The Bunce-Deddens algebras as crossed products by partial automorphisms
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- Exel, R. Bol. Soc. Bras. Mat (1994) 25: 173. doi:10.1007/BF01321306
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.