  We use results and techniques from Werner's ``quantum harmonic analysis'' to
show that $G$-invariant Toeplitz operators are norm dense in $G$-invariant
Toeplitz algebras for all subgroups $G$ of the affine unitary group $U_n\ltimes
\mathbb{C}^n$. Additionally, we prove that the quasi-radial Toeplitz operators
are dense in the quasi-radial Toeplitz algebra over the Bergman space
$\mathcal{A}^2(\mathbb{B}^n)$ and provide a constructive proof of SOT density
of Toeplitz operators in the space of all bounded operators.
