Numerical index of Banach spaces and duality

Abstract

We estimate the polynomial numerical indices of the spaces C(K) and L 1 (μ).

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

“…To finish our arguments, we need to use a class of lush spaces which was also introduced in the aforementioned paper [5], the so-called C-rich subspaces of C(K). Definition 3.3.…”

“…Among the examples of lush spaces are the real or complex spaces C(K) and L 1 (µ) (see [5] or [9, §3] for a detailed account).…”

“…The concept of lushness was introduced recently in [5] as a geometrical property of a Banach space which ensures that the space has numerical index 1. The numerical index of a Banach space X was introduced by G. Lumer in 1968 (see [6]) and it is the best constant of equivalence between the numerical radius and the usual norm of operators on L(X).…”

“…The concept of lushness is proven to be a useful tool in the theory of numerical index of Banach spaces since in [5] it helped to construct an example showing that numerical index is not inherited in general by the dual space, a latent question in the theory from the beginning of the subject. Also, in [10] the lushness was applied for estimating the related concept of polynomial numerical index in some real spaces like c 0 or 1 .…”

“…In [14], it is shown that many of these properties are actually equivalent to numerical index 1 for Banach spaces with the Radon-Nikodým property. But this is not true for general Banach spaces (see [5,Example 3.4]). In this paper we prove that Asplund spaces with numerical index 1 are lush (section 2).…”