Structural Chirality from Inverse Semigroups to Twisted Groupoid $C^{\ast }$-Algebras

Abstract

We develop a structural theory of chirality for inverse semigroups and show how it propagates canonically to étale groupoids and twisted groupoid $C^{\ast }$-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 $C^{\ast }$-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.

arXiv comment26 pages. Structural chirality for inverse semigroups is developed via Paterson's universal groupoid and shown to induce an obstruction to mirror self-duality of twisted étale groupoid C-star-algebras