Some applications of free Fisher information on frame theory

Meng, Bo; Guo, Maozheng; Cao, X.

doi:10.1016/j.jmaa.2005.02.053

Cited by 3 publications

(8 citation statements)

References 16 publications

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“…In [10], we introduced the operator-valued free Fisher information of one variable with respect to a linear map which is a generalization of D.Shlyakhtenko's notion in [15]. Such a notion can be generalized to several random variables'setting easily.…”

Section: The Free Fisher Information Of Random Matricesmentioning

confidence: 99%

“…In the case of states, the non-tracial framework was worked out by D. Shlyakhtenko in [17]. In [9,10], we generalized the notion of free Fisher information to the operator-valued setting and this work is done in the general von Neumann algebra framework and found that the free Fisher information of an operator-valued semicircular variable with conditional expectation covariance is closely related to modular frames.…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, based on the notion of operator-valued free Fisher information with respect to a map which introduced in [10], we study the free Fisher information of random matrices and obtain a formula of operator-valued free Fisher information of random matrices, which generalized Theorem 1.2, Proposition 4.1 in [12], and the method we prove it is quite different from that of Theorem 1.2 in [12], since the conditional expectations are not "tracial" in general. Furthermore, we find that the free Fisher information of circular variables is closely related to modular frames and the index of conditional expectations.…”

Section: Introductionmentioning

confidence: 99%

“…Our main method to prove the results is R.Speicher's cumulant function technique (see [14]), just as in [9,10]. We use k B := (k…”

Section: Introductionmentioning

confidence: 99%

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Operator-valued free Fisher information of random matrices

Meng

2010

Acta Mathematica Scientia

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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.

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Section: The Free Fisher Information Of Random Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Our main method to prove the results is R.Speicher's cumulant function technique (see [14]), just as in [9,10]. We use k B := (k…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Operator-valued free Fisher information of random matrices

Meng

2010

Acta Mathematica Scientia

View full text Add to dashboard Cite

show abstract

“…In [6], we have generalized the notion of free Fisher information to the operator-valued setting. (M, E, B) be a B-valued noncommutative probability space, and let X ∈ M be a self-adjoint random variable and η : B → B be a linear map.…”

Section: Rank Preserving Module Maps On L(h a )mentioning

confidence: 99%