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

Abstract

For $0<θ<π$, let $K_θ=\{e^{is}:|s|\leθ\}$ and $E_θ=\operatorname{span}_{\mathbb C}\{1,z\}\subset C(K_θ)$. We prove that every contractive linear map from $E_θ$ into $\mathcal B(H)$ admits a factorization through a unitary operator whose spectrum is contained in $K_θ$, and is therefore completely contractive. Consequently, $E_θ$ has a unique operator space structure. For $0<θ,φ<π$, we also show that $E_θ$ and $E_φ$ are isometrically isomorphic if and only if $θ=φ$, and that none of these spaces is isometrically isomorphic to $\ell_1^2$ or $\ell_\infty^2$. 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.