Journal of Operator Theory

Volume 56, Issue 2, Fall 2006 pp. 225-247.

$C^*$-algebras associated with self-similar sets

Authors: Tsuyoshi Kajiwara (1) and Yasuo Watatani (2)
Author institution: (1) Department of Environmental and Mathematical Sciences, Okayama University, Tsushima, 700-8530, Japan
(2) Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan

Summary: Let $\gamma = (\gamma_1,\dots,\gamma_N)$, $N \geqslant 2$, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset $K$. We consider the union ${\mathcal G} = \bigcup\limits_{i=1}^N \{(x,y) \in K^2 ; x = \gamma _i(y)\}$ of the cographs of $\gamma _i$. Then $X = C({\mathcal G})$ is a Hilbert bimodule over $A = C(K)$. We associate a $C^*$-algebra ${\mathcal O}_{\gamma}(K)$ with them as a Cuntz-Pimsner algebra ${\mathcal O}_X$. We show that if a system of proper contractions satisfies the open set condition in $K$, then the $C^*$-algebra ${\mathcal O}_{\gamma}(K)$ is simple, purely infinite and, in general, not isomorphic to a Cuntz algebra.

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