Mathematical Proceedings of the Cambridge Philosophical Society

Table of Contents - January 2013 - Volume 154, Issue 01  

Research Article

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

SIEGFRIED ECHTERHOFFa1 and MARCELO LACAa2

a1 Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany. e-mail: echters@uni-muenster.de

a2 Department of Mathematics and Statistics, P.O.B. 3060, University of Victoria, Victoria, B.C. Canada V8W 3R4. e-mail: laca@uvic.ca

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 C 0() ⋊ 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.

(Received January 31 2012)

(Revised June 25 2012)

(Online publication October 01 2012)