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I've seen several questions and answers on the Gelfand transform for commutative $C^*$-algebras leading to a characterization of commutative Von Neumann algebras as those whose spectrum is hyperstonean (see among others https://mathoverflow.net/questions/23408/reference-for-the-gelfand-neumark-theorem-for-commutative-von-neumann-algebras, Examples of hyperstonean space). I cannot find a reference for the definition of this type of spaces, I understand they are extremely disconnected compact spaces, but just a tiny part of them, what is one of their possible definitions as a subcategory of the extremely disconnected compact spaces?

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curious on mathematics
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  • Carefully read through this discussion – Norbert Feb 16 '15 at 13:25
  • Thanks, the discussion is quite interesting and to some extent enlightening, but I still miss the definition of hyperstonean space. Could you point a reference where I can find this definition? The article of Zaharov seems not available and I have trouble also to find the two books (Conway and Dales-Lau & Strauss). All the answers point to a categorical treatment of the question or define the spectrum of the commutative $C^$-algebras $C(X)^{*}$ as Hyperstonean covers of the compact space $C(X)$. Before hand I want to understand right away what is the meaning of hyperstonean space... – curious on mathematics Feb 16 '15 at 15:40
  • You may consider this rude, but the first part of your comment should be read as follows: google this for me. I will not. I worked on that question a long time ago I found everything you mentioned. As for the second part, there is no explicit or tractable definition of hyperstonean space. The definition of Zaharov use the notion of Kelly ideals which is quite complicated and won't shed any light on what hyperstonean space is. Hyperstonean space is the spectrum of $C(X)^{**}$. Deal with it. – Norbert Feb 16 '15 at 19:10
  • ok, thanks, I understand your point, I'm surprised that a concept which seems quite natural turn out to be so elusive. – curious on mathematics Feb 16 '15 at 19:28
  • For a definition of hyperstonean spaces see Definition III.1.14 in Takesaki's Theory of operator algebras. In fact, the whole section III.1 is to a significant degree about hyperstonean spaces. – Dmitri P. Feb 17 '15 at 12:26
  • thanks a lot! I'll follow your advice. – curious on mathematics Feb 17 '15 at 21:57
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    Please read Chapter 2 of this memoir:

    https://www1.maths.leeds.ac.uk/~pmt6hgd/preprints/Dissertatmeasurealgebrafinal.pdf

     – Tomasz Kania Feb 18 '15 at 22:19
  • thanks, I will also take a look at it! – curious on mathematics Feb 20 '15 at 20:23
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    @TomekKania Your link is dead :( – მამუკა ჯიბლაძე Mar 04 '19 at 15:56

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