Contractive and completely contractive maps over planar algebras

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