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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