Communications in Mathematical Physics

, Volume 112, Issue 4, pp 691–719 | Cite as

Dynamical entropy ofC* algebras and von Neumann algebras

  • A. Connes
  • H. Narnhofer
  • W. Thirring
Article

Abstract

The definition of the dynamical entropy is extended for automorphism groups ofC* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.

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References

  1. 1.
    Kolmogorov, A.N.: Dokl. Akad. Nauk119, 861 (1958)Google Scholar
  2. 1a.
    Sinai, Yu.: Dokl. Akad. Nauk124, 768 (1959)Google Scholar
  3. 2.
    Ruelle, D.: Thermodynamic formalism. Reading, Ma: Addison-Wesley 1978Google Scholar
  4. 3.
    Ruelle, D., Bowen, L.: Invent. Math.29, 181 (1975)Google Scholar
  5. 4.
    Lanford, O.E., Robinson, D.W.: Mean entropy states in quantum statistical mechanics. J. Math. Phys.9, 120 (1968)Google Scholar
  6. 5.
    Aizenman, M., Goldstein, S., Gruber, C., Lebowitz, J.L., Martin, P.: On the equivalence between KMS-states and equilibrium states for classical systems. Commun. Math. Phys.53, 209 (1977)Google Scholar
  7. 6.
    Lindblad, G.: Quantum ergodicity and chaos. In: Fundamental aspects of quantum theory, p. 199. Gorini, V., Frigerio, A. (eds.). New York: Plenum Press 1986Google Scholar
  8. 7.
    Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys.14, 1938 (1973)Google Scholar
  9. 8.
    Connes, A., Störmer, E.: Acat Math.134, 289 (1975)Google Scholar
  10. 9.
    Emch, G.: Acta Phys. Austr. [Suppl.]XV, 79 (1976)Google Scholar
  11. 9a.
    Moore, S.M.: Rev. Colomb. Mat.X, 57 (1976)Google Scholar
  12. 10.
    Narnhofer, H., Thirring, W.: Fizika17, 257 (1985)Google Scholar
  13. 11.
    Connes, A.: C.R. Acad. Sci. Paris t301, I, 1 (1985)Google Scholar
  14. 12.
    Lieb, E.H.: Adv. Math.11, 267 (1973)Google Scholar
  15. 13.
    Araki, H.: Pub. Res. Inst. Math. Sci.9, 165 (1973)Google Scholar
  16. 14.
    Kosaki, H.: Interpolation theory and the Wigner-Yanase-Dyson-Lieb concavity. Commun. Math. Phys.87, 315 (1982)Google Scholar
  17. 14a.
    Pusz, W., Woronowicz, S.: Form convex functions and the WYDL and other inequalities. Lett. Math. Phys.2, 505 (1978)Google Scholar
  18. 15.
    Choi, M.D., Effros, E.G.: Ann. Math.104, 585 (1976)Google Scholar
  19. 16.
    Choi, M.D., Effros, E.G.: J. Funct. Anal.24, 156 (1977)Google Scholar
  20. 17.
    Connes, A., Störmer, E.: J. Funct. Anal.28, 187 (1978)Google Scholar
  21. 18.
    Accardi, L., Cecchini, C.: Conditional expectations in von Neumann algebras and a theorem of Takesaki. J. Funct. Anal.45, 245–273 (1982)Google Scholar
  22. 19.
    Araki, H.: Gibbs states of a one dimensional quantum lattice. Commun. Math. Phys.14, 120 (1969)Google Scholar
  23. 20.
    Narnhofer, H.: Thermodynamical phases and surface effects. Acta Phys. Austr.54, 221 (1982)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • A. Connes
    • 1
  • H. Narnhofer
    • 2
  • W. Thirring
    • 2
  1. 1.IHESBures-sur-YvetteFrance
  2. 2.Institut für Theoretische PhysikUniversität WienAustria

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