James boundaries and σ-fragmented selectors

B. Cascales, M. Muñoz, and J. Orihuela. "James boundaries and σ-fragmented selectors." Studia Mathematica 188.2 (2008): 97-122. <http://eudml.org/doc/286066>.

@article{B2008,
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 ⋃ 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.},
author = {B. Cascales, M. Muñoz, J. Orihuela},
journal = {Studia Mathematica},
keywords = {Asplund spaces; -fragmented maps; selectors; -topology},
language = {eng},
number = {2},
pages = {97-122},
title = {James boundaries and σ-fragmented selectors},
url = {http://eudml.org/doc/286066},
volume = {188},
year = {2008},
}

TY - JOUR
AU - B. Cascales
AU - M. Muñoz
AU - J. Orihuela
TI - James boundaries and σ-fragmented selectors
JO - Studia Mathematica
PY - 2008
VL - 188
IS - 2
SP - 97
EP - 122
AB - 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.
LA - eng
KW - Asplund spaces; -fragmented maps; selectors; -topology
UR - http://eudml.org/doc/286066
ER -