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

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

2004-10-22

arxiv.org/abs/math/0410480v2

Mathematics

Operator Algebras

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

Scientific paper

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.

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