Model theory and Connes' bicentralizer problem

TL;DR

This paper advances the model-theoretic understanding of Connes' bicentralizer problem in von Neumann algebras, establishing axiomatizability results and new characterizations for trivial bicentralizer classes.

Contribution

It extends the equivalence between trivial bicentralizer and selfless spaces to all diffuse W*-probability spaces and provides concrete axioms for these classes.

Findings

The class of selfless W*-probability spaces is $orallorall$-axiomatizable.

The equivalence between trivial bicentralizer and selfless spaces is extended beyond separable cases.

Concrete axioms for type III_1 factors with trivial bicentralizer are developed.

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.