Fragmentability and sigma-fragmentability of Banach spaces

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 Cech-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 = boolean OR t greater than or equal to 1 T-i and(C(T-i),p), i greater than or equal to 1, is sigma-fragmentable by the norm. This answers a question of R. Haydon. Finally, a simple proof is given that l(infinity) does not have any of the properties (a)-(c) and that (l(infinity)(Gamma), weak) is not sigma-fragmentable by any metric, provided that the cardinality of Gamma is bigger than the continuum.