Free Independence is not definable
Operator Algebras
2026-02-25 v2 Logic
Abstract
Free independence is an important tool for studying the structure of operator algebras. It is natural to ask from the model-theoretic standpoint whether free independence is captured well in first-order model theory via the notion of a definable set. We prove that pairs of freely independent elements do not form a definable set in the sense of continuous model theory, relative to the theory of both C-probability spaces and tracial von Neumann algebras.
Cite
@article{arxiv.2510.04836,
title = {Free Independence is not definable},
author = {William Boulanger and Jakub Curda and Emma Harvey and Yizhi Li and Jennifer Pi},
journal= {arXiv preprint arXiv:2510.04836},
year = {2026}
}
Comments
Author-accepted manuscript; to appear in "Involve, a Journal of Mathematics"