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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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[7] J. M. G. Fell The dual spaces of C*-algebras. Trans. Amer. Math. Soc. 94 (1960), 365403.

[8] E. Guentner , N. Higson and S. Weinberger The Novikov conjecture for linear groups. Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243268.

[9] E. C. Gootman and J. Rosenberg The structure of crossed product C*-algebras: a proof of the generalized Effros–Hahn conjecture. Invent. Math. 52 (1979), no. 3, 283298.

[10] P. Green The local structure of twisted covariance algebras. Acta Math. 140 (1978), no. 3-4, 191250.

[11] M. Laca and I. Raeburn The ideal structure of the Hecke C*-algebra of Bost and Connes. Math. Ann. 318 (2000), 433451.

[12] M. Laca and I. Raeburn Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers. Adv. Math. 225 (2010), 643688.

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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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