伪闭W*-概率空间
Existentially closed W*-probability spaces
摘要：
本文研究了W $^*$ -概率空间的几个模型论问题，即具有忠实正规态的$σ$ -有限von Neumann代数.本文首先研究了存在闭的W $^*$ -空间，证明了关于这类空间的几个结构性结果，包括它们是张量吸收Araki-Woods因子$R_\infty$的III型$_1$因子.我们还研究了具有Kirchberg QWEP性质的W $^*$ -概率空间的限制类中的存在闭对象，证明了$R_\infty$本身就是这类空间中的存在闭对象.我们关于存在闭概率空间的结果意味着III型$_1$因子类形成了一个$\forall_2$ -可公理化类.我们发现，对于$λ\in (0,1)$，类的第三$_λ$因素是不是$\forall_2$ -axiomatizable，但$\forall_3$ -axiomatizable，后者的结果使用一个版本的凯斯勒的三明治定理适应连续逻辑。最后，我们讨论了III类$_λ$因子的初等等价性。利用Boutonnet，Chifan和Ioana的结果，我们证明了，对于任何$λ\in (0,1)$，有一个家庭的两两非elementarily等价的III $_λ$因素的大小连续。虽然我们不能证明相同的结果III $_1$因素，我们表明，至少有三个成对的非elementarily等价的III $_1$因素表明，类的充分的因素下保持初等等价。
英文摘要：
We study several model-theoretic aspects of W$^*$-probability spaces, that is, $σ$-finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W$^*$-spaces and prove several structural results about such spaces, including that they are type III$_1$ factors that tensorially absorb the Araki-Woods factor $R_\infty$. We also study the existentially closed objects in the restricted class of W$^*$-probability spaces with Kirchberg's QWEP property, proving that $R_\infty$ itself is such an existentially closed space in this class. Our results about existentially closed probability spaces imply that the class of type III$_1$ factors forms a $\forall_2$-axiomatizable class. We show that for $λ\in (0,1)$, the class of III$_λ$ factors is not $\forall_2$-axiomatizable but is $\forall_3$-axiomatizable; this latter result uses a version of Keisler's Sandwich theorem adapted to continuous logic. Finally, we discuss some results around elementary equivalence of III$_λ$ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any $λ\in (0,1)$, there is a family of pairwise non-elementarily equivalent III$_λ$ factors of size continuum. While we cannot prove the same result for III$_1$ factors, we show that there are at least three pairwise non-elementarily equivalent III$_1$ factors by showing that the class of full factors is preserved under elementary equivalence.
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