Structural Chirality from Inverse Semigroups to Twisted Groupoid $C^*$-Algebras

TL;DR

This paper introduces a structural theory of chirality for inverse semigroups and demonstrates how it influences the properties of associated twisted groupoid $C^*$-algebras, revealing obstructions to mirror self-duality.

Contribution

It develops a new framework linking inverse semigroup chirality to twisted groupoid $C^*$-algebras, identifying obstructions to mirror symmetry in these structures.

Findings

Identifies a structural obstruction to mirror self-duality in twisted groupoid $C^*$-algebras.

Shows how chirality propagates from inverse semigroups to groupoid $C^*$-algebras.

Provides a representation-independent framework compatible with germ groupoid models.

Abstract

We develop a structural theory of chirality for inverse semigroups and show how it propagates canonically to \'{e}tale groupoids and twisted groupoid $C^*$-algebras. Starting from inverse semigroup data equipped with admissible twist information, we construct a canonical twisted universal groupoid in the sense of Paterson and introduce a mirror correspondence encoding intrinsic asymmetry. Our main result identifies a structural obstruction to mirror self-duality at the level of twisted universal groupoids and shows that this obstruction descends to an obstruction for the associated reduced twisted groupoid $C^*$-algebra to be isomorphic to its opposite. The framework is representation-independent, yet compatible with concrete germ groupoid models, and provides a unified bridge between partial symmetries, groupoid structures, and analytic invariants in noncommutative operator algebras.