Properties of lush spaces and applications to Banach spaces with numerical index 1

Kostyantyn Boyko, et al. "Properties of lush spaces and applications to Banach spaces with numerical index 1." Studia Mathematica 190.2 (2009): 117-133. <http://eudml.org/doc/286539>.

@article{KostyantynBoyko2009,
abstract = {The concept of lushness, introduced recently, is a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to having numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c₀ can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (ℝⁿ,||·||) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.},
author = {Kostyantyn Boyko, Vladimir Kadets, Miguel Martín, Javier Merí},
journal = {Studia Mathematica},
keywords = {numerical index; lush Banach space; C-rich subspace; ultrapower; unconditional sum; renorming},
language = {eng},
number = {2},
pages = {117-133},
title = {Properties of lush spaces and applications to Banach spaces with numerical index 1},
url = {http://eudml.org/doc/286539},
volume = {190},
year = {2009},
}

TY - JOUR
AU - Kostyantyn Boyko
AU - Vladimir Kadets
AU - Miguel Martín
AU - Javier Merí
TI - Properties of lush spaces and applications to Banach spaces with numerical index 1
JO - Studia Mathematica
PY - 2009
VL - 190
IS - 2
SP - 117
EP - 133
AB - The concept of lushness, introduced recently, is a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to having numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c₀ can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (ℝⁿ,||·||) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.
LA - eng
KW - numerical index; lush Banach space; C-rich subspace; ultrapower; unconditional sum; renorming
UR - http://eudml.org/doc/286539
ER -