A non-commutative
non-self adjoint random variable z is called R-diagonal, if its ∗-distribution is
invariant under multiplication by free unitaries: if a unitary w is ∗-free from z,
then the ∗-distribution of z is the same as that of wz. Using Voiculescu’s
microstates definition of free entropy, we show that the R-diagonal elements are
characterized as having the largest free entropy among all variables y with a fixed
distribution of y∗y. More generally, let Z be a d × d matrix whose entries are
non-commutative random variables Xij, 1 ≤ i,j ≤ d. Then the free entropy of
the family {Xij}ij of the entries of Z is maximal among all Z with a fixed
distribution of Z∗Z, if and only if Z is R-diagonal and is ∗-free from the
algebra of scalar d × d matrices. The results of this paper are analogous to the
results of our paper [Nica, Alexandru; Shlyakhtenko, Dimitri and Speicher,Roland, Some minimization problems for the free analogue of the Fisher
information. Adv. Math. 141 (1999), no. 2, 282–321], where we considered
the same problems in the framework of the non-microstates definition of
entropy.
