Banach

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Abstract of a paper by Trond Abrahamsen, Vegard Lima, and Olav Nygaard
by alspach＠math.okstate.edu 07 May '13

07 May '13

This is an announcement for the paper "Remarks on diameter 2 properties" by Trond Abrahamsen, Vegard Lima, and Olav Nygaard. Abstract: If $X$ is an infinite-dimensional uniform algebra, if $X$ has the Daugavet property or if $X$ is a proper $M$-embedded space, every relatively weakly open subset of the unit ball of the Banach space $X$ is known to have diameter 2, i.e., $X$ has the diameter 2 property. We prove that in these three cases even every finite convex combination of relatively weakly open subsets of the unit ball have diameter 2. Further, we identify new examples of spaces with the diameter 2 property outside the formerly known cases; in particular we observe that forming $\ell_p$-sums of diameter 2 spaces does not ruin diameter 2 structure. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B22 Remarks: To appear in Journal of Convex Analysis Submitted from: veli(a)hials.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.7068 or http://arXiv.org/abs/1304.7068

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Abstract of a paper by Konrad J. Swanepoel
by alspach＠math.okstate.edu 07 May '13

07 May '13

This is an announcement for the paper "Equilateral sets and a Sch\"utte Theorem for the 4-norm" by Konrad J. Swanepoel. Abstract: A well-known theorem of Sch\"utte (1963) gives a sharp lower bound for the ratio between the maximum distance and minimum distance between n+2 points in n-dimensional Euclidean space. In this brief note we adapt B\'ar\'any's elegant proof of this theorem to the space $\ell_4^n$. This gives a new proof that the largest cardinality of an equilateral set in $\ell_4^n$ is n+1, and gives a constructive bound for an interval $(4-\epsilon_n,4+\epsilon_n)$ of values of p close to 4 for which it is guaranteed that the largest cardinality of an equilateral set in $\ell_p^n$ is n+1. Archive classification: math.MG math.FA Mathematics Subject Classification: Primary 46B20, Secondary 52A21, 52C17 Remarks: 5 pages Submitted from: konrad.swanepoel(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.7033 or http://arXiv.org/abs/1304.7033

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Abstract of a paper by David Preiss and Gareth Speight
by alspach＠math.okstate.edu 07 May '13

07 May '13

This is an announcement for the paper "Differentiability of Lipschitz functions in Lebesgue null sets" by David Preiss and Gareth Speight. Abstract: We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of sigma-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof. Archive classification: math.FA math.CA Mathematics Subject Classification: 46G05, 46T20 Remarks: 33 pages Submitted from: G.Speight(a)Warwick.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.6916 or http://arXiv.org/abs/1304.6916

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Abstract of a paper by K.K. Kampoukos and S.K. Mercourakis
by alspach＠math.okstate.edu 07 May '13

07 May '13

This is an announcement for the paper "On a certain class of $\K$ Banach spaces" by K.K. Kampoukos and S.K. Mercourakis. Abstract: Using a strengthening of the concept of $\K$ set, introduced in this paper, we study a certain subclass of the class of $\K$ Banach spaces; the so called strongly $\K$ Banach spaces. This class of spaces includes subspaces of strongly weakly compactly generated (SWCG) as well as Polish Banach spaces and it is related to strongly weakly $\mathcal{K}$--analytic (SWKA) Banach spaces as the known classes of $\K$ and weakly $\mathcal{K}$--analytic (WKA) Banach spaces are related. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, 54H05, 03E15, Secondary 46B26 Remarks: Topology and its Applications (to appear, 28 pages) 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/1304.6577 or http://arXiv.org/abs/1304.6577

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Abstract of a paper by Daniel Carando, Andreas Defant and Pablo Sevilla-Peris
by alspach＠math.okstate.edu 07 May '13

07 May '13

This is an announcement for the paper "Bohr's absolute convergence problem for $\mathcal{H}_p$-Dirichlet in Banach spaces" by Daniel Carando, Andreas Defant and Pablo Sevilla-Peris. Abstract: The Bohr-Bohnenblust-Hille Theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series $\sum_n a_n n^{-s}$ converges uniformly but not absolutely is less than or equal to $1/2$, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space $\mathcal{H}_\infty$ equals $1/2$. By a surprising fact of Bayart the same result holds true if $\mathcal{H}_\infty$ is replaced by any Hardy space $\mathcal{H}_p$, $1 \le p < \infty$, of Dirichlet series. For Dirichlet series with coefficients in a Banach space $X$ the maximal width of Bohr's strips depend on the geometry of $X$; Defant, Garc\'ia, Maestre and P\'erez-Garc\'ia proved that such maximal width equal $1- 1/\ct(X)$, where $\ct(X)$ denotes the maximal cotype of $X$. Equivalently, the supremum over the absolute convergence abscissas of all Dirichlet series in the vector-valued Hardy space $\mathcal{H}_\infty(X)$ equals $1- 1/\ct(X)$. In this article we show that this result remains true if $\mathcal{H}_\infty(X)$ is replaced by the larger class $\mathcal{H}_p(X)$, $1 \le p < \infty$. Archive classification: math.FA Mathematics Subject Classification: 30B50, 32A05, 46G20 Submitted from: dcarando(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.5377 or http://arXiv.org/abs/1304.5377

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Abstract of a paper by Tomasz Kania and Niels Jakob Laustsen
by alspach＠math.okstate.edu 07 May '13

