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Abstract

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, $0<\epsilon<1$, 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 $0<\epsilon<1$. 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.