JournalsprimsVol. 29, No. 5pp. 841–856

Asymptotic Behaviour of Time-Inhomogeneous Evolutions on von Neumann Algebras

  • Alberto Frigerio

    Università di Roma Tor Vergata, Italy
  • Gabriele Grillo

    Politecnica di Milano, Italy
Asymptotic Behaviour of Time-Inhomogeneous Evolutions on von Neumann Algebras cover
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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_→ ∞|| φnωn||=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