Fejér property and Galois correspondence for groupoid $C^{\ast }$-algebras

We introduce a notion of the Fejér property for topological étale groupoids. As a consequence, we show that when $\mathcal{G}$ is a principal étale second countable groupoid satisfying the Fejér property, every closed $C_0({\mathcal{G}}^0)$-bimodule $M\subset C_r^{\ast }(\mathcal{G})$ 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 $C^{\ast }$-algebra $\mathcal{B}$ with $C_0({\mathcal{G}}^0)\subseteq \mathcal{B}\subseteq C_r^{\ast }(\mathcal{G})$ is of the form $C_r^{\ast }(\mathcal{H})$ for some open subgroupoid $\mathcal{H}\le \mathcal{G}$.