Reversible part of quantum dynamical systems: A review

Abstract

In this work a quantum dynamical system (M, Φ, φ) is constituted by a von Neumann algebra M, a unital Schwartz map Φ : M → M and a Φ-invariant normal faithful state φ on M. We will prove that the ergodic properties of a quantum dynamical system are determined by its reversible part (D∞, Φ∞, φ∞); i.e. by a von Neumann sub-algebra D∞ of M, with an automorphism Φ∞ and a normal state φ∞, as the restrictions on D∞. Moreover, if D∞ is a trivial algebra, then the quantum dynamical system is ergodic. Furthermore, we will show some properties of reversible part of the quantum dynamical system, finally we will study its relations with the canonical decomposition of Nagy-Fojas of linear contraction related to a quantum dynamical system.