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.
DOI: http://dx.doi.org/10.7900/jot.2015feb23.2069
Keywords: ideals, core, self-similar maps, $C^*$-correspondences
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