Contractive and completely contractive maps over planar algebras

We consider contractive homomorphisms of a planar algebra ${\mathcal A}(\Omega)$ over a finitely connected bounded domain $\Omega \subseteq \C$ and ask if they are necessarily completely contractive. We show that a homomorphism $\rho:{\mathcal A}(\Omega) \to {\mathcal B}(\mathcal H)$ for which $\dim({\mathcal A}(\Omega)/\ker \rho) = 2$ is the direct integral of homomorphisms $\rho_T$ induced by operators on two dimensional Hilbert spaces via a suitable functional calculus $\rho_T: f \mapsto f(T), f\in {\mathcal A}(\Omega)$. It is well-known that contractive homomorphisms $\rho_T$, induced by a linear transformation $T:\C^2 \to \C^2$ 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 ${\mathcal A}(\Omega)$ 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 ${\mathcal A}(\Omega)$, we construct a dilation.