In this paper we survey some recent results concerning the numerical index
n(β ) for large classes of Banach spaces, including vector valued
βpβ-spaces and βpβ-sums of Banach spaces where 1β€p<β. 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)=mlimβn(Xmβ). For the case in which 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(Lpβ(ΞΌ,X))=n(βpβ(X))=mlimβn(βpmβ(X)) \cite
{aga-ed-kham}.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1106.482