Combinatorics of Bi-Freeness with Amalgamation

Charlesworth, Ian; Nelson, Brent; Skoufranis, Paul

doi:10.1007/s00220-015-2326-8

Cited by 38 publications

(129 citation statements)

References 9 publications

Supporting

Mentioning

129

Contrasting

Order By: Relevance

“…Due to the recent work ( [10], [7], [1], [2], [8], [3], [5]) we already have a much better understanding of bi-free probability.…”

Section: Introductionmentioning

confidence: 99%

Free probability for pairs of faces III: 2-Variables bi-free partial S- and T-transforms

Voiculescu

2016

Journal of Functional Analysis

View full text Add to dashboard Cite

Abstract. We introduce two 2-variables transforms: the partial bi-free S-transform and the partial bi-free T -transform. These transforms are the analogues for the bi-multiplicative and, respectively, for the additive-multiplicative bi-free convolution of the 2-variables partial bi-free R-transform in our previous paper in this series. IntroductionIn the first paper in this series [9] we proposed an extension of free probability to systems with left and right variables, based on the notion of bi-freeness. Due to the recent work ([10]we already have a much better understanding of bi-free probability.Here, we complement our paper [10] on the 2-variables bi-free Rtransform with two further 2-variables partial transforms adapted to the bi-multiplicative operation ⊠⊠ and to the additive-multiplicative operation ⊞⊠. In view of free-probability it was obvious to use the letter S for the first transform, while for the second which corresponds to passing from bi-free pairs (a 1 , b 1 ), (a 2 , b 2 ) to (a 1 + a 2 , b 1 b 2 ) we went for T the next letter in the alphabet.Like in the case of addition [10] we found that also for multiplication instead of our original approach to the one-variable S-transform

show abstract

“…Due to the recent work ( [10], [7], [1], [2], [8], [3], [5]) we already have a much better understanding of bi-free probability.…”

Section: Introductionmentioning

confidence: 99%

Free probability for pairs of faces III: 2-Variables bi-free partial S- and T-transforms

Voiculescu

2016

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…However, this is not true as illustrated by the following example. Consequently, the bi-Boolean Lévy-Hinčin formula (7) best characterizes the class of ⊎⊎-infinitely divisible Borel probability measures on R 2 .…”

Section: (6)mentioning

confidence: 99%

“…It is thus natural to expect that a similar property holds for bB. However, it turns out that bB is not a homomorphism with respect to the 'usual' multiplicative bi-free convolution ⊠⊠ in the literature (see [21,28]) but with respect to the one in [7,8]. Consequently, we consider the following operation.…”

Section: 2mentioning

confidence: 99%

Bi-Boolean Independence for Pairs of Algebras

Skoufranis

2017

Complex Anal. Oper. Theory

Self Cite

View full text Add to dashboard Cite

In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to pairs of algebras. The notion of B-(ℓ, r)cumulants is defined via a bi-Boolean moment-cumulant formula over the lattice of bi-interval partitions, and it is demonstrated that bi-Boolean independence is equivalent to the vanishing of mixed B-(ℓ, r)-cumulants. Furthermore, some of the simplest bi-Boolean convolutions are considered, and a bi-Boolean partial ηtransform is constructed for the study of limit theorems and infinite divisibility with respect to the additive bi-Boolean convolution. In particular, a bi-Boolean Lévy-Hinčin formula is derived in perfect analogy with the bi-free case, and some Bercovici-Pata type bijections are provided. Additional topics considered include the additive bi-Fermi convolution, some relations between the (ℓ, r)-and B-(ℓ, r)-cumulants, and bi-Boolean independence in an amalgamated setting.The last section of this paper also includes an errata that will be published with this copy of the paper. Date

show abstract

“…In this note, we demonstrate an appropriate bi-free analogue of the freeness condition. We also examine the extension of these techniques to conditional bi-free probability as studied by Gu and Skoufranis [7], and to certain operator-valued bi-free settings as considered in [4,14].Moreover, we consider a bi-free analogue of Biane's free multiplicative Brownian motion, which is a free stochastic process that may be constructed as the solution to a free stochastic equation involving free (additive) Brownian motion, or as a limit of the Markov process arising from the heat semi-group on finite dimensional unitary matrices [2]. A free multiplicative Brownian motion U (t) converges in moments to a Haar unitary operator as t → ∞; we introduced bi-free multiplicative Brownian motion, a pair of stochastic processes which behave similarly and converge in moments to a bi-Haar unitary in the sense of [4].…”

mentioning

confidence: 99%

An alternating moment condition for bi-freeness

Charlesworth

2019

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

In this note we demonstrate an equivalent condition for bi-freeness, inspired by the well-known "vanishing of alternating centred moments" condition from free probability. We show that all products satisfying a centred condition on maximal monochromatic χ-intervals have vanishing moments if and only if the family of pairs of faces they come from is bi-free, and show that similar characterisations hold for the amalgamated and conditional settings. In addition, we construct a bi-free unitary Brownian motion and show that conjugation by this process asymptotically creates bi-freeness; these considerations lead to another characterisation of bi-free independence. Introduction.Bi-free probability was introduced by Voiculescu in [14] as a generalisation of free probability studying simultaneously left and right actions of algebras on a reduced free product space. Voiculescu demonstrated that many notions from free probability generalise with appropriate care to this bi-free setting. In particular, [14] demonstrated the existence of bi-free cumulant polynomials but did not produce explicit formulae for them. Soon after, Mastnak and Nica in [9] proposed a family of cumulant functionals, which the author together with Nelson and Skoufranis in [5] showed to agree with those abstractly given by Voiculescu. In particular, [5] demonstrated that bi-freeness was equivalent to the vanishing of mixed cumulants.Since then, many more techniques from free probability have been generalised to the bi-free setting: bi-free partial transforms were studied in [8,10,12,15,16]; infinite divisibility and a bi-free Lévy-Hinčin formula in [6]; bi-matrix models in [11]; and so on. One major difficulty in generalising results to bi-free probability, however, has been that the condition defining bi-freeness is somewhat unwieldy. Free independence is equivalent to saying that alternating products of centred elements are themselves centred, but in bi-free probability it has been necessary to work either with cumulants (and hence compute the Möbius function for the lattice of bi-non-crossing partitions) or to compute moments through abstract bi-free products. In this note, we demonstrate an appropriate bi-free analogue of the freeness condition. We also examine the extension of these techniques to conditional bi-free probability as studied by Gu and Skoufranis [7], and to certain operator-valued bi-free settings as considered in [4,14].Moreover, we consider a bi-free analogue of Biane's free multiplicative Brownian motion, which is a free stochastic process that may be constructed as the solution to a free stochastic equation involving free (additive) Brownian motion, or as a limit of the Markov process arising from the heat semi-group on finite dimensional unitary matrices [2]. A free multiplicative Brownian motion U (t) converges in moments to a Haar unitary operator as t → ∞; we introduced bi-free multiplicative Brownian motion, a pair of stochastic processes which behave similarly and converge in moments to a bi-Haar unitary in the sense of [4]...

show abstract

Combinatorics of Bi-Freeness with Amalgamation

Cited by 38 publications

References 9 publications

Free probability for pairs of faces III: 2-Variables bi-free partial S- and T-transforms

Free probability for pairs of faces III: 2-Variables bi-free partial S- and T-transforms

Bi-Boolean Independence for Pairs of Algebras

An alternating moment condition for bi-freeness

Contact Info

Product

Resources

About