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

Abstract

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

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

“…Obviously our result here complements the recent work on operations on bi-free bi-partite hermitian two-faced pairs ( [10], [7], [5], [11], [8]). …”

On bi-free multiplicative convolution

“…(1 − S 1 )(1 − S 2 ) = 0, for (z, w) ∈ D r = {(z, w) : |z| < r, |w| < r} for some r > 0. By Proposition 4.1 in [DV3], S 1 and S 2 are holomorphic functions of (z, w) in a neighborhood of (0, 0). If S 1 (z, w) is not the constant function 1 in D r , by Lemma 24 in [PG], {(z, w) ∈ D r : 1 − S 1 (z, w) = 0} is a nowhere dense subset of D r .…”

“…Let (a, b) be a pair of random variables in a non-commutative probability space (A, ϕ). Voiculescu [DV3] defined the following formal power series…”

“…for z, w = 0, and z, w near 0 (Definition 2.1 in [DV3]). Huang and Wang [HW] defined the following transforms (the original transforms were defined as integral transforms of measures on the 2-dimensional torus T 2 )…”

“…Section 1 is devoted to the study of subordination properties of bi-free multiplicative convolution. Using Voiculescu's multiplicative formula for partial bi-free S-transforms (Theorem 2.1 in [DV3]), we derive a subordination equation for bi-free multiplicative convolution (Theorem 1.1), which could be regarded as a multiplicative analogue of above (0.1). If a i b i = b i a i , for i = 1, 2, then the joint distribution µ i of a i , b i is determined by its two-band moments {ϕ(a m i b n i ) : m, n = 1, 2, } for i = 1, 2.…”