Uniqueness of the von Neumann Continuous Factor

Ara, Pere ; Claramunt, Joan

Canadian journal of mathematics, Tome 70 (2018) no. 5, pp. 961-982

Résumé

For a division ring $D$ , denote by ${{\mathcal{M}}_{D}}$ the $D$ -ring obtained as the completion of the direct limit $\underset{\to n}{\mathop \lim }\,{{M}_{{{2}^{n}}}}(D)$ 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 $\text{F}$ with positive definite involution, where the algebra ${{\mathcal{M}}_{\text{F}}}$ is endowed with its natural involution coming from the $*$ -transpose involution on each of the factors ${{M}_{{{2}^{n}}}}\,(F)$ .

Détail

DOI : 10.4153/CJM-2018-010-3

Mots-clés : 16E50, 16D70, rank function, von Neumann regular ring, completion, factor, ultramatricial