AI 中文总结
该研究证明超有限$II_1$因子上满足特定Kruglov性质的对称函数空间对应的对称算子空间，与超有限$II_\text{infty}$因子上的对称空间存在线性拓扑同构，建立了经典结果的非交换版本并回答了相关问题。
AI 中文摘要
本文的主要目标是证明超有限$II_1$因子$\text{mathcal{R}}$上的一大类对称空间（元素）与超有限$II_\text{infty}$因子$\text{mathcal{R}\bar{\bigotimes}\text{mathcal{L}}(H)$上的某些对称算子空间之间存在线性拓扑同构。具体而言，我们证明，对于任意满足$E(0,1)$及其Köthe对偶均具有Kruglov性质的对称函数空间$E(0,1)$（按Lindenstrauss与Tzafriri的定义），对称算子空间$E(\text{mathcal{R})$同构于某一对称空间$Z_E^2(\text{mathcal{R}\bar{\bigotimes}\text{mathcal{L}}(H))$。该结果建立了Johnson、Maurey、Schechtman与Tzafriri著名结果的非交换版本，并回答了Mityagin提出的问题的非交换版本。
英文摘要
The primary aim of this paper is to show a linear topological isomorphism between (elements of a wide class {of}) symmetric spaces over the hyperfinite $II_1$ factor $\mathcal{R}$ and certain symmetric operator space over the hyperfinite $II_\infty $ factor $\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. Precisely, we show that for any symmetric function space $E(0,1)$ (in the sense of Lindenstrauss and Tzafriri) such that both $E(0,1)$ and its Köthe dual have the Kruglov property, the symmetric operator space $E(\mathcal{R})$ is isomorphic to some symmetric space $Z_E^2(\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. This result establishes a noncommutative version of a well-known result due to Johnson, Maurey, Schechtman and Tzafriri, and answers the noncommutative version of a question due to Mityagin.
