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This research reveals that free independence is not captured in first-order model theory, suggesting limitations in existing frameworks.
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.
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Boulanger et al. (2025) studied this question.
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