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.
Hyuga Ito (Thu,) studied this question.
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