Reference for the Gelfand duality theorem for commutative von Neumann algebras

I think we established that the literature is lacking on this question. But I think the "correct" definition of morphisms between hyperstonean spaces can be puzzled together from G. Bezhanishvili's paper "Stone duality and Gleason covers through de Vries duality" (Topology and its Applications 157:1064-1080, 2010), especially section 6.

He proves in detail a duality between the category of complete Boolean algebras and complete Boolean algebra homomorphisms, and the category of extremally disconnected compact Hausdorff spaces and continuous open maps. But commutative von Neumann algebras and normal *-homomorphisms form a full subcategory of the former (via taking projections), which corresponds to the full subcategory of the latter consisting of hyperstonean spaces.

So Gelfand duality really restricts quite cleanly: commutative von Neumann algebras and normal *-homomorphisms are dual to hyperstonean spaces and open continuous maps.

As far as I know the only purely point-set theoretic (avoiding measure theory) description of the hyperstonian cover (and morphisms thereof) was done by Zakharov in terms of so called Kelley ideals:

V. K. Zaharov, Hyperstonean cover and second dual extension, Acta Mathematica Hungarica Volume 51, Numbers 1-2, 125-149

I tried to read that paper but I failed. Good luck!

This is only an historical comment. As far as I know the equivalence between (1) and (2) is not an easy consequence of Gelfand-Neu(ai)mark theorem. One implication (I do not remember which one) was proved by Dixmier and the other one by Grothendiek. I am quite sure that Dixmier used explicitely the word von Neumann algebra. I have never read Grothendieck's paper but it is likely that he did not use this name and he just proved one of the two implications of the following theorem: $C(K)$ is a dual Banach space iff $K$ is hyperstonean.

J.Dixmier, Sur certains espaces consideres par M.H. Stone, Summa Brasil. Math. 2, 151-182 (1951)

Grothendieck, Sur les applications lineaire faiblement compactes d'espace du type C(K), Canad. J. Math. 5, (1953) 129-173.