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Free Independence is not Definable

William Boulanger, Jakub Curda, Emma Harvey, Yizhi Li, Jennifer Pi

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.

Free Independence is not Definable

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.

Paper Structure

This paper contains 6 sections, 3 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Definable Sets in Continuous Logic
  4. Non-definability of Freeness
  5. Proof for $\mathrm{C}^*$-probability spaces
  6. Proof for tracial von Neumann algebras

Key Result

Lemma 3.2

For $\mathcal{U}$-almost all $n$, the $\mathrm{C}^*$-algebra $\mathrm{C}^*(x_n)$ generated by $x_n$ contains a unital copy of $\mathbb{C}^3$ and $\mathrm{C}^*(y_n)\neq\mathbb{C}$.

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof : Sketch of proof
  • Lemma 3.4
  • proof
  • Remark 3.5