Grothendieck ordered Banach spaces with an interpolation property

Polyrakis, Ioannis; Xanthos, Foivos

doi:10.1090/S0002-9939-2012-11437-6

Grothendieck ordered Banach spaces with an interpolation property
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by Ioannis A. Polyrakis and Foivos Xanthos PDF

Proc. Amer. Math. Soc. 141 (2013), 1651-1661 Request permission

Abstract:

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

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