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FUNCTIONS ATTAINING THE SUPREMUM AND ISOMORPHIC PROPERTIES OF A BANACH SPACE
D. Acosta, Maria,Becerra Guerrero, Julio,Ruiz Galan, Manuel Korean Mathematical Society 2004 대한수학회지 Vol.41 No.1
We prove that a Banach space that is convex-transitive and such that for some element u in the unit sphere, and for every subspace Μ containing u, it happens that the subset of norm attaining functionals on Μ is second Baire category in $M^{*}$ is, in fact, almost-transitive and superreflexive. We also obtain a characterization of finite-dimensional spaces in terms of functions that attain their supremum: a Banach space is finite-dimensional if, for every equivalent norm, every rank-one operator attains its numerical radius. Finally, we describe the subset of norm attaining functionals on a space isomorphic to $\ell$$_1$, where the norm is the restriction of a Luxembourg norm on $L_1$. In fact, the subset of norm attaining functionals for this norm coincides with the subset of norm attaining functionals for the usual norm.m.
Bishop-Phelps-Bollobas property for certain spaces of operators
Acosta, M.D.,Becerra Guerrero, J.,Garcia, D.,Kim, S.K.,Maestre, M. Academic Press 2014 Journal of mathematical analysis and applications Vol.414 No.2
We characterize the Banach spaces Y for which certain subspaces of operators from L<SUB>1</SUB>(μ) into Y have the Bishop-Phelps-Bollobas property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop-Phelps-Bollobas property.
Functions attaining the supremum and isomorphic properties of a Banach space
Mar\'{\i}a D. Acosta,Julio Becerra Guerrero,Manuel Ruiz 대한수학회 2004 대한수학회지 Vol.41 No.1
We prove that a Banach space that is convex-transitive and such that for some element u in the unit sphere, and for every subspace M containing u, it happens that the subset of norm attaining functionals on M is second Baire category in M∗ is, in fact, almost-transitive and superreflexive. We also obtain a characterization of finite-dimensional spaces in terms of functions that attain their supremum: a Banach space is finite-dimensional if, for every equivalent norm, every rank-one operator attains its numerical radius. Finally, we describe the subset of norm attaining functionals on a space isomorphic to 1, where the norm is the restriction of a Luxembourg norm on L1. In fact, the subset of norm attaining functionals for this norm coincides with the subset of norm attaining functionals for the usual norm.