Strongly graded groupoids and strongly graded Steinberg algebras

Abstract

We study strongly graded groupoids, which are topological groupoids G equipped with a continuous, surjective functor κ:G→Γ, to a discrete group Γ, such that κ−1(γ)κ−1(δ)=κ−1(γδ), for all γ,δ∈Γ. We introduce the category of graded G-sheaves, and prove an analogue of Dade's Theorem: G is strongly graded if and only if every graded G-sheaf is induced by a Gε-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.