$B$-valued semi-circular system and the free Poincaré inequality

We characterize $B$-valued semi-circular system in terms of $B$-valued free probabilistic analogue of Poincaré 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 Poincaré inequality in \cite{aim06} is not in the affirmative as it is.