College of Liberal Arts & Sciences

# Operator Theory Seminar

**Abstract: **

Given a not-necessarily Hausdorﬀ, topologically free, twisted ´etale groupoid $(\mathcal{G},\mathcal{L})$, we consider its essential groupoid $C^*$-algebra, denoted $C^{*}_{ess}(\mathcal{G},\mathcal{L})$, obtained by completing $\mathcal{C}_{c}(\mathcal{G},\mathcal{L})$ with the smallest among all $C^*$-seminorms coinciding with the uniform norm on $C_{c}(\mathcal{G}^{(0)})$. The inclusion of $C^*$-algebras "$C_{0}(\mathcal{G}^{(0)})\subseteq C^{*}_{ess}(\mathcal{G},\mathcal{L})$" is then proven to satisfy a list of properties characterizing it as what we call a *weak Cartan* inclusion. We then prove that every weak Cartan inclusion "$A\subseteq B$", with $B$ separable, is modeled by a twisted ´etale groupoid, as above. This talk is based on joint work with D. Pitts.