JAMES BOUNDARIES AND σ-FRAGMENTED SELECTORS

ABSTRACT. We study the boundary structure for w∗-compact subsets of dual Banach spaces. Being more precise, for a Banach space X, 0 < ε < 1 and T a subset of the dual space X ∗ such that S{B(t, ε) : t ∈ T} contains a James boundary for BX ∗ we study different kind of conditions on T, besides T being countable, which ensure that (S) X ∗ = spanT We analyze two different non separable cases where the equality (S) holds: (a) if J: X → 2BX ∗ is the duality mapping and there exists a σ-fragmented map f: X → X ∗ such that B(f(x), ε) ∩ J(x) 66 = ∅ for every x ∈ X, then (S) holds for T = f(X) and in this case X is Asplund; (b) if T is weakly count-ably K-determined then (S) holds, X ∗ is weakly countably K-determined and moreover for every James boundaryB ofBX ∗ we haveBX ∗ = co(B). Both approaches use Simons ’ inequality and ideas exploited by Godefroy in the sep-arable case (i.e., when T is countable). While proving (a) we prove that X is Asplund if, and only if, the duality mapping has an ε-selector, 0 < ε < 1, that sends separable sets into separable ones. A consequence of the above is that the dual unit ball BX ∗ is norm fragmented if, and only if, it is norm ε-fragmented for some fixed 0 < ε < 1. Our analysis is completed by offering a characteri-zation of those Banach spaces (non necessarily separable) without copies of `1 via the structure of the boundaries of w∗-compact sets of their duals. Several ap-plications and complementary results are proved. Our results extend to the non separable case results by Rodé, Godefroy and Contreras-Payá. 1