Descriptive properties of elements of biduals of Banach spaces

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 in [2, p. 1047]. Also, several examples witnessing natural limits of our positive results are presented.2010 Mathematics Subject Classification. 46B99,46A55,26A21.

The minimum principle for affine functions and isomorphisms of continuous affine function spaces

“…Since 2X and −2X are affinely homeomorphic to X, f is of the first Borel class on 2X ∪ −2X. Now we can use [11,Theorem 3.5(b)] to conclude that f is of the first Borel class on 2B (A c (X)) * = co(2X ∪ −2X). As above we obtain that T * * h x is of the first Borel class on Y .…”

Baire classes of complex L 1-preduals

“…P r o o f. The proof is analogous to the proof of Lemma 2.23, we only use instead of [15], Theorem 1.2, as the starting point of transfinite induction the following fact from [15], Theorem 1.3: If ext B X * is a Lindelöf H-set and h ∈ X * * is a strongly affine function on B X * whose restriction on ext B X * is Baire-1, then h is Baire-1 on B X * . Any such function is then in X * * 1 by Proposition 2.22.…”

“…Let f : ext K → R be bounded and continuous. Then there exist a lower semicontinuous convex Baire function l : K → R and upper semicontinuous concave Baire function u : K → R such that l u and l = u = f on ext K. P r o o f. Using [15], Lemma 4.5, we find sequences (u n ) and (l n ) such that ⊲ the functions u n are continuous concave on K, l n are continuous convex on K,…”

“…Complex L 1 -preduals were studied, e.g., in [4], [6], [11], [18], [20] or recently in [17]. Our contribution to the subject of L 1 -preduals can be found in [13], [14] and [15]. After intensive studies of real L 1 -preduals, the investigation of its complex version came more into focus.…”

A characterization of complex $L_1$-preduals via a complex barycentric mapping

“…in [6], [16], [9], [17], [4] and recently in [15]. The second author's contribution to this subject can be found in [13], [11], [12] and [10].…”