In quantum mechanics we have position and momentum operators $P$ and $Q$ acting on $L^2(\mathbb{R})$ in the usual way. I'm wondering what the von Neumann algebra generated by the bounded functions of $P$ and $Q$ is. I.e., what is
$$
\{f(Q), f(P)\,\vert\,f\in L^\infty(\mathbb{R})\}''?
$$
I have a hunch that the answer may have something to do with the [Wigner-Weyl transform](https://en.wikipedia.org/wiki/Wigner%E2%80%93Weyl_transform), but (to my surprise) I have not been able to find any reference treating this question.



von Neumann algebra of canonical commutation relations
The C$^\*$-algebra generated by the exponentials $e^{isQ}$ and $e^{itP}$, $s,t \in \mathbb{R}$, is the CCR algebra. The von Neumann algebra they generate in this representation is all of $B(L^2(\mathbb{R}))$. That is because $e^{isQ}$ is multiplication by $e^{isx}$ and $e^{itP}$ is translation by $t$. Any operator that commutes with multiplication by $e^{isx}$ for all $s$ is a multiplication operator, and the only multiplication operators that commute with all translations are scalar multiples of the identity. So the result follows from the double commutant theorem.


