The Caratheodory-Fejer Interpolation Problems and the Von-Neumann inequality

Abstract

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carathéodory-Fejérinterpolation problem on the polydisc$\D^n. $ in the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. We discuss an alternative approach to the Carathéodory-Fejérinterpolation problem, in the special case of $n=2$, adapting a theorem of Korányi and Pukánzsky. As a consequence, a class of polynomials are isolated for which a complete solution to the Carathéodory-Fejér interpolation problem is easily obtained. Many of our results remain valid for any $n\in \mathbb N$, however the computations are somewhat cumbersome. Recall the well known inequality due to Varopoulos, namely, $\lim{n\to \infty}C_2(n)\leq 2 K^\C_G$, where $K^\C_G$ is the complex Grothendieck constant and \[C_2(n)=sup\{\|p(\boldsymbolT)\|:\|p\|_{\D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1\}.\] Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples$\boldsymbolT:=(T_1,\ldots,T_n)$ of contractions. We show that \[\lim_{n\to \infty} C_2 (n)\leq \frac{3\sqrt{3}}{4} K^\C_G\] obtaining a slight improvement in the inequality of Varopoulos. We also discuss several finite and infinite dimensional operator space structures on $\ell^1(n) $, $n>1. $