Index and flow of weights of factors of type III
type: Item
instance of: scholarly article
title: Index and flow of weights of factors of type III
zbMATH Open document ID: 0643.46044
author: Toshihiro Hamachi
author: Hideki Kosaki
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal M}\) be a factor of type III and \({\mathcal N}^ a \)subfactor of \({\mathcal M}\) of the same type. Let E be a conditional expectation (Index E\(<\infty)\) from \({\mathcal M}\) into \({\mathcal N}\). The authors investigate to show that \({\mathcal M}\) and \({\mathcal N}\) possess some similar properties in the type III set-up. In fact, they sketch a proof of the following theorem: Theorem. There exists a flow \((X,\{T_ t\}_{t\in R})\) satisfying (i) X is isomorphic to \(X_{{\mathcal M}}\times \{1,2,...,m\}\) (resp. \(X_{{\mathcal N}}\times \{1,2,...,n\})\) as a measure space for some positive integer m, \(m\leq Index E\) (resp. positive integer n, \(n\leq Index E)\), (ii) the projection map \(\pi_{{\mathcal M}}\) (resp. \(\pi_{{\mathcal M}})\) from X onto \(X_{{\mathcal M}}\) (resp. \(X_{{\mathcal N}})\) intertwines \(T_ t\) and \(T_ t^{{\mathcal M}}\) (resp. \(T_ t\) and \(T_ t^{{\mathcal N}})\): \[ T_ t^{{\mathcal M}}\circ \pi_{{\mathcal M}}=\pi_{{\mathcal M}}\circ T_ t,\quad T_ t^{{\mathcal N}}\circ \pi_{{\mathcal N}}=\pi_{{\mathcal N}}\circ T_ t,\quad t\in R. \]
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 4048402
zbMATH Keywords: factor of type III
zbMATH Keywords: subfactor
zbMATH Keywords: conditional expectation
MaRDI profile type: Publication
full work available at URL: pjaa.64.11
OpenAlex ID: W2073160273
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DOI: 10.3792/PJAA.64.11
description: scientific article; zbMATH DE number 4048402