Uniqueness of the von Neumann Continuous Factor
Uniqueness of the von Neumann Continuous Factor
复制标题
冯诺依曼连续因子的独特性
DOI：
--
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
J. Claramunt
中科院分区：
文献类型：
--
作者：
P. Ara;J. Claramunt
Abstract For a division ring $D$ , denote by ${{mathcal{M}}_{D}}$ the $D$ -ring obtained as the completion of the direct limit $underset{ o n}{mathop lim },{{M}_{{{2}^{n}}}}(D)$ with respect to themetric induced by its unique rank function. We prove that, for any ultramatricial $D$ -ring $B$ and any non-discrete extremal pseudo-rank function $N$ on $B$ , there is an isomorphism of $D$ -rings $overline{B},cong ,{{mathcal{M}}_{D}}$ , where $overline{B}$ stands for the completion of $B$ 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 $ ext{F}$ with positive definite involution, where the algebra ${{mathcal{M}}_{ ext{F}}}$ is endowed with its natural involution coming from the $*$ -transpose involution on each of the factors ${{M}_{{{2}^{n}}}},(F)$ .
Abstract For a division ring $D$ , denote by ${{mathcal{M}}_{D}}$ the $D$ -ring obtained as the completion of the direct limit $underset{ o n}{mathop lim },{{M}_{{{2}^{n}}}}(D)$ with respect to themetric induced by its unique rank function. We prove that, for any ultramatricial $D$ -ring $B$ and any non-discrete extremal pseudo-rank function $N$ on $B$ , there is an isomorphism of $D$ -rings $overline{B},cong ,{{mathcal{M}}_{D}}$ , where $overline{B}$ stands for the completion of $B$ 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 $ ext{F}$ with positive definite involution, where the algebra ${{mathcal{M}}_{ ext{F}}}$ is endowed with its natural involution coming from the $*$ -transpose involution on each of the factors ${{M}_{{{2}^{n}}}},(F)$ .