Pullbacks of Groupoid $C^*$-Algebras over Disjoint Invariant Sets

We establish a pullback theorem for \(C^*\)-algebras of locally compact Hausdorff étale groupoids. The theorem shows that, when the unit space is covered by two closed invariant subsets whose complements are disjoint open invariant subsets, the corresponding groupoid \(C^*\)-algebras form a pullback