Characterization conditions and the numerical index

In this paper we survey some recent results concerning the numerical index n(.) for large classes of Banach spaces, including vector valued l(p)-spaces and l(p)-sums of Banach spaces where 1 <= p <= infinity. 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) = lim(m)n(X-m). 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(L-p(mu, X)) = n(l(p)(X)) = lim(m)n(l(p)(m)(X)).