The Commutation Theorem for Tensor Products of von Neumann Algebras

It is shown that the commutation theorem for tensor products of general von Neumann algebras follows trivially from the case of von Neumann algebras with a separating and cyclic vector. Let A/i and M2 be von Neumann algebras acting in Hilbert spaces %x and %2 respectively. Then the commutation theorem for tensor products of von Neumann algebras states (Mx ® M2)' = M\ ® M'2. The proof of this theorem in full generality was first obtained by Tomita in 1967 (see [3]). Later a number of simpler versions have been obtained (see e.g. [2]). Usually standard techniques are used to reduce the general case to the case of von Neumann algebras having a cyclic and separating vector. In this note we show that this reduction is almost a triviality. Assume that the commutation theorem has been proved for von Neumann algebras with a separating and cyclic vector. To show that (Mx ® M2)' = M\ ® M'2 it is sufficient to show that (Mx ® M2)' and (M\ ® M'2)' commute since obviously M\ ® M'2 C (Mx ® M2)'. If W] G %x and w2 G %2 we define ex = [Af. to], e\ = [Mx w] and similarly e2 — [M'2w], e'2 = [M2u>] where [Mu] denotes the projection onto the closed subspace generated by xw with xGM.lt is well known that ex G Mx, e\ G M\ and that the von Neumann algebras ex A/. ex e\ and e\ M\ e\ e. as acting on ex e\ % are commutants of each other and have wi as a separating and cyclic vector. A similar statement is true for M2. Denote % = %x ® %2, M = Mx® M2, A = M\ ® M'2, e = ex® e2 and e' = e\ ® e'2. From the commutation theorem for von Neumann algebras with a separating Received by the editors March 9, 1975. AMS (MOS) subject classifications (1970). Primary 46L10. Copyright © 1977, American Mathematical Society