Galois Correspondence and Fourier Analysis on Local Discrete 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 [BDVG21], 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. Contents1. Introduction 2. Preliminaries 2.1. The canonical endomorphism 2.2. Conditional expectations 2.3. Braided and local subfactors 2.4. Compact hypergroups 2.5. Duality theorem and dominated UCP maps 3. α-induction for discrete subfactors 4. Intermediate inclusions 5. Galois correspondence 6. Fourier transform 6.1. Extension of the Fourier transform to UCP maps 6.2. The local discrete case: Fourier transform on measures 6.3. L p spaces and Fourier inequalities 6.4. Involutions, convolutions and products 6.5. Convolution inequalities 6.6. Inversion formula and uncertainty principles References

A planar algebraic description of conditional expectations

“…in the study, classification and constructions of finite index extensions of Quantum Field Theories in the algebraic setting [LR95], [LR04], [KL04], [BKL15], [BKLR16]. For generalizations to infinite index subfactors and inclusions, which are relevant also in QFT, we refer to [FI99], [DVG18], [JP19], [BDVG21a], [BDVG21b].…”

Quantum convolution inequalities on Frobenius von Neumann algebras

“…In [2], Beckner remarkably proved the sharp Young's inequality [29]. Recently, quantum Young's inequality for convolution has been established on quantum symmetries, such as subfactors [11,7], fushion bi-algebras [21], Kac algebras [22], locally compact quantum groups [12], etc., see further discussions in the framework of quantum Fourier analysis [10]. Moreover, the extremal pairs of quantum Young's inequality are characterized on subfactor planar algebras [13] and kac algebras [22].…”