Reversible part of a quantum dynamical system

In this work a quantum dynamical system $(\mathfrak M,\Phi, \varphi)$ is constituted by a von Neumann algebra $\mathfrak M$, by a unital Schwartz map $\Phi:\mathfrak{M\rightarrow M}$ and by a $\Phi$-invariant normal faithful state $\varphi$ on $\mathfrak M$. The ergodic properties of a quantum dynamical system, depends on its reversible part $(\mathfrak{D}_\infty,\Phi_\infty, \varphi_\infty)$. It is constituted by a von Neumann sub-algebra $\mathfrak{D}_\infty$ of $\mathfrak M$ by an automorphism $\Phi_\infty$ and a normal state $\varphi_\infty$, the restrictions of $\Phi$ and $\varphi$ on $\mathfrak{D}_\infty$ respectively. Moreover, if $\mathfrak{D}_\infty$ is a trivial algebra the quantum dynamical system is ergodic. Furthermore we will give some properties of the reversible part of quantum dynamical system, in particular, we will study its relationships with the canonical decomposition of Nagy-Fojas of linear contraction related to the quantum dynamical system.