A Structural Result of Irreducible Inclusions of Type III_a FACTORS, λ ∈ (0,1)

Given an irreducible inclusion of factors with finite index N ⊂ M , where M is of type III _{λ1/ m} , N of type III _{λ1/ n} , 0 < λ < 1, and m , n are relatively prime positive integers, we will prove that if N ⊂ M satisfies a commuting square condition, then its structure can be characterized by using fixed point algebras and crossed products of automorphisms acting on the middle inclusion of factors associated with N ⊂ M . Relations between N ⊂ M and a certain G -kernel on subfactors are also discussed.