Dilation on an annulus and von Neumann's inequality on certain varieties in the biball

We give an alternative proof to Agler’s famous result on success of rational dilation on an annulus by an application of a result due to Dritschel and McCullough. We show interplay between operators associated with an annulus, $C_{1,r}$ or quantum annulus and operator pairs living on a certain variety in ${\mathbb{C}}^2$ and its intersection with the biball. It is shown that the minimal spectral sets and von Neumann’s inequality for these classes $C_{1,r}$, quantum annulus can also be studied via appropriate operator pairs associated with the biball.