Unique operator space structures on a family of two dimensional Banach spaces

For $0<\theta <\pi$, let $K_{\theta }=\{e^{is}:\mathrm{\mid }s\mathrm{\mid }\le \theta \}$ and $E_{\theta }={\mathrm{span}}_{\mathbb{C}}\{1,z\}\subset C(K_{\theta })$. We prove that every contractive linear map from $E_{\theta }$ into $\mathcal{B}(H)$ admits a factorization through a unitary operator whose spectrum is contained in $K_{\theta }$, and is therefore completely contractive. Consequently, $E_{\theta }$ has a unique operator space structure. For $0<\theta ,\phi <\pi$, we also show that $E_{\theta }$ and $E_{\phi }$ are isometrically isomorphic if and only if $\theta =\phi$, and that none of these spaces is isometrically isomorphic to ${\mathrm{\ell }}_1^2$ or ${\mathrm{\ell }}_{\mathrm{\infty }}^2$. Thus we obtain uncountably many mutually non-isometric two-dimensional complex Banach spaces with a unique operator space structure, giving a negative answer to a question of Pisier.