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.Comment: Page proof corrections included in this version