On the c_0-extension property

TL;DR

This paper explores the c_0-extension property in Banach spaces, generalizing Sobczyk's theorem, and investigates its relation to dual space properties and the independence of certain cases from ZFC axioms.

Contribution

It establishes a sufficient condition involving weak-star monolithic dual balls and examines the property in C(K) spaces, including independence results from ZFC.

Findings

A Banach space with a weak-star monolithic dual ball has the c_0-extension property.

Existence of certain C(K) spaces lacking the property is independent of ZFC.

The work generalizes Sobczyk's theorem to nonseparable Banach spaces.

Abstract

In this work we investigate the c_0-extension property. This property generalizes Sobczyk's theorem in the context of nonseparable Banach spaces. We prove that a sufficient condition for a Banach space to have this property is that its closed dual unit ball is weak-star monolithic. We also present several results about the c_0-extension property in the context of C(K) Banach spaces. An interesting result in the realm of C(K) spaces is that the existence of a Corson compactum K such that C(K) does not have the c_0-extension property is independent from the axioms of ZFC.