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arXiv · 2507.10168

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

Gaute Schwartz
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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.

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Gaute Schwartz. 2025-07-14. Boundary quotients of C$^*$-algebras of left cancellative monoids and their groupoid models. https://arxiv.org/abs/2507.10168

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