Banach July 2016

banach@mathdept.okstate.edu

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Abstract of a paper by Miek Messerschmidt
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “A strengthening of Wickstead's Theorem: Ordered Banach spaces in which every precompact set is order bounded” by Miek Messerschmidt<http://arxiv.org/find/math/1/au:+Messerschmidt_M/0/1/0/all/0/1>. Abstract: A theorem of Wickstead from 1975 characterizes the ordered Banach spaces with order bounded precompact sets in terms of a geometric property, "coadditivity", relating the space's order with its topology. We strengthen Wickstead's Theorem by showing for an ordered Banach space to have all its precompact sets be order bounded, it is necessary and sufficient for the space to have all its null sequences be order bounded. To establish our strengthening of Wickstead's Theorem, we first prove an Open Mapping Theorem for cone-valued correspondences, which is then employed to prove a Klee-And type theorem for coadditivity (the classical Klee-And Theorem concerns another geometric property, namely "conormality"). By employing this Klee-And type theorem for coadditivity, we establish the equivalence of an ordered Banach space having the coadditivity property from Wickstead's original result with the space having all its null sequences be order bounded. Finally, for the purpose of illustration, we briefly investigate the natural order structures of the James space and the Tsirelson space. The James space is not a Banach lattice, but all its precompact sets are order bounded. The Tsirelson space is a Banach lattice, but not all its precompact sets are order bounded. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00249

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Abstract of a paper by Trond A. Abrahamsen, Vegard Lima, Olav Nygaard, Stanimir Troyanski
by Bentuo Zheng (bzheng) 01 Jul '16

01 Jul '16

This is an announcement for the paper “Diameter two properties, convexity and smoothness” by Trond A. Abrahamsen<http://arxiv.org/find/math/1/au:+Abrahamsen_T/0/1/0/all/0/1>, Vegard Lima<http://arxiv.org/find/math/1/au:+Lima_V/0/1/0/all/0/1>, Olav Nygaard<http://arxiv.org/find/math/1/au:+Nygaard_O/0/1/0/all/0/1>, Stanimir Troyanski<http://arxiv.org/find/math/1/au:+Troyanski_S/0/1/0/all/0/1>. Abstract: We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter $2$ properties. We prove that the strong diameter $2$ property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter $2$ property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to $2$ as we want." Spaces with this property are shown to have non-smooth bidual.. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00221

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Banach July 2016