Numerical index and renorming

Abstract

We study the numerical index of a Banach space from the isomorphic point of view, that is, we investigate the values of the numerical index which can be obtained by renorming the space. The set of these values is always an interval which contains $[0,1/3[$ in the real case and $[e^{-1},1/2[$ in the complex case. Moreover, for “most” Banach spaces the least upper bound of this interval is as large as possible, namely $1$.