It is known from the work of
Bade and Curtis that if A is a Banach subalgebra of C(Ω),Ω a compact
Hausdorff space, and if Ω is an F-space in the sense of Gillman and Hendriksen
then A = C(Ω). This paper is concerned with the extension of this and
similar results to the setting of Grothendieck spaces ( G-spaces for short).
An important feature of the extension is that emphasis is shifted from the
underlying topological structure of Ω to the linear topological character of
C(Ω).