PreprintCC BY 4.0arXiv
Unique operator space structures on a family of two dimensional Banach spaces
Abstract
For 0<θ<π, let Kθ={eis:|s|≤θ} and Eθ=spanℂ{1,z}⊂ C(Kθ). We prove that every contractive linear map from Eθ into ℬ(H) admits a factorization through a unitary operator whose spectrum is contained in Kθ, and is therefore completely contractive. Consequently, Eθ has a unique operator space structure. For 0<θ,φ<π, we also show that Eθ and Eφ are isometrically isomorphic if and only if θ=φ, and that none of these spaces is isometrically isomorphic to ℓ12 or ℓ∞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.
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BibTeX
@misc{ray2026unique,
title = {{Unique operator space structures on a family of two dimensional Banach spaces}},
author = {Samya Kumar Ray},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04608},
eprint = {2610.04608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04608},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unique operator space structures on a family of two dimensional Banach spaces
AU - Samya Kumar Ray
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04608
UR - https://arxiv.org/abs/2610.04608
AB - For 0<θ<π, let K_θ={e^is:|s|≤θ} and E_θ=span_ℂ{1,z}⊂ C(K_θ). We prove that every contractive linear map from E_θ into 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 ℓ_1^2 or ℓ_∞^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.
ER -