Uniqueness of the von Neumann Continuous Factor

Abstract

AbstractFor a division ring D, denote by 𝓜D the D-ring obtained as the completion of the direct limit with respect to themetric 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 , where 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 МF is endowed with its natural involution coming from the *-transpose involution on each of the factors .