Elements of Tomita-Takesaki theory for embeddable \(AW^*\)-algebras
type: Item
instance of: scholarly article
title: Elements of Tomita-Takesaki theory for embeddable \(AW^*\)-algebras
zbMATH Open document ID: 0647.46048
DOI: 10.1007/BF00127862
author: M. Frank
published in: Annals of Global Analysis and Geometry
publication date: 1989-01-01T00:00:00Z
review text: The subject of the paper is to describe elements of Tomita-Takesaki theory for embeddable AWß*-algebras M, possessing in one of their ``normal'' representations \(\pi\) (i.e., \(\pi (M)=\pi (M)'')\) on a faithful self-dual Hilbert Cß*-module \({\mathcal H}\) over a commutative AWß*-algebra A a cyclic-separating element \=x\(\in {\mathcal H}\). The proving technics are derived mainly from apaper of \textit{M. A. Rieffel} and \textit{A. van Daele} [Pac. J. Math. 69,187-221 (1976; Zbl 0347.46073)] with changes in the proofs and generalizations of the basic definitions. The classical Tomita-Takesaki theory on Hilbert space is included as a special case. The crucial rule in the proofs play special real subspaces \({\mathcal K}\) of \({\mathcal H}\) generated by the self-adjoint part of \(\pi\) (M) and by \=x\(\in {\mathcal H}\), as well as related to \({\mathcal K}\) operators. From them an order continuous A-linear unitary one-parameter group \(\{\Delta ^{it}:t\in {\mathbb{R}}\}\) and a conjugate-A-linear involution J arise being definedon \({\mathcal H}\) and satisfying the Tomita-Takesaki conditions \(J\pi (M)J=\pi (M)'\), \(\Delta ^{it}\pi (M)\Delta ^{- it}=\pi (M)\) (t\(\in {\mathbb{R}})\). Moreover, this one-parameter group fulfils a generalized K.M.S. condition with respect to \({\mathcal K}\). Additionally, a geometrical interpretation of these results in terms of locally trivial Hilbert bundles on stoneian basic spaces is given.
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46H25
zbMATH DE Number: 4056329
zbMATH Keywords: Tomita-Takesaki theory for embeddable \(AW^*\)-algebras
zbMATH Keywords: faithful self-dual Hilbert \(C^*\)-module
zbMATH Keywords: one-parameter group
zbMATH Keywords: generalized K.M.S. condition
zbMATH Keywords: locally trivial Hilbert bundles on Stoneian basic spaces
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00127862
OpenAlex ID: W2003112564
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description: scientific article; zbMATH DE number 4056329