Optimal square size for separating the operator and completely bounded norms of Schur multipliers on $S^4$

Abstract

We determine the smallest square size at which the operator norm and the completely bounded norm of a Schur multiplier on $S^4$ can differ. More precisely, for $A=\left[\begin{array}{ccc}1& 2& 0\\ 1& -2& 0\\ 2& 2\mathrm{i}& 0\end{array}\right]$ we prove \[ \|M_A\|_{S^4_3\to S^4_3}^2\leq\frac{1119}{250} <\left(\frac{18983532}{947485}\right)^{1/2} \leq \|\mathrm{Id}_{S^4_2}\otimes M_A\|_{S^4_6\to S^4_6}^2. \] We also prove that no $2\times 2$ Schur multiplier can have unequal operator and completely bounded norms on $S^4$, so square size three is optimal. In addition, we determine the exact ordinary norm of the previous multiplier analytically and give an independent computer-assisted certification of a rational upper bound.

arXiv comment32 pages