Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps

We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum _{k=1}^{\mathrm{\infty }}⟨u,y_k⟩x_k,$$ then there exist nonzero scalars $\{{\alpha }_k{\}}_{k=1}^{\mathrm{\infty }}$ such that both sequences $\{{\alpha }_k{x}_k{\}}_{k=1}^{\mathrm{\infty }}$ and $\{{\bar{\alpha }}_k^{-1}{y}_k{\}}_{k=1}^{\mathrm{\infty }}$ are frames. Our result has the following equivalent reformulation: if $\mathrm{\Phi }:{\mathrm{\ell }}^{\mathrm{\infty }}\to B(H)$ is a bounded linear map such that for every element of the unit vector basis $e_k$ in ${\mathrm{\ell }}^{\mathrm{\infty }}$ the operator $\mathrm{\Phi }(e_k)$ has rank one, then $\mathrm{\Phi }$ is completely bounded.