Normal coactions extend to the C*-envelope

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.

27 pages. v2: New applications for operator algebras of non group-embmeddable cancellative right LCM monoids. v3: Reworked abstract and some of the introduction, fixed minor issues with Proposition 2.3 and Lemma 3.4, added acknowledgements, added Remark 3.7 which directly compares the work of Sehnem to ours, updated and added references. Accepted version

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