Hilbert C*-bimodules and continuous Cuntz-Krieger algebras

Abstract

We consider certain correspondences on disjoint unions $\Omega$ of circles which naturally give Hilbert $C^{*}$-bimodules $X$ over circle algebras $A$. The bimodules $X$ generate $C^{*}$-algebras $Ox$ 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 $\left(II\right)\text{-}$ freeness previously discussed by the authors. More precisely, we have a bijective correspondence between the set of closed two sided ideals of ${\mathcal{O}}_X$ and saturated hereditary open subsets of $\Omega$. We also note that a formula of $K$-groups given by Deaconu is given without any minimality condition by just applying Pimsner's result.