Banach spaces with projectional skeletons

TL;DR

This paper explores Banach spaces with projectional skeletons, expanding the class beyond Plichko spaces, and provides characterizations, properties, and preservation results related to these structures.

Contribution

It introduces a broader class of Banach spaces with projectional skeletons, characterizes their existence via elementary substructures, and links them to Plichko spaces through commutative skeletons.

Findings

Every space with a projectional skeleton has a projectional resolution of the identity.

Spaces with a projectional skeleton have a norming space with properties similar to Sigma-spaces.

A Banach space is Plichko if and only if it has a commutative projectional skeleton.

Abstract

A projectional skeleton in a Banach space is a sigma-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Sigma-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.