Relative Hamiltonians which are not bounded from above
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.
scientific article; zbMATH DE number 4165728
scientific article