On cardinal invariants and generators for von Neumann algebras

TL;DR

This paper explores how key cardinal invariants of von Neumann algebras can be derived from the decomposability number and minimal generating set size, linking the generator problem to these invariants.

Contribution

It establishes that most invariants are computable from dec(M) and gen(M), and connects the generator problem to the monotonicity of gen(M).

Findings

Dec(M) and gen(M) determine all common invariants.

The generator problem is equivalent to questions about gen(M).

Range of (gen(M), dec(M)) is characterized by dec(M) ≤ c^{gen(M)}.

Abstract

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)}.