The Wigner distribution of an operator $A$ is given by
$$ W_A(x,p) :=\frac{1}{2\pi} \int_\mathbb{R} \! dy \, \langle x+y/2| A | x-y/2 \rangle \, e^{ipy}, $$
and associates a function in phase space to an operator on a Hilbert space. See for example Wikipedia page.
I used the convention that $\hbar=1$ although in issues about quantization it is standard to re-insert it. However my question is for fixed $\hbar$.
Namely, what is the (a) largest sub-algebra of operators for which the Wigner transform is exactly an homomorphism. Can one get more than the abelian algebra of differential operators (or multiplication operators)?