# 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 *τn* of dynamical maps of a von Neumann algebra *M* into itself, each of which has a faithful normal invariant state *ωn*, and we investigate conditions under which the time-evolved *φn*=*φ_0°_τ_1⋯°_τn* of an arbitrary normal initial state *φ0* is such that lim_n_→ ∞|| *φn*—*ωn*||=0. This is proved under conditions on the spectral gap of *τn* extended to a contraction on the GNS space of (*M*, *ωn*), and on the difference (in a sense to be made precise below) between *ωn* and *ω__n*-1, we do not require detailed balance of *τn* w. r. t. *ωn*. We also give conditions on the sequence of relative Hamiltonians *hn* between *ωn* 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