On a Question of N. Th. Varopoulos and the constant $C_2(n)$
Let $\mathbb C_k[Z_1,\ldots, Z_n]$ denote the set of all polynomials of degree at most $k$ in $n$ complex variables and $\mathscr{C}_n$ denote the set of all $n$ - tuple $\boldsymbol T=(T_1,\ldots,T_n)$ of commuting contractions on some Hilbert space $\mathbb{H}.$ The interesting inequality $$K_{G}^{\mathbb C}\leq \lim_{n\to \infty}C_2(n)\leq 2 K^\mathbb C_G,$$ where \[C_k(n)=\sup\big\{\|p(\boldsymbol T)\|:\|p\|_{\mathbb D^n,\infty}\leq 1, p\in \mathbb C_k[Z_1,\ldots,Z_n],\boldsymbol T\in\mathscr{C}_n \big\}\] 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 $\lim_{n\to\infty} \frac{C_2(n)}{K^\mathbb C_G}$ is strictly bigger than $1.$ Let $\mathbb C_2^s[Z_1,\ldots , Z_n]$ denote the set of all complex valued homogeneous polynomials $p(z_1,\ldots,z_n)$ $=\sum_{j,k=1}^{n}a_{jk}z_jz_k$ of degree two in $n$ - variables, where $(\!(a_{jk})\!)$ is a $n\times n$ complex symmetric matrix. For each $n\in\mathbb{N},$ define the linear map $\mathscr{A}_n:\big (\mathbb C_2^s[Z_1,\ldots , Z_n],\|\cdot\|_{\mathbb D^n, \infty}\big ) \to \big (M_n, \|\cdot \|_{\infty \to 1}\big )$ to be $\mathscr{A}_n\big (p) = (\!(a_{jk})\!).$ We show that the supremum (over $n$) of the norm of the operators $\mathscr{A}_n;\,n\in\mathbb{N},$ is bounded below by the constant $π^2/8.$ 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 $n \to \infty.$