Descriptive properties of elements of biduals of Banach spaces
Volume 209 / 2012
Studia Mathematica 209 (2012), 71-99
MSC: 46B99, 46A55, 26A21.
DOI: 10.4064/sm209-1-6
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 a 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 a result on the 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 (2003). Also, several examples witnessing natural limits of our positive results are presented.
