Irreducible inclusions of factors and multiplicative unitaries (Q1270219)

This work is an extension of the author and \textit{R. Nest} [J. Funct. Anal. 137, No. 2, 466-543 (1996; Zbl 0847.22003)]. Let \(M_0\subset M_1\) be an irreducible depth 2 inclusion of factors and \(T_1\) a faithful semi-finite normal operator-valued weight from \(M_1\) to \(M_0\), and \((M_i)_{i\in \mathbb{N}}\) the associated canonical tower of factors. Let \(T_i\) be the associated weight from \(M_i\) to \(M_{i-1}\) and assume that \(T_2\) restricted to \(M_2\cap M_0'\) and \(T_3\) restricted to \(M_3\cap M_1'\) are semifinite. Then the \(M_{i+1}\cap M_{i-1}'\) are Woronowicz algebras \(W_i\), \(W_{i+1}\) being isomorphic to the commutant of the dual of \(W_i\). Moreover there exists an outer action \(\alpha_i\) of \(W_{i+1}\) on \(M_i\) such that \(M_{i-1}\) is the fixed-point algebra and \(M_{i+1}\) is the crossed product. Then the tower is isomorphic to the tower of successive crossed products.