Rational dilation problems associated with constrained algebras

TL;DR

This paper investigates the failure of rational dilation on certain constrained subalgebras of the disk algebra, providing new test functions, an interpolation theorem, and examples of non-contractive representations.

Contribution

It demonstrates rational dilation failure on broad classes of constrained algebras, introduces minimal test functions, and constructs non-contractive representations with contractions.

Findings

Rational dilation fails on distinguished varieties associated with constrained subalgebras.

A minimal set of test functions for these algebras is identified.

Existence of non-contractive unital representations with contraction generators is proved.

Abstract

It is shown that rational dilation fails on broad collection of distinguished varieties associated to constrained subalgebras of the disk algebra of the form C + B A(D), where B is a finite Blaschke product with two or more zeros. This is accomplished in part by finding a minimal set of test functions. In addition, an Agler-Pick interpolation theorem is given and it is proved that there exist Kaijser-Varopoulos style examples of non-contractive unital representations where the generators are contractions.