Type III factor representation 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?
 A: I seem to recall that work of Wright and others contains constructions of type III
AW* factors that are not W*. The lattice of projections of the former is also a complete orthomodular lattice, and these probably (?) characterize the AW* algebras (i.e., if $A$ and $B$ are AW*-factors, and their projection lattices are isomorphic, then presumably $A$ and $B$ would be isomorphic; I don't know whether this is true, but it looks plausible). So if there were a condition for representation in W*-factors, it would have to include some additional constraint referring to normal states, which would not be very natural. 
Perhaps this is not the most concise analogy, but consider what happens in the commutative case. Commutative AW* algebras are $C(X)$ where $X$ is extremally disconnected, but commutative W* algebras are $C(X)$ where $X$ is hyperstonean, a rather artificial** concept, because it requires a separating family of normal states.
**Not the right word; I mean that it cannot be easily made more abstract.
