Functional Dependence between the Hamiltonian and the Modular Operator Associated with a Faithful Invariant State of a $W^*$-Dynamical System

Abstract

Let H and A be the hamiltonian, resp. the modular operator associated with an invariant faithful normal state w of a W^*-dynamical system (A, a). Then J =/(//) for some decreasing function f if and only if (roughly speaking) w is 2-passive with respect to a. It follows that under certain conditions a 3-passive state is an equilibrium (i.e. KMS) state.