Couniversality and controlled maps on product systems over right LCM semigroups

Kakariadis, E. T.A., Katsoulis, E. G., Laca, M. and Li, X. (2023) Couniversality and controlled maps on product systems over right LCM semigroups. Analysis and PDE, 16(6), pp. 1433-1483. (doi: 10.2140/apde.2023.16.1433)

Abstract

We study the structure of C∗-algebras associated with compactly aligned product systems over group embeddable right LCM semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine these with applications of the C∗-envelope theory for cosystems of nonselfadjoint operator algebras recently produced. We derive several applications of these methods that generalize results on single C∗-correspondences. First we show that if the controlling group is exact then the couniversal C∗-algebra of the product system coincides with the quotient of the Fock C∗-algebra by the ideal of strong covariance relations. We show that if the controlling group is amenable then the product system is amenable. In particular if the controlling group is abelian then the couniversal C∗-algebra is the C∗-envelope of the tensor algebra. Secondly we give necessary and sufficient conditions for the Fock C∗-algebra to be nuclear and exact. When the controlling group is amenable we completely characterize nuclearity and exactness of any equivariant injective Nica-covariant representation of the product system. Thirdly we consider controlled maps that enjoy a saturation property. In this case we induce a compactly aligned product system over the controlling pair that shares the same Fock representation, and preserves injectivity. By using couniversality, we show that they share the same reduced covariance algebras. If in addition the controlling pair is a total order then the fixed point algebra of the controlling group induces a super product system that has the same reduced covariance algebra and is moreover reversible.

• Item Type: Articles
• Additional Information: Kakariadis was partially supported by EPSRC grant EP/T02576X/1 and LMS grant 41908. Katsoulis was partially supported by NSF grant 2054781. Laca was partially supported by NSERC Discovery Grant #RGPIN-2017-04052. Li has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 817597).
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Li, Professor Xin
• Authors: Kakariadis, E. T.A., Katsoulis, E. G., Laca, M., and Li, X.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Analysis and PDE
• Publisher: Mathematical Sciences Publishers
• ISSN: 2157-5045
• ISSN (Online): 1948-206X
• Copyright Holders: Copyright © 2023 MSP (Mathematical Sciences Publishers)
• First Published: First published in Analysis and PDE 16(6):1433–1483
• Publisher Policy: Reproduced under a Creative Commons licence