James boundaries and $\sigma$-fragmented selectors

B. Cascales¹ ; M. Muñoz² ; J. Orihuela¹

¹ Departamento de Matemáticas Universidad de Murcia 30100 Espinardo, Murcia, Spain
² Departamento de Matemática Aplicada Universidad Politécnica de Cartagena 30202 Cartagena, Murcia, Spain

Studia Mathematica, Tome 188 (2008) no. 2, pp. 97-122

Résumé

We study the boundary structure for $w^*$-compact subsets of dual Banach spaces. To be more precise, for a Banach space $X$, $0\epsilon 1$ and a subset $T$ of the dual space $X^*$ such that $\bigcup\{B(t,\epsilon): t\in T\}$ contains a James boundary for $B_{X^\ast}$ we study different kinds of conditions on $T$, besides $T$ being countable, which ensure that $$ X^*=\overline{\mathop{\rm span} T}^{\|\cdot\|}.\tag*{(SP)} $$ We analyze two different non-separable cases where the equality (SP) holds: (a) if $J:X\to 2^{B_{X^*}}$ is the duality mapping and there exists a $\sigma$-fragmented map $f:X\to X^*$ such that $B(f(x),\epsilon)\cap J(x)\not \not =\emptyset$ for every $x\in X$, 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 $B_{X^*}=\overline{{\rm co} (B)}^{\parallel\cdot\parallel}$. 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 $\epsilon$-selector, $0\epsilon1$, that sends separable sets into separable ones. A consequence is that the dual unit ball $B_{X^\ast}$ is norm fragmented if, and only if, it is norm $\epsilon$-fragmented for some fixed $0\epsilon1$. Our analysis is completed by a characterization of those Banach spaces (not necessarily separable) without copies of $\ell^1$ 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.

Détail

DOI : 10.4064/sm188-2-1

Keywords: study boundary structure * compact subsets dual banach spaces precise banach space epsilon subset dual space * bigcup epsilon contains james boundary ast study different kinds conditions besides being countable which ensure * overline mathop span cdot tag* analyze different non separable cases where equality holds nbsp * duality mapping there exists sigma fragmented map * epsilon cap emptyset every holds asplund nbsp weakly countably k determined holds * weakly countably k determined moreover every james boundary * have * overline parallel cdot parallel approaches simons inequality ideas exploited godefroy separable countable while proving asplund only duality mapping has epsilon selector epsilon sends separable sets separable consequence dual unit ball ast norm fragmented only norm epsilon fragmented fixed epsilon analysis completed characterization those banach spaces necessarily separable without copies ell via structure boundaries * compact sets their duals several applications complementary results proved results extend non separable results godefroy contreras pay rod