Dilations and constrained algebras

TL;DR

This paper investigates the properties of contractive representations of a subalgebra of the disk algebra, revealing cases where they are not completely contractive and providing a characterization for such representations.

Contribution

It demonstrates the existence of contractive but not completely contractive representations for a specific subalgebra and offers a characterization of when contractive representations are completely contractive.

Findings

Existence of contractive but not completely contractive representations for the algebra A.

A characterization of contractive representations of A that are completely contractive.

Unital contractive representations of rational functions with poles off a certain variety are completely contractive.

Abstract

It is well known that unital contractive representations of the disk algebra are completely contractive. Let A denote the subalgebra of the disk algebra consisting of those functions f whose first derivative vanishes at 0. We prove that there are unital contractive representations of A which are not completely contractive, and furthermore provide a Kaiser and Varopoulos inspired example for A and present a characterization of those contractive representations of A which are completely contractive. In the positive direction, for the algebra of rational functions with poles off the distinguished variety V in the bidisk determined by (z-w)(z+w)=0, unital contractive representations are completely contractive.