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

Published online by Cambridge University Press:  01 October 2012

Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany. e-mail:
Department of Mathematics and Statistics, P.O.B. 3060, University of Victoria, Victoria, B.C. CanadaV8W 3R4. e-mail:


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.

Research Article
Copyright © Cambridge Philosophical Society 2012

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[1]Archbold, R. J. and Spielberg, J. S.Topologically free actions and ideals in discrete C*-dynamical systems. Proc. Edinburgh Math. Soc. 37 (1994), 119124.CrossRefGoogle Scholar
[2]Cuntz, J.C*-algebras associated with the ax + b-semigroup over N. Cortiñas, G., (ed.) et al., K-theory and noncommutative geometry. Proceedings of the ICM 2006 satellite conference (Valladolid, Spain, August 31-September 6, 2006). Zürich: European Mathematical Society (EMS). Series of Congress Reports (2008), 201215.Google Scholar
[3]Cuntz, J. and Li, X.The regular C*-algebra of an integral domain. Clay Mathematics Proceedings 10 (2010), 149170.Google Scholar
[4]Cuntz, J. and Li, X.C*-algebras associated with integral domains and crossed products by actions on adele spaces. J. Noncommut. Geom. 5 (2011), 137.CrossRefGoogle Scholar
[5]Cuntz, J., Deninger, C. and Laca, M. C*-algebras of Toeplitz type associated with algebraic number fields. Math. Ann. (2012), doi: 10.1007/s00208-012-0826-9.CrossRefGoogle Scholar
[6]Dixmier, J.C*-algebras. North-Holland Mathematical Library, Vol. 15 (North-Holland Publishing Co., 1977).Google Scholar
[7]Fell, J. M. G.The dual spaces of C*-algebras. Trans. Amer. Math. Soc. 94 (1960), 365403.Google Scholar
[8]Guentner, E., Higson, N. and Weinberger, S.The Novikov conjecture for linear groups. Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243268.CrossRefGoogle Scholar
[9]Gootman, E. C. and Rosenberg, J.The structure of crossed product C*-algebras: a proof of the generalized Effros–Hahn conjecture. Invent. Math. 52 (1979), no. 3, 283298.CrossRefGoogle Scholar
[10]Green, P.The local structure of twisted covariance algebras. Acta Math. 140 (1978), no. 3-4, 191250.CrossRefGoogle Scholar
[11]Laca, M. and Raeburn, I.The ideal structure of the Hecke C*-algebra of Bost and Connes. Math. Ann. 318 (2000), 433451.CrossRefGoogle Scholar
[12]Laca, M. and Raeburn, I.Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers. Adv. Math. 225 (2010), 643688.CrossRefGoogle Scholar
[13]Renault, J.The ideal structure of groupoid crossed product C*-algebras. With an appendix by Georges Skandalis. J. Operator Theory 25 (1991), no. 1, 336.Google Scholar
[14]Sierakowski, A.The ideal structure of reduced crossed products. Münster J. of Math. 3 (2010), 237262.Google Scholar