Simplicity of algebras associated to étale groupoids

We prove that the full C*-algebra of a second-countable, Hausdorff, etale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G has totally disconnected unit space, then the complex *-algebra of its inverse semigroup of compact open bisections, as introduced by Steinberg, is simple if and only if G is both effective and minimal.