Schur multipliers of Schatten--von Neumann classes

We study in this paper properties of Schur multipliers of Schatten von Neumann classes ${\mathit{S}}_p$. We prove that for $p\le 1$, Schur multipliers of ${\mathit{S}}_p$ are necessarily completely bounded. We also introduce for $p\le 1$ a scale ${\mathcal{W}}_p$ of tensor products of ${\mathrm{\ell }}^{\mathrm{\infty }}$ and prove that matrices in ${\mathcal{W}}_p$ are Schur multipliers of ${\mathit{S}}_p$. We compare this sufficient condition with the sufficient condition of membership in the $p$-tensor product of ${\mathrm{\ell }}^{\mathrm{\infty }}$ spaces.