대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제41권1호
  • /
  • Pages.189-198
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  • 2004
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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QUANTUM DYNAMICAL SEMIGROUP AND ITS ASYMPTOTIC BEHAVIORS

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

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.

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참고문헌

  1. Operator algebras and quantum statistical mechanics I. O.Bratteli;D.W.Robinson
  2. One-parameter semigroups E.B.Davies
  3. Comm. Math. Phys. v.54 no.3 Irreducible quantum dynamical semigroups D.E.Evans https://doi.org/10.1007/BF01614091
  4. Math Z. v.180 no.2 Long-time asymptotic properties of dynamical semigroups on $W^{*}$ - algebras A.Frigerio;M.Verri https://doi.org/10.1007/BF01318911
  5. Lecture Notes in Math. v.1184 Positive semigroups on $C^{*}$ - and $W^{*}$ -algebras, In: One-parameter Semigroups of positive Operators U.Groh https://doi.org/10.1007/BFb0074935
  6. Math. Proc. Cambridge Philos. Soc. v.111 no.1 Invariant states and ergodic dynamical systems on $W^{*}$ -algebras A.Luczak https://doi.org/10.1017/S0305004100075265
  7. J. Math. Anal. Appl. v.221 no.1 Eigenvalues and eigenspaces of quantum dynamical systems and their tensor products https://doi.org/10.1006/jmaa.1995.4987
  8. Math. Z. v.235 no.3 Mixing and asymptotic properties of Markov semigroups on von Neumann algebras https://doi.org/10.1007/s002090000159
  9. Harmonic analysis of operators on hilbert space B.Sz-Nagy;C.Foias
  10. Comm. Math. Phys. v.85 no.1 Strongly positive semigroups and faithful invariant ststes D.W.Robinson https://doi.org/10.1007/BF02029138
  11. Theory of operator algebras I. M.Takesaki
  12. Studia Math. v.81 no.3 Invariant states for positive operator semigroups K.E.Thomsen
  13. J. Math. Anal. Appl. v.86 no.2 Asymptotic behavior and eigenvalus of dynamical semigroups on operator algevras S.Watanabe https://doi.org/10.1016/0022-247X(82)90231-1

피인용 문헌

  1. A Survey on Invariance and Ergodicity of Quantum Markov Semigroups vol.32, pp.3, 2014, https://doi.org/10.1080/07362994.2014.897136
  2. Recurrence and Transience of Quantum Markov Semigroups vol.33, pp.1, 2015, https://doi.org/10.1080/07362994.2014.968287