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.
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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
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DOI: https://doi.org/10.1007/BF01321306