The recent successes in the characterizations of C$^\ast$-simplicity of discrete groups and their crossed products by compact spaces have largely been enabled by a new description of an old tool: The Furstenberg boundary of the group, recast as Hamana's equivariant injective envelope of the action in question. By providing a new method of induction for the action of a (locally compact Hausdorff) étale groupoid with compact unit space, we construct groupoid-equivariant injective envelopes in the appropriate category. This yields a notion of Furstenberg boundary of such groupoids, with which we prove a new sufficient criterion for groupoid C$^\ast$-simplicity. As an application, we discuss simplicity of the Elek algebras.

This work is part of the speaker's PhD thesis under the supervision of Mikael Rørdam and Magdalena Musat and supported by a grant from the Danish Council for Independent Research,