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Weak* polynomial density, diffuseness and proper infinitude

  • Alexandru Chirvasitu
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Abstract

The unitary group of a von Neumann algebra on an ℵ0-dimensional Hilbert space is polynomially weak*-dense (equivalently, Gdelta dense or residual) in the normal unit ball if and only if the canonical finite atomic quotient of the von Neumann algebra is trivial. The analogous result holds for the usual weak* topology, in which case one can drop the normality constraint. In the latter form, this answers a question of T. Eisner and generalizes the counterpart for full bounded-operator algebras.

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Cite

BibTeXRIS
BibTeX
@misc{chirvasitu2026weak,
  title = {{Weak$^*$ polynomial density, diffuseness and proper infinitude}},
  author = {Alexandru Chirvasitu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23724},
  eprint = {2609.23724},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23724},
}
RIS
TY  - UNPB
TI  - Weak^* polynomial density, diffuseness and proper infinitude
AU  - Alexandru Chirvasitu
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23724
UR  - https://arxiv.org/abs/2609.23724
AB  - The unitary group of a von Neumann algebra on an ℵ_0-dimensional Hilbert space is polynomially weak^*-dense (equivalently, G_delta dense or residual) in the normal unit ball if and only if the canonical finite atomic quotient of the von Neumann algebra is trivial. The analogous result holds for the usual weak^* topology, in which case one can drop the normality constraint. In the latter form, this answers a question of T. Eisner and generalizes the counterpart for full bounded-operator algebras.
ER  -