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C*-Algebras Associated with Mauldin–Williams Graphs

Published online by Cambridge University Press:  20 November 2018

Marius Ionescu  and
Yasuo Watatani
Marius Ionescu
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH, U.S.A.andDepartment of Mathematics, Cornell University, Ithica, NY, U.S.A.
Yasuo Watatani
Affiliation:
Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan
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Abstract

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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.

Keywords

46L3546L0846L8037B10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 4 , 01 December 2008 , pp. 545 - 560
Copyright
Copyright © Canadian Mathematical Society 2008

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