超有限II_1因子理论的小片段性质
Properties expressible in small fragments of the theory of the hyperfinite II_1 factor
摘要：
我们证明了任何具有与超有限II $_1$因子$\mathcal{R}$相同的4-量词理论的II $_1$因子满足Popa阶乘换位嵌入问题（FCEP）的结论并具有Brown性质.这些结果改进了最近的结果证明相同的结论下，更强的假设，该因素实际上是基本等价于$\mathcal{R}$。本着同样的精神，我们改进了第一作者最近的一个结果，该结果表明，如果（1）在一个性质（T）基上的可嵌入因子的合并自由积再次是可嵌入的，并且（2）$\mathcal{R}$是一个无限一般的可嵌入因子，则FCEP对所有性质（T）因子都成立。本文证明了第（2）项可以弱化为假设$\mathcal{R}$具有与无限类属可嵌入因子相同的3-量词理论。
英文摘要：
We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $\mathcal{R}$. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $\mathcal{R}$ is an infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that $\mathcal{R}$ has the same 3-quantifier theory as an infinitely generic embeddable factor.
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