FRAGMENTABILITY AND SIGMA-FRAGMENTABILITY OF BANACH SPACES

Abstract

It is proved that, for a Banach space X, the following properties are equivalent: (a) (X, weak) is fragmentable by a metric d(., .) that majorizes the norm topology (that is, the topology generated by d contains the norm topology), (b) (X, weak) is fragmentable by a metric d(., .) that majorizes the weak topology, and (c) (X, weak) is sigma-fragmentable by the norm. The paper gives a game characterization of these equivalent properties and uses it to show that a large class of Banach spaces (including the spaces that are Čech-analytic in their weak topology) enjoy the properties (a)–(c). In particular, if the unit ball B of X is a Borel subset of the second dual ball (B**, weak*) then X possesses the properties (a)–(c). This provides an alternative approach to the proofs of some of the results of J. E. Jayne, I. Namioka and C. A. Rogers on sigma-fragmentability of Banach spaces. It is also proved that C(T), with the topology p of pointwise convergence, is sigma-fragmentable by the norm whenever T=∪_{i[ges ]1}T_i and (C(T_i), p), i[ges ]1, is sigma-fragmentable by the norm. This answers a question of R. Haydon. Finally, a simple proof is given that l^∞ does not have any of the properties (a)–(c) and that (l^∞(Γ), weak) is not sigma-fragmentable by any metric, provided that the cardinality of Γ is bigger than the continuum.