The set of projections of a von Neumann algebra forms an orthomodular lattice.
Is there some kind of correspondence between such lattices and von Neumann algebras? More specifically, does there exist a theorem that states
"Let $\mathcal{H}$ be some Hilbert space and $\mathcal{P}$ be some family of projections on $\mathcal{H}$. Then the following are equivalent:
-The set $\mathcal{P}$ is an orthomodular lattice.
-The set $\mathcal{P}$ is the set of all projections of some von Neumann algebra $\mathcal{M}$. "?
No, sets of projections that are the projections of a von Neumann algebra are much more special than just being orthomodular lattices (or even more specifically, sub-orthomodular complete lattices of the orthomodular complete lattice of all projections on $\mathcal{H}$). For a really simple example, consider $\mathcal{H}=\mathbb{C}^2$ and let $p$ and $q$ be two rank $1$ projections on $\mathcal{H}$ that are neither equal nor orthogonal complements of each other. Then $\{0,1,p,q,1-p,1-q\}$ is a subset of the lattice of all projections on $\mathcal{H}$ that is closed under joins, meets, and complements. However, it is not the projections of any von Neumann algebra: the von Neumann algebra it generates is the entire matrix algebra $M_2(\mathbb{C})$ which has many more projections.