Dilation on an annulus and von Neumann's inequality on certain varieties in the biball
Sourav Pal, Nitin Tomar
TL;DR
The paper provides an alternative route to Agler’s rational dilation on the annulus by exploiting Dritschel–McCullough's dilation framework and Arveson’s dilation theory, and then transfers the problem to the principal variety $Z(q)\cap\overline{\mathbb B}_2$ via the κ-map to the biball. It shows an $\mathbb A_r$-contraction $T$ is characterized by $\kappa(T)$ having a spectral set on the principal variety, and it develops a robust correspondence between $\mathbb A_r$-contractions, $C_\alpha$, and $C_{1,r}$ with spherical and $\mathbb B_2$-contractions, including multiple proofs of Cohen–Bello–Yakubovich type results. The work then analyzes minimal spectral and von Neumann sets, revealing that the natural annulus sets are not minimal for several classes, while the unit disk and principal varieties supplied by the κ-map yield minimal von Neumann sets in the corresponding frameworks. Parallel theories are developed for the quantum annulus with $SA_r, PA_r, QA_r$, extending the κ-map approach via $\kappa_0$ and a related polynomial variety $Z(q_0)\cap\overline{\mathbb B}_2$, and establishing minimal spectral sets and dilation principles in that setting. Overall, the paper ties dilation theory on annuli to polyball and spectral-set geometry, offering unified proofs and broader minimal-set results across classical and quantum annulus contexts.
Abstract
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.