Contractive and Completely Contractive Homomorphisms of Planar Algebras

We consider contractive homomorphisms of a planar algebra A(Ω) over a finitely connected bounded domain Ω ⊆ C and ask if they are necessarily completely contractive. We show that a homomorphism ρ : A(Ω) → B(H) for which dim(A(Ω)/ ker ρ) = 2 is the direct integral of homomorphisms ρT induced by operators on two dimensional Hilbert spaces via a suitable functional calculus ρT : f 7→ f(T ), f ∈ A(Ω). It is well-known that contractive homomorphisms ρT , induced by a linear transformation T : C → C are necessarily completely contractive. Consequently, using Arveson’s dilation theorem for completely contractive homomorphisms, one concludes that such a homomorphism ρT 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 ρT 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 ρT of 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 ρT of the planar algebra A(Ω), we construct a dilation.