On James boundaries in dual Banach spaces

Let X be a Banach space, K ⊂ X∗ a w∗-compact subset and B a boundary of K. We study when the fact co(B) = cow∗(K) allows to “localize” inside K, even inside B, a copy of the basis of 1(c) and a structure that we call a w∗-N-family. Among other things, we prove that: (i) if either K is w∗-metrizable or B is a w∗-countable determined boundary of K, the fact co(B) = cow∗(K) implies that K contains a w∗-N-family and a copy of the basis of 1(c); (ii) if either B = Ext(K) or B is a w∗-K analytic boundary of K, then K contains a copy of the basis of 1(c) (resp., a w∗-N-family) if and only if B does. © 2012 Elsevier Inc. All rights reserved