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Uniqueness of the von Neumann Continuous Factor
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- Journal:
- Canadian Journal of Mathematics / Volume 70 / Issue 5 / 01 October 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 961-982
- Print publication:
- 01 October 2018
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- Article
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For a division ring
$D$, denote by 
${{\mathcal{M}}_{D}}$ the $D$-ring obtained as the completion of the direct limit 
$\underset{\to n}{\mathop \lim }\,{{M}_{{{2}^{n}}}}(D)$ 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 
$\overline{B}\,\cong \,{{\mathcal{M}}_{D}}$, where 
$\overline{B}$ 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 
$\text{F}$ with positive definite involution, where the algebra 
${{\mathcal{M}}_{\text{F}}}$ is endowed with its natural involution coming from the $*$-transpose involution on each of the factors 
${{M}_{{{2}^{n}}}}\,(F)$.
