An Algebraic Approach to the Weyl Groupoid

TL;DR

This paper presents an algebraic framework unifying two constructions of Weyl groupoids, extending their applicability to broader algebraic structures like *-rings while preserving key properties.

Contribution

It introduces a unified algebraic approach to Weyl groupoids using only *-semigroup reducts, broadening the scope beyond C*-algebras.

Findings

Unified the Kumjian-Renault and Lawson-Lenz Weyl groupoid constructions.

Extended local compactness property to general classes of *-rings.

Demonstrated the applicability of the approach to broader algebraic structures.

Abstract

We unify the Kumjian-Renault Weyl groupoid construction with the Lawson-Lenz version of Exel's tight groupoid construction. We do this by utilising only a weak algebraic fragment of the C*-algebra structure, namely its *-semigroup reduct. Fundamental properties like local compactness are also shown to remain valid in general classes of *-rings.