Local perturbations of the dynamics of of infinite systems

V. Ya. Golodets

Teoretičeskaâ i matematičeskaâ fizika, Tome 23 (1975) no. 3, pp. 300-309

Résumé

Systems, the dynamics of which is locally perturbed, are studied. Observables of the system under consideration are supposed to generate a $C^*$-algebra $A$, and unperturbed $\sigma_t$ and perturbed $\sigma_t^p$ evolutions are represented as one-parameter groups of automorphisms on $A$. If $\omega$ is $\sigma_t^p$-KMS-state and $A$ is asymptotically abelian then $\lim\limits_{t\to\pm\infty}\omega(\sigma_t(a))=\omega_{\pm}(a)$ $(a\in A)$ exists, $\omega_+=\omega_-$ and $\omega_{\pm}$ is $\sigma_t$-KMS-state. If moreover $\lim\limits_{s\to\pm\infty}\sigma_s^p\sigma_s=\gamma_{\pm}$ exists and determines epimorphisms $\gamma_{\pm}$ (not necessarily invertible) of $A$ intertwining $\sigma_t$ and $\sigma_t^p$ $(\gamma_{\pm}\sigma_t=\sigma_t^p\gamma_{\pm})$ then $\gamma_{\pm}$ can be extended to automorphisms of von Neumann algebra $M=\pi_{\omega}(A)''$ where $\pi_{\omega}$ is the representation of $A$ corresponding to the state $\omega$. Therefore if $\gamma_{\pm},\sigma_t$ and $\sigma_t^p$ are considered as automorphisms of $M$ then $\gamma_{\pm}^{-1}\sigma_t^p=\sigma_t\gamma_{\pm}^{-1}$. With the aid of this result we prove that $\lim\limits_{|t|\to\infty}\omega_{\pm}(\sigma_t^p(a))$ exists and is equal to $\omega(a)$ $(a\in A)$. We also prove that $M=\pi_{\omega}(A)''$ is asymptotically abelian with respect to the extension of $\sigma_t$ to the automorphisms of $M$ and that $M$ is of the type III.