Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A REMARK ON INVARIANCE OF QUANTUM MARKOV SEMIGROUPS

  • Published : 2008.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In [3, 9], using the theory of noncommutative Dirichlet forms in the sense of Cipriani [6] and the symmetric embedding map, authors constructed the KMS-symmetric Markovian semigroup $\{S_t\}_{t{\geq}0}$ on a von Neumann algebra $\cal{M}$ with an admissible function f and an operator $x\;{\in}\;{\cal{M}}$. We give a sufficient and necessary condition for x so that the semigroup $\{S_t\}_{t{\geq}0}$ acts separately on diagonal and off-diagonal operators with respect to a basis and study some results.

Keywords

References

  1. L. Accardi, F. Fagnola, and S. Hachicha, Generic q-Markov semigroups and speed of convergence of q-algorithms, Inf. Dim. Anal. Quantum Probab. Related Topics 9 (2006), 567-594 https://doi.org/10.1142/S0219025706002548
  2. S. Albeverio and R. Hoegh-Krohn, Dirichlet forms and Markovian semigroups on C* - algebras, Comm. Math. Phys. 56 (1977), 173-187 https://doi.org/10.1007/BF01611502
  3. C. Bahn, C. K. Ko, and Y. M. Park, Dirichlet forms and symmetric Markovian semigroups on CCR algebras with quasi-free states, J. Math. Phys. 44 (2003), 723-753 https://doi.org/10.1063/1.1532770
  4. O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics, Springer-Verlag, New York-Heidelberg-Berlin, vol I (1979), vol. II (1981)
  5. R. Carbone, F. Fagnola, and S. Hachicha, Generic quantum Markov semigroups: the Gaussian gauge invariant case, priprint
  6. F. Cipriani, Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras, J. Funct. Anal. 147 (1997), 259-300 https://doi.org/10.1006/jfan.1996.3063
  7. F. Cipriani, F. Fagnola, and J. M. Lindsay, Spectral Analysis and Feller Properties for Quantum Ornstein-Uhlenbeck Semigroups, Comm. Math. Phys. 210 (2000), 85-105 https://doi.org/10.1007/s002200050773
  8. F. Fagnola and R. Quezada, Two-photon and emission process, Inf. Dim. Anal. Quantum Probab. Related Topics 8 (2005), 573-591 https://doi.org/10.1142/S0219025705002116
  9. Y. M. Park, Construction of Dirichlet forms on standard forms of von Neumann algebras, Inf. Dim. Anal. Quantum Probab. Related Topics 3 (2000), 1-14 https://doi.org/10.1142/S0219025700000029
  10. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhauser, Basel, 1992
  11. V. S. Sunder, An Invitation to von Neumann Algebras, Springer-Verlag, Newyork Berlin Heidelberg London Paris Tokyo, 1986