AI 中文总结
本文证明一族二维复Banach空间$E_\theta$具有唯一算子空间结构，并构造出不可数多个互不等距同构的此类空间，否定回答了Pisier的一个问题。
AI 中文摘要
对于$0<\theta<\pi$，令$K_\theta=\{e^{is}:|s|\le\theta\}$且$E_\theta=\operatorname{span}_{\mathbb C}\{1,z\}\subset C(K_\theta)$。我们证明从$E_\theta$到$\mathcal B(H)$的每个压缩线性映射都可通过一个谱包含在$K_\theta$中的酉算子进行分解，因此是完全压缩的。由此，$E_\theta$具有唯一的算子空间结构。对于$0<\theta,\varphi<\pi$，我们还证明$E_\theta$与$E_\varphi$等距同构当且仅当$\theta=\varphi$，并且这些空间均不与$\ell_1^2$或$\ell_\infty^2$等距同构。因此我们得到不可数多个互不等距同构的二维复Banach空间，它们具有唯一的算子空间结构，从而对Pisier的一个问题给出否定回答。
英文摘要
For $0<θ<π$, let $K_θ=\{e^{is}:|s|\leθ\}$ and $E_θ=\operatorname{span}_{\mathbb C}\{1,z\}\subset C(K_θ)$. We prove that every contractive linear map from $E_θ$ into $\mathcal B(H)$ 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 $\ell_1^2$ or $\ell_\infty^2$. 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.
Comments13 pages. Comments are welcome
