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Given an irreducible inclusion of factors with finite index N C M, where M is of type III,X1/, N of type III,1/nX 0 < A < 1, and m,n are relatively prime positive integers, we will prove that if N C 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 C M. Relations between N C M and a certain G-kernel on subfactors are also discussed.

A structural result of irreducible inclusions of type 𝐼𝐼𝐼_{𝜆} factors, 𝜆∈(0,1)