Contractive and completely contractive maps over planar algebras

Abstract

We consider contractive homomorphisms of a planar algebra $\mathcal{A}(\Omega )$ over a finitely connected bounded domain $Ω\subseteq \C$ 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.

15 pages