James boundaries and σ-fragmented selectors

B. Cascales ; M. Muñoz ; J. Orihuela

Studia Mathematica, Tome 187 (2008), p. 97-122 / Harvested from The Polish Digital Mathematics Library

Résumé

We study the boundary structure for w*-compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < ϵ < 1 and a subset T of the dual space X* such that ⋃ B(t,ϵ): t ∈ T contains a James boundary for $B_{X*}$ we study different kinds of conditions on T, besides T being countable, which ensure that $X* = \overline{span T}^{||·||}$. (SP) We analyze two different non-separable cases where the equality (SP) holds: (a) if $J: X → 2^{B_{X*}}$ is the duality mapping and there exists a σ-fragmented map f: X → X* such that B(f(x),ϵ) ∩ J(x) ≠ ∅ for every x ∈ X, then (SP) holds for T = f(X) and in this case X is Asplund; (b) if T is weakly countably K-determined then (SP) holds, X* is weakly countably K-determined and moreover for every James boundary B of $B_{X*}$ we have $B_{X*} = \overline{co(B)}^{||·||}$. Both approaches use Simons’ inequality and ideas exploited by Godefroy in the separable case (i.e., when T is countable). While proving (a) we show 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 is that the dual unit ball $B_{X*}$ is norm fragmented if, and only if, it is norm ϵ-fragmented for some fixed 0 < ϵ < 1. Our analysis is completed by a characterization of those Banach spaces (not necessarily separable) without copies of ℓ¹ via the structure of the boundaries of w*-compact sets of their duals. Several applications and complementary results are proved. Our results extend to the non-separable case results by Godefroy, Contreras-Payá and Rodé.

Détail

Publié le : 2008-01-01

Zbl 1154.54008

EUDML-ID : urn:eudml:doc:286066