Contractive homomorphisms and tensor product norms

Abstract

For any complex domain Ω, one can ask if all contractive algebra homomorphisms ofA(Ω) (into the algebra of Hilbert space operators) are completely contractive or not. By Ando's Theorem, this has an affirmative answer for Ω =D 2, the bi-disc-while the answer is unknown for Ω =D 2, the unit ball of ℂ2 with ℓ1-norm. In this paper, we consider a special class of homomorphisms associated with any bounded complex domain; this well known construct generalizes Parrott's example. Our question has an affirmative answer for homomorphisms in this class with Ω = (ℓ1(2))1. We show that there are many domains in ℓ2 for which the question can be answered in the affirmative by reducing it to that of Ω =D 2 or (ℓ1(2))1. More generally, the question for an arbitrary Ω can often be reduced to the case of the unit ball of an associated finite dimensional Banach space. If we restrict attention to a smaller subclass of homomorphisms the question for a Banach ball becomes equivalent to asking whether in the analogue of Grothendieck's inequality, in this Banach space, restricted to positive operators, the best constant is = 1 or not. We show that this is indeed the case for Ω =D 2,D 3 or the dual balls, but not forD n or its dual forn≥4. Thus we isolate a large class of homomorphisms ofA(D 3) for which contractive implies completely contractive. This has many amusing relations with injective and projective tensor product norms and with Parrott's example.