07 May '13

This is an announcement for the paper "Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals" by Tomasz Kania and Niels Jakob Laustsen. Abstract: Denote by $[0,\omega_1)$ the set of countable ordinals, equipped with the order topology, let $L_0$ be the disjoint union of the compact ordinal intervals $[0,\alpha]$ for $\alpha$ countable, and consider the Banach spaces $C_0[0,\omega_1)$ and $C_0(L_0)$ consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces $[0,\omega_1)$ and~$L_0$, respectively, and which vanish eventually. Our main result states that a bounded operator $T$ between any pair of these two Banach spaces fixes a copy of $C_0(L_0)$ if and only if the identity operator on $C_0(L_0)$ factors through $T$, if and only if the Szlenk index of $T$ is uncountable. This implies that the set $\mathscr{S}_{C_0(L_0)}(C_0(L_0))$ of $C_0(L_0)$-strictly singular operators on $C_0(L_0)$ is the unique maximal ideal of the Banach algebra $\mathscr{B}(C_0(L_0))$ of all bounded operators on $C_0(L_0)$, and that $\mathscr{S}_{C_0(L_0)}(C_0[0,\omega_1))$ is the second-largest proper ideal of $\mathscr{B}(C_0[0,\omega_1))$. Moreover, it follows that the Banach space $C_0(L_0)$ is primary and complementably homogeneous. Archive classification: math.FA Submitted from: t.kania(a)lancaster.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.4951 or http://arXiv.org/abs/1304.4951

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Abstract of a paper by Julio Becerra Guerrero, Gines Lopez Perez and Abraham Rueda Zoido
by alspach＠math.okstate.edu 07 May '13

07 May '13

This is an announcement for the paper "Big slices versus big relatively weakly open subsets in Banach spaces" by Julio Becerra Guerrero, Gines Lopez Perez and Abraham Rueda Zoido. Abstract: We study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in Banach spaces. We show that every Banach space containing isomorphic copies of $c_0$ can be equivalently renormed so that every slice of its unit ball has diameter 2 and satisfying that its unit ball contains nonempty relatively weakly open subsets with diameter strictly less than 2, which answers by the negative an open problem. As a consequence a Banach space is constructed satisfying that every slice of its unit ball has diameter 2 and containing nonempty relatively weakly open subsets of its unit ball with diameter arbitrarily small, which stresses the differences between the size of slices and relatively weakly open subsets of the unit ball of Banach spaces. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B22 Remarks: 12 pages Submitted from: glopezp(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.4397 or http://arXiv.org/abs/1304.4397

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Abstract of a paper by Kevin Beanland and Daniel Freeman
by alspach＠math.okstate.edu 07 May '13

07 May '13

This is an announcement for the paper "Uniformly factoring weakly compact operators" by Kevin Beanland and Daniel Freeman. Abstract: Let $X$ and $Y$ be separable Banach spaces. Suppose $Y$ either has a shrinking basis or $Y$ is isomorphic to $C(2^\nn)$ and $\aaa$ is a subset of weakly compact operators from $X$ to $Y$ which is analytic in the strong operator topology. We prove that there is a reflexive space with a basis $Z$ such that every $T \in \aaa$ factors through $Z$. Likewise, we prove that if $\aaa \subset \llll(X, C(2^\nn))$ is a set of operators whose adjoints have separable range and is analytic in the strong operator topology then there is a Banach space $Z$ with separable dual such that every $T \in \aaa$ factors through $Z$. Finally we prove a uniformly version of this result in which we allow the domain and range spaces to vary. Archive classification: math.FA Remarks: 19 pages, comments welcome Submitted from: kbeanland(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.3471 or http://arXiv.org/abs/1304.3471

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Abstract of a paper by Jan-David Hardtke
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "K\"othe-Bochner spaces and some geometric properties related to rotundity and smoothness" by Jan-David Hardtke. Abstract: In 2000 Kadets et al. introduced the notions of acs, luacs and uacs spaces, which form common generalisations of well-known rotundity and smoothness properties of Banach spaces. In a recent preprint the author introduced some further related notions and investigated the behaviour of these geometric properties under the formation of absolute sums. This paper is in a sense a continuation of the previous work. Here we will study the behaviour of said properties under the formation of K\"othe-Bochner spaces, thereby generalising some results of Sirotkin on the acs, luacs and uacs properties of $L^p$-Bochner spaces. Archive classification: math.FA Mathematics Subject Classification: 46B20 46B42 46E30 Remarks: 40 pages, 4 figures, partial text overlap with arXiv:1201.2300 Submitted from: hardtke(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.2950 or http://arXiv.org/abs/1304.2950

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Abstract of a paper by H. Garth Dales, Matthew Daws, Hung Le Pham, and Paul Ramsden
by alspach＠math.okstate.edu 12 Apr '13

12 Apr '13

This is an announcement for the paper "Equivalence of multi-norms" by H. Garth Dales, Matthew Daws, Hung Le Pham, and Paul Ramsden. Abstract: The theory of multi-norms was developed by H.\ G.\ Dales and M.\ E.\ Polyakov in a memoir that was published in \emph{Dissertationes Mathematicae}. In that memoir, the notion of `equivalence' of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of multi-norms are mutually equivalent. Archive classification: math.FA Mathematics Subject Classification: Primary 46B28, Secondary 46M05, 47L05 Submitted from: hung.pham(a)vuw.ac.nz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.2096 or http://arXiv.org/abs/1304.2096

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