Boundary quotients of C$^*$-algebras of left cancellative monoids and their groupoid models

TL;DR

This paper explores boundary quotients of semigroup C*-algebras for left cancellative monoids, proposing groupoid models and conditions for their realization, including examples that challenge existing regularity notions.

Contribution

It introduces new conditions for groupoid models of boundary quotients and connects them to C*-regularity, with an example of a non-C*-regular monoid satisfying these conditions.

Findings

Two groupoid models for boundary quotients are proposed.

Conditions are formulated to determine when these models are valid.

An example monoid satisfies new conditions but is not C*-regular.

Abstract

For a left cancellative monoid $S$ we consider a quotient of the reduced semigroup C$^*$-algebra $C_r^*(S)$ known as the boundary quotient. We present two potential groupoid models for this boundary quotient, obtained as reductions of Paterson and Spielberg's groupoids associated to $S$, and formulate conditions on $S$ which guarantees that either is a groupoid model. We outline how these conditions are related to the notions (strong) C$^*$-regularity introduced in a previous paper, and construct an example of a left cancellative monoid which is not C$^*$-regular, but satisfies both of the new conditions.