Uniqueness of the von Neumann continuous factor

Abstract

For a division ring $D$, denote by ${\mathcal{M}}_D$ the $D$-ring obtained as the completion of the direct limit ${\underrightarrow{\lim }}_n{M}_{2^n}(D)$ 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 $\ast$-algebras over fields $F$ with positive definite involution, where the algebra ${\mathcal{M}}_F$ is endowed with its natural involution coming from the $\ast$-transpose involution on each of the factors $M_{2^n}(F)$.

23 pages. Final version

Cited by in corpus (3)

• Bivariant and extended Sylvester rank functions
• Sylvester rank functions for amenable normal extensions
• Group von Neumann algebras, inner amenability, and unit groups of continuous rings