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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.

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# Maximal localizations of von Neumann algebras

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, elements with left and right support being equal to 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)).

Von Neumann proved that if M is finite, then L and A are canonically isomorphic.

What can be said about the relationship of L and A when M has type III?