Ideals of the core of \(C^\ast\)-algebras associated with self-similar maps (Q2835234)

A self-similar map on a compact metric space \(K\) is a family of proper contractions \(\gamma = (\gamma_1, \dots, \gamma_N)\) on \(K\) such that \(K=\displaystyle{\bigcup_{i=1}^N}\gamma_i(K)\). The authors of the present paper in [Complex Anal. Oper. Theory 8, No. 1, 243--254 (2014; Zbl 1302.46054)] introduced \(C^\ast\)-algebras associated with self-similar maps on compact metric spaces as Cuntz-Pimsner algebras and showed that the associated \(C^\ast\)-algebras are simple and purely infinite. Recall that the fixed point subalgebra of the gauge action of the \(C^\ast\)-algebras is called the core.NEWLINENEWLINEIn this paper, the authors use a matrix representation of the \(n\)-th core (that is, an isometric \(C^\ast\)-homomorphism from the \(n\)-th core \(\mathcal F^{(n)}\) to a matrix algebra over \(C(K)\)) and apply the Rieffel correspondence of ideals between Morita equivalent \(C^\ast\)-algebras to give a complete classification of the ideals of the core of the \(C^\ast\)-algebras associated with self-similar maps under a certain condition, by showing that any ideal \(I\) of the core is completely determined by the closed subset of the self-similar set which corresponds to the ideal \(C(K)\cap I\). They also show that the core is simple if and only if the self-similar map has no branch point, and describe all primitive ideals of the core \(\mathcal F^{(\infty)}\).