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May 2013

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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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