Infinite divisibility and a non-commutative Boolean-to-free Bercovici–Pata bijection

Belinschi, Serban; Popa, Mihai; Vinnikov, Victor

doi:10.1016/j.jfa.2011.09.006

Cited by 31 publications

(36 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Brown measure µ x as defined in (7) could be a priori a Schwartz distribution but one can show that the map λ → log ∆(x − λ) is subharmonic and hence µ x is a positive measure on C. In fact µ x is a probability measure supported on a subset of the spectrum of x. One can show that for the examples from Sections 2.1.2-2.1.4 the above definition gives the correct values for (3), (5), and (6).…”

Section: 13mentioning

confidence: 99%

Eigenvalues of non-Hermitian random matrices and Brown measure of non-normal operators: Hermitian reduction and linearization method

Belinschi

Śniady

Speicher

2018

Linear Algebra and its Applications

View full text Add to dashboard Cite

We study the Brown measure of certain non-Hermitian operators arising from Voiculescu's free probability theory. Usually those operators appear as the limit in ⋆-moments of certain ensembles of non-Hermitian random matrices, and the Brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices. A prominent class for our operators is given by polynomials in ⋆-free variables. Other explicit examples include R-diagonal elements and elliptic elements, for which the Brown measure was already known, and a new class of triangular-elliptic elements. Our method for the calculation of the Brown measure is based on a rigorous mathematical treatment of the Hermitian reduction method, as considered in the physical literature, combined with subordination and the linearization trick.1991 Mathematics Subject Classification. 15B52 (Primary), 46L54, 46L10 (Secondary).

show abstract

Section: 13mentioning

confidence: 99%

Eigenvalues of non-Hermitian random matrices and Brown measure of non-normal operators: Hermitian reduction and linearization method

Belinschi

Śniady

Speicher

2018

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…Voiculescu extended free probability to amalgamated free products of C * -algebras in [Voi95], replacing states acting on these algebras with conditional expectations onto a distinguished subalgebra. This theory has achieved remarkable growth in recent years and we refer to [Spe98] for an overview of the combinatorial approach to this subject and [BPV12], [PV13], [BMS13] and [AW14b] for some of the recent advances in this field.…”

Section: Introductionmentioning

confidence: 99%

B-Valued free convolution for unbounded operators

Williams¹

2017

Indiana Univ. Math. J.

View full text Add to dashboard Cite

Abstract. Consider the B-valued probability space (A, E, B), where A is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operatorsÃ. For a random variable X ∈Ã sa we study the Cauchy transform G X and show that the von Neumann algebra (B ∪ {X})′′ can be recovered from this function. In the case where B is finite dimensional, we show that, when X, Y ∈Ã sa are assumed to be B-free, the R-transforms are defined on universal subsets of the resolvent and satisfyExamples indicating a failure of the theory for infinite dimensional B are provided. Lastly, we show that the functions that arise as the Cauchy transform of affiliated operators are the limit points of the Cauchy transforms of bounded operators in a suitable topology.

show abstract

“…Theorem 2.4. A family {(A k,ℓ , A k,r )} k∈K of pairs of algebras in a two-state non-commutative probability space (A, ϕ, ψ) is c-bi-free with respect to (ϕ, ψ) if and only if (1) ψ(a 1 · · · a n ) =…”

Section: Preliminariesmentioning

confidence: 99%

Conditionally Bi-Free Independence with Amalgamation

Skoufranis

2017

International Mathematics Research Notices

View full text Add to dashboard Cite

In this paper, we introduce the notion of conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operatorvalued conditionally bi-free moment and cumulant functions. It is demonstrated that conditionally bifree independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. Furthermore, an operator-valued conditionally bi-free partial R-transform is constructed and various operator-valued conditionally bi-free limit theorems are studied. IntroductionThe notion of conditionally free (c-free for short) independence was introduced in [3] as a generalization of the notion of free independence to two-state systems. In our previous paper [6] we introduced the notion of conditionally bi-free (c-bi-free for short) independence in order to study the non-commutative left and right actions of algebras on a reduced c-free product simultaneously. Thus conditional bi-freeness is an extension of the notion of bi-free independence [14] to two-state systems. Moreover [6] introduced c-(ℓ, r)-cumulants and demonstrated that a family of pairs of algebras in a two-state non-commutative probability space is conditionally bi-free if and only if mixed (ℓ, r)-and c-(ℓ, r)-cumulants vanish.In [13] Voiculescu generalized his own notion of free independence by replacing the scalars with an arbitrary algebra thereby obtaining the notion of free independence with amalgamation (see also [12] for the combinatorial aspects). For c-free independence, the generalization to an amalgamated setting over a pair of algebras was done by Popa in [9] (see also [8]). On the other hand, the framework for generalizing bi-free independence to an amalgamated setting was suggested by Voiculescu in [14, Section 8] and the theory was fully developed in [4].The main goal of this paper is to extend the notion of c-bi-free independence to an amalgamated setting over a pair of algebras. Furthermore, we demonstrate that the combinatorics of conditionally bi-free probability and bi-free probability with amalgamation, which are governed by the lattice of bi-non-crossing partitions, are specific instances of more general combinatorial structures.Including this introduction this paper contains nine sections which are structured as follows. Section 2 briefly reviews some of the background material pertaining to conditionally bi-free probability and bi-free probability with amalgamation from [4][5][6]. In particular, the notions bi-non-crossing partitions and diagrams, their lateral refinements and cappings, interior and exterior blocks, B-B-non-commutative probability spaces, operator-valued bi-multiplicative functions, and the operator-valued bi-free moment and cumulant functions are recalled.Section 3 introduces the structures studied within conditionally bi-free independence with amalgamation. We define the notion of a B-B-non-commutative probability space with a pair of (B, D)-valued expectations (A, E, F, ...

show abstract

Infinite divisibility and a non-commutative Boolean-to-free Bercovici–Pata bijection

Cited by 31 publications

References 22 publications

Eigenvalues of non-Hermitian random matrices and Brown measure of non-normal operators: Hermitian reduction and linearization method

Eigenvalues of non-Hermitian random matrices and Brown measure of non-normal operators: Hermitian reduction and linearization method

B-Valued free convolution for unbounded operators

Conditionally Bi-Free Independence with Amalgamation

Contact Info

Product

Resources

About