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

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

2009-06-26

arxiv.org/abs/0906.4825v2

Proc. London Math. Soc. (3) 103 (2011), no. 4, 563-600

Mathematics

Operator Algebras

40 pages, 2 figures; v2: minor changes to the introduction, references added and updated

Scientific paper

10.1112/plms/pdq028

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.

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