A Fock space model for addition and multiplication of c-free random variables

Popa, Mihai

doi:10.1090/s0002-9939-2014-11786-2

Cited by 2 publications

(2 citation statements)

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“…An important application of our construction is the following: It is assumed in several sources (see [12] and [13], for example) that conditionally free algebras with respect to the states ϕ and ψ, see Definition 1.1, are also free with respect to ψ -see Definition 2.1. Although a general characterization of when c. free algebras are also free is still an open question, in this paper we give a satisfactory answer in the following sense: given c. free algebras we can always find, under mild assumptions, copies of them that are conditionally free and also free (in the way it is specified above), see Theorem 4.4. This paper is divided in five sections.…”

Section: Introductionmentioning

confidence: 99%

Conditionally free reduced products of Hilbert spaces

Arizmendi¹,

Ballesteros²,

Torres-Ayala³

2020

Studia Math.

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We present a product of pairs of pointed Hilbert spaces that, in the context of Bozėjko, Leinert and Speicher's theory of conditionally free probability, plays the role of the reduced free product of pointed Hilbert spaces, and thus gives a unified construction for the natural notions of independence defined by Muraki.We additionally provide important applications of this construction. We prove that, assuming minor restrictions, for any pair of conditionally free algebras there are copies of them that are conditionally free and also free, a property that is frequently assumed (as hypothesis) to prove several results in the literature. Finally, we give a short proof of the linearization property of the c R-transform (the analog of Voiculescu's R-transform in the context of conditionally free probability).

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Section: Introductionmentioning

confidence: 99%

Conditionally free reduced products of Hilbert spaces

Arizmendi¹,

Ballesteros²,

Torres-Ayala³

2020

Studia Math.

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show abstract

“…There, for X a non-commutative random variable, we define an analytic function c T X (z), inspired by Voiculescu's S-transform, such that if X and Y are c-free , then c T XY (z) = c T X (z) · c T Y (z). Alternate proofs of this result were given in [19] and [22].…”

Section: Introductionmentioning

confidence: 99%