Operator-valued free Fisher information of random matrices

Abstract

Abstract. We study the operator-valued free Fisher information of random matrices in an operator-valued noncommutative probability space. We obtain a formula for Φ * M 2 (B) (A, A * , M 2 (B), η), where A ∈ M 2 (B) is a 2 × 2 operator matrix on B, and η is linear operators on M 2 (B). Then we consider a special setting: A is an operator-valued semicircular matrix with conditional expectation covariance, and find that Φ * B (c, c * : B, id) = 2Index(E), where E is a conditional expectation of B onto D and c is a circular variable with covariance E. Introduction and preliminariesFree probability theory is a noncommutative probability theory where the classical concept of independence is replaced by the notion of "freeness". This theory, due to D.Voiculescu, has very important applications on operator algebras (see [6, 21]) Originally, a noncommutative probability space is a pair (A, τ ), where A is a C * − or von Neumann algebra and τ is a state on A. Free independence is defined in terms of reduced free product relation given by τ . This notion was generalized by D.Voiculescu and others to an algebra valued noncommutative probability space where τ is replaced by a conditional expectation E B onto a subalgebra B of A, and freeness is replaced by freeness with amalgamation. Definition 1.[20] Let A be a unital algebra over C, and let B be a subalgebra of A, 1 ∈ B. E B : A −→ B is a conditional expectation, i.e. a linear map such thatan operator-valued (or B− valued) noncommutative probability space and elements in A are called B− random variables.The algebra freely generated by B and an indeterminate X will be denoted bybe subalgebras. The family (A i ) i∈I will be called free with amalgamation over B(or B−free), if E B (a 1 a 2 · · · a n ) = 0 whenever a j ∈ A ij with i 1 = i 2 = · · · = i n and E B (a j ) = 0, 1 ≤ j ≤ n. A sequence {a i } i∈I ⊆ A will be called free with amalgamation over B if the family of subalgebras generated by (B {a i }) i∈I is B−free.2000 Mathematics Subject Classification. Primary 46L54, 42C15.