# To what extend can a von Neumann algebra be determined by its projection lattice structure?

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 $$\varphi_L : P \to Q$$ of lattices. My question is, what can we say about the relation between $$M$$ and $$N$$, knowing only that their projection lattice structures are isomorphic?

For example,

1. Suppose $$M$$ is a factor, and $$P$$ is isomorphic to $$Q$$ as lattices, is $$N$$ a factor as well? If so, do they have the same type?

2. Are there any examples where $$P$$ and $$Q$$ are isomorphic as lattices, but $$M$$ and $$N$$ are not isomorphic as involutive algebras?

Here is some of my thoughts about question 1.

I believe that we can conclude $$N$$ is also a factor in this case. If I remember correctly, we can identify factors as von Neumann algebras which can not be further decomposed as the direct sum of two (smaller) von Neumann algebras (can anyone give a reference for a proof of this fact, or in the case I am wrong, a counterexample?). Such a direct sum decomposition of a von Neumann algebra would be reflected on its projection lattice structure.

As for the types of $$M$$ and $$N$$. By the fundamental theorem of projective geometry, it is easy to see that if $$M$$ is a type $$I_n$$ factor, $$N$$ a type $$I_m$$ factor, and $$P$$ is isomorphic to $$Q$$ as lattices, then $$m = n$$. This naturally raises the question of whether the same can be said for factors of other types.

Question 2 seems more intractable to me. But it sure is interesting in its own right. I think maybe some easy examples can be constructed, as it seems too wild to conjecture that $$M$$ is isomorphic to $$N$$ if $$P$$ is isomorphic to $$Q$$. Yet, I can not produce such an example after some effort.

Perhaps answers to these questions are well-known among experts, as they seem rather basic to me. In this case, please kindly point out the relevant references.

• This is close to a duplicate of: mathoverflow.net/questions/256062/… In particular, counterexamples to 2 are given by Connes' examples of von Neumann algebras not *-isomorphic to their opposites: jstor.org/stable/1970940 – Robert Furber Oct 18 '18 at 11:50
• @RobertFurber Thanks for the link. Am I right in assuming question 1 is still unanswered? The type of the factor depends not only on the projection lattice, but also on the von Neumann-Murray comparaison theory. I don't see exactly how the Jordan $\ast$-homomorphism can be used to "detect" the comparaison of projections. – Rick Sternbach Oct 18 '18 at 12:35
• Question 1 is answered here (plus the fact that type I factors are characterized by having minimal projections). – Nik Weaver Oct 18 '18 at 13:59