Abstract
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)≠co¯w⁎(K) allows to “localize” inside K, even inside B, a copy of the basis of ℓ1(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)≠co¯w⁎(K) implies that K contains a w⁎-N-family and a copy of the basis of ℓ1(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 ℓ1(c) (resp., a w⁎-N-family) if and only if B does.
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