## 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 language | English |
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Article number | 108286 |

Journal | Advances in Mathematics |

Volume | 400 |

DOIs | |

State | Published - 14 May 2022 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2022 Elsevier Inc.

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics