C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

Dor-On, A., Kakariadis, E.T.A., Katsoulis, E., Laca, M. and Li, X. (2022) C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems. Advances in Mathematics, 400, 108286. (doi: 10.1016/j.aim.2022.108286)

Abstract

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify with the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.

• Item Type: Articles
• Additional Information: Adam Dor-On was supported by the NSF grant DMS-1900916 and by the European Union’s Horizon 2020 Marie Sklodowska-Curie grant No 839412. Evgenios Kakariadis acknowledges support from EPSRC as part of the programme “Operator Algebras for Product Systems” (EP/T02576X/1). Elias Katsoulis was partially supported by the NSF grant DMS-2054781. Marcelo Laca was partially supported by NSERC Discovery Grant RGPIN-2017-04052. Xin 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: Dor-On, A., Kakariadis, E.T.A., Katsoulis, E., Laca, M., and Li, X.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Advances in Mathematics
• Publisher: Elsevier
• ISSN: 0001-8708
• ISSN (Online): 1090-2082
• Published Online: 28 February 2022
• Copyright Holders: Copyright © 2022 Elsevier Inc.
• First Published: First published in Advances in Mathematics 400: 108286
• Publisher Policy: Reproduced in accordance with the publisher copyright policy