KMS states on the -algebras of Fell bundles over étale groupoids
Volume 279 / 2024
Abstract
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.
