Graded Steinberg algebras and their representations

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff

groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital

left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this,

we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation

of the Cohen–Montgomery smash product of the Steinberg algebra of the underlying groupoid with the

grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in

the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator

ideals of these minimal representations, and effectiveness of the groupoid.

Specialising our results, we produce a representation of the monoid of graded finitely generated

projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the K0-group

of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate

the graded monoid for Kumjian–Pask algebras of row-finite k-graphs with no sources. We prove that

these algebras are graded von Neumann regular rings, and record some structural consequences of this.