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

Published:2008-12-01
Printed: Dec 2008
• Marius Ionescu
• Yasuo Watatani
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## 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.
 MSC Classifications: 46L35 - Classifications of $C^*$-algebras 46L08 - $C^*$-modules 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 37B10 - Symbolic dynamics [See also 37Cxx, 37Dxx]

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