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, i.e., elements whose left and right support 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), 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 we say about the relationship of L and A when M has type III?
I am also interested in the properly infinite semifinite case.