Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids

Abstract: Given a not-necessarily Hausdorff, topologically free, twisted \'etale groupoid $(G, L)$, we consider its "essential groupoid C*-algebra", denoted $C^*_{ess}(G, L)$, obtained by completing $C_c(G, L)$ with the smallest among all C*-seminorms coinciding with the uniform norm on $C_c(G^0)$. The inclusion of C*-algebras $(C_0(G^0), C^*_{ess}(G, L))$ 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 $(A, B)$, with $B$ separable, is modeled by a topologically free, twisted \'etale groupoid, as above. In our second main result we give a necessary and sufficient condition for an inclusion of C*-algebras $(A, B)$ to be modeled by a twisted \'etale groupoid based on the notion of "canonical states". A simplicity criterion for $C^*_{ess}(G, L)$ is proven and many examples are provided.