Asymptotic Behaviour of Time-Inhomogeneous Evolutions on von Neumann Algebras

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.