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 $(Q_\gamma:\ \gamma\in\Gamma)$ on $X$ such that $(Q_\gamma{}^*:\ \gamma\in\Gamma)$ is a commutative 1-projectional skeleton on $X^*$. We consider both, real and also complex, Banach spaces.