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