KMS states on the $\mathrm C^*$-algebras of Fell bundles over étale groupoids

Let $p\colon \mathcal A \to G$ be a saturated Fell bundle over a locally compact, Hausdorff, second countable, étale groupoid $G$, and let $\mathrm {C}^*(G;\mathcal {A})$ denote its full $\mathrm {C}^*$-algebra. We prove an integration-disintegration theorem for KMS states on $\mathrm {C}^*(G;\mathcal {A})$ by establishing a one-to-one correspondence between such states and fields of measurable states on the $\mathrm {C}^*$-algebras of the Fell bundles over the isotropy groups. This correspondence is also established for certain states on $\mathrm {C}^*(G;\mathcal {A})$. While proving this main result, we construct an induction $\mathrm {C}^*$-correspondence between $\mathrm {C}^*(G;\mathcal {A})$ and the $\mathrm {C}^*$-algebra of an isotropy Fell bundle. We illustrate our results through many examples, such as groupoid crossed products, twisted groupoid crossed products and matrix algebras $\mathrm {M}_n(\mathrm {C}(X))\otimes A$.