The Mackey Machine for Crossed Products by Regular Groupoids. II

We prove that given a regular groupoid $G$ whose isotropy subgroupoid $S$ has a Haar system, along with a dynamical system $(A,G,\alpha )$, there is an action of $G$ on the spectrum of $A⋊S$ such that the spectrum of $A⋊G$ is homeomorphic to the orbit space of this action via induction. In addition, we give a strengthening of these results in the case where the crossed product is a groupoid algebra.