On the duality between Boolean-valued analysis and reduction theory under the assumption of separability
type: Item
instance of: scholarly article
title: On the duality between Boolean-valued analysis and reduction theory under the assumption of separability
zbMATH Open document ID: 0779.47034
DOI: 10.1007/BF00673354
author: Hirokazu Nishimura
published in: International Journal of Theoretical Physics
publication date: 1994-01-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 46S20
Mathematics Subject Classification ID: 03C90
zbMATH DE Number: 205373
zbMATH Keywords: Stone's duality theorem
zbMATH Keywords: Boolean-valued models
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: reduction theory
zbMATH Keywords: Boolean-valued analysis
reviewed by: W. A. J. Luxemburg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 205373