The ideal structure of Exel-Pardo algebras and their higher rank analogues

Abstract

Given a pseudo-free self-similar action of a countable group $G$ on a countable directed graph $E$ with amenable stabilizers of the vertices, we identify the exact conditions under which these stabilizers do not contribute to the ideal structure of the corresponding Exel-Pardo algebra $\mathcal{O}_{G,E}$. Under these conditions, we give a complete description of the primitive ideal space of $\mathcal{O}_{G,E}$ in graph-theoretic terms. Our results apply in particular to certain crossed products $\mathcal{O}_E\rtimes G$, where $G$ acts on $E$ by graph automorphisms. When $G$ is trivial, this recovers Hong-Szymanski's description of the ideal structure of the Cuntz-Krieger algebras $\mathcal{O}_E$. Similar results are then obtained for self-similar actions of groups on row-finite higher rank graphs without sources. In order to obtain these results we formalize the notion of a graded groupoid with essentially central isotropy, which generalizes essentially principal groupoids and groupoids injectively graded by abelian groups. Under the amenability and second countability assumptions, we describe the primitive ideal spaces of the corresponding C$^*$-algebras as topological spaces.