In this talk, we explore étale groupoids $G$ with a locally compact Hausdorff unit space $X$, where $G$ itself may not be globally Hausdorff. For such groupoids, the essential $C^*$-algebra $C_{\textrm{ess}}^*(G)$ offers a more suitable framework than the reduced $C^*$-algebra $C_r^*(G)$, as it captures additional structural nuances. Specifically, $C_{\textrm{ess}}^*(G)$ arises as a proper quotient of $C_r^*(G)$.

We introduce the concept of essential amenability for groupoids, a condition that is strictly weaker than (topological) amenability yet sufficient to guarantee the nuclearity of $C_{\textrm{ess}}^*(G)$. To establish this, we define a maximal version of the essential $C^*$-algebra and show that any function with dense cosupport must be supported within the set of "dangerous arrows”, that is, arrows that cannot be topologically separated.

This essential amenability framework characterizes the nuclearity of $C_{\textrm{ess}}^*(G)$ and establishes its isomorphism to the maximal essential $C^*$-algebra. Our results offer new insights into the interplay between groupoid structure and operator algebras, extending the utility of $C_{\textrm{ess}}^*(G)$ in non-Hausdorff settings. This is based on joint work with Diego Martinez.