  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.
