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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.

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    $\begingroup$ extremely disconnected -> extremally disconnected ?? $\endgroup$
    – Gerald Edgar
    Commented Jan 7, 2012 at 1:01
  • $\begingroup$ Just a quick comment, based on a rather hazy memory (I may expand on it when I have more time and can check for myself): look inside Haskell Rosenthal's Acta Mathematica paper on injective Banach spaces (late 1960s?) for an injective $C(K)$ space that is not a dual space; I believe that such a space is discussed there... maybe the $K$ there is extremally disconnected? $\endgroup$
    – Philip Brooker
    Commented Jan 8, 2012 at 6:41
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    $\begingroup$ Further to my earlier comment concerning Rosenthal's paper, see Section 4 of that paper (i.e., Section 4 of 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
    Commented Jan 9, 2012 at 9:48
  • $\begingroup$ Dixmier's paper is available here: dmitripavlov.org/scans/dixmier.pdf $\endgroup$
    – Dmitri Pavlov
    Commented Feb 27, 2015 at 17:21
  • $\begingroup$ I also had trouble tracking down this paper a while back. As Choi mentions, the counter-example consists of bounded Borel functions modulo those with meagre support. I guessed at the proof (basically as outlined in the book mentioned by Dashiell it seems) and posted it in answer to another question here - math.stackexchange.com/questions/856804/…. $\endgroup$
    – Tristan Bice
    Commented May 31, 2015 at 18:06

2 Answers 2

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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.

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While I am not familiar with the details of the proof, I think the example being referred to is given by taking the algebra of bounded Baire functions on the real line, and quotienting out by the ideal of functions which vanish on a nowhere dense set. Historically this was of note because it is an example of an $AW^\ast$-algebra in the sense of Kaplansky that is not a von Neumann algebra.

If you look at the last sentence of the MathReview which says

Also, a stonian space is characterised as the spectrum of the totality of the self-adjoint algebra of the totality of bounded Borel functions (modulo those vanishing on a set of first category) on a Hausdorff space. In this way, an example is given of a stonian space which is not hyperstonian.

then this would seem to bear out my hazy recollection.

I have a feeling that the example can be found either in Dixmier's C*-algebra book or in his von Neumann algebra book - admittedly that is not too helpful given the density of those texts, but you might be able to find it. See also an article by Kadison:

MR0861016 (88f:46114) Kadison, Richard V.(1-PA) The von Neumann algebra characterization theorems. Exposition. Math. 3 (1985), no. 3, 193–227.

which apparently describes the Dixmier example (I don't have electronic access to the paper).

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