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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 ${\textstyle \frac{X_1+\cdots+X_{n}}{\sqrt{n}}}$,\break $H\Big({\textstyle \frac{X_1+\cdots+X_{n}}{\sqrt{n}}}\Big)$, 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 {\it not} a strictly increasing function of $n$, then $X_1$ ($x_1$, respectively) must be Gaussian (semicircular, respectively).

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