An alternating moment condition for bi-freeness

Abstract

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]...

Bi-Free Entropy with Respect to Completely Positive Maps

“…. , s χ (6)) = (1,2,3,6,5,4) and the partition given by π = {{1, 4}, {2, 5}, {3, 6}} is bi-non-crossing with respect to χ even though π / ∈ NC (6). This may also be seen via the following diagrams:…”

“…The proof of positivity in the operator-valued free case requires the characterization of the vanishing of alternating centred moments in [13,Proposition 3.3.3] to ensure positivity in the end. One may attempt to use the bi-free analogue of 'alternating centred moments vanish' from [1], however the bi-free formulae generalization of [13,Proposition 3.3.3] is far more complicated. In particular, the proof from [13] will not immediately generalize, as Example 3.4 shows E will not be positive and the traciality of τ B will need to come into play.…”

“…As Ξ ∈ alg (X, Y, M 2 (C) ℓ , M 2 (C) r ) τ2 , it is not difficult to see that ξ i,j ∈ alg(x, x * , y, y * ) • ϕ . To see that ξ 2,1 and ξ 1,2 satisfy the appropriate left bi-free conjugate variable relations, one need only use equation (1), choose b k = E i k ,j k satisfying the above required relations for a non-zero value and expand both sides of equation ( 1) in an identical way to that above. The resulting equations are exactly the left bi-free conjugate variable relations required.…”

“…1 , which are then self-adjoint elements of A 2 . The pair (X, Y ) is intimately related to whether or not (x, y) is bi-R-diagonal ([7, Definition 2.12]).…”

Bi-monotonic Independence for Pairs of Algebras

“…If furthermore K = {a, b}, ω −1 ({a}) = {1, 2, 3, 4, 6, 8, 11, 12} and ω −1 ({b}) = {5, 7, 9, 10}, then the partition π χ,ω is given by The above definition was used in [10] to define bi-Boolean independence. The relevance here is that the partition π ω,χ was also used in the paper [3] to provide another characterization of c-bi-free independence given as follows.…”