© Canadian Mathematical Society 2018. For a division ring D, denote by MD the D-ring obtained as the completion of the direct limit lim M2n (D) with respect to the metric induced by its unique rank function. We prove that, for any ultramatricial D-ring B and any non-discrete extremal pseudo-rank function N on B, there is an isomorphism of D-rings B = MD, where B stands for the completion of 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 MF is endowed with its natural involution coming from the ∗-transpose involution on each of the factors M2n (F).
- Rank function
- Von Neumann regular ring