Groupoid fell bundles for product systems over quasi-lattice ordered groups

Consider a product system over the positive cone of a quasi-lattice ordered group. We construct a Fell bundle over an associated groupoid so that the cross-sectional algebra of the bundle is isomorphic to the Nica–Toeplitz algebra of the product system. Under the additional hypothesis that the left actions in the product system are implemented by injective homomorphisms, we show that the cross-sectional algebra of the restriction of the bundle to a natural boundary subgroupoid coincides with the Cuntz–Nica–Pimsner algebra of the product system. We apply these results to improve on existing sufficient conditions for nuclearity of the Nica–Toeplitz algebra and the Cuntz–Nica–Pimsner algebra, and for the Cuntz–Nica–Pimsner algebra to coincide with its co-universal quotient.