Analytic subordination for bi-free convolution

Belinschi, Serban; Bercovici, Hari; Gu, Yinzheng; Skoufranis, Paul

doi:10.1016/j.jfa.2018.03.003

Cited by 10 publications

(34 citation statements)

References 35 publications

(98 reference statements)

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“…holds for (s, t) ∈ R 2 and |y|, |v| ≥ M, it follows that for all µ ∈ F , (2) : µ ∈ F } can be obtained in a similar way. Now the sufficiency follows since for any r > 0,…”

Section: Preliminariesmentioning

confidence: 95%

“…Throughout the remaining part of the paper, points (s, t) in R 2 will be denoted by the bold letter x and the origin (0, 0) will be written as 0. We will also denote by the real numbers v (1) and v (2) the s-and t-coordinate of a given vector v ∈ R 2 .…”

Section: Bi-free Infinite Divisibility and Lévy-hinčin Representationmentioning

confidence: 99%

“…Since the introduction of bi-free probability by Voiculescu in 2013, combinatorial and analytical approaches have been the main research focuses so far [7][8] [11] [13][17] [18]. Given a two-faced pair (a, b) in a C * -probability space (A, ϕ), its bi-free partial R-transform is defined through its Cauchy transform G (a,b) (z, w) = ϕ((z − a) −1 (w − b) −1 ) as the left faces a, c and the freeness of the right ones b, d. The reader is referred to [2] [12] for some recent developments.…”

Section: Introductionmentioning

confidence: 99%

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Limit theorems in bi-free probability theory

Huang

Wang

2018

Probab. Theory Relat. Fields

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In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bi-freely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bi-free probability theory. Complete descriptions of bi-free stability and fullness of planar probability distributions are also set down. All these results reveal one important feature about the theory of bi-free probability that it parallels the classical theory perfectly well. The emphasis in the whole work is not on the tool of bi-free combinatorics but only on the analytic machinery.for complex values z and w in a neighborhood of zero, where R a and R b are the usual R-transforms of a and b, respectively [20]. This transform plays the same role as the usual R-transform does in the free case. As in the classical situation, the bi-freeness of (a, b) and (c, d) yields the freeness of

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“…holds for (s, t) ∈ R 2 and |y|, |v| ≥ M, it follows that for all µ ∈ F , (2) : µ ∈ F } can be obtained in a similar way. Now the sufficiency follows since for any r > 0,…”

Section: Preliminariesmentioning

confidence: 95%

Section: Bi-free Infinite Divisibility and Lévy-hinčin Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Limit theorems in bi-free probability theory

Huang

Wang

2018

Probab. Theory Relat. Fields

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“…where τ = 15δ (1,1) +15δ (−1,1) +15δ (1,−1) . Then K m,n = 15(−1) m +15(−1) n +15 for m, n ≥ 0, (m, n) = (0, 0).…”

Section: 2mentioning

confidence: 99%

“…Assume that (a 1 , b 1 ) and (a 2 , b 2 ) have the same probability distribution µ = 1 2 (δ (0,1) + δ (1,0) ). Notice that Hence X 1 is clearly a self-adjoint matrix.…”

mentioning

confidence: 99%

Bi-monotonic Independence for Pairs of Algebras

Skoufranis

2019

J Theor Probab

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In this article, the notion of bi-monotonic independence is introduced as an extension of monotonic independence to the two-faced framework for a family of pairs of algebras in a non-commutative space. The associated cumulants are defined and a moment-cumulant formula is derived in the bi-monotonic setting. In general the bi-monotonic product of states is not a state and the bi-monotonic convolution of probability measures on the plane is not a probability measure. This provides an additional example of how positivity need not be preserved under conditional bi-free convolutions.

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On operator-valued bi-free distributions

Skoufranis

2016

Advances in Mathematics

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In this paper, operator-valued bi-free distributions are investigated. Given a subalgebra D of a unital algebra B, it is established that a two-faced family Z is bi-free from (B, B op ) over D if and only if certain conditions relating the B-valued and D-valued bi-free cumulants of Z are satisfied. Using this, we verify that a two-faced family of matrices is R-cyclic if and only if they are bi-free from the scalar matrices over the scalar diagonal matrices. Furthermore, the operator-valued bi-free partial R-, S-, and T -transforms are constructed. New proofs of results from free probability are developed in order to facilitate many of these bi-free results.for all T ∈ L(M); that is, apply T to 1 M and take the expectation of M onto N. Further define *homomorphisms L : M → L(M) and R : M op → L(M) by L(X)(A) = XA and R(X)(A) = AX for all X, A ∈ M. For this discussion we will call L(X) a left operator and R(X) a right operator. If M = M 1 * N M 2 , the amalgamated free product of von Neumann algebras M 1 and M 2 over N, then the Date: July 9, 2018. 2010 Mathematics Subject Classification. 46L54, 46L53. Key words and phrases. bi-free probability, operator-valued distributions, amalgamating over a subalgebra, R-cyclic matrices, R-transform, S-transform. PAUL SKOUFRANIS Given χ : {1, . . . , n} → {ℓ, r} withdefine the permutation s χ on {1, . . . , n} by s χ (q) = k q . The only differences between the combinatorial aspects of free and bi-free probability arise from dealing with s χ . Using s χ , define the total ordering ≺ χ on {1, . . . , n} by k 1 ≺ χ k 2 if and only if s −1 χ (k 1 ) < s −1 χ (k 2 ). Instead of reading {1, . . . , n} in the traditional order, ≺ χ corresponds to reading χ −1 ({ℓ}) in increasing order followed by reading χ −1 ({r}) in decreasing order.A subset V ⊆ {1, . . . , n} is said to be a χ-interval if V is an interval with respect to the ordering ≺ χ . In addition, min ≺χ (V ) and max ≺χ (V ) denote the minimal and maximal elements of V with respect to the ordering ≺ χ .Definition 2.1. A partition π ∈ P(n) is said to be bi-non-crossing with respect to χ if the partition s −1 χ · π (the partition formed by applying s −1 χ to each entry of each block of π) is non-crossing. Equivalently π is bi-non-crossing if whenever there are blocks U, V ∈ π with u 1

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Analytic subordination for bi-free convolution

Cited by 10 publications

References 35 publications

Limit theorems in bi-free probability theory

Limit theorems in bi-free probability theory

Bi-monotonic Independence for Pairs of Algebras

On operator-valued bi-free distributions

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