|ALEXANDRU NICA, Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada|
|Some minimization problems for the free analogue of the Fisher information|
We consider the free non-commutative analogue , introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value of , if a is a non-commutative random variable subject to the constraint that the distribution of is prescribed. More generally, we obtain the minimal possible value of , if is a family of non-commutative random variables such that the distribution of is prescribed, where A is the matrix (aij)i,j=1d. The -generalization is obtained from the case d=1 via a result of independent interest, concerning the minimal value of when the matrix A = (aij)i,j=1d and its adjoint have a given joint distribution. (A version of this result describes the minimal value of when the matrix B = (bij)i,j=1d is selfadjoint and has a given distribution.)
We then show how the minimization results obtained for lead to maximization results concerning the free entropy , also defined by Voiculescu.
This is joint work with Dimitri Shlyakhtenko and Roland Speicher.