Every $\mathrm{AW}^*$-Algebra is Normal

Jananan Arulseelan; James E. Hanson

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arXiv · 2609.10218

Every $\mathrm{AW}^*$-Algebra is Normal

Jananan Arulseelan·James E. Hanson

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Abstract

Using set-theoretic methods, we prove that every $\mathrm{AW}^*$-algebra is normal, resolving a question of Wright that has stood open for 46 years. This was previously known in the case of $\mathrm{AW}^*$-factors, by work of Sait\^o and Wright. We also show that if there is a model $V$ of $\mathrm{ZFC}$ with an $\mathrm{AW}^*$-algebra that fails to be monotone complete, then in some forcing extension of $V$ there is an $\mathrm{AW}^*$-factor that fails to be monotone complete, implying that any $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-factors are monotone complete yields a $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-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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Jananan Arulseelan, James E. Hanson. 2026-09-09. Every $\mathrm{AW}^*$-Algebra is Normal. https://arxiv.org/abs/2609.10218

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