Boundary quotient C*-algebras of semigroups
Evgenios T.A. Kakariadis, Elias G. Katsoulis, Marcelo Laca, Xin Li

TL;DR
This paper investigates boundary quotient C*-algebras associated with semigroups, establishing their universal properties, connections to group C*-algebras, and conditions under which they coincide with C*-envelopes, covering various algebraic structures.
Contribution
It introduces new universal and co-universal characterizations of boundary quotient C*-algebras for semigroups, linking them to group C*-algebras and boundary actions.
Findings
Boundary quotient C*-algebra $ ext{C}^*_ ext{env}( ext{A}(P))$ coincides with $ ext{C}^*_ ext{lambda}(G)$ under certain conditions.
$ ext{A}(P)$ is hyperrigid when conditions are met.
Characterization of Ore semigroups via boundary quotient C*-algebras.
Abstract
We study two classes of operator algebras associated with a unital subsemigroup of a discrete group : 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 that reflect the space 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 and we consider the boundary quotient related to the minimal boundary space. We show that is co-universal in two different classes: (a) with respect to the equivariant constructible isometric representations…
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