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Optimal transport for types and convex analysis for definable predicates in tracial $\mathrm{W}^*$-algebras

David Jekel

Abstract

We investigate the connections between continuous model theory, free probability, and optimal transport/convex analysis in the context of tracial von Neumann algebras. In particular, we give an analog of Monge-Kantorovich duality for optimal couplings where the role of probability distributions on $\mathbb{C}^n$ is played by model-theoretic types, the role of real-valued continuous functions is played by definable predicates, and the role of continuous function $\mathbb{C}^n \to \mathbb{C}^n$ is played by definable functions. In the process, we also advance the understanding of definable predicates and definable functions by showing that all definable predicates can be approximated by "$C^1$ definable predicates" whose gradients are definable functions. As a consequence, we show that every element in the definable closure of $\mathrm{W}^*(x_1,\dots,x_n)$ can be expressed as a definable function of $(x_1,\dots,x_n)$. We give several classes of examples showing that the definable closure can be much larger than $\mathrm{W}^*(x_1,\dots,x_n)$ in general.

Optimal transport for types and convex analysis for definable predicates in tracial $\mathrm{W}^*$-algebras

Abstract

We investigate the connections between continuous model theory, free probability, and optimal transport/convex analysis in the context of tracial von Neumann algebras. In particular, we give an analog of Monge-Kantorovich duality for optimal couplings where the role of probability distributions on is played by model-theoretic types, the role of real-valued continuous functions is played by definable predicates, and the role of continuous function is played by definable functions. In the process, we also advance the understanding of definable predicates and definable functions by showing that all definable predicates can be approximated by " definable predicates" whose gradients are definable functions. As a consequence, we show that every element in the definable closure of can be expressed as a definable function of . We give several classes of examples showing that the definable closure can be much larger than in general.
Paper Structure (18 sections, 27 theorems, 154 equations)

This paper contains 18 sections, 27 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on operator algebras
  4. Languages and structures
  5. Syntax: Terms, formulas, and sentences
  6. Theories, models, and axioms
  7. Types, definable predicates, and definable functions
  8. Elementary extensions, saturation, and strong homogeneity
  9. Definable and algebraic closures
  10. Definition and Properties
  11. Examples in von Neumann algebras
  12. Monge-Kantorovich duality for types and definable predicates
  13. Proof of duality
  14. Properties of convex definable predicates
  15. Semiconcave and semiconvex definable predicates
  16. ...and 3 more sections

Key Result

Theorem 1.1

Fix a complete theory $\mathrm{T}$ of a tracial von Neumann algebra. Let $\mu$ and $\nu \in \mathbb{S}_n(\mathrm{T})$ be types. Then there exist convex $\mathrm{T}_{\mathop{\mathrm{tr}}\nolimits}$-definable predicates $\phi$ and $\psi$ such that and such that equality is achieved when $(\mathbf{x},\mathbf{y})$ is an optimal coupling of $(\mu,\nu)$. Hence, $C(\mu,\nu)$ is the infimum of $\mu[\phi]

Theorems & Definitions (79)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.4
  • Definition 2.11
  • Definition 2.13
  • Definition 2.14
  • Definition 2.15
  • ...and 69 more