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Does there exist any theorem which permits, under suitable hypotheses, to represent a particular complete orthomodular lattice as the projection lattice of a Type III von Neumann factor?

Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \...