$B$-valued semi-circular system and the free Poincar\'{e} inequality

TL;DR

This paper characterizes B-valued semi-circular systems using a B-valued free Poincaré inequality and demonstrates that Voiculescu's conjecture on this inequality does not hold as previously conjectured.

Contribution

It provides a B-valued generalization of Biane's theorem and disproves Voiculescu's conjecture on the free Poincaré inequality.

Findings

Characterization of B-valued semi-circular systems via free Poincaré inequality

Disproof of Voiculescu's conjecture on B-valued free Poincaré inequality

Extension of Biane's theorem to B-valued setting

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