On a Question of N. Th. Varopoulos and the constant $C_2(n)$

Let ${\mathbb{C}}_k[Z_1,\dots ,Z_n]$ denote the set of all polynomials of degree at most $k$ in $n$ complex variables and ${\mathcal{C}}_n$ denote the set of all $n$ - tuple $\mathit{T}=(T_1,\dots ,T_n)$ of commuting contractions on some Hilbert space $\mathbb{H}\mathrm{.}$ The interesting inequality $$K_G^{\mathbb{C}}\le \underset{n\to \mathrm{\infty }}{\lim }{C}_2(n)\le 2K_G^{\mathbb{C}},$$ where $$C_k(n)=\sup \{\mathrm{\parallel }p(\mathit{T})\mathrm{\parallel }:\mathrm{\parallel }p{\mathrm{\parallel }}_{{\mathbb{D}}^n,\mathrm{\infty }}\le 1,p\in {\mathbb{C}}_k[Z_1,\dots ,Z_n],\mathit{T}\in {\mathcal{C}}_n\}$$ and $K_G^{\mathbb{C}}$ is the complex Grothendieck constant, is due to Varopoulos. We answer a long–standing question by showing that the limit ${\lim }_{n\to \mathrm{\infty }}\frac{C_2(n)}{K_G^{\mathbb{C}}}$ is strictly bigger than $1.$ Let ${\mathbb{C}}_2^s[Z_1,\dots ,Z_n]$ denote the set of all complex valued homogeneous polynomials $p(z_1,\dots ,z_n)$ $={\sum }_{j,k=1}^n{a}_{jk}{z}_j{z}_k$ of degree two in $n$ - variables, where $(\text{\hspace{0.17em}⁣}(a_{jk})\text{\hspace{0.17em}⁣})$ is a $n\times n$ complex symmetric matrix. For each $n\in \mathbb{N},$ define the linear map $\mathscr{A}n:\big (\mathbb C_2^s[Z_1,\ldots , Z_n],|\cdot|{\mathbb D^n, \infty}\big ) \to \big (M_n, |\cdot |_{\infty \to 1}\big )$tobe$\mathscr{A}n\big (p) = (!(a{jk})!).$Weshowthatthesupremum(over$n$) of the norm of the operators ${\mathcal{A}}_n;n\in \mathbb{N},$ is bounded below by the constant ${\pi }^2\mathrm{/}8.$ Using a class of operators, first introduced by Varopoulos, we also construct a large class of explicit polynomials for which the von Neumann inequality fails. We prove that the original Varopoulos–Kaijser polynomial is extremal among a, suitably chosen, large class of homogeneous polynomials of degree two. We also study the behaviour of the constant $C_k(n)$ as $n\to \mathrm{\infty }\mathrm{.}$