Banach spaces having the Radon-Nikodym property and numerical index 1

Abstract:

Let $X$ be a Banach space with the Radon-Nikodỳm property. Then, the following are equivalent. (i) $X$ has numerical index 1. (ii) $|x^{**}(x^*)|=1$ for all $x^*\in \mathrm {ex}(B_{X^*})$ and $x^{**}\in \mathrm {ex}(B_{X^{**}})$. (iii) $X$ is an almost-CL-space. (iv) There are a compact Hausdorff space $K$ and a linear isometry $J:X \to C(K)$ such that $|x^{**}(J^*\delta _s)|=1$ for all $s\in K$ and $x^{**}\in \mathrm {ex}(B_{X^{**}})$. If $X$ is a real space, the above conditions are equivalent to being semi-nicely embedded in some space $C(K)$.