Maximality of the microstates free entropy for R-diagonal elements

Abstract

A non-commutative non-selfadjoint 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 X_{ij}. Then the free entropy of the family {X_{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 Some minimization problems for the free analogue of the Fisher information, where we considered the same problems in the framework of the non-microstates definition of free entropy.