Cuntz-Nica-Pimsner algebras of product systems over groupoids

Abstract

Let $X$ be a product system over a quasi-lattice ordered groupoid $(G,P)$. Under mild hypotheses, we associate to $X$ a $C^*$-algebra which is couniversal for injective Nica covariant Toeplitz representations of $X$ which preserve the gauge coaction. When $(G,P)$ is a quasi-lattice ordered group this couniversal $C^*$-algebra coincides with the Cuntz-Nica-Pimsner algebra introduced by Carlsen-Larsen-Sims-Vittadello, and under some mild amenability conditions with that of Sims and Yeend. We prove related gauge invariant uniqueness theorems in this general setup.