Strongly graded groupoids and strongly graded Steinberg algebras

Abstract

We study strongly graded groupoids, which are topological groupoids $\mathcal{G}$ equipped with a continuous, surjective functor $\kappa :\mathcal{G}\to \Gamma$, to a discrete group $\Gamma$, such that ${\kappa }^{-1}(\gamma ){\kappa }^{-1}(\delta )={\kappa }^{-1}(\gamma \delta )$, for all $\gamma ,\delta \in \Gamma$. We introduce the category of graded $\mathcal{G}$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal{G}$ is strongly graded if and only if every graded $\mathcal{G}$-sheaf is induced by a ${\mathcal{G}}_{\epsilon }$-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. Applying this result, we obtain a complete graphical characterisation of strongly graded Leavitt path and Kumjian-Pask algebras.