The Bunce-Deddens Algebras as Crossed Products by Partial Automorphisms

TL;DR

This paper characterizes Bunce-Deddens algebras and their Toeplitz versions as crossed products of commutative C*-algebras by partial automorphisms, linking their structure to odometer dynamics on the Cantor set.

Contribution

It provides a new description of Bunce-Deddens algebras as crossed products by partial automorphisms involving the odometer map and a space combining the Cantor set with natural numbers.

Findings

Bunce-Deddens algebras are isomorphic to crossed products by odometer maps.

The Toeplitz versions involve a spectrum combining the Cantor set and natural numbers.

The structure is described via partial automorphisms acting on a specific topological space.

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.