Cuntz-Nica-Pimsner algebras of product systems over groupoids
Massoud Amini, Mahdi Moosazadeh
TL;DR
This work extends the Cuntz-Nica-Pimsner framework to product systems indexed by subsemigroupoids of quasi-lattice ordered groupoids, constructing a couniversal $C^*$-algebra for gauge-compatible injective Nica covariant representations. Using a gauge coaction and core analysis, it defines both the reduced and full CN-P algebras $ ext{NO}_X^r$ and $ ext{NO}_X$, and proves a gauge-invariant uniqueness theorem under mild hypotheses, including amenability-type conditions for the associated Fell bundles. The paper connects to established theories by recovering Carlsen–Larsen–Sims–Vittadello’s CN-P algebra and Sims–Yeend’s results in amenable/group settings, and extends to étale groupoids, groupoid crossed products, and multilayered topological $k$-graphs, offering a robust toolkit for analyzing $C^*$-algebras from groupoid-indexed product systems. These results deepen the understanding of how groupoid structure, Fell bundles, and gauge actions govern the structure and invariants of CN-P algebras, with potential applications to higher-rank graph algebras and dynamical systems.
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.