C*-Algebras Associated with Mauldin–Williams Graphs

Ionescu, Marius ; Watatani, Yasuo

Canadian mathematical bulletin, Tome 51 (2008) no. 4, pp. 545-560

Résumé

A Mauldin–Williams graph $M$ 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 ${{O}_{M}}\left( K \right)$ with a Mauldin–Williams graph $M$ 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 $M$ satisfies the open set condition in $K$ , and $G$ is irreducible and is not a cyclic permutation, then the associated ${{C}^{*}}$ -algebra ${{O}_{M}}\left( K \right)$ is simple and purely infinite. We calculate the $K$ -groups for some examples including the inflation rule of the Penrose tilings.

Détail

DOI : 10.4153/CMB-2008-054-0

Mots-clés : 46L35, 46L08, 46L80, 37B10

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