The commutation theorem for tensor products over von Neumann algebras (Q1296772)

Given a \(W^*\)-algebra \(M\), a \(W^*\)-algebra \(N\subset M\) containing \(1_M\) and two \(W^*\)-algebras \(M_1,M_2\subset N\), \(M\) is the tensor product over \(N\) of \(M_1\) and \(M_2\) if \(M= M_1\vee M_2\) and, in some spatial representation of \(M\), \(M_1\subset N\vee N_1\) and \(M_2\subset N\vee N_2\) for some commuting type \(I\) von Neumann subalgebras \(N_1,N_2\subset N\). The main result of this paper: \((M_1\vee M_2)'= (M_1'\cap N_1)\vee (M_2'\cap N_2)\vee (N\vee N_1\vee N_2)'\).