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The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers


The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup RR× over R and as a full corner of a crossed product C0() ⋊ KK*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of KK* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of KK* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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