An unbounded generalization of the Tomita-Takesaki theory. II
type: Item
instance of: scholarly article
title: An unbounded generalization of the Tomita-Takesaki theory. II
zbMATH Open document ID: 0641.47050
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1987-01-01T00:00:00Z
review text: We have continued our study of an unbounded generalization of the Tomita- Takesaki theory begun in a previous paper [ibid. 22, 725-765 (1986; Zbl 0624.47044)]. We first define the notion of modular vectors for a closed \(0\) \(*_ p\)-algebra (\({\mathcal M},{\mathcal D})\) and showed that if \(\xi_ 0\) is a modular vector for (\({\mathcal M},{\mathcal D})\) then a one-parameter group \(\{\sigma_ t^{\xi_ 0}\}\) of *-automorphisms of an unbounded biscommutant (\({\mathcal M}| {\mathcal D}_{\xi_ 0})''_{wc}\) of the \(0\) \(*_ p\)-algebra \({\mathcal M}| {\mathcal D}_{\xi_ 0}\) on a dense subspace \({\mathcal D}_{\xi_ 0}\) of \({\mathcal D}\) is defined and the vector state \(\omega_{\xi_ 0}\) on (\({\mathcal M}| {\mathcal D}_{\xi_ 0})''_{wc}\) satisfies the KMS-condition with respect to \(\{\sigma_ t^{\xi_ 0}\}.\) We have applied the unitary Radon-Nikodym cocycle introduced by Connes to the unbounded case. We have studied Radon-Nikodym theorems and Lebesgue decomposition theorems for \(0\) \(*_ p\)-algebras.
Mathematics Subject Classification ID: 47L60
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L60
zbMATH DE Number: 4044551
zbMATH Keywords: modular vectors
zbMATH Keywords: closed \(0\) \(*_ p\)-algebra
zbMATH Keywords: one-parameter group
zbMATH Keywords: *- automorphisms of an unbounded biscommutant
zbMATH Keywords: KMS-condition
zbMATH Keywords: unitary Radon- Nikodym cocycle
zbMATH Keywords: Radon-Nikodym theorems and Lebesgue decomposition theorems for \(0\) \(*_ p\)-algebras
author: A. Inoue
reviewed by: Ayumu Inoue
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195176253
OpenAlex ID: W3148373397
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DOI: 10.2977/PRIMS/1195176253
description: scientific article; zbMATH DE number 4044551