GROUPOID ALGEBRAS AS COVARIANCE ALGEBRAS

Suppose G is a second-countable locally compact Hausdorff etale 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.