Isotropy fibers of ideals in groupoid C$^{*}$-algebras

TL;DR

This paper investigates the structure of ideals in groupoid C*-algebras, showing how they relate to isotropy groups and providing classifications and descriptions of primitive and maximal ideals.

Contribution

It introduces a method to analyze ideals via isotropy group C*-algebras and characterizes primitive and maximal ideals in this context.

Findings

Every proper ideal is contained in an induced primitive ideal

Maximal ideals are explicitly described

Primitive ideals are classified for certain graded groupoids

Abstract

Given a locally compact \'etale groupoid and an ideal $I$ in its groupoid C$^*$-algebra, we show that $I$ defines a family of ideals in group C$^*$-algebras of the isotropy groups and then study to which extent $I$ is determined by this family. As an application we obtain the following results: (a) prove that every proper ideal is contained in an induced primitive ideal; (b) describe the maximal ideals; (c) classify the primitive ideals for a class of graded groupoids with essentially central isotropy.