A characterization of Banach spaces with numerical index one

Abstract

We investigate the extremal properties of the unit ball of $L(X{)}_w^{\ast }$, the dual space of bounded linear operators defined on a Banach space $X$ equipped with the numerical radius norm. As an application of the present study, we obtain a geometric characterization of Banach spaces with numerical index one, which extends the well-known McGregor's characterization of finite-dimensional Banach spaces with numerical index one. We also present refinements of several earlier results in this direction, including an explicit description of the extreme points of $B_{L(X{)}_w^{\ast }}$, the unit ball of $L(X{)}_w^{\ast }$, for any finite-dimensional Banach space $X$. This allows us to obtain an independent and elementary proof of McGregor's characterization of finite-dimensional Banach spaces with numerical index one.