Journal of Operator Theory

Volume 60, Issue 1, Summer 2008 pp. 125-136.

Semicircularity, gaussianity and monotonicity of entropy

Authors: Hanne Schultz
Author institution: Department of Mathematics and Computer Science, University of Sourthern Denmark, Denmark

Summary: S. Artstein, K. Ball, F. Barthe, and A. Naor have shown that if $(X_j)_{j=1}^\infty$ are i.i.d.\ random variables, then the entropy of ${\textstyle \frac{X_1+\cdots+X_{n}}{\sqrt{n}}}$, $H\Big({\textstyle \frac{X_1+\cdots+X_{n}}{\sqrt{n}}}\Big)$, increases as $n$ increases. The free analogue was recently proven by D. Shlyakhtenko. That is, if $(x_j)_{j=1}^\infty$ are freely independent, identically distributed, self-adjoint elements in a noncommutative probability space, then the free entropy of ${\textstyle \frac{x_1+\cdots+x_{n}}{\sqrt{n}}}$, $\chi\Big({\textstyle \frac{x_1+\cdots+x_{n}}{\sqrt{n}}}\Big)$, increases as $n$ increases. In this paper we prove that if $H(X_1)>-\infty$ $(\chi(x_1)>-\infty$, respectively$)$, and if the entropy (the free entropy, respectively) is not a strictly increasing function of $n$, then $X_1$ $($$x_1$, respectively$)$ must be Gaussian
$($semicircular, respectively$)$.

Keywords: Shannon entropy, free entropy

Contents Full-Text PDF