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Let ( G ,  P ) be a quasi‐lattice ordered group, and let  X  be a product system over  P  of Hilbert bimodules. 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 realize 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.

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