Semicircularity, Gaussianity and Monotonicity of Entropy

S. Artstein, K. Ball, F. Barthe, and A. Naor have shown that if (X_j) are i.i.d. random variables, then the entropy of n-1/2(X_1+....+X_n) increases as n increases. The free analogue was recently proven by D. Shlyakhtenko. That is, if (x_j) are freely independent, identically distributed, self-adjoint elements in a noncommutative probability space, then the free entropy of n-1/2(x_1+....+x_n) increases as n increases. In this paper we prove that if X_1 (x_1, resp.) has finite entropy (free entropy, resp.), and if the entropy (the free entropy, resp.) is not a strictly increasing function of n, then X_1 (x_1, resp.) must be Gaussian (semicircular, resp.).