Let $H$ be a Hilbert space and let $M$ be a [densely defined operator][1] $D(M) \subset H \to H$. 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][2] that $\overline{M}$ is [affiliated][3] 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][4] 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 $\mathcal{M}$?

  [1]: https://en.wikipedia.org/wiki/Densely_defined_operator
  [2]: https://en.wikipedia.org/wiki/Von_Neumann_algebra
  [3]: https://en.wikipedia.org/wiki/Affiliated_operator
  [4]: https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life