Generalization of Longo–Rehren Construction to Subfactors of Infinite Depth and Amenability of Fusion Algebras

We extend the Longo–Rehren construction to an infinite system of bimodules (of sectors). The construction is similar to the crossed product. We also show that the Longo–Rehren construction gives an isomorphic subfactor to Popaʹs symmetric enveloping algebra. We discuss the relation between the Longo–Rehren subfactors and amenability of fusion algebras in the sense of Hiai and Izumi.