KMS states on the $\mathrm{C}^*$-algebras of Fell bundles over {\'e}tale groupoids

Abstract

Let $p\colon \mathcal{A} \to G$ be a saturated Fell bundle over a locally compact, Hausdorff, second countable, {\'e}tale 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 established for certain states on $\mathrm{C}^*(G;\mathcal{A})$ also. 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 demonstrate our results through many examples such as groupoid crossed products, twisted groupoid crossed products, $G$-spaces and matrix algebras~$\mathrm{M}_n(\mathrm{C}(X))\otimes A$. While studying the matrix algebra~$\mathrm{M}_n(\mathrm{C}(X))$, we propose a groupoid model for it. While demonstrating our main result for this groupoid model, we provide a solution to the Radon--Nikodym problem for the groupoid used in this model.