A conjecture of Godefroy concerning James' theorem

TL;DR

This paper explores the relationship between James' theorem and the boundary problem, proving a variant of James' sup-theorem for C(K)-spaces that simplifies weak compactness testing to measures with countable support.

Contribution

It introduces a new variant of James' sup-theorem for C(K)-spaces, confirming Godefroy's conjecture and advancing understanding of weak compactness criteria.

Findings

A variant of James' sup-theorem for C(K)-spaces is proven.

Weakly closed subsets are weakly compact if sup-attainment holds for countably supported measures.

The paper confirms Godefroy's conjecture on the boundary problem.

Abstract

In this note we look at the interdependences between James' theorem and the boundary problem. To do so we show a variant of James' sup-theorem for C(K)-spaces conjectured by Godefroy: in order to know that a bounded weakly closed subset of a C(K)- space is weakly compact it is enough to test the sup-attainment only for measures with countable support.