The commutation theorem for tensor products over von Neumann algebras

A general commutation theorem is proved for tensor products of von Neumann algebras over common von Neumann subalgebras. Roughly speaking, if the non- common parts of two von Neumann algebras M-1 and M-2 on the same Hilbert sp ace are appropriately separated by commuting type I von Neumann algebras N- 1 and N-2, then the commutant of the von Neumann algebra generated by M-1 a nd M-2 is generated by the relative commutants M-1' boolean AND N-1 and M-2 ' boolean AND N-2, as well as by the intersection of the commutants of all concerned von Neumann algebras. This theorem extends both Tomita's classica l commutation theorem and a splitting result in tensor products, proved rec ently in the factor case by L. Ge and R. V. Kadison. Applications are given to a decomposition criterion in ordinary tensor products and to a partial solution of a conjecture of S. Popa concerning the maximal injectivity of t ensor products of maximal injective von Neumann subalgebras. (C) 1999 Acade mic Press.