Phase transitions with spontaneous symmetry breaking on Hecke C*-algebras from number fields

  • Marcelo Laca
  • Machiel van Frankenhuijsen
Part of the Aspects of Mathematics book series (ASMA)


When one attempts to generalize the results of Bost and Connes [BC] to algebraic number fields, one has to face sooner or later the fact that in a number field there is no unique factorization in terms of primes. As is well known, this failure is twofold: the ring of integers has nontrivial units, and even if one considers integers modulo units, (equivalently the principal integral ideals), it turns out that factorization in terms of these can fail too, essentially because ‘irreducible’ does not mean ‘prime’. The first difficulty, with the units, already arises in the situation of [BC], Remark 33.b], but is easily dealt with by considering elements fixed by a symmetry corresponding to complex conjugation. In the existing generalizations the lack of unique factorization has been dealt with in various ways. It has been eliminated, through replacing the integers by a principal ring that generates the same field [HLe], it has been sidestepped, by basing the construction of the dynamical system on the additive integral adeles with multiplication by (a section of) the integral ideles [Coh], and it has been ignored, by considering an almost normal subgroup that makes no reference to multiplication [ALR]. These simplifications make the construction and analysis of interesting dynamical systems possible, but they come at a price. Indeed, the noncanonical choices introduced in [HLe] and [Coh] lead to phase transitions with groups of symmetries that are not obviously isomorphic to actual Galois groups of maximal abelian extensions, and have slightly perturbed zeta functions in the case of [HLe], while the units not included in the almost normal subgroup in [ALR], reappear as a (possibly infinite) group of symmetries under which KMS states have to be invariant, which causes severe difficulties in their computation.


Normal Subgroup Galois Group Class Number Spontaneous Symmetry Breaking Double Coset 


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  1. [ALR]
    J. Arledge, M. Laca, I. Raeburn, Semigroup crossed products and Hecke algebras arising from number fields, Doc. Math. 2 (1997) 115–138.MATHMathSciNetGoogle Scholar
  2. [Bi]
    M.W. Binder, Induced factor representations of discrete groups and their types, J. Functional Analysis 115 (1993), 294–312.MATHCrossRefMathSciNetGoogle Scholar
  3. [BC]
    J.-B. Bost and A. Connes, Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory, Sel. Math. (New Series) 1 (1995), 411–457.CrossRefMathSciNetGoogle Scholar
  4. [Coh]
    P. B. Cohen, A C*-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking, Journées Arithmétiques de Limoges, 1997.Google Scholar
  5. [CM]
    A. Connes and M. Marcolli, Quantum statistical mechanics of Q-lattices, preprint (2004).Google Scholar
  6. [HLe]
    D. Harari and E. Leichtnam, Extension du phénomène de brisure spontanée de symétrie de Bost-Connes au cas de corps globaux quelconques, Sel. Math. (New Series), 3 (1997), 205–243.MATHCrossRefMathSciNetGoogle Scholar
  7. [K]
    A. Krieg, Hecke Algebras, Mem. Amer. Math. Soc. 87 (1990), No. 435.Google Scholar
  8. [L1]
    M. Laca, Semigroups of *-endomorphisms, Dirichlet series and phase transitions, J. Funct. Anal. 152 (1998), 330–378.MATHCrossRefMathSciNetGoogle Scholar
  9. [L2]
    M. Laca, From endomorphisms to automorphisms and back: dilations and full corners, J. London Math. Soc. 61 (2000), 893–904.MATHCrossRefMathSciNetGoogle Scholar
  10. [LLar1]
    M. Laca and N. S. Larsen, Hecke algebras of semidirect products, Proc. Amer. Math. Soc., 131 (2003), 2189–2199.MATHCrossRefMathSciNetGoogle Scholar
  11. [LR1]
    M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Functional Analysis, 139 (1996), 415–440.MATHCrossRefMathSciNetGoogle Scholar
  12. [LR2]
    M. Laca and I. Raeburn, A semigroup crossed product arising in number theory, J. London Math. Soc.,(2) 59 (1999), 330–344.CrossRefMathSciNetGoogle Scholar
  13. [LvF]
    M. Laca and M. van Frankenhuijsen, Phase transitions on Hecke C*-algebras and classfield theory over ℚ, preprint, 2004.Google Scholar
  14. [N]
    S. Neshveyev, Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem, Proc. Amer. Math. Soc. 130 (2002), 2999–3003.MATHCrossRefMathSciNetGoogle Scholar
  15. [Ni]
    A. Nica, C*-algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory 27 (1992), 17–52.MATHMathSciNetGoogle Scholar
  16. [Ta]
    J. T. Tate, Fourier Analysis in Number Fields and Hecke’s Zeta-Functions, Ph.D. Dissertation, Princeton University, Princeton, N. J., 1950. (Reprinted in: Algebraic Number Theory, J. W. S. Cassels and A. Fröhlich (eds.), Academic Press, New York, 1967, pp. 305–347.)Google Scholar
  17. [Tz]
    K. Tzanev, Hecke C*-algebras and amenability, J. Operator Theory 50 (2003), 169–178.MATHMathSciNetGoogle Scholar
  18. [We]
    A. Weil, Basic Number Theory, 3rd edition, Springer-Verlag, New York, 1974.MATHGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

Authors and Affiliations

  • Marcelo Laca
    • 1
  • Machiel van Frankenhuijsen
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of MathematicsUtah Valley State CollegeOremUSA

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