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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.

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  • $\begingroup$ 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 $\endgroup$ – Robert Furber Oct 18 '18 at 11:50
  • $\begingroup$ @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. $\endgroup$ – Rick Sternbach Oct 18 '18 at 12:35
  • $\begingroup$ Question 1 is answered here (plus the fact that type I factors are characterized by having minimal projections). $\endgroup$ – Nik Weaver Oct 18 '18 at 13:59

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