Groupoids and $C^*$-algebras for left cancellative small categories

TL;DR

This paper generalizes the construction of $C^*$-algebras from categories of paths to the broader context of left cancellative small categories, developing the theory and analyzing key algebraic structures.

Contribution

It extends existing techniques to a more general class of categories, providing a comprehensive framework for their associated $C^*$-algebras.

Findings

Derived the structure of $C^*$-algebras for left cancellative small categories

Analyzed the regular representation and Wiener-Hopf algebra in this context

Established the applicability of techniques to a broader class of categories

Abstract

Categories of paths are a generalization of several kinds of oriented discrete data that have been used to construct $C^*$-algebras. The techniques introduced to study these constructions apply almost verbatim to the more general situation of left cancellative small categories. We develop this theory and derive the structure of the $C^*$-algebras in the most general situation. We analyze the regular representation, and the Wiener-Hopf algebra in the case of a subcategory of a groupoid.