Maximality of the Microstates Free Entropy for R-diagonal Elements

An 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 variablesXij , 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 [3], where we considered the same problems in the framework of the non-microstates definition of entropy.