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On Cardinal Invariants and Generators for von Neumann Algebras
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- Journal:
- Canadian Journal of Mathematics / Volume 64 / Issue 2 / 16 April 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 455-480
- Print publication:
- 16 April 2012
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- Article
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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 
$\text{dens}\left( \mathcal{M} \right)\,\le {{\mathfrak{C}}^{\text{gen}\left( \mathcal{M} \right)}}$.
