Projections in duals to Asplund spaces made without Simons' lemma

TL;DR

This paper revisits the construction of projectional resolutions in duals of Asplund spaces, simplifying Stegall's approach and extending it to include projectional skeletons without relying on Simons' lemma.

Contribution

It provides a clearer, more detailed proof of Stegall's method and demonstrates how to obtain projectional skeletons without using Simons' lemma.

Findings

Simplified proof of Stegall's projectional resolution construction.

Extension of the method to produce projectional skeletons.

Elimination of the need for Simons' lemma in these constructions.

Abstract

G. Godefroy and the second author of this note proved in 1988 that in duals to Asplund spaces there always exists a projectional resolution of the identity. A few years later, Ch. Stegall succeeded to drop from the original proof a deep lemma of S. Simons. Here, we rewrite the condensed argument of Ch. Stegall in a more transparent and detailed way. We actually show that this technology of Ch. Stegall leads to a bit stronger/richer object ---the so called projectional skeleton--- recently constructed by W. Kubi\'s, via S. Simons' lemma and with help of elementary submodels from logic.