KMS states on the ${\mathrm{C}}^{\ast }$-algebras of Fell bundles over {é}tale groupoids

Abstract

Let $p:\mathcal{A}\to G$ be a saturated Fell bundle over a locally compact, Hausdorff, second countable, {é}tale groupoid~$G$, and let ${\mathrm{C}}^{\ast }(G;\mathcal{A})$ denote its full ${\mathrm{C}}^{\ast }$-algebra. We prove an integration-disintegration theorem for KMS states on ${\mathrm{C}}^{\ast }(G;\mathcal{A})$ by establishing a one-to-one correspondence between such states and fields of measurable states on the ${\mathrm{C}}^{\ast }$-algebras of the Fell bundles over the isotropy groups. This correspondence is established for certain states on ${\mathrm{C}}^{\ast }(G;\mathcal{A})$ also. While proving this main result, we construct an induction ${\mathrm{C}}^{\ast }$-correspondence between~${\mathrm{C}}^{\ast }(G;\mathcal{A})$ and the ${\mathrm{C}}^{\ast }$-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.

42 pages