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On Cardinal Invariants and Generators for von Neumann Algebras

Published online by Cambridge University Press:  20 November 2018

David Sherman
David Sherman*
Affiliation:
Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA 22904, USA email: dsherman@virginia.edu
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Abstract

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We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal{M}$ can be computed from the decomposability number, $\text{dens}\left( \mathcal{M} \right)$, and the minimal cardinality of a generating set, $\text{gen}\left( \mathcal{M} \right)$. Applications include the equivalence of the well-known generator problem, “Is every separably-acting von Neumann algebra singly-generated?”, with the formally stronger questions, “Is every countably-generated von Neumann algebra singly-generated?” and “Is the gen invariant monotone?” Modulo the generator problem, we determine the range of the invariant $\left( \text{gen}\left( \mathcal{M} \right),\,\text{dens}\left( \mathcal{M} \right) \right)$ , which is mostly governed by the inequality $\text{dens}\left( \mathcal{M} \right)\,\le {{\mathfrak{C}}^{\text{gen}\left( \mathcal{M} \right)}}$.

Keywords

46L10von Neumann algebracardinal invariantgenerator problemdecomposability numberrepresentation density
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 2 , 16 April 2012 , pp. 455 - 480
Copyright
Copyright © Canadian Mathematical Society 2012

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