Contractive homomorphisms and tensor product norms
For any complex domain \(\Omega\), one can ask if all contractive algebra homomorphisms of \({\mathcal A}(\Omega)\) (into the algebra of Hilbert space operators) are completely contractive or not. By Ando's Theorem, this has an affirmative answer for \(\Omega= \mathbb{D}^2\), the bi-disc -- while the answer is unknown for \(\Omega= (\ell^1(2))_1\), the unit ball of \(\mathbb{C}^2\) with \(\ell^1\)-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 \(\Omega= (\ell^1(2))_1\). We show that there are many domains in \(\mathbb{C}^2\) for which the question can be answered in the affirmative by reducing it to that of \(\Omega= \mathbb{D}^2\) or \((\ell^1(2))_1\). More generally, the question for an arbitrary \(\Omega\) 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 \(\Omega= \mathbb{D}^2\), \(\mathbb{D}^3\) or the dual balls, but not for \(\mathbb{D}^n\) or its dual for \(n\geq 4\). Thus we isolate a large class of homomorphisms of \({\mathcal A}(\mathbb{D}^3)\) for which contractive implies completely contractive. This has many amusing relations with injective and projective tensor product norms and with Parrott's example.
injective and projective tensor product norms
scientific article; zbMATH DE number 765284