Projectional skeletons and Markushevich bases

TL;DR

This paper investigates Banach spaces with projectional skeletons, establishing their properties, characterizations, and the existence of Markushevich bases, while comparing commutative and non-commutative cases and presenting open problems.

Contribution

It proves that spaces with a 1-projectional skeleton form a lass, shows they admit strong Markushevich bases, and provides characterizations and analysis of commutative versus non-commutative cases.

Findings

Spaces with a 1-projectional skeleton form a lass.

Such spaces admit strong Markushevich bases.

Comparison of behavior between commutative and non-commutative skeletons.

Abstract

We prove that Banach spaces with a $1$-projectional skeleton form a $\mathcal{P}$-class and deduce that any such space admits a strong Markushevich basis. We provide several equivalent characterizations of spaces with a projectional skeleton and of spaces having a commutative one. We further analyze known examples of spaces with a non-commutative projectional skeleton and compare their behavior with the commutative case. Finally, we collect several open problems.