A two-dimensional subspace of $\ell_\infty^3$ has a unique operator space structure if and only if it is isometric to $\ell_\infty^2$
Year: 2026
Type: preprint
Abstract: We show that a two-dimensional complex subspace of $\ell_\infty^3$ carries a unique operator space structure if and only if it is isometric to $\ell_\infty^2$; 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 $\ell_\infty^3$ has the $2$-summing property, the $2$-summing property does not imply uniqueness of the operator space structure for two dimensional complex Banach spaces.
Source: arXiv (Cornell University)
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8749306478
OpenAlex ID: W7223067973