# Gelfand duality restricted to the category of Stone spaces

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.