Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

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 Bischoff et al. (J Funct Anal 281(1):109004, 2021), 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 α-induction and σ-restriction for braided subfactors previously known in the finite index case.