Functional dependence between the Hamiltonian and the modular operator associated with a faithful invariant state of a \(W^*\)-dynamical system
type: Item
instance of: scholarly article
title: Functional dependence between the Hamiltonian and the modular operator associated with a faithful invariant state of a \(W^*\)-dynamical system
zbMATH Open document ID: 0573.46039
author: Jean De Cannière
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: Let (A,\(\alpha)\) be a \(W^*\)-dynamical system. An \(\alpha\)-invariant normal state \(\theta\) on A is said strongly n-spectrally passive if \(x_ i\in M[\lambda_ i,+\infty)\) \((i=1,2,...,n)\) implies \(\prod^{n}_{i=1}\omega (x_ ix^*_ i)\leq \prod^{n}_{i=1}\) \((x^*_ ix_ i)\) whenever \(\sum^{n}_{i=1}\lambda_ i\geq 0\), where \(M[\lambda_ i,+\infty)\) is the Arveson spectral subspace of A. If \(n=1\), this concept is very close to the concept of passivity introduced by Pusz and Woronowicz, while it is equivalent to the KMS condition if it is true for all n (see the list of references). It is shown that this remain true under the \(n=3\) condition only provided that \(Sp\alpha ={\mathbb{R}}\), and that the \(n=2\) condition is equivalent to the relation \(\Delta =f(H)\) where \(\Delta\) and H are, respectively, the modular operator and the hamiltonian associated with \(\omega\), and f a decreasing function.
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 46N99
zbMATH DE Number: 3915000
zbMATH Keywords: \(W^*\)-dynamical system
zbMATH Keywords: strongly n-spectrally passive
zbMATH Keywords: Arveson spectral subspace
zbMATH Keywords: KMS condition
zbMATH Keywords: modular operator
reviewed by: G. Loupias
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181829
OpenAlex ID: W2114507495
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cites work: Q4163925
cites work: Tomita's theory of modular Hilbert algebras and its applications
DOI: 10.2977/PRIMS/1195181829
description: scientific article; zbMATH DE number 3915000