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arxiv: 2409.16498 · v3 · pith:S6JH2VILnew · submitted 2024-09-24 · 🧮 math.OA · math.RA

B-valued semi-circular system and the free Poincar\'{e} inequality

classification 🧮 math.OA math.RA
keywords valuedfreeinequalitypoincarcitesemi-circularsystemtheorem
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We characterize $B$-valued semi-circular system in terms of $B$-valued free probabilistic analogue of Poincar\'{e} inequality. This is a $B$-valued generalization of Biane's theorem \cite[Theorem 5.1]{b03}. Moreover, we prove that Voiculescu's conjecture on $B$-valued free Poincar\'{e} inequality in \cite{aim06} is not in the affirmative as it is.

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