Advertisement

Communications in Mathematical Physics

, Volume 90, Issue 2, pp 251–262 | Cite as

A Markov dilation of a non-quasifree Bloch evolution

  • B. Kümmerer
  • W. Schröder
Article

Abstract

We construct a new minimal dilation of a dynamical system governed by a Bloch equation. In contrast to a dilation of the same dynamical system recently obtained by Varilly [13] our dilation satisfies a Markov property. This presents the first example of a Markov dilation for a non-commutative dynamical system which is not equivalent to a quasifree evolution. Furthermore the dilation turns out to be a generalizedK-system.

Keywords

Neural Network Dynamical System Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. New York, Heidelberg, Berlin: Springer 1981Google Scholar
  2. 2.
    Emch, G.G.: GeneralizedK-flows. Commun. Math. Phys.49, 191–215 (1976)Google Scholar
  3. 3.
    Emch, G.G.: Minimal dilations ofCP-flows. In: Lecture Notes in Mathematics, Vol.650, pp. 156–159. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  4. 4.
    Emch, G.G., Albeverio, S., Eckmann, J.-P.: Quasi-free generalizedK-flows. Rep. Math. Phys.13, 73–85 (1978)Google Scholar
  5. 5.
    Emch, G.G., Varilly, J.C.: On the standard form of the Bloch equation. Lett. Math. Phys.3, 113–116 (1979)Google Scholar
  6. 6.
    Evans, D.E.: Completely positive quasi-free maps on the CAR algebra. Commun. Math. Phys.70, 53–68 (1979)Google Scholar
  7. 7.
    Kern, M., Nagel, R., Palm, G.: Dilations of positive operators: Construction and ergodic theory. Math. Z.156, 256–277 (1977)Google Scholar
  8. 8.
    Kümmerer, B.: Markov dilations onW*-algebras (to appear)Google Scholar
  9. 9.
    Kümmerer, B., Schröder, W.: A theory of Markov dilations for the spin-1/2-relaxation (to appear)Google Scholar
  10. 10.
    Parry, W.: Topics in Ergodic theory. Cambridge: Cambridge University Press 1981Google Scholar
  11. 11.
    Schröder, W.:W*-K-systems and their mixing properties (to appear)Google Scholar
  12. 12.
    Sz.-Nagy, B., Foias, C.: Harmonic analysis of operators on Hilbert space. Amsterdam: North Holland 1970Google Scholar
  13. 13.
    Varilly, J.C.: Dilation of a non-quasifree dissipative evolution. Lett. Math. Phys.5, 113–116 (1981)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • B. Kümmerer
    • 1
  • W. Schröder
    • 1
  1. 1.Mathematisches InstitutUniversität TübingenTübingenFederal Republic of Germany

Personalised recommendations