Which complete orthomodular lattices arise from von Neumann algebras?

Question

Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice.

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Question 1: Is the construction $A \mapsto \Pi(A)$ a functor from von Neumann algebras to complete orthomodular lattices?

For this to make sense, I should say what a morphism of von Neumann algebras is -- but I'm not sure what the appropriate choice of morphism is. I should also say what a morphism of complete orthomodular lattices is, and here there is a natural guess -- a morphism $f: L \to M$ should be a function which preserves $(-)^\perp$ and sups (equivalently, infs).

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The construction $A \mapsto \Pi(A)$ makes sense even when $A$ is just a $C^\ast$-algebra, except that we only know that $\Pi(A)$ is an orthoposet (maybe an ortholattice?).

Question 2: Let $A$ be a $C^\ast$-algebra, and suppose that $\Pi(A)$ is a complete orthomodular lattice. Does it follow that $A$ is a von Neumann algebra?

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Question 3: Assuming the answer to Question 1 is "yes", let $\Pi: vNA \to COML$ be the above functor from von Neumann algebras to complete orthomodular lattices.

(a.) Is the functor $\Pi$ fully faithful?

(b.) Does the functor $\Pi$ have a left or right adjoint?

(c.) What is the essential image of the functor $\Pi$?

(3c) is the title question, of course. It seems there was substantial interest in this question in the '60's and '70's. I've come across work by Holland, Topping, and Fillmore identifying various properties of orthomodular lattices of the form $\Pi(A)$ not shared by all complete orthomodular lattices. I believe there is a characterization those lattices of the form $\Pi(A)$ where $A = B(H)$ is the algebra of all bounded operators on a Hilbert space $H$ (though I'm having trouble tracking down where I read this). But the trail seems to go cold after this period. Did the question just go out of fashion?

I do have the sense that there might be a characterization which has something to do with "having a full set of states". For instance, suppose we define a state on a complete orthomodular lattice $L$ to be a function $f: L \to [0,1]$ which preserves order and directed sups, is additive ($f(0) = 0$, $f(a \vee b) = f(a) + f(b)$ if $a \perp b$), and is normalized so that $f(1) = 1$ (I think maybe this should really be called a "normal state" or something like that?). Say that $L$ has a full set of states if for $a, b \in L$ we have $a \leq b \Leftrightarrow f(a) \leq f(b)$ for all states $f$. Then I believe that if $L = \Pi(A)$, then $L$ has a full set of states. Is the converse true?

Answer 1

Question 1: Yes, if you take the von Neumann algebra morphisms to be normal $*$-homomorphisms. Restricting any such map to the projections will preserve sups and orthocomplements.

Question 2: No, this already fails in the commutative case. Look up "abelian AW*-algebra" or see the discussion of Stonean spaces in volume one of Kadison-Ringrose.

Question 3: (a) Any von Neumann algebra is generated by its projections, so the functor is faithful. It is not full, just look at $M_2(\mathbb{C})$ where the lattice of projections contains a $0$ and a $1$ and $2^{\aleph_0}$ incomparable elements between those two. There are all kinds of morphisms from this lattice to itself that don't extend linearly.

Question 3: (b) I'm weak on category theory, but I'd imagine the $M_2(\mathbb{C})$ example falsifies both possibilities.

Question 3: (c) I don't think there's any good answer to this question, and that's the reason research in this direction petered out. The idea of asking for a full set of normal states is good, and this does answer the question in the abelian case (again, see Kadison-Ringrose), but I doubt it will work in general. There's just no way to linearize these states on lattices. But counterexamples will take some work.

Answer 2

I'm actually starting to think that the situation is not as bad as indicated in Nik Weaver's answer. Some notation:

• If $\mathcal A$ is a $C^\ast$-algebra or JB-algebra (resp. von Neumann algebra or JBW-algebra), let $\Pi(\mathcal A)$ be its ortholattice of projections. Let $State(\mathcal A)$ (resp. $State_n(\mathcal A)$) be its convex space of states (resp. normal states). If $L$ is a complete orthomodular lattice, let $State_n(L)$ be the convex space of normal states on $L$. If $K$ is a convex space, let $Aff(K)$ be the space of bounded ($\mathbb R$-valued) affine functionals on $K$.

We are led to the following considerations:

1. By Gleason's Theorem, if $\mathcal A$ is a von Neumann algebra without a direct summand of $M_2(\mathbb C)$, the natural restriction map $State_n(\mathcal A) \to State_n(\Pi(\mathcal A))$ is an isomorphism.

So in order to recover $\mathcal A$ from $\Pi(\mathcal A)$, it typically suffices to recover it from $State_n(\mathcal A)$.

2. Let $\mathcal A$ be a JB-algebra (resp. JBW-algebra). Then as Banach spaces, there is a natural isomorphism $Aff(State(\mathcal A)) \cong \mathcal A$ (resp. $Aff(State_n(\mathcal A)) \cong \mathcal A$). Moreover, Alfsen and Schultz have shown (see Geometry of State Spaces of Operator Algebras and predecessor works) how to recover the Jordan multiplication on $\mathcal A$ purely from the convex structure on $State(\mathcal A)$ (resp. $State_n(\mathcal A)$), including giving a characterization of those convex spaces for which this can be done. (Briefly, they identify conditions under which a functional calculus can be developed. Since a Jordan multiplication is determined by its squaring operation, this yields a candidate Jordan multiplication, and they identify further conditions ensuring that the candidate multiplication is bilinear and hence actually is a Jordan multiplication.)

Thus if $\mathcal A$ is a $C^\ast$-algebra (resp. von Neumann algebra), then the self-adjoint part of $\mathcal A$ and its Jordan multiplication can be reconstructed from $State_n(\mathcal A)$. But because $\mathcal A$ is not typically isomorphic to its opposite algebra (a fact which taking $State_n(\mathcal A)$ forgets), there is no hope in general to recover the algebra multiplication on $\mathcal A$. However,

3. Alfsen and Schultz have shown that any JB-algebra (resp. JBW-algebra) $\mathcal A$ is "locally" the self-adjoint part of at most two different $C^\ast$-algebras (resp. von Neumann algebras), which are opposite to each other. Moreover, they have characterized in terms of "orientation" data on the convex space $State(\mathcal A)$ (resp. $State_n(\mathcal A)$) exactly when a global algebra multiplication can be found, and how to distinguish between the different possible choices.

Putting this together, we see that the functor $\mathcal A \mapsto State_n(\mathcal A)$ is "almost injective" on isomorphism classes, its essential image can be explicitly characterized, and the additional data to make the functor 1-to-1 on isomorphism classes has been described. This leaves me suspecting that with the right notion of morphism of oriented convex space, the functor becomes fully faithful.

This is not quite the same as characterizing things in terms of $\Pi(\mathcal A)$, but if we exclude von Neumann algebras with summands of $M_2(\mathbb C)$, then $State_n$ factors through $\Pi$. In particular, we have a description of the essential image of $\Pi$, in terms of the normal state space of a complete orthomodular lattice $L$. It might be nice to have a more direct description, but the state space of $L$ is a pretty natural invariant to consider, so one might also be satisfied with this.