Limit theorems for the numerical index

We improve on a limit theorem (see Martin et al. (2011) [13], Th. 5.1) for numerical index n(⋅) for large classes of Banach spaces including vector valued ℓp-spaces and ℓp-sums of Banach spaces where 1≤p<∞. We introduce two conditions on a Banach space X, a local characterization condition (LCC) and a global characterization condition (GCC). We prove that if a norm on X satisfies the (LCC), then n(X)=limmn(Xm). An analogous result, in which N will be replaced by a directed, infinite set S will be proved for X satisfying the (GCC). We also present examples of Banach spaces satisfying the above mentioned conditions.