Model theory and Connes' bicentralizer problem

Abstract

We make a series of model-theoretic contributions to Connes' bicentralizer problem, one of the most prominent open problems in the theory of von Neumann algebras. Our work builds on the recent result of Houdayer and Marrakchi who show that, for separable diffuse W$^*$-probability spaces, having trivial bicentralizer is equivalent to being selfless, that is, having the first factor inclusion into the free product be an existential embedding. We first show that the class of selfless $W^*$-probability spaces is $\forall\exists$-axiomatizable. We then extend the Houdayer-Marrakchi equivalence to all diffuse W$^*$-probability spaces, removing the separability hypothesis. Combining these results, we show that for any axiomatizable class of diffuse $W^*$-probability spaces, those with trivial bicentralizer form an $\forall\exists$-axiomatizable class; in particular, the class of type $\mathrm{III}_1$ factors with trivial bicentralizer is $\forall\exists$-axiomatizable. We give concrete axioms for this class using totally bounded variants of Haagerup's characterization of the bicentralizer, which we develop here and believe to be of independent interest. We also introduce the notion of pseudoperiodic $\mathrm{III}_1$ factors and show that any such factor has trivial bicentralizer. In the final section, we prove that the bicentralizer problem has a positive solution if and only if the bicentralizer functor is a zeroset relative to the theory of $\mathrm{III}_1$ factors. We use this result to give an equivalent formulation of the bicentralizer problem in terms of a uniformity condition on Haagerup's Dixmier-type characterization of the bicentralizer.