Suppose M is a von Neumann algebra. Denote by L its maximal noncommutative localization, i.e., the Ore localization with respect to the set of all left and right regular elements(equivalently, , i.e., elements with whose left and right support being equal to 1)equals 1.
Denote by A the set of all closed unbounded operators with dense domain affiliated with the standard representation of M on a Hilbert space(i.e., L^2(M)), i.e., L^2(M), also known as the standard form of M.
Von Neumann proved that if M is finite, then L and A are canonically isomorphic.
What can be said we say about the relationship of L and A when M has type III?
I am also interested in the properly infinite semifinite case.