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The Bunce-Deddens Algebras as Crossed Products by Partial Automorphisms

Ruy Exel
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Abstract

We describe both the Bunce-Deddens C*-algebras and their Toeplitz versions, as crossed products of commutative C*-algebras by partial automorphisms. In the latter case, the commutative algebra has, as its spectrum, the union of the Cantor set and a copy 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-Deddens C*-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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Ruy Exel. 1993-02-09. The Bunce-Deddens Algebras as Crossed Products by Partial Automorphisms. https://arxiv.org/abs/funct-an/9302001

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