Boundary quotients of C*-algebras of right LCM semigroups

TL;DR

This paper investigates boundary quotients of C*-algebras derived from right LCM semigroups, establishing their structure as groupoid C*-algebras and analyzing conditions for simplicity and pure infiniteness.

Contribution

It introduces a boundary quotient construction for right LCM semigroups and characterizes these quotients as groupoid C*-algebras, providing new insights into their properties.

Findings

Boundary quotients are isomorphic to tight C*-algebras of inverse semigroups.

Conditions for simplicity and pure infiniteness of boundary quotients are established.

Applications include analysis of self-similar groups and Zappa-Szép products.

Abstract

We study C*-algebras associated to right LCM semigroups, that is, semigroups which are left cancellative and for which any two principal right ideals are either disjoint or intersect in another principal right ideal. If $P$ is such a semigroup, its C*-algebra admits a natural boundary quotient $\mathcal{Q}(P)$. We show that $\mathcal{Q}(P)$ is isomorphic to the tight C*-algebra of a certain inverse semigroup associated to $P$, and thus is isomorphic to the C*-algebra of an \'etale groupoid. We use this to give conditions on $P$ which guarantee that $\mathcal{Q}(P)$ is simple and purely infinite, and give applications to self-similar groups and Zappa-Sz\'ep products of semigroups.