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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-2,
author = {Kostyantyn Boyko and Vladimir Kadets and Miguel Mart\'\i n and Javier Mer\'\i },
title = {Properties of lush spaces and applications to Banach spaces with numerical index 1},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {117-133},
zbl = {1168.46005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-2}
}
Kostyantyn Boyko; Vladimir Kadets; Miguel Martín; Javier Merí. Properties of lush spaces and applications to Banach spaces with numerical index 1. Studia Mathematica, Tome 192 (2009) pp. 117-133. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-2/