Proof that $C(K)$ is a Grothendieck space for $K$ an extremely disconnected compact space.

Question

I am looking for a proof, other than the original article by Grothendieck which is in French, that the space $C(K)$ is Grothendieck when $K$ is extremely disconnected.

Answer 1

You will find one proof here. Here's a possible different route.

Show that for every compact Hausdorff space $K$, the Banach space $C(K)$ has Pełczyński's property (V):

A. Pełczyński, Banach spaces in which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. 10 (1962), 641-648.

Show that dual spaces with property (V) are Grothendieck. Conclude that $\ell_\infty(\Gamma)$ is Grothendieck for any index set $\Gamma$.

Use the fact that $C(K)$ are injective for $K$ extremely disconnected, hence complemented in $\ell_\infty(\Gamma)$ for some $\Gamma$. Conclude the Grothendieck property.

You may also apply

T. Andô, Convergent sequences of finitely additive measures, Pacific J. Math. 11 (1961), 395-404.

where it is proved that $C(K)$ is Grothendieck for a compact $\sigma$-Stonean space $K$. Extremely disconnected spaces are certainly $\sigma$-Stonean spaces. You may also want to see the proof for compact F-spaces:

G.L. Seever, Measures on F-spaces, Trans. Amer. Math. Soc. 133 (1968), 267-280.