On Generators for W^*-bundles — Draconitic Research

Abstract

We give an explicit example of a W^*-bundle M over a compact, metrizable space K, which has each fiber M_p a II₁-factor with separable predual, and which satisfies the following negation of the generator problem: given any finite family a₁,⋯,a_n∈ M of continuous sections, there is a p∈ K (depending upon that family) so that a_1,p,⋯,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 a₁,⋯,a_n∈ M of continuous sections, there is a p∈ K (depending upon that family) so that a_1,p,⋯,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.

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BibTeX RIS

BibTeX

@misc{hayes2026generators,
  title = {{On Generators for $W^*$-bundles}},
  author = {Ben Hayes},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16428},
  eprint = {2609.16428},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16428},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On Generators for W^*-bundles
AU  - Ben Hayes
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.16428
UR  - https://arxiv.org/abs/2609.16428
AB  - We give an explicit example of a W^*-bundle M over a compact, metrizable space K, which has each fiber M_p a II_1-factor with separable predual, and which satisfies the following negation of the generator problem: given any finite family a_1,⋯,a_n∈ M of continuous sections, there is a p∈ K (depending upon that family) so that a_1,p,⋯,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,⋯,a_n∈ M of continuous sections, there is a p∈ K (depending upon that family) so that a_1,p,⋯,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.
ER  -

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