For any complex valued functions over any topological space F there exists a relation in von Neumann algebras of *-graded that is bounded on compact Hausdorff where for category- I, II, III there exists a commutative form of 〖AW〗^* algebras such that to satisfy a monotone complete C^* algebra suffice an isomorphic factor f on the same 〖AW〗^* tamed as W^* having the generators η for a generic group η(G) for 2-groups G_+ and G_- for the former being additive integers generating the later free group for 〖AW〗^* algebras where compact Hausdorff CH a Borel measure β exists in compact set C norms the associated Hausdorff space over a locally finite σ-algebra via β(C)⋖∞.