Exotic $C^*$-completions of Étale Groupoids

Abstract: We generalize the ideal completions of countable discrete groups, as introduced by Brown and Guentner, to second countable Hausdorff \'etale groupoids. Specifically, to every pair consisting of an algebraic ideal in the algebra of bounded Borel functions on the groupoid and a non-empty family of quasi-invariant measures on the unit space, we construct a $C^*$-algebra in a way which naturally encapsulates the constructions of the full and reduced groupoid $C^*$-algebras. We investigate the connection between these constructions and the Haagerup property, and use the construction to show the existence of many exotic groupoid $C^*$-algebras for certain classes of groupoids.