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# On Cardinal Invariants and Generators for von Neumann Algebras

Published:2011-07-15
Printed: Apr 2012
• David Sherman,
Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA 22904, USA
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## Abstract

We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal M$ can be computed from the decomposability number, $\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating set, $\operatorname{gen}(\mathcal M)$. 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 $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M), \operatorname{dens}(\mathcal M) \bigr)$, which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq \mathfrak C^{\operatorname{gen}(\mathcal M)}$.
 Keywords: von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation density
 MSC Classifications: 46L10 - General theory of von Neumann algebras