Model theory of quasitracial AW^*-algebras and the absoluteness of monotone completeness

Abstract

We examine various applications of logic to AW^*-algebras, particularly the monotone completeness conjecture. We develop a model theory for AW^*-algebras equipped with a faithful normalized quasitrace. Using this language, we provide a separable coding which allows us to reduce the finite case of the monotone completeness conjecture to monotone sequential completeness of II₁ AW^*-factors. Combining an explicit recursive axiomatization with the completeness theorem for continuous logic, we show that this case of the conjecture, and Kaplansky's problem whether every AW^*-factor of type II₁ is a von Neumann algebra, are each equivalent in ZFC to a Π⁰₂ sentence of arithmetic, so that their answers are the same in every ω-model of ZFC. We also consider real AW^*-algebras. Here the Baer-*-ring notion of an abelian projection must be distinguished from commutativity; following the real-operator-algebra terminology we call the former semi-abelian. Handelman's matrix theorem implies that a real AW^*-algebra with no nonzero central semi-abelian projection has AW^* complexification. We answer a number of other open problems in real AW^*-algebras.

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BibTeX RIS

BibTeX

@misc{arulseelan2026model,
  title = {{Model theory of quasitracial $\mathrmAW^*$-algebras and the absoluteness of monotone completeness}},
  author = {Jananan Arulseelan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17435},
  eprint = {2610.17435},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17435},
}

RIS

TY  - UNPB
TI  - Model theory of quasitracial AW^*-algebras and the absoluteness of monotone completeness
AU  - Jananan Arulseelan
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17435
UR  - https://arxiv.org/abs/2610.17435
AB  - We examine various applications of logic to AW^*-algebras, particularly the monotone completeness conjecture. We develop a model theory for AW^*-algebras equipped with a faithful normalized quasitrace. Using this language, we provide a separable coding which allows us to reduce the finite case of the monotone completeness conjecture to monotone sequential completeness of II_1 AW^*-factors. Combining an explicit recursive axiomatization with the completeness theorem for continuous logic, we show that this case of the conjecture, and Kaplansky's problem whether every AW^*-factor of type II_1 is a von Neumann algebra, are each equivalent in ZFC to a Π^0_2 sentence of arithmetic, so that their answers are the same in every ω-model of ZFC. We also consider real AW^*-algebras. Here the Baer-*-ring notion of an abelian projection must be distinguished from commutativity; following the real-operator-algebra terminology we call the former semi-abelian. Handelman's matrix theorem implies that a real AW^*-algebra with no nonzero central semi-abelian projection has AW^* complexification. We answer a number of other open problems in real AW^*-algebras.
ER  -

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