Universally Lusin-measurable and Baire-$1$ projections

Abstract:

It is obvious that a dual Banach space ${E^\ast }$ is reflexive if and only if the natural projection P from ${E^{ \ast \ast \ast }}$ to ${E^\ast }$ is ${\text {weak}^\ast }$ to weak continuous. In this paper it is proved that the next best condition on P, namely that P is ${\text {weak}^\ast }$ to weak universally Lusin-measurable is necessary and sufficient for ${E^\ast }$ to have the Radon-Nikodým property. In addition we prove that if E is any Banach space that is complemented in its second dual by a ${\text {weak}^\ast }$ to weak Baire-1 projection, then E has the Radon-Nikodým property. We also prove that if E is a Banach space that is complemented in its second dual ${E^{ \ast \ast }}$ by a projection $P:{E^{\ast \ast }} \to E$ with $F = {P^{ - 1}}(0)$ weakly K-analytic; then saying that ${E^{ \ast \ast }}$ has the Radon-Nikodým property is equivalent to saying that P is ${\text {weak}^\ast }$ to weak universally Lusin-measurable.