Model theory of quasitracial ${\mathrm{A}\mathrm{W}}^{\ast }$-algebras and the absoluteness of monotone completeness

We examine various applications of logic to ${\mathrm{A}\mathrm{W}}^{\ast }$-algebras, particularly the monotone completeness conjecture. We develop a model theory for ${\mathrm{A}\mathrm{W}}^{\ast }$-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 ${\mathrm{I}\mathrm{I}}_1$ ${\mathrm{A}\mathrm{W}}^{\ast }$-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 ${\mathrm{A}\mathrm{W}}^{\ast }$-factor of type ${\mathrm{I}\mathrm{I}}_1$ is a von Neumann algebra, are each equivalent in ZFC to a ${\mathrm{\Pi }}_2^0$ sentence of arithmetic, so that their answers are the same in every $\omega$-model of ZFC. We also consider real ${\mathrm{A}\mathrm{W}}^{\ast }$-algebras. Here the Baer-$\ast$-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 ${\mathrm{A}\mathrm{W}}^{\ast }$-algebra with no nonzero central semi-abelian projection has ${\mathrm{A}\mathrm{W}}^{\ast }$ complexification. We answer a number of other open problems in real ${\mathrm{A}\mathrm{W}}^{\ast }$-algebras.