B -valued Chebyshev polynomials and the free Poincaré inequality for B -valued semi-circular systems

We characterize [Formula: see text]-valued semi-circular systems in terms of a [Formula: see text]-valued free probabilistic analogue of Poincaré inequality. This is a [Formula: see text]-valued generalization of Biane’s theorem [B03, Theorem 5.1]. Moreover, we give a partial answer to the question “when does the free Poincaré inequality hold?” asked by Voiculescu [AIM06].