A characterization of Banach spaces with numerical index one

TL;DR

This paper characterizes Banach spaces with numerical index one by analyzing the extremal properties of the dual space of bounded linear operators with the numerical radius norm, extending McGregor's finite-dimensional results.

Contribution

It provides a geometric characterization of Banach spaces with numerical index one and refines existing results, including an explicit description of extreme points.

Findings

Characterization of Banach spaces with numerical index one

Explicit description of extreme points of the unit ball in the dual space

Elementary proof of McGregor's finite-dimensional characterization

Abstract

We investigate the extremal properties of the unit ball of $L(X)_w^*$, 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 $B_{L(X)_w^*}$, the unit ball of $L(X)_w^*$, 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.