Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

TL;DR

This paper establishes a Galois correspondence for local discrete subfactors, linking intermediate von Neumann algebras with subhypergroups, and extends Fourier analysis and induction/restriction results to this setting.

Contribution

It introduces a Galois correspondence for local discrete subfactors and extends Fourier analysis and induction/restriction theories beyond finite index cases.

Findings

Proves a Galois correspondence between intermediate algebras and subhypergroups.

Develops a subfactor Fourier transform in the local discrete setting.

Extends $eta$-induction and $eta$-restriction results to infinite index subfactors.

Abstract

Discrete subfactors include a particular class of infinite index subfactors and all finite index ones. A discrete subfactor is called local when it is braided and it fulfills a commutativity condition motivated by the study of inclusion of Quantum Field Theories in the algebraic Haag-Kastler setting. In [BDG21], we proved that every irreducible local discrete subfactor arises as the fixed point subfactor under the action of a canonical compact hypergroup. In this work, we prove a Galois correspondence between intermediate von Neumann algebras and closed subhypergroups, and we study the subfactor theoretical Fourier transform in this context. Along the way, we extend the main results concerning $\alpha$-induction and $\sigma$-restriction for braided subfactors previously known in the finite index case.