Maximality of the microstates free entropy for R-diagonal elements

A non-commutative non-self adjoint random variable a is called R-diagonal, if its *-distribution is invariant under multiplication by free unitaries: if a unitary zu 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 x d matrix whose entries are non-commutative random variables X-ij, 1 less than or equal to i,j less than or equal to d. Then the free entropy of the family {X-ij}(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 x d matrices. The results of this paper are analogous to the results of our paper [3], where we considered the same problems in the framework of the non-microstates definition of entropy.