Exchange relations and Frobenius subalgebras

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 Karoubian 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.

15 pages. Revised version. The Abelian assumption is weakened to Karoubian. Comments are welcome!