B-valued semi-circular system and the free Poincar\'e inequality
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 [Theorem 5.1]b03. Moreover, we prove that Voiculescu's conjecture on B-valued free Poincar\'e inequality in aim06 is not in the affirmative as it is.
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