Ergodic properties of automorphisms on the rotation algebra
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)].
generalized crossed product construction
scientific article; zbMATH DE number 1091057
scientific article