W^*-categories are von Neumann — Draconitic Research

Abstract

A C^*-category is to a C^*-algebra what a groupoid is to a group. An example is the category of Hilbert spaces and bounded linear maps, and indeed, every C^*-category can be represented as a sub-*-category of this. Analogously, a W^*-category is a C^*-category where every object has a predual, and we might expect a representation theory on Hilbert spaces where the resulting morphism spaces are σ-weakly closed. We identify a gap in the literature here, stemming from the (perhaps surprising) fact that a C^*-algebra can have a non-isometric predual and yet not be a W^*-algebra. Using the theory of dual TROs (ternary rings of operators) we repair this gap, and along the way, also give further justification to arguments in the literature about dual TROs. Motivated by the theory of Dual Banach Algebras, we offer an alternative axiomatisation of W^*-categories where we allow non-isometric preduals of the hom spaces, but require that they are additionally bimodules in a certain sense. This involves the theory of self-dual Hilbert C^*-modules.

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BibTeX RIS

BibTeX

@misc{daws2026wcategories,
  title = {{$W^*$-categories are von Neumann}},
  author = {Matthew Daws},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.30015},
  eprint = {2609.30015},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30015},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - W^*-categories are von Neumann
AU  - Matthew Daws
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.30015
UR  - https://arxiv.org/abs/2609.30015
AB  - A C^*-category is to a C^*-algebra what a groupoid is to a group. An example is the category of Hilbert spaces and bounded linear maps, and indeed, every C^*-category can be represented as a sub-*-category of this. Analogously, a W^*-category is a C^*-category where every object has a predual, and we might expect a representation theory on Hilbert spaces where the resulting morphism spaces are σ-weakly closed. We identify a gap in the literature here, stemming from the (perhaps surprising) fact that a C^*-algebra can have a non-isometric predual and yet not be a W^*-algebra. Using the theory of dual TROs (ternary rings of operators) we repair this gap, and along the way, also give further justification to arguments in the literature about dual TROs. Motivated by the theory of Dual Banach Algebras, we offer an alternative axiomatisation of W^*-categories where we allow non-isometric preduals of the hom spaces, but require that they are additionally bimodules in a certain sense. This involves the theory of self-dual Hilbert C^*-modules.
ER  -

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