柔顺性、管状性和嵌入$\mathcal R^ω$
Amenability, tubularity, and embeddings into $\mathcal R^ω$
摘要：
设$M$是可嵌入到超有限$II_1$ -因子的超积$\mathcal R^ω$中的迹von Neumann代数，$X$是$M$的自伴生成元的$n$ -元组.用$Γ(X;m,k,γ)$表示$(m,k,γ)$阶$X$的微观状态空间。我们说$X$是管状的，如果对于任何$ε>0$，存在$m \in \mathbb N$和$γ>0$，使得如果$(x_1,..., x_n), (y_1, ..., y_n) \in Γ(X;m,k,γ),$，则存在$k \times k$酉$u$，对于每个$1 \leq i \leq n.$，满足$|ux_iu^* - y_i|_2 < ε$。我们证明了以下条件是等价的：1）$M$是顺从的（即，单射）。2)$X$是管状的; 3）$M$嵌入到$\mathcal R^ω$中的任何两个嵌入被$\mathcal R^ω$中的酉u共轭。
英文摘要：
Suppose $M$ is a tracial von Neumann algebra embeddable into $\mathcal R^ω$ (the ultraproduct of the hyperfinite $II_1$-factor) and $X$ is an $n$-tuple of selfadjoint generators for $M$. Denote by $Γ(X;m,k,γ)$ the microstate space of $X$ of order $(m,k,γ)$. We say that $X$ is tubular if for any $ε>0$ there exist $m \in \mathbb N$ and $γ>0$ such that if $(x_1,..., x_n), (y_1, ..., y_n) \in Γ(X;m,k,γ),$ then there exists a $k \times k$ unitary $u$ satisfying $|ux_iu^* - y_i|_2 < ε$ for each $1 \leq i \leq n.$ We show that the following conditions are equivalent: 1) $M$ is amenable (i.e., injective). 2) $X$ is tubular; 3) Any two embeddings of $M$ into $\mathcal R^ω$ are conjugate by a unitary u in $\mathcal R^ω$.
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