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

• Published in: arXiv.org 2024-03
• Main Author: Jekel, David
• Format: Article
• Language: English
• Subjects: Continuity (mathematics); Couplings
• ISSN: 2331-8422

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.