PreprintCC BY 4.0arXiv
A two-dimensional subspace of ℓ∞3 has a unique operator space structure if and only if it is isometric to ℓ∞2
Abstract
We show that a two-dimensional complex subspace of ℓ∞3 carries a unique operator space structure if and only if it is isometric to ℓ∞2; the ``if'' half is the Andô's theorem. For each of the remaining spaces we exhibit an explicit non-normal Y ∈ ℳ2, 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 ℓ∞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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BibTeX
@misc{gupta2026twodimensional,
title = {{A two-dimensional subspace of $\ell_\infty^3$ has a unique operator space structure if and only if it is isometric to $\ell_\infty^2$}},
author = {Rajeev Gupta and Arpita Mal and Gadadhar Misra and Samya Kumar Ray},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26738},
eprint = {2610.26738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26738},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A two-dimensional subspace of ℓ_∞^3 has a unique operator space structure if and only if it is isometric to ℓ_∞^2
AU - Rajeev Gupta
AU - Arpita Mal
AU - Gadadhar Misra
AU - Samya Kumar Ray
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26738
UR - https://arxiv.org/abs/2610.26738
AB - We show that a two-dimensional complex subspace of ℓ_∞^3 carries a unique operator space structure if and only if it is isometric to ℓ_∞^2; the ``if'' half is the Andô's theorem. For each of the remaining spaces we exhibit an explicit non-normal Y ∈ 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 ℓ_∞^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.
ER -