Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems

TL;DR

This paper constructs a co-universal C*-algebra for product systems over quasi-lattice groups, establishing key uniqueness theorems and linking to Cuntz-Nica-Pimsner algebras and crossed products.

Contribution

It introduces a co-universal C*-algebra for product systems that generalizes existing constructions and proves new gauge-invariant uniqueness theorems.

Findings

The co-universal algebra coincides with the Cuntz-Nica-Pimsner algebra under amenability.

Established gauge-invariant uniqueness theorems for these algebras.

Connected the co-universal algebra framework to reduced crossed products.

Abstract

Let X be a product system over a quasi-lattice ordered group. Under mild hypotheses, we associate to X a C*-algebra which is co-universal for injective Nica covariant Toeplitz representations of X which preserve the gauge coaction. Under appropriate amenability criteria, this co-universal C*-algebra coincides with the Cuntz-Nica-Pimsner algebra introduced by Sims and Yeend. We prove two key uniqueness theorems, and indicate how to use our theorems to realise a number of reduced crossed products as instances of our co-universal algebras. In each case, it is an easy corollary that the Cuntz-Nica-Pimsner algebra is isomorphic to the corresponding full crossed product.