Let $ H$ be a Hilbert space and let $ M$ be a densely defined operator $ D(M) \subset H \to H$ . It is closable iff its adjoint $ M^{\star}$ is densely defined, and then its closure $ \overline{M}$ is $ M^{\star \star}$ . Let $ \mathcal{M}$ be the smallest von Neumann algebra that $ \overline{M}$ is affiliated with; it is called the von Neumann algebra generated by $ M$ .

Question 1: Is there a bounded operator $ X \in B(H)$ such that $ W^{\star}(X) = \mathcal{M}$ ?

In other words: Can a von Neumann algebra generated by a densely defined closable operator, be also generated by a bounded operator?

Question 2: Is there a way to generalize the generation of a von Neumann algebra to any densely defined operator (i.e. non necessarily closable)?

If an answer to Question 1 gives a process defining $ X$ from $ M$ and if this process works for any densely defined operator, that would also answer Question 2.

The motivation comes from the densely defined operator associated to an integer map $ m: \mathbb{N} \to \mathbb{N}$ (i.e. $ M: \mathbb{C}[\mathbb{N}] \subset H \to H$ with $ Me_n = e_{m(n)}$ ) such that $ \exists n \in \mathbb{N}$ with $ m^{-1}(\{ n\})$ infinite.

As a non-obvious example, consider Conway’s game of life and let $ S$ be the set of states of the grid with only finitely many alive cells. Then the application of the rules produces a map $ r:S \to S$ , which can be reformulated into a map $ m: \mathbb{N} \to \mathbb{N}$ because $ S$ is countable infinite. A problem is that the vacuum state (i.e. all cells dead) has an infinite pre-image (and so is any state of $ r(S)$ ). A positive answer to Question 2 would give a way to generate a von Neumann algebra $ \mathcal{M}$ from Conway’s game of life (or any other cellular automaton); if so, what is $ \mathcal{M}$ ?