  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
Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc $\mathbb D^n.$
In the special case of $n=2$ (which follows from Ando's theorem as well), this
necessary condition is made explicit. An alternative approach to the
Carath\'{e}odory-Fej\'{e}r interpolation problem, in the special case of $n=2,$
adapting a theorem of Kor\'{a}nyi and Puk\'{a}nzsky is given. As a consequence,
a class of polynomials are isolated for which a complete solution to the
Carath\'{e}odory-Fej\'{e}r interpolation problem is easily obtained. A natural
generalization of the Hankel operators on the Hardy space of $H^2(\mathbb T^2)$
then becomes apparent. Many of our results remain valid for any $n\in \mathbb
N,$ however, the computations are somewhat cumbersome for $n>2$ and are
omitted. The inequality $\lim_{n\to \infty}C_2(n)\leq 2 K^\mathbb C_G$, where
$K_G^\mathbb C$ is the complex Grothendieck constant and
\[C_2(n)=\sup\big\{\|p(\boldsymbol T)\|:\|p\|_{\mathbb D^n,\infty}\leq 1,
\|\boldsymbol T\|_{\infty} \leq 1 \big\}\] is due to Varopoulos. Here the
supremum is taken over all complex polynomials $p$ in $n$ variables of degree
at most $2$ and commuting $n$ - tuples $\boldsymbol T:=(T_1,\ldots,T_n)$ of
contractions. We show that \[\lim_{n\to \infty}C_2(n)\leq \frac{3\sqrt{3}}{4}
K^\mathbb C_G\] obtaining a slight improvement in the inequality of Varopoulos.
We show that the normed linear space $\ell^1(n),$ $n>1,$ has no isometric
embedding into $k\times k$ complex matrices for any $k\in \mathbb N$ and
discuss several infinite dimensional operator space structures on it.

Functional Analysis