  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.
