Rich families and projectional skeletons in Asplund WCG spaces

TL;DR

This paper introduces a new method for constructing projectional skeletons in Banach spaces using rich families, and characterizes Asplund WCG spaces via the existence of specific commutative 1-projectional skeletons.

Contribution

It provides a novel construction technique for projectional skeletons and characterizes Asplund WCG spaces through the existence of compatible commutative 1-projectional skeletons.

Findings

Construction of projectional skeletons via rich families in Banach spaces.

Characterization of Asplund WCG spaces using commutative 1-projectional skeletons.

Applicable to both real and complex Banach 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.