Direct Integral - Direct Integrals of Von Neumann Algebras

Direct Integrals of Von Neumann Algebras

Let {H_x}_{x ∈ X} be a measurable family of Hilbert spaces. A family of von Neumann algebras {A_x}_{x ∈ X} with

is measurable if and only if there is a countable set D of measurable operator families that pointwise generate {A_x} _{x ∈ X}as a von Neumann algebra in the following sense: For almost all x ∈ X,

where W*(S) denotes the von Neumann algebra generated by the set S. If {A_x}_{x ∈ X} is a measurable family of von Neumann algebras, the direct integral of von Neumann algebras

consists of all operators of the form

for T_x ∈ A_x.

One of the main theorems of von Neumann and Murray in their original series of papers is a proof of the decomposition theorem: Any von Neumann algebra is a direct integral of factors. We state this precisely below.

Theorem. If {A_x}_{x ∈ X} is a measurable family of von Neumann algebras and μ is standard, then the family of operator commutants is also measurable and