The Radon-Nikodým property in ordered Banach spaces (Q1895133)

Let \(X\) be an ordered Banach space. In this paper, the following problem is studied: If the positive cone \(X_+\) of \(X\) is almost generating (i.e., the space \(X_+- X_+\) is dense in \(X\)) and \(X_+\) has the Radon-Nikodým property (RNP) does \(X\) have the RNP? In this paper, it is shown that if the positive cone of \(L_p(\mu, X)^*\) is almost generating and \(X^*_+\) has the RNP then the spaces \(X^*\) and \(L_p(\mu, X)^*\) have the RNP. It is also proved that the cone \(X^*_+\) has the RNP if and only if the positive cones of \(L_p(\mu, X)^*\) and \(L_q(\mu, X^*)\) coincide. Finally, by an example it is shown that the answer to our problem is not positive in general.