Relative Hamiltonians which are not bounded from above
type: Item
instance of: scholarly article
title: Relative Hamiltonians which are not bounded from above
zbMATH Open document ID: 0709.46030
DOI: 10.1016/0022-1236(90)90050-U
author: Matthew J. Donald
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: Let \({\mathcal A}\) be a von Neumann algebra, let \(\omega\) be a normal state on \({\mathcal A}\) and let h be a selfadjoint operator affiliated with A which is bounded below. The aim of this paper is to define the perturbed state \(\omega^ h\) (a possibility already noted by Araki) and to prove three fundamental properties of it. The first property says that if \(\rho\leq X\omega\) for some real X, then there exists h such that \(\rho =\omega^ h\). The second one asserts that \((\omega^ h)^ H=\omega^{h+H}\) for all pairs (h,H) of lower-bounded operators, provided the right-hand side exists. Finally, the relationship between the set of states which have the same modular group as \(\omega\) and the set of states which have the same modular group as \(\omega^ h\) is established, a result important for potential applications.
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 4165728
zbMATH Keywords: entropy
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: normal state
zbMATH Keywords: perturbed state
zbMATH Keywords: modular group
reviewed by: Florian-Horia Vasilescu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4165728