Asymptotic Behaviour of Time-Inhomogeneous Evolutions on von Neumann Algebras

Abstract

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 ${\phi }_n={\phi }_0\circ {\tau }_1\circ \cdots \circ {\tau }_n$ of an arbitrary normal initial state ${\phi }_0$ is such that ${\lim }_{n\to \infty }||{\phi }_n-{\omega }_n||=0$. This is proved under conditions on the spectral gap of ${\tau }_n$ extended to a contraction on the GNS space of $(\mathcal{M},{\omega }_n)$, and on the difference (in a sense to be made precise below) 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 $C^{\ast }$-algebras and non-normal states.