The Bunce-Deddens algebras as crossed products by partial automorphisms

  • Ruy Exel


We describe both the Bunce-DeddensC*-algebras and their Toeplitz versions, as crossed products of commutativeC*-algebras by partial automorphisms. In the latter case, the commutative algebra has, as its spectrum, the union of the Cantor set and a copy of the set of natural numbers ℕ, fitted together in such a way that ℕ is an open dense subset. The partial automorphism is induced by a map that acts like the odometer map on the Cantor set while being the translation by one on ℕ. From this we deduce, by taking quotients, that the Bunce-DeddensC*-algebras are isomorphic to the (classical) crossed product of the algebra of continuous functions on the Cantor set by the odometer map.


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  1. [1]
    R. J. Archbold, “An averaging process forC *-algebras related to weighted shifts”,Proc. London Math. Soc.,35: (1977), 541–554.Google Scholar
  2. [2]
    B. Blackadar, “K-theory for operator algebras”, MSRI Publications, Springer-Verlag, 1986.Google Scholar
  3. [3]
    B. Blackadar and A. Kumjian, “Skew products of relations and the structure of simpleC *-algebras”,Math. Z.,189: (1989), 55–63.Google Scholar
  4. [4]
    J. W. Bunce and J. A. Deddens, “C *-algebras generated by weighted shifts”,Indiana Univ. Math. J.,23: (1973), 257–271.Google Scholar
  5. [5]
    J. W. Bunce and J. A. Deddens, “A family of simpleC *-algebras related to weighted shift operators”,J. Funct. Analysis,19: (1975), 13–24.Google Scholar
  6. [6]
    E. E. Effros and J. Rosenberg, “C *-algebras with approximate inner flip”,Pacific J. Math.,77: (1978), 417–443.Google Scholar
  7. [7]
    R. Exel, “Circle actions onC *-algebras, partial automorphisms and a generalized Pimsner-Voiculescu exact sequence”,J. Funct. Analysis,122: (1994) 361–401.Google Scholar
  8. [8]
    R. Exel, “Approximately finiteC *-algebras and partial automorphisms”,Math. Scand, to appear.Google Scholar
  9. [9]
    P. Green, “The local structure of twisted covariance algebras”,Acta Math.,140: (1978), 191–250.Google Scholar
  10. [10]
    G. K. Pedersen, “C *-Algebras and their Automorphism Groups”, Academic Press, 1979.Google Scholar
  11. [11]
    S. C. Power, “Non-self-adjoint operator algebras and inverse systems of simplicial complexes”,J. Reine Angew. Math.,421: (1991), 43–61.Google Scholar
  12. [12]
    I. F. Putnam, “TheC *-algebras associated with minimal homeomorphisms of the Cantor set”,Pacific J. Math.,136: (1989), 329–353.Google Scholar
  13. [13]
    N. Riedel, “Classification of theC *-algebras associated with minimal rotations”,Pacific J. Math.,101: (1982), 153–161.Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 1994

Authors and Affiliations

  • Ruy Exel
    • 1
  1. 1.Departamento de MatemáticaUniversidade de São PauloSão PauloBrazil

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