Characterization Conditions and the Numerical Index

TL;DR

This paper surveys recent results on the numerical index of Banach spaces, introducing local and global characterization conditions that relate the numerical index of complex spaces to their finite-dimensional components.

Contribution

It introduces the Local and Global Characterization Conditions (LCC and GCC) and proves their implications for the numerical index of Banach spaces, extending known results to broader classes.

Findings

LCC implies n(X) equals the limit of n(X_m)

GCC extends the result to infinite directed sets

The approach generalizes known formulas for L_p spaces

Abstract

In this paper we survey some recent results concerning the numerical index $n(\cdot)$ for large classes of Banach spaces, including vector valued $\ell_p$-spaces and $\ell_p$-sums of Banach spaces where $1\leq p < \infty$. In particular by defining two conditions on a norm of a Banach space $X$, namely a Local Characterization Condition (LCC) and a Global Characterization Condition (GCC), we are able to show that if a norm on $X$ satisfies the (LCC), then $n(X) = \displaystyle\lim_m n(X_m).$ For the case in which $ \mathbb{N}$ is replaced by a directed, infinite set $S$, we will prove an analogous result for $X$ satisfying the (GCC). Our approach is motivated by the fact that $ n(L_p(\mu, X))= n(\ell_p(X)) = \displaystyle \lim_m n(\ell_p^m (X))$ \cite {aga-ed-kham}.