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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https://www1.maths.leeds.ac.uk/~pmt6hgd/preprints/Dissertatmeasurealgebrafinal.pdf
– Tomasz Kania Feb 18 '15 at 22:19