Abstract

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 [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.

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