Characterization Conditions and the Numerical Index

Abstract

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 $1\le p<\infty$. 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 $n(X)=\underset{m}{\lim }n(X_m).$ 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 $n(L_p(\mu ,X))=n({\ell }_p(X))=\underset{m}{\lim }n({\ell }_p^m(X))$ \cite {aga-ed-kham}.

17 pages. arXiv admin note: text overlap with arXiv:1106.4822