On Generators for $W^{\ast }$-bundles

Abstract

We give an explicit example of a $W^{\ast }$-bundle $M$ over a compact, metrizable space $K$, which has each fiber $M_p$ a ${\text{II}}_1$-factor with separable predual, and which satisfies the following negation of the generator problem: given any finite family $a_1,\cdots ,a_n\in M$ of continuous sections, there is a $p\in K$ (depending upon that family) so that $a_{1,p},\cdots ,a_{n,p}$ do not generate $M_p$ as a von Neumann algebra. More generally, if $N$ is a sub-bundle of $M$ with the property that each fiber is hyperfinite, or has a Cartan, or is generated by two commuting diffuse subalgebras, or has diffuse central sequence algebra, or is generated by a single sequential commutation orbit, then given any finite family $a_1,\cdots ,a_n\in M$ of continuous sections, there is a $p\in K$ (depending upon that family) so that $a_{1,p},\cdots ,a_{n,p}$ together with $N_p$ do not generate $M_p$. We discuss implications for the generator problem for von Neumann algebras: e.g. there is no ``continuous" way to take countably many generators for a tracial von Neumann algebra with separable predual and produce a single generator, at least if such a procedure works for all von Neumann algebras simultaneously.

arXiv comment42 pages, no figures. No LLM's were used at any stage during this project. Comments welcome!