A two-dimensional subspace of ${\mathrm{\ell }}_{\mathrm{\infty }}^3$ has a unique operator space structure if and only if it is isometric to ${\mathrm{\ell }}_{\mathrm{\infty }}^2$

We show that a two-dimensional complex subspace of ${\mathrm{\ell }}_{\mathrm{\infty }}^3$ carries a unique operator space structure if and only if it is isometric to ${\mathrm{\ell }}_{\mathrm{\infty }}^2$; the ``if’’ half is the Andô’s theorem. For each of the remaining spaces we exhibit an explicit non-normal $Y\in {\mathcal{M}}_2$, with $(I,Y)$ an extreme point of the associated matrix ball, for which the induced linear map is contractive but not completely contractive. Since every two-dimensional subspace of ${\mathrm{\ell }}_{\mathrm{\infty }}^3$ has the $2$-summing property, the $2$-summing property does not imply uniqueness of the operator space structure for two dimensional complex Banach spaces.