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Phase Transition in O 2

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Abstract

The paper presents an example of a one-parameter group of automorphisms on the Cuntz algebra O 2 for which the KMS states exhibit a phase transition. The factor type of the extremal KMS states are determined.

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References

  1. Anantharaman-Delaroche C.: Purely infinite C*-algebras arising from dynamical systems. Bull. Soc. Math. France 125, 199–225 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics I + II, Texts and Monographs in Physics, Springer Verlag, New York, Heidelberg, Berlin (1979, 1981)

  3. Connes A.: Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup. 6, 133–252 (1973)

    MathSciNet  MATH  Google Scholar 

  4. Cuntz J.: Simple C*-algebras generated by isometries. Commun. Math. Phys. 57, 173–185 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Deaconu V.: Groupoids associated with endomorphisms. Trans. Am. Math. Soc. 347, 1779–1786 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Denker M., Urbanski M.: On the existence of conformal measures. Trans. Am. Math. Soc. 328, 563–587 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Exel R.: KMS states for generalized gauge actions on Cuntz-Krieger algebras (an application of the Ruelle-Perron-Frobenius theorem). Bull. Braz. Math. Soc. (N.S) 35, 1–12 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Haagerup U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type III 1. Acta Math. 158, 95–148 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hofbauer F.: Examples for the non-uniqueness of the equilibrium state. Trans. Am. Math. Soc. 228, 223–241 (1977)

    Article  MATH  Google Scholar 

  10. Kadison R.V., Ringrose J.R.: Fundamentals of the Theory of Operator Algebras II. Academic, London (1986)

    MATH  Google Scholar 

  11. Neshveyev S.: KMS states on the C*-algebras of non-principal groupoids. J. Oper. Theory 70, 513–530 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Olesen D., Pedersen G.K.: Some C*-dynamical systems with a single KMS-state. Math. Scand. 42, 111–118 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pedersen G.K.: C*-Algebras and their automorphism groups. Academic, London (1979)

    MATH  Google Scholar 

  14. Renault J.: A Groupoid Approach to C*-algebras, LNM 793. Springer Verlag, Berlin, Heidelberg, New York (1980)

    Book  Google Scholar 

  15. Renault, J.: Cuntz-like algebras, Operator theoretical methods (Timisoara, 1998), pp. 371–386, Theta Found., Bucharest (2000)

  16. Thomsen, K.: KMS-and conformal measures. Commun. Math. Phys. 316, 615–640 (2012) doi:10.1007/s00220-012-1591-z

  17. Thomsen K.: KMS weights on groupoid and graph C*-algebras. J. Funct. Anal. 266, 2959–2988 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Thomsen, K.: KMS weights on graph C*-algebras II. Factor types and ground states. arXiv:1412.6762

  19. Walters P.: A natural space of functions for the Ruelle operator theorem. Ergod. Theory Dyn. Syst. 27, 1323–1348 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Klaus Thomsen.

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Communicated by Y. Kawahigashi

I mindet om Uffe Haagerup

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Thomsen, K. Phase Transition in O 2 . Commun. Math. Phys. 349, 481–492 (2017). https://doi.org/10.1007/s00220-016-2742-4

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  • DOI: https://doi.org/10.1007/s00220-016-2742-4

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