James boundaries and σ-fragmented selectors

Abstract

Abstract. 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 S {B(t, ε) : t ∈ T } contains a James boundary for BX * we study different kinds of conditions on T , besides T being countable, which ensure thatWe 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 BX * we have BX * = 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 BX * 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 1 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.

Boundaries of Asplund spaces

“…The topology γ was used in [28] to characterize Asplund spaces which are those for which (X * , γ ) is Lindelöf. Other papers where topology γ has been studied are [2,3] and [4].…”

“…This class of maps is wide enough as to include all Borel measurable maps between complete metric spaces: σ -fragmentable maps were introduced in [16] and they have been extensively studied in [19,25,2]. Let us introduce them with the following property, see [2, Section 2] for a complete characterization:…”

“…Clearly, this property is very strong, and it is held for polyhedral spaces only (just take all σ n 's are equal to the same finite set σ ). However, a small modification of this property (which we call FSP, see Definition 3.1) allows us to prove the following: To prove the "if" part of Theorem 3.9 we use a separable reduction (similar to ones used in [2,4,8,11]); in the proof of the "only if" part we use the existence of the so-called Jayne-Rogers selector for the duality mapping in Asplund spaces.…”

“…The main tool here is a new notion we introduce, namely, the notion of finitely self-predictable set, FSP in short, see Definition 3.1. The definition of FSP-set is inspired by properties of boundaries of polyhedral spaces [10], and by properties of boundaries described with σ -fragmentable maps [2,4,8,16]. The discussion on σ -fragmentable maps needs some background which is done in Section 4.…”

Duality in spaces of p-integrable functions with respect to a vector measure.

“…Other cases where the previous equality holds can be found in [11]. Note that (2.5) implies that σ(Y, C) coincides with the weak topology on any bounded subset of Y.…”