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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.

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