Fejér property and Galois correspondence for groupoid $C^*$-algebras

TL;DR

This work introduces the Fejér property for topological étale groupoids and develops a Fourier-analytic framework for reduced groupoid C*-algebras. It proves that weak amenability implies a bounded Fejér property and, crucially, that Fejér property enforces inner exactness. For principal étale groupoids with Fejér property, it establishes a Galois correspondence: every intermediate C*-algebra between $C_0(\mathcal{G}^{(0)})$ and $C_r^*(\mathcal{G})$ is of the form $C_r^*(\mathcal{H})$ for an open subgroupoid $\mathcal{H}$, and closed $C_0(\mathcal{G}^{(0)})$-bimodules are generated by open sets via $\overline{C_c(U)}^r$. These results extend known group-case correspondences to groupoids and connect bimodule spectral theory with open-subgroupoid structures, enriching the structural understanding of étale groupoid C*-algebras and their intermediate subalgebras.

Abstract

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 $M\subset C_r^*(\mathcal{G})$ 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 $\mathcal{H}\leq \mathcal{G}$.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Example 2.1: Transformation Groupoids
• Definition 3.1
• Proposition 3.2
• proof
• Example 3.3
• Theorem 3.4