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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

SIEGFRIED ECHTERHOFF  and
MARCELO LACA
SIEGFRIED ECHTERHOFF
Affiliation:
Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany. e-mail: [email protected]
MARCELO LACA
Affiliation:
Department of Mathematics and Statistics, P.O.B. 3060, University of Victoria, Victoria, B.C. CanadaV8W 3R4. e-mail: [email protected]
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Abstract

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 R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, 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 K ⋊ K* 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 K ⋊ K* 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 1 , January 2013 , pp. 119 - 126
Copyright
Copyright © Cambridge Philosophical Society 2012

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