Grothendieck ordered Banach spaces with an interpolation property

Abstract

In this paper we prove that if E E is an ordered Banach space with the countable interpolation property, E E has an order unit and E + E_+ is closed and normal, then E E is a Grothendieck space; i.e. any weak-star convergent sequence of E ∗ E^* is weakly convergent. By the countable interpolation property we mean that for any A , B ⊆ E A,B\subseteq E countable, with A ≤ B A\leq B , we have A ≤ { x } ≤ B A\leq \{x\}\leq B for some x ∈ E x\in E .