Properties of lush spaces and applications to Banach spaces with numerical index 1
Volume 190 / 2009
Studia Mathematica 190 (2009), 117-133
MSC: Primary 46B20; Secondary 46B03, 46B04, 47A12.
DOI: 10.4064/sm190-2-2
Abstract
The concept of lushness, introduced recently, is a Banach space property, which ensures that the space has numerical index . 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 c_0 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 =(\mathbb R^n, \|\cdot\|) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.
