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

TL;DR

This paper introduces the concept of C*-envelopes for cosystems, proves their existence, and shows their relation to co-universal C*-algebras for product systems, extending previous results in the field.

Contribution

It defines C*-envelopes for cosystems, establishes their existence, and demonstrates their role as co-universal C*-algebras for certain product systems, generalizing prior work.

Findings

C*-envelopes for cosystems always exist.

The C*-envelope is co-universal for specific product systems.

Identification of the C*-envelope with reduced cross-sectional algebra.

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 to 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.