On the duality between Boolean-valued analysis and reduction theory under the assumption of separability

Abstract

It is well known that the real and complex numbers in the Scott-Solovay universeV (B) of ZFC based on a complete Boolean algebraB are represented by the real-valued and complex-valued Borel functions on the Stonean spaceΩ ofB. The main purpose of this paper is to show that the separable complex Hilbert spaces and the von Neumann algebras acting on them inV (B) can be represented by reasonable classes of families of complex Hilbert spaces and of von Neumann algebras overΩ. This could be regarded as the duality between Boolean-valued analysis developed by Ozawa, Takeuti, and others and the traditional reduction theory based not on measure spaces but on Stonean spaces. With due regard to Ozawa, this duality could pass for a sort of reduction theory forAW *-modules over commutativeAW *-algebras and embeddableAW *-algebras. Under the duality we establish several fundamental correspondence theorems, including the type correspondence theorems of factors.