remarks: 22 pages Submitted from: aminsofi(a)kashmiruniversity.ac.in The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.7165 or http://arXiv.org/abs/1310.7165

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Abstract of a paper by S. J. Dilworth and B. Randrianantoanina
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Almost transitive and maximal norms in Banach spaces" by S. J. Dilworth and B. Randrianantoanina. Abstract: We prove that the spaces $\ell_p$, $1<p<\infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This answers a problem posed by Deville, Godefroy and Zizler in 1993. We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of $\ell_2^2$ belongs to the two-dimensional asymptotic structure and we obtain some information about the asymptotic structure in higher dimensions. We also obtain several other results about properties of classical, Tsirelson type and non-commutative Banach spaces with almost transitive norms. Further, we prove that the spaces $\ell_p$, $1<p<\infty$, $p\ne 2$, have continuum different renormings with 1-unconditional bases each with a different maximal isometry group, and that every symmetric space other than $\ell_2$ has at least a countable number of such renormings. On the other hand we show that the spaces $\ell_p$, $1<p<\infty$, $p\ne 2$, have continuum different renormings each with an isometry group which is not contained in any maximal bounded subgroup of the group of isomorphisms of $\ell_p$. This answers a question of Wood. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B03, 22F50 Submitted from: randrib(a)miamioh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.7139 or http://arXiv.org/abs/1310.7139

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Abstract of a paper by Eve Oja
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Principle of local reflexivity respecting subspaces" by Eve Oja. Abstract: We obtain a strengthening of the principle of local reflexivity in a general form. The added strength makes local reflexivity operators respect given subspaces. Applications are given to bounded approximation properties of pairs, consisting of a Banach space and its subspace. Archive classification: math.FA Submitted from: eve.oja(a)ut.ee The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.6232 or http://arXiv.org/abs/1310.6232

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Abstract of a paper by Bunyamin Sari and Konstantinos Tyros
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "On the structure of the set of the higher order spreading models" by Bunyamin Sari and Konstantinos Tyros. Abstract: We generalize some results concerning the classical notion of a spreading model for the spreading models of order $\xi$. Among them, we prove that the set $SM_\xi^w(X)$ of the $\xi$-order spreading models of a Banach space $X$ generated by subordinated weakly null $\mathcal{F}$-sequences endowed with the pre-partial order of domination is a semi-lattice. Moreover, if $SM_\xi^w(X)$ contains an increasing sequence of length $\omega$ then it contains an increasing sequence of length $\omega_1$. Finally, if $SM_\xi^w(X)$ is uncountable, then it contains an antichain of size the continuum. Archive classification: math.FA Mathematics Subject Classification: 46B06, 46B25, 46B45 Remarks: 23 pages Submitted from: chcost(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.5429 or http://arXiv.org/abs/1310.5429

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Abstract of a paper by Masato Mimura
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Sphere equivalence, Banach expanders, and extrapolation" by Masato Mimura. Abstract: We study the Banach spectral gap lambda_1(G;X,p) of finite graphs G for pairs (X,p) of Banach spaces and exponents. We introduce the notion of sphere equivalence between Banach spaces, and study behavior of lambda_1(G;X,p) for fixed p in terms of this equivalence. We further study behavior of lambda_1(G;X,p) for fixed X. As a byproduct, we show a generalization of Matousek's extrapolation to that for any Banach space which is sphere equivalent to a uniformly convex Banach space. We as well prove that expanders are expanders with respects to (X,p) for any X sphere equivalent to a uniformly curved Banach space and for any finite p strictly bigger than 1. Archive classification: math.GR math.CO math.FA math.MG Remarks: 23 pages, no figure Submitted from: mimura-mas(a)m.tohoku.ac.jp The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.4737 or http://arXiv.org/abs/1310.4737

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Abstract of a paper by Anna Novikova
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Lyapunov theorem for q-concave Banach spaces" by Anna Novikova. Abstract: Generalization of Lyapunov convexity theorem is proved for vector measure with values in Banach spaces with unconditional bases, which are q-concave for some $q<\infty.$ Archive classification: math.FA Mathematics Subject Classification: 46E30 Remarks: 7 pages Submitted from: anna.novikova(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.4663 or http://arXiv.org/abs/1310.4663

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Abstract of a paper by Niushan Gao
by alspach＠math.okstate.edu 31 Oct '13

31 Oct '13

This is an announcement for the paper "Unbounded order convergence in dual spaces" by Niushan Gao. Abstract: A net $(x_\alpha)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\in X$ if the net $(\abs{x_\alpha-x}\wedge y)$ converges to $0$ in order for all $y\in X_+$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let $X$ be a Banach lattice. We prove that every norm bounded uo-convergent net in $X^*$ is $w^*$-convergent iff $X$ has order continuous norm, and that every $w^*$-convergent net in $X^*$ is uo-convergent iff $X$ is atomic with order continuous norm. We also characterize among $\sigma$-order complete Banach lattices the spaces in whose dual space every simultaneously uo- and $w^*$-convergent sequence converges weakly/in norm. Archive classification: math.FA Submitted from: niushan(a)ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.4438 or http://arXiv.org/abs/1310.4438

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