A Necessary and Sufficient Condition for a Von Neumann Algebra to be in Standard Form
Ronny Rousseau, Alfons Van Daele, Lutgarde Vanheeswijck · Journal of the London Mathematical Society · 1977
If M is a von Neumann algebra acting on a Hilbert space H, then M is said to be in standard form if there is a conjugate linear involution J on H such that JMJ = M′ and JaJ = a* for all a ∈ M ∩ M′. It is proved that M is in standard form if and only if there exists a family {ξα} in H which is cyclic both for M and M′ and such that the spaces {Mξα} are mutually orthogonal, as well as the spaces {M′ξα}. This is a generalization of the corresponding result for σ-finite von Neumann algebras in which case the family {ξα} is replaced by a single vector ξ.