Descriptive properties of elements of biduals of Banach spaces

TL;DR

This paper investigates the descriptive properties of elements in the bidual of Banach spaces, showing how these properties relate to behavior on extreme points and exploring connections with Baire classes in $L_1$-preduals.

Contribution

It generalizes previous results by linking descriptive properties of bidual elements to their behavior on extreme points and examines the relationship between Baire classes in $L_1$-preduals.

Findings

Descriptive properties often determined by behavior on extreme points.

Established links between Baire classes and intrinsic Baire classes.

Provided examples illustrating limits of the positive results.

Abstract

If $E$ is a Banach space, any element $x^{**}$ in its bidual $E^{**}$ is an affine function on the dual unit ball $B_{E^*}$ that might possess variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of $x^{**}$ are quite often determined by the behaviour of $x^{**}$ on the set of extreme points of $B_{E^*}$, 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.