Asymptotic behaviour of time-inhomogeneous evolutions on von Neumann algebras
type: Item
instance of: scholarly article
title: Asymptotic behaviour of time-inhomogeneous evolutions on von Neumann algebras
zbMATH Open document ID: 0795.46046
author: Alberto Frigerio
author: Gabriele Grillo
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1994-05-03T00:00:00Z
review text: Summary: We consider a sequence \(\tau_ n\) of dynamical maps of a von Neumann algebra \({\mathcal M}\) into itself, each of which has a faithful normal invariant state \(\omega_ n\), and we investigate conditions under which the time-evolved \(\varphi_ n= \varphi_ 0\circ \tau_ 1\circ \dots\circ \tau_ n\) of an arbitrary normal initial state \(\varphi_ 0\) is such that \(\lim_{n\to\infty} \| \varphi_ n- \omega_ n\| =0\). This is proved under conditions on the spectral gap of \(\tau_ n\) extended to a contraction on a GNS space of \(({\mathcal M}, \omega_ n)\), and on the difference between \(\omega_ n\) and \(\omega_{n-1}\); we do not require detailed balance of \(\tau_ n\) w.r.t. \(\omega_ n\). We also give conditions on the sequence of relative Hamiltonians \(h_ n\) between \(\omega_ n\) and \(\omega_{n-1}\) 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.
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L57
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 548931
zbMATH Keywords: dynamical maps of a von Neumann algebra
zbMATH Keywords: faithful normal invariant state
zbMATH Keywords: spectral gap
zbMATH Keywords: GNS space
zbMATH Keywords: relative Hamiltonians
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: non- normal states
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195166577
OpenAlex ID: W2059650387
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DOI: 10.2977/PRIMS/1195166577
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description: scientific article; zbMATH DE number 548931