Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T14:24:41.038Z Has data issue: false hasContentIssue false
  • English
  • Français

CONTINUOUS AND DISCRETE FLOWS ON OPERATOR ALGEBRAS

Part of: Connections with other structures, applications Selfadjoint operator algebras

Published online by Cambridge University Press:  01 April 2009

BENJAMÍN A. ITZÁ-ORTIZ
BENJAMÍN A. ITZÁ-ORTIZ*
Affiliation:
Centro de Investigación en Matemáticas, Universidad Autónoma del Estado de Hidalgo, Carretera Pachuca-Tulancingo Km. 4.5, Pachuca, Hidalgo, 42184, México (email: itza@uaeh.edu.mx)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let (N,ℝ,θ) be a centrally ergodic W* dynamical system. When N is not a factor, we show that for each nonzero real number t, the crossed product induced by the time t automorphism θt is not a factor if and only if there exist a rational number r and an eigenvalue s of the restriction of θ to the center of N, such that rst=2π. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if (A,ℝ,α) is a minimal unital C* dynamical system and A is not simple, then, for each nonzero real number t, the crossed product induced by the time t automorphism αt is not simple if there exist a rational number r and an eigenvalue s of the restriction of α to the center of A, such that rst=2π. The converse is true if, in addition, A is commutative or prime.

Keywords

minimal flowtime t automorphism

MSC classification

Secondary: 46L55: Noncommutative dynamical systems 54H20: Topological dynamics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 2 , April 2009 , pp. 169 - 176
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G., Ergodic Theory, Grundl. Math. Wiss., 245 (Springer, Berlin, 1982).CrossRefGoogle Scholar
[2]Glasner, E., Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
[3]Itzá-Ortiz, B., ‘Eigenvalues, K-theory and minimal flows’, Canad. J. Math. 59 (2007), 596–613.CrossRefGoogle Scholar
[4]Kishimoto, A., ‘Simple crossed products of C*-algebras by locally compact abelian groups’, Yokohama Math. J. 28 (1980), 69–85.Google Scholar
[5]Kishimoto, A. and Pedersen, G., ‘Quasi-product flows on a C*-algebra’, Comm. Math. Phys. 229 (2002), 397–413.CrossRefGoogle Scholar
[6]Olesen, D. and Pedersen, G., ‘Applications of the Connes spectrum to C*-dynamical systems’, J. Funct. Anal. 30 (1978), 179–197.CrossRefGoogle Scholar
[7]Olesen, D. and Pedersen, G., ‘Applications of the Connes spectrum to C*-dynamical systems II’, J. Funct. Anal. 36 (1980), 18–32.CrossRefGoogle Scholar
[8]Olesen, D. and Pedersen, G., ‘Applications of the Connes spectrum to C*-dynamical systems III’, J. Funct. Anal. 45 (1982), 357–390.Google Scholar
[9]Pedersen, G. K., C*-algebras and their Automorphism Groups, London Mathematical Society Monographs, 14 (Academic Press, London, 1979).Google Scholar
[10]Størmer, E., ‘Spectra of ergodic transformations’, J. Funct. Anal. 15 (1974), 202–215.CrossRefGoogle Scholar
[11]Strătilă, S., Modular Theory in Operator Algebras (Editura Academiei and Abacus Press, Bucharest and Tunbridge Wells, 1981).Google Scholar
[12]Takesaki, M., ‘Duality for crossed products and the structure of von Neumann algebras of type III’, Acta Math. 131 (1973), 249–310.CrossRefGoogle Scholar
[13]Takesaki, M., Theory of Operator Algebras II, Encyclopedia of Mathematical Sciences, 125 (Springer, Heidelberg, 2002).Google Scholar
[14]Walters, P., An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79 (Springer, New York, 1982).CrossRefGoogle Scholar