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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, 46L08, 46L80, 37B10 show english descriptions Classifications of $C^*$-algebras
$C^*$-modules
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Symbolic dynamics [See also 37Cxx, 37Dxx]
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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