The Tensor Product Problem for Reflexive Algebras

It was observed by Gilfeather, Hopenwasser, and Larson in [1] that Tomita's commutation formula for tensor products of von Neumann algebras can be rewritten in a way that makes sense for tensor products of arbitrary reflexive algebras. The tensor product problem for reflexive algebras is to decide for which pairs of reflexive algebras this tensor product formula is valid. Recall that a subalgebra jtf of the algebra B(3P) of all bounded operators on a Hubert space %? is said to be a von Neumann algebra if it is closed in the weak operator topology, contains the identity operator I, and is self-adjoint (i.e., A t ^ implies A* e J?). The commutant J? of Jt is the set of all operators B e B{^) such that BA = AB for all A e Jf. The commutant of a von Neumann algebra is again a von Neumann algebra. Moreover, it follows from von Neumann's double commutant theorem that a self-adjoint subalgebra Jf of B(J%f) is a von Neumann algebra if and only if Jf = Jf" . Let Jf c B(jr) and JV c B[X) be von Neumann algebras, and let %? ® 3t denote the Hilbert space tensor product of %f and 3?. If A e Jf and B € JV, there is a unique operator A ® B in Bffi ® 3t) such that (A ® B)(x ®y) = Ax®By for all x e %? and J / G J . The von Neumann algebra generated by {A ® B\A e J? and B e J^} is denoted by J? ® J^. Tomita's commutation theorem asserts that for any pair of von Neumann algebras J? and Jf the following commutation formula is valid: