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arXiv · 2610.26738

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

Rajeev Gupta·Arpita Mal·Gadadhar Misra·Samya Kumar Ray
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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.

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Rajeev Gupta, Arpita Mal, Gadadhar Misra, Samya Kumar Ray. 2026-10-19. A two-dimensional subspace of $\ell_\infty^3$ has a unique operator space structure if and only if it is isometric to $\ell_\infty^2$. https://arxiv.org/abs/2610.26738

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