Asymptotic behaviour of time-inhomogeneous evolutions on von Neumann algebras
Summary: We consider a sequence \(\tau_ n\) of dynamical maps of a von Neumann algebra \({\mathcal M}\) into itself, each of which has a faithful normal invariant state \(\omega_ n\), and we investigate conditions under which the time-evolved \(\varphi_ n= \varphi_ 0\circ \tau_ 1\circ \dots\circ \tau_ n\) of an arbitrary normal initial state \(\varphi_ 0\) is such that \(\lim_{n\to\infty} \| \varphi_ n- \omega_ n\| =0\). This is proved under conditions on the spectral gap of \(\tau_ n\) extended to a contraction on a GNS space of \(({\mathcal M}, \omega_ n)\), and on the difference between \(\omega_ n\) and \(\omega_{n-1}\); we do not require detailed balance of \(\tau_ n\) w.r.t. \(\omega_ n\). We also give conditions on the sequence of relative Hamiltonians \(h_ n\) between \(\omega_ n\) and \(\omega_{n-1}\) ensuring that the result holds. Finally, we prove that the techniques of the present paper do not admit a simple generalization to \(C^*\)-algebras and non-normal states.
scientific article; zbMATH DE number 548931
scientific article