On the theories of McDuff's II$_1$ factors

TL;DR

This paper investigates the logical theories of McDuff's II$_1$ factors, using Ehrenfeucht-Fraisse games to analyze their elementary equivalence and distinguishability.

Contribution

It introduces a method to bound the quantifier-depth needed to differentiate McDuff's II$_1$ factors using model-theoretic techniques.

Findings

Ultrapowers of distinct McDuff factors are non-isomorphic.

Provides bounds on the logical complexity needed to distinguish these factors.

Advances understanding of the model theory of operator algebras.

Abstract

Recently, Boutonnet, Chifan, and Ioana proved that McDuff's family of continuum many pairwise nonisomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two distinct members of the family are nonsiomorphic. We use Ehrenfeucht-Fraisse games to provide an upper bound on the quantifier-depth of sentences which distinguish these theories.