Contractive and Completely Contractive Homomorphisms of Planar Algebras

We consider contractive homomorphisms of a planar algebra $A(\Omega)$ over a nitely connected bounded domain $\Omega \subseteq C$ and ask if they are necessarily completely contractive. We show that a homomorphism $\rho : A(\Omega) \rightarrow B(H)$ for which $dim(A( \Omega )= ker \rho) = 2$ is the direct integral of homomorphisms $\rho T$ induced by operators on two-dimensional Hilbert spaces via a suitable functional calculus $\rho T : f \mapsto f(T)$; $f \in A(\Omega)$. It is well known that contractive homomorphisms $\rho T$ induced by a linear transformation $T : C^2 \rightarrow C^2$ are necessarily completely contractive. Consequently, using Arveson's dilation theorem for completely contractive homomorphisms, one concludes that such a homomorphism $\rho T$ possesses a dilation. In this paper, we construct this dilation explicitly. In view of recent examples discovered by Dritschel and McCullough, we know that not all contractive homomorphisms $\rho T$ are completely contractive even if T is a linear transformation on a finite-dimensional Hilbert space. We show that one may be able to produce an example of a contractive homomorphism $\rho T$ of $A(\Omega)$ which is not completely contractive if an operator space which is naturally associated with the problem is not the MAX space. Finally, within a certain special class of contractive homomorphisms $\rho T$ of the planar algebra $A(\Omega)$, we construct a dilation