Uniqueness theorems for combinatorial C*-algebras

TL;DR

This paper proves new uniqueness theorems for a broad class of combinatorial C*-algebras using groupoid models and inverse semigroup theory, enhancing previous results and extending to more general cases.

Contribution

It introduces generalized uniqueness theorems for combinatorial C*-algebras, improving and extending prior results to broader classes.

Findings

Proved a generalized uniqueness theorem for boundary quotient C*-algebras of right LCM monoids.

Extended the row-finite higher-rank graph uniqueness theorem to finitely aligned cases.

Utilized groupoid models and Exel's inverse semigroup theory to establish these results.

Abstract

Spielberg's construction of C*-algebras from left cancellative small categories is a common generalization for most C*-algebras one would consider to come from ``combinatorial data,'' including graph and $k$-graph C*-algebras, Li's semigroup C*-algebras, Nekrashevych's self-similar action algebras, and more. We use known groupoid models of these algebras and Exel's theory of tight representations of inverse semigroups to prove uniqueness theorems for these C*-algebras. As applications, we improve on our previous uniqueness theorem for the boundary quotient C*-algebras of right LCM monoids, and we also generalize the uniqueness theorem of Brown, Nagy, and Reznikoff for row-finite higher-rank graphs to the finitely aligned case.