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 (see Theorem 5.1 in [P. Biane, Logarithmic Sobolev inequalities, matrix models and free entropy, Acta Math. Sin. (Engl. Ser.) 19(3) (2003) 497–506]). Moreover, we give a partial answer to the question “when does the free Poincaré inequality hold?” asked by Voiculescu [Problems posed in workshop “Free Analysis” at American Institute of Mathematics (2006), https://aimath.org/WWN/freeanalysis/freeanalysis.pdf ].