Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not *hyperstonean*). Everyone points out the following reference:

J. Dixmier, Sur certains espaces consideres par M. H. Stone, *Summa Bras. Math.* 2 (1951), 151–182.

Unfortunately, my French does not exist but this not a real obstacle. The problem is that this article seems to be unavailable. Can one describe me please how does this counter-example look like? Are there any other known such counter-examples?

I am aware that there are certain topological characterisations of hyperstonean spaces by several Russian mathematicians but they are not easy to verify. This is why I am just interested in *concrete* spaces rather than general theorems.

On injective banach spaces and the spaces $L_\infty(\mu)$ for finite measures $\mu$, Acta Math. **124**(1) (1970), 205-248). There it is discussed how work of Haim Gaifman yields a stonean compact $K$ such that $C(K)$ is not a dual space. I think the relevant papers are quite easily accessible. $\endgroup$ – Philip Brooker Jan 9 '12 at 9:48