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H. A. Dye showed that a type II or III factor $R$ is determined, up to *-algebraic isomorphism or anti-isomorphism, by the ortholattice-isomorphism type of its projection lattice ("ortholattice-isomorphism" meaning bijection between projection lattices that preserves order and orthogonality), but has anyone figured out how much we can determine about $R$ just from the order-isomorphism type of its projection lattice?

Dye says as of 1955 it was open whether "order-isomorphic" is equivalent to "ortholattice-isomorphic" in this context. Has progress been made on this question? I've been told that the Levy collapse poset* embeds densely into the projection lattice of any non-commutative non-type-I factor with separable predual, from which I think one could show that the lattice's order type tells you nothing more about such a factor. But I don't know if this was meant in the sense of "this is an established fact" or "this seems very likely."


[I asked this recently on math.stackexchange but it's likely too specialized ... any help hugely appreciated!]

[Also, I initially mentioned a paper by C. Heunen and M. Reyes, but as @BertLindenhovius explains below, their example gave only a pair of order-isomorphic projection lattices belonging to algebras that are not *-isomorphic; to be relevant to my question, the algebras would need to be neither *-isomorphic nor *-anti-isomorphic.]

*more precisely the Levy collapse of $2^\omega$ to $\omega$, i.e. the set of finite sequences of ordinals $< 2^\omega$, ordered by reverse inclusion.

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I cannot answer your question, but maybe the following remarks are useful. Firstly, it is important to specify what is meant with "up to isomorphism." I assume that you mean *-isomorphisms if you mention isomorphisms between von Neumann algebras.

Dye did not show that the ortholattice-isomorphism type of the projections of each type II or III factor $M$ determines $M$ up to *-isomorphism, but that it determines $M$ up to *-isomorphism or *-anti-isomorphism (i.e., a map that reverses the multiplication). In orther words, if the projections of two von Neumann factors $M$ and $N$ are ortholattice-isomorphic, then there is either a *-isomorphism $M\to N$ or a *-anti-isomorphism $M\to N$, but it is not necessarily the case that there is both a *-isomorphism and a *-anti-isomorphism $M\to N$.

The type III factor $M$ in the article by Connes as cited in Heunen & Reyes is a counterexample. Connes showed that $M$ is not *-anti-isomorphic to itself. This is equivalent with saying that $M$ is not *-isomorphic to its opposite algebra N, i.e., the von Neumann factor with the same underlying vector space, but with multiplication defined by $(a,b)\mapsto ba$ instead of $(a,b)\mapsto ab$. By definition of $N$, it is *-anti-isomorphic to $M$, but apparently not *-isomorphic. Clearly the projection lattices of both algebras are ortholattice-isomorphic.

Secondly, from what I understood, Heunen and Reyes do not discuss order-isomorphisms between projection lattices, only ortholattice morphisms. Since the type III factors $M$ and $N$ of Connes have ortholattice-isomorphic projection lattices, they do not provide an example of algebras whose projections are order-isomorphic but not ortholattice-isomorphic.

Edit: I came across this overview article, "Orthomodular Lattices and Quadratic Spaces: a Survey", by Piziak, where, below Theorem 4.2, he discusses a counterexample by Birkhoff of a lattice atmitting two orthocomplementations making the lattice an orthomodular lattice, but such that the two resulting orthomodular lattices are not ortho-isomorphic. This still does not answer your question whether or not there are von Neumann algebras with isomorphic projection lattices, which are not ortholattice isomorphic, but at least it leaves open the open that such an example can be found.

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  • $\begingroup$ Thank you very much for taking the time to clarify this for me! I have edited the question accordingly. $\endgroup$ Nov 3, 2016 at 14:04
  • $\begingroup$ Edit to my post: if you generalize your question to the case of all orthomodular lattices, then indeed there are examples of non-isomorphic orthomodular lattices whose underlying lattices are isomorphic, see the link. $\endgroup$ Nov 5, 2016 at 20:57
  • $\begingroup$ Thought I'd mention, Michiya Mori seems to have answered this question, at least for type III algebras: arxiv.org/abs/2006.08959 . Here's Cor. 4.3: Let $M, N$ be von Neumann algebras of type I$_\infty$ or III. Suppose that $\Phi: P(M) \rightarrow P(N)$ is a lattice isomorphism. Then there exist a real ∗-isomorphism $\psi:M \rightarrow N$ and an invertible element $y \in LS(N)$ such that $\Phi(p) =l(y \psi(p)), p \in P(M).$ Here P(M) is M's proj. latt., l(x) is x's left support projection, and LS(M) is the alg. of "locally measurable operators" of M. In type III factor case, LS(M) = M. $\endgroup$ Dec 11, 2020 at 18:33

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