Type: Article
Publication Date: 1999-02-01
Citations: 13
DOI: https://doi.org/10.2140/pjm.1999.187.333
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 X ij , 1 ≤ i, j ≤ 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 × 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.