Standard generalized vectors for algebras of unbounded operators
type: Item
instance of: scholarly article
title: Standard generalized vectors for algebras of unbounded operators
zbMATH Open document ID: 0884.47023
published in: Journal of the Mathematical Society of Japan
publication date: 1998-03-18T00:00:00Z
review text: The notion of (full) standard generalized vectors which makes it possible to develop the Tomita-Takesaki theory in \(\text{O}^*\)-algebras is defined and studied. Let \(\lambda\) be a standard generalized vector for a generalized von Neumann algebra \(\mathcal M\) on \(\mathcal D\) which is a generalization of von Neumann algebras to \(\text{O}^*\)-algebras. A one-parameter group \(\{\sigma^\lambda_t\}_{t\in R}\) of \(*\)-automorphisms of \(\mathcal M\) is defined and the quasi-weight \(\varphi_\lambda\) defined by \(\lambda\) satisfies the KMS-condition with respect to \(\{\sigma^\lambda_t\}_{t\in R}\). The Connes cocycle theorem and the Pedersen-Takesaki Radon-Nikodým theorem for von Neumann algebras are generalized to for generalized von Neumann algebras. Let \(\lambda\) and \(\mu\) be full standard generalized vectors for \(\mathcal M\). There uniquely exists a strongly continuous map \(t\in R\to U_t\in{\mathcal M}_u\equiv \{U\in{\mathcal M};\;\overline U\) is unitary\} such that \(U_{s+t}= U_t\sigma^\lambda_s(U_s)\) and \(\sigma^\mu_t(X)= U_t\sigma^\lambda_t(X) U^*_t\) for all \(s,t\in R\) and \(X\in{\mathcal M}\). It is shown that \(\{U_t\}_{t\in R}\) is a one-parameter group if and only if the domain \(D(\mu)\) of \(\mu\) is \(\{\sigma^\lambda_t\}\)-invariant and \(|\mu(\sigma^\lambda_t(X))|= |\mu(X)|\) for all \(X\in D(\mu)\) and \(t\in R\) if and only if \(\{U_t\}_{t\in R}\subset{\mathcal M}^{\sigma^\lambda}_b\equiv \{A\in{\mathcal M};\;\overline A\) is bounded and \(\sigma^\lambda_t(A)= A\) for all \(t\in R\}\). Furthermore, \(\mu\) satisfies the KMS-condition with respect to \(\{\sigma^\lambda_t\}\) if and only if \(\sigma^\mu_t= \sigma^\lambda_t\) for all \(t\in R\).
Mathematics Subject Classification ID: 47L60
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
zbMATH DE Number: 844456
zbMATH Keywords: standard generalized vectors
zbMATH Keywords: Tomita-Takesaki theory in \(\text{O}^*\)-algebras
zbMATH Keywords: generalized von Neumann algebra
zbMATH Keywords: one-parameter group
zbMATH Keywords: \(*\)-automorphisms
zbMATH Keywords: quasi-weight
zbMATH Keywords: KMS-condition
zbMATH Keywords: Connes cocycle theorem
zbMATH Keywords: Pedersen-Takesaki Radon-Nikodým theorem
author: A. Inoue
reviewed by: A. Inoue
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04720329
OpenAlex ID: W2018354388
DOI: 10.2969/JMSJ/04720329
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description: scientific article; zbMATH DE number 844456