Geodesic
On Cardinal Invariants and Generators for von Neumann Algebras
Canadian journal of mathematics, Tome 64 (2012) no. 2, pp. 455-480

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DOI

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 $\text{dens}\left( \mathcal{M} \right)\,\le {{\mathfrak{C}}^{\text{gen}\left( \mathcal{M} \right)}}$ .
DOI : 10.4153/CJM-2011-048-2
Mots-clés : 46L10, von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation density 
Sherman, David. On Cardinal Invariants and Generators for von Neumann Algebras. Canadian journal of mathematics, Tome 64 (2012) no. 2, pp. 455-480. doi: 10.4153/CJM-2011-048-2
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     title = {On {Cardinal} {Invariants} and {Generators} for von {Neumann} {Algebras}},
     journal = {Canadian journal of mathematics},
     pages = {455--480},
     year = {2012},
     volume = {64},
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     doi = {10.4153/CJM-2011-048-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-048-2/}
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