Alexandru Nica - Some minimization problems for the free analogue of the Fisher information

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Alexandru Nica - Some minimization problems for the free analogue of the Fisher information

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 $\Phi^{\ast}$ , introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value of $\Phi^{\ast} (a,a^{\ast})$ , if a is a non-commutative random variable subject to the constraint that the distribution of $a^{\ast}a$ is prescribed. More generally, we obtain the minimal possible value of $\Phi^{\ast} (\{ a_{ij},a_{ij}^{\ast}\}_{i,j})$ , if $\{a_{ij}\}_{1 \leq i,j \leq d}$ is a family of non-commutative random variables such that the distribution of $A^{\ast}A$ is prescribed, where A is the matrix (a_ij)_i,j=1^d. The $d \times d$ -generalization is obtained from the case d=1 via a result of independent interest, concerning the minimal value of $\Phi^{\ast} (\{ a_{ij},a_{ij}^{\ast}\}_{i,j})$ when the matrix A = (a_ij)_i,j=1^d and its adjoint have a given joint distribution. (A version of this result describes the minimal value of $\Phi^{\ast} (\{ b_{ij} \}_{ij})$ when the matrix B = (b_ij)_i,j=1^d is selfadjoint and has a given distribution.)

We then show how the minimization results obtained for $\Phi^{\ast}$ lead to maximization results concerning the free entropy $\chi^{\ast}$ , also defined by Voiculescu.

This is joint work with Dimitri Shlyakhtenko and Roland Speicher.

Next: John Phillips - Spectral Up: Operator Algebras / Algèbres Previous: Michael Lamoureux - Crossed