Asymptotic Behaviour of Time-Inhomogeneous Evolutions on von Neumann Algebras
Alberto Frigerio, Gabriele Grillo · Publications of the Research Institute for Mathematical Sciences · 1993
We consider a sequence τ_n of dynamical maps of a von Neumann algebra \mathcal 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\circ τ_1\circ ⋯\circ 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 (\mathcal 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.