Hilbert C^*-bimodules and continuous Cuntz-Krieger algebras

Abstract

We consider certain correspondences on disjoint unions Ω of circles which naturally give Hilbert C^*-bimodules X over circle algebras A. The bimodules X generate C^*-algebras \mathcal{O}_X which are isomorphic to a continuous version of Cuntz-Krieger algebras introduced by Deaconu using groupoid method. We study the simplicity and the ideal structure of the algebras under some conditions using (I)-freeness and (II)-freeness previously discussed by the authors. More precisely, we have a bijective correspondence between the set of closed two sided ideals of \mathcal{O}_{\bm{X}} and saturated hereditary open subsets of Ω. We also note that a formula of K-groups given by Deaconu is given without any minimality condition by just applying Pimsner's result.