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.



ED compact $K$ such that $C(K)$ is not a dual Banach space
Your desired space is discussed in the book "Topics in Banach Space Theory", by Albiac and Kalton. Springer 2006. See Remark 4.3.9, p. 85 and Problems 4.8 and 4.9, p. 99.


