article

Title: Finite-dimensional Banach spaces with numerical index zero

Authors: Miguel Martin, Javier Meri and Angel Rodriguez-Palacios

Issue: Volume 53 (2004), Issue 5, 1279-1289

Abstract:

$We prove that a finite-dimensional Banach space $X$ has numerical index $0$ if and only if it is the direct sum of a real space $X_0$ and nonzero complex spaces $X_1, \dots, X_n$ in such a way that the equality $\|x_0 + \mathrm{e}^{iq_1\rho}x_1 + \cdots + \mathrm{e}^{iq_n\rho}x_n\| = \|x_0 + \cdots + x_n\|$ holds for suitable positive integers $q_1, \dots, q_n$, and every $\rho \in \mathbb{R}$ and every $x_j \in X_j$ ($j=0$, $1, \dots, n$). If the dimension of $X$ is two, then the above result gives $X = \mathbb{C}$, whereas $\dim(X)=3$ implies that $X$ is an absolute sum of $\mathbb{R}$ and $\mathbb{C}$. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.$