Characterization Conditions and the Numerical Index

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}.