# Uniqueness of the von Neumann continuous factor

## TL;DR

This paper proves the uniqueness of the von Neumann continuous factor as a completion of certain ultramatricial rings, extending von Neumann's classical result to a broader algebraic setting.

## Contribution

It generalizes von Neumann's uniqueness theorem to ultramatricial D-rings and their completions, including a corresponding result for *-algebras over fields with involution.

## Key findings

- Any ultramatricial D-ring with a non-discrete extremal pseudo-rank function completes to the von Neumann continuous factor.
- The result extends to *-algebras over fields with positive definite involution, with a natural involution structure.
- The paper establishes a uniqueness theorem for the von Neumann continuous factor in a generalized algebraic context.

## Abstract

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 $M_{2^n}(F)$.