Normal coactions extend to the C*-envelope

TL;DR

This paper proves that normal coactions of discrete groups on operator algebras extend to their C*-envelopes, solving an open problem and clarifying the structure of certain operator algebra C*-envelopes.

Contribution

It demonstrates the extension of normal coactions to C*-envelopes and identifies these envelopes for specific classes of operator algebras, including non-group-embeddable monoids.

Findings

Normal coactions extend to C*-envelopes for discrete groups.

Resolved an open problem in the theory of coactions.

Identified C*-envelopes for operator algebras of certain monoids.

Abstract

We show that a normal coaction of a discrete group on an operator algebra extends to a normal coaction on the C*-envelope. This resolves an open problem attempted by several experts in the area, and provides a more direct proof of a prominent result of Sehnem. As an application, we resolve a question of X. Li, where we identify the C*-envelopes of the operator algebras of groupoid-embeddable categories and of cancellative right LCM monoids. This latter class includes many examples of monoids that are not group-embeddable.