On Certain Classes of Baire-1 Functions with Applications to Banach Space Theory

Certain subclasses of B 1 (K), the Baire-1 functions on a compact metric space K, are defined and characterized. Some applications to Banach spaces are given. Let X be a separable infinite dimensional Banach space and let K denote its dual ball, Ba(X *), with the weak* topology. K is compact metric and X may be naturally identified with a closed subspace of C(K). X * * may also be identified with a closed subspace of A ∞ (K), the Banach space of bounded affine functions on K in the sup norm. Our general objective is to deduce information about the isomorphic structure of X or its subspaces from the topological nature of the functions F ∈ X * * ⊆ A ∞ (K). A classical example of this type of result is: X is reflexive if and only if X * * ⊂ C(K). A second example is the following theorem. (B 1 (K) is the class of bounded Baire-1 functions on K and DBSC(K) is the subclass of differences of bounded semicontinuous functions on K. The precise definitions appear below in §1.) We write Y ֒→ X if Y is isomorphic to a subspace of X. Theorem A. Let X be a separable Banach space and let K = Ba(X *) with the weak* topology. a) [35] ℓ 1 ֒→ X iff X * * \ B 1 (K) = ∅. b) [7] c 0 ֒→ X iff [X * * ∩ DBSC(K)] \ C(K) = ∅. Theorem A provides the motivation for this paper: What can be said about X if X * * ∩ [B 1 (K) \ DBSC(K)] = ∅? To study this problem we consider various subclasses of