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It is well known that the category of unital commutative $C^\ast$-algebras is dually equivalent to the category $\mathbf{KHaus}$ of compact Hausdorff spaces.

I have found that restricting the Gelfand duality to the full subcategory of Stonean spaces, i.e the compact Hausdorff spaces which moreover are extremally disconnected, yields the subcategory of $AW^\ast$-algebras. Futhermore since under the Stone duality between Stone spaces and Boolean algebras the Stonean spaces corresponds to complete Boolean algebras, we should have an equivalence between the subcategory of complete Boolean algebras and the subcategory of $AW^\ast$-algebras.

Given this, I would like to know if anything is know about the restriction of the Gelfand duality to the full subcategory of $\mathbf{KHaus}$ consisting of Stone spaces. That is what is the correspondent subcategory of the category of unital commutative $C^\ast$-algebras, or equivalently which unital commutative $C^\ast$-algebras corresponds to Boolean algebras, under these dualities?

Any help is much appreciated.

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Here is a clean and short way to prove Stone duality:

  • show that the category of Stone spaces is the category of pro-objects in finite sets (profinite sets),
  • show that the category of Boolean rings is the category of ind-objects in finite Boolean rings,
  • show that the category of finite Boolean rings is equivalent to the opposite of the category of finite sets.

Now, under Gelfand duality, the opposite of the category of finite sets is also equivalent to the category of finite dimensional unital commutative C*-algebras. So it follows that the opposite of the category of profinite sets is also equivalent to the category of ind-finite dimensional unital commutative C*-algebras.

According to Wikipedia this condition is equivalent to being a (unital, commutative) AF C*-algebra. (A priori this condition is slightly more restrictive but Wikipedia claims it's equivalent to the spectrum being totally disconnected, so...)

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