$R$-Bunce-Deddens Algebras

Abstract

We construct an $R$-algebraic analog of the Bunce-Deddens algebra using the theory of Leavitt labelled path algebras. In particular, we will construct a labelled space whose associated partial action on tight filters is exactly the odometer action on the Cantor set that induces the classical Bunce-Deddens algebra. After defining $R$-Bunce-Deddens algebras, we will prove $R$-algebraic analogs of various results for classical Bunce-Deddens algebras. As our primary application of these results, we will show that, for any field $K$, a $K$-Bunce-Deddens algebra is not Morita equivalent to any Leavitt path algebra.