Model theory of quasitracial $\mathrm{AW}^*$-algebras and the absoluteness of monotone completeness

Abstract

We examine various applications of logic to $\mathrm{AW}^*$-algebras, particularly the monotone completeness conjecture. We develop a model theory for $\mathrm{AW}^*$-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 $\mathrm{II}_1$ $\mathrm{AW}^*$-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 $\mathrm{AW}^*$-factor of type $\mathrm{II}_1$ 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 $\mathrm{AW}^*$-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 $\mathrm{AW}^*$-algebra with no nonzero central semi-abelian projection has $\mathrm{AW}^*$ complexification. We answer a number of other open problems in real $\mathrm{AW}^*$-algebras.