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Journal of Operator Theory

Volume 75, Issue 1, Winter 2016  pp. 225-255.

Ideals of the core of $C^*$-algebras associated with self-similar maps

Authors:  Tsuyoshi Kajiwara (1) and Yasuo Watatani (2)
Author institution: (1) Department of Environmental and Mathematical Sciences, Okayama University, Tsushima, Okayama, 700-8530, Japan
(2) Department of Mathematical Sciences, Kyushu University, Motooka, Fukuoka, 819-0395, Japan

Summary: We give a complete classification of the ideals of the core of the $C^*$-algebras associated with self-similar maps under a certain condition. Any ideal is completely determined by the intersection with the coefficient algebra $C(K)$ of the self-similar set $K$. The corresponding closed subset of $K$ is described by the singularity structure of the self-similar map. In particular the core is simple if and only if the self-similar map has no branch point. A matrix representation of the core is essentially used to prove the classification.

Keywords: ideals, core, self-similar maps, $C^*$-correspondences

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