Groupoid algebras as covariance algebras

Suppose G is a second-countable locally compact Hausdorff \'{e}tale groupoid, G is a discrete group containing a unital subsemigroup P, and c:G→G is a continuous cocycle. We derive conditions on the cocycle such that the reduced groupoid C∗-algebra C∗r(G) may be realised as the covariance algebra of a product system over P with coefficient algebra C∗r(c−1(e)). When (G,P) is a quasi-lattice ordered group, we also derive conditions that allow C∗r(G) to be realised as the Cuntz--Nica--Pimsner algebra of a compactly aligned product system.