On cardinal invariants and generators for von Neumann algebras

Details On cardinal invariants and generators for von Neumann algebras On cardinal invariants and generators for von Neumann algebras

2009-08-31

arxiv.org/abs/0908.4565v2

Mathematics

Operator Algebras

22 pages; the main additions are Theorem 3.8 and Section 8

Scientific paper

We demonstrate how virtually all common cardinal invariants associated to a von Neumann algebra M can be computed from the decomposability number, dec(M), and the minimal cardinality of a generating set, gen(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 gen invariant monotone?" Modulo the generator problem, we determine the range of the invariant (gen(M), dec(M)), which is mostly governed by the inequality dec(M) leq c^{gen(M)}.

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