$C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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.

14 pages, Latex, 4 figures. We added new references and did minor changes in the introduction