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

TL;DR

This paper constructs a Fell bundle over a groupoid associated with a product system over a quasi-lattice ordered group, linking its cross-sectional algebra to key operator algebra structures and improving conditions for nuclearity and algebraic equivalences.

Contribution

It introduces a new Fell bundle construction for product systems over quasi-lattice groups, connecting to Nica-Toeplitz and Cuntz-Nica-Pimsner algebras, and refines nuclearity criteria.

Findings

Cross-sectional algebra is isomorphic to the Nica-Toeplitz algebra.

Under injective homomorphisms, the boundary restriction yields the Cuntz-Nica-Pimsner algebra.

Improved sufficient conditions for nuclearity and algebraic coincidences.

Abstract

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.