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Kakutani Dichotomy on Free States

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Abstract

Two quasi-free states on a CAR or CCR algebra are shown to generate quasi-equivalent representations unless they are disjoint.

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References

  1. Alberti P.M., Uhlmann A.: On Bures distance and *-algebraic transition probability between inner derived positive linear forms over W*-algebras. Acta Appl. Math. 60, 1–37 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araki H.: On quasifree states of CAR and Bogoliubov automorphisms. Publ. RIMS 6, 385–442 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  3. Araki H., Yamagami S.: On quasi-equivalence of quasifree states of the canonical commutation relations. Publ. RIMS 18, 703–758 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, Berlin (1979)

    Google Scholar 

  5. Goderis D., Vets P.: Central limit theorem for mixing quantum systems and the CCR-algebra of fluctuations. Commun. Math. Phys. 122, 249–265 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Haagerup, U.: L p-spaces associated with an arbitrary von Neumann algebra. Colloques internationaux du CNRS, No. 274 (1977)

  7. Kakutani S.: On equivalence of infinite product measures. Ann. Math. 49, 214–224 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  8. Matsui T.: Factoriality and quasi-equivalence of quasifree states for Z 2 and U(1) invariant CAR algebras. Rev. Roumaine Math. Pures Appl. 32, 693–700 (1987)

    MathSciNet  MATH  Google Scholar 

  9. Matsui T.: On the algebra of fluctuation in quantum spin chains. Ann. Inst. H. Poincar 4, 63–83 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Power R.T., Størmer E.: Free states of the canonical anticommutation relations. Commun. Math. Phys. 16, 1–33 (1970)

    Article  ADS  Google Scholar 

  11. Pusz W., Woronowicz S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8, 159–170 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Raggio G.A.: Comparison of Uhlmann’s transition probability with the one induced by the natural cone of von Neumann algebras in standard form. Lett. Math. Phys. 6, 233–236 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Shale D.: Linear symmetries of free Boson fields. Trans. Am. Math. Soc. 103, 149–167 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shale D., Stinespring W.F.: States on the Clifford algebra. Ann. Math. 80, 365–381 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  15. Van Daele A.: Quasi-equivalence of quasi-free states on the Weyl algebra. Commun. Math. Phys. 21, 171–191 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Yamagami S.: Algebraic aspects in modular theory. Publ. RIMS 28, 1075–1106 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yamagami S.: Geometric mean of states and transition amplitudes. Lett. Math. Phys. 84, 123–137 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Yamagami S.: Geometry of quasi-free states of CCR-algebras. Int. J. Math. 21, 875–913 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, Fukuoka, Japan

    Taku Matsui

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Shigeru Yamagami

Authors
  1. Taku Matsui
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  2. Shigeru Yamagami
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Correspondence to Shigeru Yamagami.

Additional information

S. Yamagami partially supported by KAKENHI (22540217).

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Matsui, T., Yamagami, S. Kakutani Dichotomy on Free States. Lett Math Phys 102, 285–295 (2012). https://doi.org/10.1007/s11005-012-0579-0

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  • DOI: https://doi.org/10.1007/s11005-012-0579-0

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