Ergodic properties of automorphisms on the rotation algebra
type: Item
instance of: scholarly article
title: Ergodic properties of automorphisms on the rotation algebra
zbMATH Open document ID: 0914.46057
DOI: 10.1016/S0034-4877(97)89755-8
published in: Reports on Mathematical Physics
publication date: 1999-02-01T00:00:00Z
review text: The rotation algebra \({\mathcal A}_\Theta\), \(\Theta\in R\) is a good point to start investigating ergodic properties in terms of noncommutative algebras. So that, in this paper, the author first defines the various kinds of automorphisms on the rotation algebra and discusses their possible representations, such as those in \({\mathcal B}(L^2(R))\) or in \({\mathcal B}(L^2({\mathcal T}^2))\), etc. Moreover, it is shown how cluster properties and uniqueness of the invariant states depend on the rotation parameter. Actually, for rationally depending \(\Theta\), the different algebras are related by a generalized crossed product construction, and this relation is recognized as an essential tool in the discussion how ergodic properties depend on the rotation parameter. In particular, the author concentrates on the ergodic properties of the automorphisms of the von Neumann algebra \(\pi_{\mathcal T}({\mathcal A}_\Theta)''\) obtained in the tracial representation, and the related results from \textit{F. Benatti}, \textit{H. Narnhofer} and \textit{G. L. Sewell} [Lett. Math. Phys. 21, No. 2, 157-172; correction 22, No. 1, 81 (1991; Zbl 0722.46033)] are also introduced. In the last part of the paper, one open problem about the possibility of other invariant states and his conjecture on erratic behaviours of the Connes-Størmer entropy \(h\), in connection with the ergodic theory, are stated as well. For other related works, e.g. see \textit{H. Narnhofer} [J. Math. Phys. 33, No. 4, 1502-1510 (1992; Zbl 0755.58035)] and \textit{H. Narnhofer} and \textit{W. Thirring} [Lett. Math. Phys. 35, No. 2, 145-154 (1995; Zbl 0833.46051)].
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L40
zbMATH DE Number: 1091057
zbMATH Keywords: rotation algebra
zbMATH Keywords: ergodic properties
zbMATH Keywords: noncommutative algebras
zbMATH Keywords: cluster properties
zbMATH Keywords: invariant states
zbMATH Keywords: generalized crossed product construction
zbMATH Keywords: automorphisms
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: tracial representation
zbMATH Keywords: Connes-Størmer entropy
author: Heide Narnhofer
reviewed by: Isamu Dôku
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1091057