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Limit $$C^*$$-algebras associated with an automorphism. (English) Zbl 1067.46059
Summary: 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.

##### MSC:
 46L40 Automorphisms of selfadjoint operator algebras 46L55 Noncommutative dynamical systems 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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