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

A. Dor-On, E. T.A. Kakariadis, E. Katsoulis, M. Laca, X. Li

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number108286
JournalAdvances in Mathematics
Volume400
DOIs
StatePublished - 14 May 2022
Externally publishedYes

Bibliographical note

Funding Information:
Elias Katsoulis was partially supported by the NSF grant DMS-2054781 .

Funding Information:
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 ).

Funding Information:
Evgenios Kakariadis acknowledges support from EPSRC as part of the programme “Operator Algebras for Product Systems” ( EP/T02576X/1 ).

Funding Information:
The authors are thankful to the anonymous referee whose comments and suggestions improved the exposition in the paper. Part of the research was carried out during the Focused Research Group 20frg248: Noncommutative Boundaries for Tensor Algebras at the Banff International Research Station. 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).

Funding Information:
Marcelo Laca was partially supported by NSERC Discovery Grant RGPIN-2017-04052 .

Funding 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 .

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

  • C*-envelope
  • Co-universal algebra
  • Coaction
  • Covariance algebra
  • Nica-Pimsner algebras
  • Product systems

ASJC Scopus subject areas

  • Mathematics (all)

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