Contractive and completely contractive maps over planar algebras

Abstract

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