$C^{\ast}$-Algebras associated with Mauldin-Williams Graphs

Abstract

A Mauldin-Williams graph $\mathcal{M}$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ with a Mauldin-Williams graph $\mathcal{M}$ and the invariant set $K$ laying emphasis on the singular points. We assume that the underlying graph $G$ has no sinks and no sources. If $\mathcal{M}$ satisfies the open set condition in $K$ and $G$ is irreducible and is not a cyclic permutation, then the associated $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ is simple and purely infinite. We calculate the K-groups for some examples including the inflation rule of the Penrose tilings.