Left regular representations of Garside categories I. C*-algebras and groupoids

TL;DR

This paper explores the construction and analysis of C*-algebras and groupoids derived from Garside categories, linking algebraic structures from braid groups to operator algebras and providing classification and ideal structure results.

Contribution

It introduces a framework connecting Garside categories with C*-algebras and groupoids, offering classification criteria and analyzing ideal structures, extending understanding of higher rank graph and Artin-Tits monoid C*-algebras.

Findings

Classification of closed invariant subspaces of the groupoids

Criteria for topological freeness and local contractiveness

Complete analysis of ideal structures in related C*-algebras

Abstract

We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin-Tits monoids.