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

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Proceedings of the London Mathematical Society, ISSN: 0024-6115, Vol: 103, Issue: 4, Page: 563-600

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Carlsen, Toke Meier; Larsen, Nadia S.; Sims, Aidan; Vittadello, Sean
Oxford University Press (OUP)
Mathematics; Physical Sciences and Mathematics; Mathematics - Operator Algebras; 46L05
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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. © 2011 London Mathematical Society.