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