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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.

A CONJECTURE OF GODEFROY CONCERNING JAMES’ THEOREM