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Boundary quotient C*-algebras of semigroups

Lookup NU author(s): Dr Evgenios Kakariadis



This is the of an article that has been published in its final definitive form by John Wiley and Sons Ltd, 2022.

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© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence. We study two classes of operator algebras associated with a unital subsemigroup (Formula presented.) of a discrete group (Formula presented.) : one related to universal structures and one related to co-universal structures. First we provide connections between universal C*-algebras that arise variously from isometric representations of (Formula presented.) that reflect the space (Formula presented.) of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation (Formula presented.), and we consider the boundary quotient (Formula presented.) related to the minimal boundary space. We show that (Formula presented.) is co-universal in two different classes: (a) with respect to the equivariant constructible isometric representations of (Formula presented.); and (b) with respect to the equivariant C*-covers of the reduced non-selfadjoint semigroup algebra (Formula presented.). If (Formula presented.) is an Ore semigroup, or if (Formula presented.) acts topologically freely on the minimal boundary space, then (Formula presented.) coincides with the usual C*-envelope (Formula presented.) in the sense of Arveson. This covers total orders, finite type and right-angled Artin monoids, the Thompson monoid, multiplicative semigroups of non-zero algebraic integers, and the (Formula presented.) -semigroups over integral domains that are not a field. In particular, we show that (Formula presented.) is an Ore semigroup if and only if there exists a canonical (Formula presented.) -isomorphism from (Formula presented.), or from (Formula presented.), onto (Formula presented.). If any of the above holds, then (Formula presented.) is shown to be hyperrigid.

Publication metadata

Author(s): Kakariadis ETA, Katsoulis EG, Laca M, Li X

Publication type: Article

Publication status: Published

Journal: Journal of the London Mathematical Society

Year: 2022

Pages: Epub ahead of print

Online publication date: 15/02/2022

Acceptance date: 06/08/2021

Date deposited: 08/08/2021

ISSN (print): 0024-6107

ISSN (electronic): 1469-7750

Publisher: John Wiley and Sons Ltd


DOI: 10.1112/jlms.12557


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