Assume that
$M$ is a paracompact Hausdorff space (second countable, if necessary)
$\mu$ is a Borel measure on $M$ (Radon or regular, if necessary)
Let $Homeo(M)$ be the group of homeomorphisms $\phi:M\to M$, and let $Homeo(M,\mu)\subset Homeo(M)$ be the monoid of such $\phi$ that the Radon-Nikodym derivative $d\mu/d\mu_\phi\in L^\infty(M,\mu)$, where $\mu_\phi=\mu\circ\phi$. Consider the regular representation of $Homeo(M,\mu)$ on the Hilbert space $L^2(M,\mu)$, $$ T_\phi f(x)=f(\phi(x)). $$ I guess all $T_\phi\in\mathcal{L}(L^2(M,\mu))$ (bounded) for $\phi\in Homeo(M,\mu)$. Unless $\phi$ is an isometry for $\mu$ in which case $\mu_\phi=\mu$ and $T_\phi$ is unitary, $T_\phi$ is hardly goig to be normal, so no reference to spectral theory can be made.
Question: Consider the complex subalgebra $\mathbb{C}[T_{Homeo(M,\mu)}]\subset\mathcal{L}(L^2(M,\mu))$ generated by the image of $Homeo(M,\mu)$ through $T$. Is this subalgebra dense in the weak operator topology?
Comments are welcome. Answers for special cases as $M$ being a manifold and $Diff(M)$ instead of $Homeo(M)$ are welcome as well. References would be most appreciated. Thank you.