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Uniqueness of the von Neumann Continuous Factor

Published online by Cambridge University Press:  20 November 2018

Pere Ara  and
Joan Claramunt
Pere Ara
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain. e-mail: para@mat.uab.cat
Joan Claramunt
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain., e-mail: jclaramunt@mat.uab.cat
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Abstract

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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 ${{M}_{{{2}^{n}}}}\,(F)$.

Keywords

16E5016D70rank functionvon Neumann regular ringcompletionfactorultramatricial
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 5 , 01 October 2018 , pp. 961 - 982
Copyright
Copyright © Canadian Mathematical Society 2018

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