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

Abstract

Using set-theoretic methods, we prove that every ${\mathrm{AW}}^{\ast }$-algebra is normal, resolving a question of Wright that has stood open for 46 years. This was previously known in the case of ${\mathrm{AW}}^{\ast }$-factors, by work of Saitô and Wright. We also show that if there is a model $V$ of $\mathrm{ZFC}$ with an ${\mathrm{AW}}^{\ast }$-algebra that fails to be monotone complete, then in some forcing extension of $V$ there is an ${\mathrm{AW}}^{\ast }$-factor that fails to be monotone complete, implying that any $\mathrm{ZFC}$ proof that all ${\mathrm{AW}}^{\ast }$-factors are monotone complete yields a $\mathrm{ZFC}$ proof that all ${\mathrm{AW}}^{\ast }$-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.

27 pages, comments welcome