Descriptive properties of elements of biduals of Banach spaces

Abstract

If $E$ is a Banach space, any element $x^{\ast \ast }$ in its bidual $E^{\ast \ast }$ is an affine function on the dual unit ball $B_{E^{\ast }}$ that might possess variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of $x^{\ast \ast }$ are quite often determined by the behaviour of $x^{\ast \ast }$ on the set of extreme points of $B_{E^{\ast }}$, generalizing thus results of J. Saint Raymond and F. Jellett. We also prove several results on relation between Baire classes and intrinsic Baire classes of $L_1$-preduals which were introduced by S.A. Argyros, G. Godefroy and H.P. Rosenthal. Also, several examples witnessing natural limits of our positive results are presented.