# Asymptotic Behaviour of Time-Inhomogeneous Evolutions on von Neumann Algebras

### Alberto Frigerio

Università di Roma Tor Vergata, Italy### Gabriele Grillo

Politecnica di Milano, Italy

## Abstract

We consider a sequence $τ_{n}$ of dynamical maps of a von Neumann algebra $M$ into itself, each of which has a faithful normal invariant state $ω_{n}$, and we investigate conditions under which the time-evolved $φ_{n}=φ_{0}∘τ_{1}∘⋯∘τ_{n}$ of an arbitrary normal initial state $φ_{0}$ is such that $lim_{n→∞}∣∣φ_{n}−ω_{n}∣∣=0$. This is proved under conditions on the spectral gap of $τ_{n}$ extended to a contraction on the GNS space of $(M,ω_{n})$, and on the difference (in a sense to be made precise below) between $ω_{n}$ and $ω_{n−1}$, we do not require detailed balance of $τ_{n}$ w. r. t. $ω_{n}$. We also give conditions on the sequence of relative Hamiltonians $h_{n}$ between $ω_{n}$ and $ω_{n−1}$ 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.

## Cite this article

Alberto Frigerio, Gabriele Grillo, Asymptotic Behaviour of Time-Inhomogeneous Evolutions on von Neumann Algebras. Publ. Res. Inst. Math. Sci. 29 (1993), no. 5, pp. 841–856

DOI 10.2977/PRIMS/1195166577