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Normal coactions extend to the C*-envelope

Kevin Aguyar Brix, Chris Bruce, Adam Dor-On

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 considered by Kakariadis, Katsoulis, Laca, and X. Li, and provides an elementary proof of a prominent result of Sehnem. As an application, we resolve a question of Li by identifying the C*-envelope of the operator algebra arising from a groupoid-embeddable category and of cancellative right LCM monoids. This latter class includes many examples of monoids that are not group-embeddable.

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 considered by Kakariadis, Katsoulis, Laca, and X. Li, and provides an elementary proof of a prominent result of Sehnem. As an application, we resolve a question of Li by identifying the C*-envelope of the operator algebra arising from a groupoid-embeddable category and of cancellative right LCM monoids. This latter class includes many examples of monoids that are not group-embeddable.
Paper Structure (10 sections, 30 theorems, 78 equations)

This paper contains 10 sections, 30 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coactions on operator algebras
  4. Operator algebras from categories
  5. Existence of coactions on C*-envelopes
  6. C*-envelopes of operator algebras from categories
  7. The boundary quotient is a C*-cover
  8. From functors to coactions
  9. Groupoid-embeddable categories
  10. Right LCM monoids

Key Result

Theorem A

Let $\mathcal{A}$ be an operator algebra with a contractive approximate identity. Then, any normal coaction of a discrete group $G$ on $\mathcal{A}$ has a (necessarily unique) extension to a normal coaction of $G$ on $C_\textup{env}^*(\mathcal{A})$.

Theorems & Definitions (63)

  • Theorem A: Theorem \ref{['thm:main']}
  • Theorem B: Theorem \ref{['thm:gpoid-embeddable']}
  • Theorem C: Theorem \ref{['thm:lcm-monoids']}
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 53 more