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.
