Limit $C^*$-algebras associated with an automorphism

Abstract

We present and study $C^*$-algebras generated by "periodic weighted creation operators" on the Fock space associated with an automorphism $\alpha$ on a $C^*$-algebra $A$. These algebras can be viewed as generalized Bunce-Deddens algebras associated with the automorphism and can be written as a certain direct limit. We prove a crossed product presentation for such an algebra and find a necessary and sufficient condition for it to be simple. In the case where the automorphism is induced by an irrational rotation (on C(T)) we compute the K-theory groups and obtain a complete classification of these algebras.