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In C *-algebras, a Bunce–Deddens algebra is a certain type of direct limit of matrix algebras over the continuous functions on the circle. They are therefore examples of simple unital AT algebras. In the inductive system defining these algebras, the connecting maps between each stage are given by embeddings between families of shift operators with periodic weights. Each inductive system defining a Bunce–Deddens algebra is associated with a supernatural number, which is a complete invariant for these algebras. In the language of K-theory, the supernatural number correspond to the ''K''0 group of the algebra. Also, Bunce–Deddens algebras can be expressed as the C *-crossed product of the Cantor set with a certain natural minimal action, so-called ''odometer action''. They also admit a unique tracial state. Together with the fact that they are AT, this implies they have real rank zero. In a broader context of the classification program for simple separable nuclear C *-algebras, AT-algebras of real rank zero were shown to be completely classified by their K-theory, the Choquet simplex of tracial states, and the natural pairing between ''K''0 and traces. The classification of Bunce–Deddens algebras is thus a precursor to the general result. It is also known that, in general, crossed products arising from minimal homeomorphism on the Cantor set are simple AT-algebras of real rank zero. == Definition and basic properties == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bunce–Deddens algebra」の詳細全文を読む スポンサード リンク
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