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 TMF, 1975, Volume 23, Number 3, Pages 300–309 (Mi tmf3808)

Local perturbations of the dynamics of of infinite systems

V. Ya. Golodets

Abstract: 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.

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English version:
Theoretical and Mathematical Physics, 1975, 23:3, 525–532

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Citation: V. Ya. Golodets, “Local perturbations of the dynamics of of infinite systems”, TMF, 23:3 (1975), 300–309; Theoret. and Math. Phys., 23:3 (1975), 525–532

Citation in format AMSBIB
\Bibitem{Gol75} \by V.~Ya.~Golodets \paper Local perturbations of the dynamics of of infinite systems \jour TMF \yr 1975 \vol 23 \issue 3 \pages 300--309 \mathnet{http://mi.mathnet.ru/tmf3808} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=489593} \zmath{https://zbmath.org/?q=an:0313.46055} \transl \jour Theoret. and Math. Phys. \yr 1975 \vol 23 \issue 3 \pages 525--532 \crossref{https://doi.org/10.1007/BF01041670} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. Ya. Golodets, “Modular operators and asymptotic commutativity in von Neumann algebras”, Russian Math. Surveys, 33:1 (1978), 47–106
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