On James boundaries in dual Banach spaces

Let X be a Banach space, K subset of x* a omega* -compact subset and B a boundary of K. We study when the fact (sic) (B) not equal (sic)(omega)* (K) allows to "localize" inside K, even inside B, a copy of the basis of l(1) (c) and a structure that we call a omega*-N-family. Among other things, we prove that: (i) if either K is omega*-metrizable or B is a omega*-countable determined boundary of K, the fact (sic) (B) not equal(omega)* (K) implies that K contains a omega*-N-family and a copy of the basis of l(1) (c); (ii) if either B = Ext (K) or B is a omega*-K analytic boundary of K, then K contains a copy of the basis of l(1) (c) (resp., a omega*-N-family) if and only if B does. (c) 2012 Elsevier Inc. All rights reserved.