Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

GROUND STATES OF A COVARIANT SEMIGROUP C∗-ALGEBRA

  • Jang, Sun Young (Department of Mathematics, University of Ulsan) ;
  • Ahn, Jieun (Department of Mathematics, University of Ulsan)
  • Received : 2020.06.10
  • Accepted : 2020.06.16
  • Published : 2020.08.15
Download PDF
⟨ Previous Next ⟩

Abstract

Let P ⋊ ℕx be a semidirect product of an additive semigroup P = {0, 2, 3, ⋯ } by a multiplicative positive natural numbers semigroup ℕx. We consider a covariant semigroup C∗-algebra 𝓣(P ⋊ ℕx) of the semigroup P ⋊ ℕx. We obtain the condition that a state on 𝓣(P ⋊ ℕx) can be a ground state of the natural C∗-dynamical system (𝓣(P ⋊ ℕx), ℝ, σ).

Keywords

Acknowledgement

S. Y. Jang was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-201807042748).

References

  1. J. E. Ahn and S. Y. Jang, Structures and KMS states of a generalized Toeplitz Algebra, arXiv 2006.03197.
  2. J. B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (New Series), 1 (1995), 411-457. https://doi.org/10.1007/BF01589495
  3. O. Bratteli and D.W. Robinson, Operator algebras and Quantum Statistical Mechanics 1, Springer-Verlag, Berlin, 1979.
  4. O. Bratteli and D.W. Robinson, Operator algebras and Quantum Statistical Mechanics 2, second ed., Springer-Verlag, Berlin, 1997.
  5. L. A. Coburn, The C*-algebra generated by an isometry, I, Bull. Amer. Math. Soc., 73 (1967), 722-226. https://doi.org/10.1090/S0002-9904-1967-11845-7
  6. J. Cuntz, Simple C*-algebras generated by isometries, Comm. Math. Phys., 57(1977), 173-185. https://doi.org/10.1007/BF01625776
  7. J. Cuntz, K-Theory for certain C*-algebras, Ann of Math., 113 (1981), 181-197. https://doi.org/10.2307/1971137
  8. J. Cuntz, C*-algebra associated with the ax + b semigroup over ℕ, in K-Theory and Noncommutative Geometry Valladolid, 2006, European Math. Soc., 2008, 201- 215.
  9. J. Cuntz and X. Li, The regular C*-algebra of an integral domain, In Quanta of Maths, in Clay Math Proc., vol 11, American Mathematical Society, Providence, RI, 2010, 149-170.
  10. D. E. Evans, On On, Publ. Res. Inst. Math. Sci., 16 (1980), 915-927. https://doi.org/10.2977/prims/1195186936
  11. R. Exel and M. Laca, Partial dynamical systems and the KMS condition, Comm. Math. Phys., 232 (2003), 223-277. https://doi.org/10.1007/s00220-002-0713-4
  12. A. A. Huef and M. Laca, I, Reaburn, and A. Sims, KMS states on the C*-algebras of reducible graphs, Cambridge University Press, 2014.
  13. S. Y. Jang, Wiener-Hopf C*-algebras of strongly perforated semigroups, Bull. Kor. Math. Soc., 47 (2010), no. 6, 1275-1283. https://doi.org/10.4134/BKMS.2010.47.6.1275
  14. M. Laca, Semigroup of *-endomorphisms, Dirichlet series and phase transition, J. Func. Anal., 152 (1998), 330-378. https://doi.org/10.1006/jfan.1997.3166
  15. M. Laca and M. van Frankenhuijsen, Phase transition on Hecke C*-algebras and class field theory over ℚ, J. Reine Angew. Math., 595 (2006), 25-53.
  16. M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Funct. Anal., 139 (1996), 415-446. https://doi.org/10.1006/jfan.1996.0091
  17. M. Laca and I. Raeburn, Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers, Adv. Math., 225 (2010), 643-688. https://doi.org/10.1016/j.aim.2010.03.007
  18. G. J. Murpy, Ordered groups and crossed products of C*-algebras, Pacific J. Math., 148 (1991), 319-349. https://doi.org/10.2140/pjm.1991.148.319
  19. A. Nica, C*-algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory, 27 (1992), 17-52.
  20. X. Li, Semigroup C*-algebras and amenability of semigroups, J. Func. Anal., 262 (2012), 4302-4340. https://doi.org/10.1016/j.jfa.2012.02.020