Rich families and projectional skeletons in Asplund WCG spaces

Abstract We show a way of constructing projectional skeletons using the concept of rich families in Banach spaces which admit a projectional generator. Our next result is that a Banach space X is Asplund and weakly compactly generated if and only if there exists a commutative 1-projectional skeleton ( Q γ : γ ∈ Γ ) on X such that ( Q γ ⁎ : γ ∈ Γ ) is a commutative 1-projectional skeleton on X ⁎ . We consider both, real and also complex, Banach spaces.