Assorted couple of Kac systems and inclusions of factors of type \(II_1\) (Q1273942)

Let \(K\subset L\) be an irreducible inclusion of type \(\text{II}_1\) factors with finite indices with \(L\) generated by intermediate subfactors \(M\) and \(N\) which are isomorphic to the crossed product \(K\) by the Kac algebras \(A\) and \(B\). The author gives conditions so that there is inversion \(T\) on \(A\) and \(B\) such that \(L\) is isomorphic to the crossed product of \(K\) by the \(T\)-tensor product of \(A\) and \(B\). The left action and the right action by \(T\) of \(B\) on \(A\) and of \(A\) on \(B\) are investigated.