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arXiv · 2409.16498

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

Hyuga Ito
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Abstract

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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Hyuga Ito. 2024-09-24. $B$-valued semi-circular system and the free Poincar\'{e} inequality. https://arxiv.org/abs/2409.16498

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