Functional dependence between the Hamiltonian and the modular operator associated with a faithful invariant state of a \(W^*\)-dynamical system (Q1063211)

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.