A STEINBERG ALGEBRA APPROACH TO ÉTALE GROUPOID C^∗-ALGEBRAS

We construct the full and reduced C∗-algebras of an ample groupoid from its complex Steinberg algebra. We also show that our construction gives the same C∗-algebras as the standard constructions. In the last section, we consider an arbitrary locally compact, second-countable, étale groupoid, possibly non-Hausdorff. Using the techniques developed for Steinberg algebras, we show that every ∗-homomorphism from Connes’ space of functions to B(H) is automatically I-norm bounded. Previously, this was only known for Hausdorff groupoids.