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 $M_{ 2^n}(F)$.
Categories: math.RA math.OA