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

Published online by Cambridge University Press:  16 March 2017

ADAM RENNIE ,
DAVID ROBERTSON  and
AIDAN SIMS
ADAM RENNIE
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia. e-mail: [email protected]
DAVID ROBERTSON
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, 2308, Australia. e-mail: [email protected]
AIDAN SIMS
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia. e-mail: [email protected]
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 561 - 580
Copyright
Copyright © Cambridge Philosophical Society 2017 

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