Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. J. Archbold, “An averaging process forC *-algebras related to weighted shifts”,Proc. London Math. Soc.,35: (1977), 541–554.
B. Blackadar, “K-theory for operator algebras”, MSRI Publications, Springer-Verlag, 1986.
B. Blackadar and A. Kumjian, “Skew products of relations and the structure of simpleC *-algebras”,Math. Z.,189: (1989), 55–63.
J. W. Bunce and J. A. Deddens, “C *-algebras generated by weighted shifts”,Indiana Univ. Math. J.,23: (1973), 257–271.
J. W. Bunce and J. A. Deddens, “A family of simpleC *-algebras related to weighted shift operators”,J. Funct. Analysis,19: (1975), 13–24.
E. E. Effros and J. Rosenberg, “C *-algebras with approximate inner flip”,Pacific J. Math.,77: (1978), 417–443.
R. Exel, “Circle actions onC *-algebras, partial automorphisms and a generalized Pimsner-Voiculescu exact sequence”,J. Funct. Analysis,122: (1994) 361–401.
R. Exel, “Approximately finiteC *-algebras and partial automorphisms”,Math. Scand, to appear.
P. Green, “The local structure of twisted covariance algebras”,Acta Math.,140: (1978), 191–250.
G. K. Pedersen, “C *-Algebras and their Automorphism Groups”, Academic Press, 1979.
S. C. Power, “Non-self-adjoint operator algebras and inverse systems of simplicial complexes”,J. Reine Angew. Math.,421: (1991), 43–61.
I. F. Putnam, “TheC *-algebras associated with minimal homeomorphisms of the Cantor set”,Pacific J. Math.,136: (1989), 329–353.
N. Riedel, “Classification of theC *-algebras associated with minimal rotations”,Pacific J. Math.,101: (1982), 153–161.
Author information
Authors and Affiliations
About this article
Cite this article
Exel, R. The Bunce-Deddens algebras as crossed products by partial automorphisms. Bol. Soc. Bras. Mat 25, 173–179 (1994). https://doi.org/10.1007/BF01321306
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01321306