An analogue of Longo's canonical endomorphism for bimodule theory and its application to asymptotic inclusions

We give an analogous characterization of Longo's canonical endomorphism in the bimodule theory, and by using this, we construct an inclusion of factors of type II₁ from a finite system of bimodules as a parallel construction to that of Longo-Rehren in a type III setting. When the original factors are approximately finite dimensional, we prove this new inclusion is isomorphic to the asymptotic inclusion in the sense of Ocneanu. This solves a conjecture of Longo-Rehren.