• Choi, Veni (Department of Mathematics, Yonsei University)
  • Published : 2004.02.01


In this study we consider quantum dynamical semi-group with a normal faithful invariant state. A quantum dynamical semigroup $\alpha\;=\;\{{\alpha}_t\}_{t{\geq}0}$ is a class of linear normal identity-preserving mappings on a von Neumann algebra M with semigroup property and some positivity condition. We investigate the asymptotic behaviors of the semigroup such as ergodicity or mixing properties in terms of their eigenvalues under the assumption that the semigroup satisfies positivity. This extends the result of [13] which is obtained under the assumption that the semi group satisfy 2-positivity.


  1. Math. Z. v.235 no.3 Mixing and asymptotic properties of Markov semigroups on von Neumann algebras
  2. Harmonic analysis of operators on hilbert space B.Sz-Nagy;C.Foias
  3. Comm. Math. Phys. v.54 no.3 Irreducible quantum dynamical semigroups D.E.Evans
  4. One-parameter semigroups E.B.Davies
  5. J. Math. Anal. Appl. v.86 no.2 Asymptotic behavior and eigenvalus of dynamical semigroups on operator algevras S.Watanabe
  6. Comm. Math. Phys. v.85 no.1 Strongly positive semigroups and faithful invariant ststes D.W.Robinson
  7. Theory of operator algebras I. M.Takesaki
  8. Math. Proc. Cambridge Philos. Soc. v.111 no.1 Invariant states and ergodic dynamical systems on $W^{*}$ -algebras A.Luczak
  9. Math Z. v.180 no.2 Long-time asymptotic properties of dynamical semigroups on $W^{*}$ - algebras A.Frigerio;M.Verri
  10. Studia Math. v.81 no.3 Invariant states for positive operator semigroups K.E.Thomsen
  11. Operator algebras and quantum statistical mechanics I. O.Bratteli;D.W.Robinson
  12. J. Math. Anal. Appl. v.221 no.1 Eigenvalues and eigenspaces of quantum dynamical systems and their tensor products
  13. Lecture Notes in Math. v.1184 Positive semigroups on $C^{*}$ - and $W^{*}$ -algebras, In: One-parameter Semigroups of positive Operators U.Groh

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