An Asplund space with norming Marku\v{s}evi\v{c} basis that is not weakly compactly generated

TL;DR

This paper constructs a specific Asplund Banach space with a norming Markushevich basis that is not weakly compactly generated, solving a long-standing open problem and introducing a novel scattered compact space example.

Contribution

It provides the first example of an Asplund space with a norming Markushevich basis that is not weakly compactly generated, addressing a question from the early 1990s.

Findings

Constructed an Asplund space with a norming Markushevich basis that is not WCG.

Developed a new example of a scattered compact space.

Solved a long-standing open problem in Banach space theory.

Abstract

We construct an Asplund Banach space $\mathcal{X}$ with a norming Marku\v{s}evi\v{c} basis such that $\mathcal{X}$ is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wies{\l}aw Kubi\'s and Arkady Leiderman.