Rich families and projectional skeletons in Asplund WCG spaces

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 (Qγ:γ∈Γ) on X such that (Qγ⁎:γ∈Γ) is a commutative 1-projectional skeleton on X⁎. We consider both, real and also complex, Banach spaces.