C^*-algebras associated with self-similar sets

Abstract

Let $\gamma =({\gamma }_1,...,{\gamma }_N)$, $N\ge 2$, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset $K$. We consider the union $\mathcal{G}={\cup }_{i=1}^N\{(x,y)\in K^2;x={\gamma }_i(y)\}$ of the cographs of γ_i$.Then$X = C({\mathcal G})$isaHilbertbimoduleover$A = C(K)$.Weassociatea$C^*$-algebra${\mathcal O}_γ(K)$withthemasaCuntz-Pimsneralgebra${\mathcal O}_X$.Weshowthatifasystemofpropercontractionssatisfiestheopensetconditionin$K$,thenthe$C^*$-algebra${\mathcal O}_γ(K)$ is simple and purely infinite, which is not isomorphic to a Cuntz algebra in general.

22 pages: Corrected Version

References in corpus (1)

• A construction of $C^{\ast }$-algebras from $C^{\ast }$-correspondences