von Neumann连续因子的唯一性
Uniqueness of the von Neumann continuous factor
摘要：
对于除环$D$，用$\mathcal M_D$表示$D$ -环，该环作为关于由其唯一秩函数导出的度量的直接极限$\varinjlim_n M_{2^n}(D)$的完备化而得到。证明了对任意超矩阵$D$ -环$\mathcal B$和$\mathcal B$上的任意非离散极值伪秩函数$N$，存在$D$ -环$\overline{\mathcal B} \cong \mathcal M_D$的同构，其中$\overline{\mathcal B}$表示$\mathcal B$关于$N$诱导的伪度量的完备化.这推广了von Neumann的一个结果。我们还证明了域$F$上具有正定对合的$*$ -代数的唯一性，其中代数$\mathcal M_F$的自然对合来自于每个因子$M_{2^n}(F)$上的$*$ -转置对合.
英文摘要：
For a division ring $D$, denote by $\mathcal M_D$ the $D$-ring obtained as the completion of the direct limit $\varinjlim_n M_{2^n}(D)$ with respect to the metric induced by its unique rank function. We prove that, for any ultramatricial $D$-ring $\mathcal B$ and any non-discrete extremal pseudo-rank function $N$ on $\mathcal B$, there is an isomorphism of $D$-rings $\overline{\mathcal B} \cong \mathcal M_D$, where $\overline{\mathcal B}$ stands for the completion of $\mathcal B$ with respect to the pseudo-metric induced by $N$. This generalizes a result of von Neumann. We also show a corresponding uniqueness result for $*$-algebras over fields $F$ with positive definite involution, where the algebra $\mathcal M_F$ is endowed with its natural involution coming from the $*$-transpose involution on each of the factors $M_{2^n}(F)$.
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