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 ${\mathrm{\ell }}_p$-spaces and ${\mathrm{\ell }}_p$-sums of Banach spaces where $1\le p<\mathrm{\infty }$. 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 $n(X)={\displaystyle \underset{m}{\lim }n(X_m)\mathrm{.}}$ 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 $n(L_p(\mu ,X))=n({\mathrm{\ell }}_p(X))={\displaystyle \underset{m}{\lim }n({\mathrm{\ell }}_p^m(X))}$ \cite {aga-ed-kham}.