Fej\'er property and Galois correspondence for groupoid $C^*$-algebras

TL;DR

This paper introduces the Fejér property for topological étale groupoids and establishes a Galois correspondence for intermediate $C^*$-algebras, linking algebraic structures to subgroupoids.

Contribution

It defines the Fejér property for groupoids and proves a Galois correspondence between intermediate $C^*$-algebras and open subgroupoids.

Findings

Fejér property characterized for principal étale groupoids

Closed bimodules correspond to open sets in the groupoid

Intermediate $C^*$-algebras correspond to open subgroupoids

Abstract

We introduce a notion of the Fej\'er property for topological \'etale groupoids. As a consequence, we show that when $\mathcal{G}$ is a principal \'etale second countable groupoid satisfying the Fej\'er property, every closed $C_0(\mathcal{G}^0)$-bimodule $M\subset C_r^*(\mathcal{G})$ is of the form $\overline{C_c(U)}^r$ for some open set $U$. Moreover, we get a Galois correspondence in the sense that every intermediate $C^*$-algebra $\mathcal{B}$ with $C_0(\mathcal{G}^0)\subseteq \mathcal{B}\subseteq C_r^*(\mathcal{G})$ is of the form $C_r^*(\mathcal{H})$ for some open subgroupoid $\mathcal{H}\leq \mathcal{G}$.