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Orbit Representations and Circle Maps

  • Carlos Correia Ramos
  • Nuno Martins
  • Paulo R. Pinto
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 181)

Abstract

We yield C*-algebras representations on the orbit spaces from the family of interval maps f(x) = βx+α (mod 1) lifted to circle maps, in which case β ∈ N.

Each orbit will encode an unitary equivalence class of an irreducible representation of: a Cuntz algebra O β if = 0 and β > 1; an irrational rotation algebra A β if α ∉ ℚ and β = 1; and a Cuntz-Krieger O Aα,β whenever β > 1 and the critical point is periodic, where A α,β is the underlying Markov transition matrix of f.

Keywords

Interval maps irrational rotation algebra Cuntz-Krieger algebra irreducible representations 

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References

  1. [1]
    M. Abe, K. Kawamura, Recursive fermion systems in Cuntz algebra, I. Commun. Math. Phys. 228 (2003), 85–101.CrossRefMathSciNetGoogle Scholar
  2. [2]
    G. Baumslag, D. Solitar, Some two-generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc. 68 (1962), 199–201.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [3]
    O. Bratteli, P.E.T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra. Mem. Amer. Math. Soc. 663 (1999), 1–89.MathSciNetGoogle Scholar
  4. [4]
    O. Bratteli, P.E.T. Jorgensen, Representation theory and numerical AF-invariants. The representations and centralizers of certain states on O d. Mem. Amer. Math. Soc. 168, no. 797, xviii+178 pp., 2004.Google Scholar
  5. [5]
    B.A. Brenken, Representations and automorphisms of the irrational rotation algebra. Pacific J. Math. 111 (1984), 257–282.MathSciNetzbMATHGoogle Scholar
  6. [6]
    C. Correia Ramos, N. Martins, P.R. Pinto, J. Sousa Ramos, Orbit equivalence and von Neumann algebras for piecewise linear unimodal maps. Grazer Math. Ber. 350 (2006), 45–54.MathSciNetzbMATHGoogle Scholar
  7. [7]
    C. Correia Ramos, N. Martins, P.R. Pinto, J. Sousa Ramos, Orbit equivalence and von Neumann algebras for expansive interval maps. Chaos, Solitons and Fractals 33 (2007) 109–117.CrossRefMathSciNetGoogle Scholar
  8. [8]
    C. Correia Ramos, N. Martins, P.R. Pinto, J. Sousa Ramos, Cuntz-Krieger algebras representations from orbits of interval maps. Preprint.Google Scholar
  9. [9]
    J. Cuntz, W. Krieger, A class of C*-algebras and topological Markov chains Inv. Math. 56 (1980), 251–268.CrossRefMathSciNetzbMATHGoogle Scholar
  10. [10]
    K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. London Math. Soc. 78 (1999), 401–430.CrossRefMathSciNetGoogle Scholar
  11. [11]
    J. Feldman, C.E. Sutherland, R.J. Zimmer, Subrelations of ergodic equivalence relations. Ergodic Th. & Dynam. Sys. 9 (1989), 239–269.MathSciNetzbMATHGoogle Scholar
  12. [12]
    R. Høegh, T. Skjelbred, Classification of C*-algebras admitting ergodic actions of the two-dimensional torus. J. Reine Angew. Math. 328 (1981), 1–8.MathSciNetGoogle Scholar
  13. [13]
    K. Kawamura, Representation of the Cuntz algebra O 2 arising from real quadratic transformations. Preprint RIMS-1396, 2003.Google Scholar
  14. [14]
    N. Martins, J. Sousa Ramos, Cuntz-Krieger algebras arising from linear mod one transformations. Fields Inst. Commun. 31 (2002), 265–273.MathSciNetGoogle Scholar
  15. [15]
    N.E. Wegge-Olsen: K-Theory and C*-Algebras, Oxford University Press, 1993.Google Scholar
  16. [16]
    V. Ostrovskyi, Yu. Samoilenko, Introduction to the theory of representations of finitely presented *-algebras, I. Representations by bounded operators. Rev. in Math. and Math. Phys., 11, Harwood Academic Publishers, Amsterdam, iv+261 pp., 1999.Google Scholar
  17. [17]
    G.K. Pedersen, C*-Algebras and their Automorphism Groups. Academic Press, London Mathematical Society Monographs, 14, ix+416, 1979.Google Scholar
  18. [18]
    M. Pimsner, D. Voiculescu, Imbedding the irrational rotation C*-algebra into an AF-algebra. J. Operator Theory 4 (1980), 201–210.MathSciNetzbMATHGoogle Scholar
  19. [19]
    M.A. Rieffel, C*-algebras associated with irrational rotations. Pacific J. Math. 93 (1981), 415–429.MathSciNetzbMATHGoogle Scholar
  20. [20]
    M. Takesaki, Theory of Operator Algebras. III. Encyclopaedia of Mathematical Sciences, 127. Operator Algebras and Non-commutative Geometry, 8. Springer-Verlag, Berlin, xxii+548 pp., 2003.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Carlos Correia Ramos
    • 1
  • Nuno Martins
    • 2
  • Paulo R. Pinto
    • 2
  1. 1.Departamento de MatemáticaUniversidade de ÉvoraÉvoraPortugal
  2. 2.Centro de Análise Matemática, Geometria e Sistemas Dinâmicos Departamento de MatemáticaInstituto Superior TécnicoLisboaPortugal

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