Every AW^*-Algebra is Normal — Draconitic Research

Abstract

Using set-theoretic methods, we prove that every AW^*-algebra is normal, resolving a question of Wright that has stood open for 46 years. This was previously known in the case of AW^*-factors, by work of Saitô and Wright. We also show that if there is a model V of ZFC with an AW^*-algebra that fails to be monotone complete, then in some forcing extension of V there is an AW^*-factor that fails to be monotone complete, implying that any ZFC proof that all AW^*-factors are monotone complete yields a ZFC proof that all AW^*-algebras are monotone complete. Both results build on transfer principles for Boolean-valued factor representations originally developed by Ozawa. This work was assisted by the Danus LLM orchestration system.

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BibTeX RIS

BibTeX

@misc{arulseelan2026every,
  title = {{Every $\mathrmAW^*$-Algebra is Normal}},
  author = {Jananan Arulseelan and James E. Hanson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10218},
  eprint = {2609.10218},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10218},
}

RIS

TY  - UNPB
TI  - Every AW^*-Algebra is Normal
AU  - Jananan Arulseelan
AU  - James E. Hanson
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10218
UR  - https://arxiv.org/abs/2609.10218
AB  - Using set-theoretic methods, we prove that every AW^*-algebra is normal, resolving a question of Wright that has stood open for 46 years. This was previously known in the case of AW^*-factors, by work of Saitô and Wright. We also show that if there is a model V of ZFC with an AW^*-algebra that fails to be monotone complete, then in some forcing extension of V there is an AW^*-factor that fails to be monotone complete, implying that any ZFC proof that all AW^*-factors are monotone complete yields a ZFC proof that all AW^*-algebras are monotone complete. Both results build on transfer principles for Boolean-valued factor representations originally developed by Ozawa. This work was assisted by the Danus LLM orchestration system.
ER  -

Abstract

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