Optimal transport for types and convex analysis for definable predicates in tracial $\mathrm{W}^*$-algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929269079277568 |
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| author | Jekel, David |
| author_facet | Jekel, David |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11058 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal transport for types and convex analysis for definable predicates in tracial $\mathrm{W}^*$-algebras Jekel, David Operator Algebras Logic Optimization and Control 46L53, 03C66, 49Q22, 49N15 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. |
| title | Optimal transport for types and convex analysis for definable predicates in tracial $\mathrm{W}^*$-algebras |
| topic | Operator Algebras Logic Optimization and Control 46L53, 03C66, 49Q22, 49N15 |
| url | https://arxiv.org/abs/2308.11058 |