On the duality between Boolean-valued analysis and reduction theory under the assumption of separability
Analogous to Stone's duality theorem between Boolean algebras and zero dimensional compact Hausdorff spaces Scott and Solovay using Boolean- valued models established a duality between the real numbers in \(V^{(B)}\), a Boolean-valued model of \(ZFC\) with respect to a complete Boolean algebra \(B\), and the Borel functions on the Stonean space \(\Omega\) of \(B\). In this paper the author establishes a duality between separable complex Hilbert spaces in \(V^{(B)}\) and an appropriate class of families of complex Hilbert spaces over \(\Omega\). In addition it is shown that the duality extends to the von Neumann algebras acting on separable Hilbert spaces and a corresponding appropriate class of such algebras over \(\Omega\). Since the latter families coincide with those studies in reduction theory the latter duality may be regarded as a duality of Boolean-valued analysis and reduction theory. For further details the reader should consult the well-written paper under review.
scientific article; zbMATH DE number 205373
scientific article