Exchange relations and Frobenius subalgebras

TL;DR

This paper generalizes a bijection between intermediate subfactors and idempotents to abelian monoidal categories using Frobenius subalgebras, and characterizes certain morphisms on C*-correspondences as averaging operators.

Contribution

It extends the concept of exchange relations and Frobenius subalgebras from subfactor theory to a broader categorical setting.

Findings

Generalization of the bijection to abelian monoidal categories

Identification of morphisms arising from intermediate C*-subalgebras as averaging operators

Application to C*-correspondences

Abstract

Bisch and Jones established a bijection between the intermediate subfactors of an irreducible subfactor and certain idempotents satisfying exchange relations. In this paper, we generalize this result to abelian monoidal categories through Frobenius subalgebras. As an application, we show that certain morphisms on a C*-correspondence arise from intermediate C*-subalgebras if and only if they are averaging operators.