Free evolution on algebras with two states

Anshelevich, Michael

doi:10.1515/crelle.2010.003

Cited by 11 publications

(28 citation statements)

References 21 publications

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“…• If m belongs to a block V of π where min(V ) < m < max(V ), then W m = " , " (just a comma). • If m forms by itself a singleton block of π, then W m = "F [1] " (no parentheses or comma besides the occurrence of F [1] ).…”

Section: Remark 32 (Nested Terms F [π]mentioning

confidence: 99%

Convolution powers in the operator-valued framework

Anshelevich¹,

Belinschi²,

Fevrier³

et al. 2012

Trans. Amer. Math. Soc.

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We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map \eta from B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever \eta is completely positive, while the free additive convolution power is defined whenever \eta - 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent \eta. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion.Comment: 33 pages, no figure

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Section: Remark 32 (Nested Terms F [π]mentioning

confidence: 99%

Convolution powers in the operator-valued framework

Anshelevich¹,

Belinschi²,

Fevrier³

et al. 2012

Trans. Amer. Math. Soc.

Self Cite

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“…In the end, we give a new proof for some Biane's results on the densities of the free multiplicative analogue of the normal distributions. the multiplicative analogue of examples studied in [1,4,6,11]. We define the set (A) = {µ ∈ M * T : m 1 (µ) = e −1/2 }.…”

mentioning

confidence: 99%

Free Brownian motion and free convolution semigroups: multiplicative case

Zhong

2014

Pacific J. Math.

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Abstract. We consider a pair of probability measures µ, ν on the unit circle such that Σ λ (η ν (z)) = z/η µ (z). We prove that the same type of equation holds for any t ≥ 0 when we replace ν by ν ⊠λ t and µ by M t (µ), where λ t is the free multiplicative analogue of the normal distribution on the unit circle of C and M t is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subordination functions in multiplicative free Brownian motion and multiplicative free convolution semigroups. We use the modified S-transform introduced by Raj Rao and Speicher to deal with the case that ν has mean zero. The same type of the result holds for convolutions on the positive real line. In the end, we give a new proof for some Biane's results on the densities of the free multiplicative analogue of the normal distributions.

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“…It is a re-formulation of Theorem 2.8 in [Zho14], and generalizes Theorem 1.6 in [BN08]. See also [Ans15].…”

Section: Asymptotically Limmentioning

confidence: 69%

“…By similar methods, we also easily obtain the following analogs of Corollary 4.13 in [Nic09] and Lemma 7 in [Ans15].…”

Section: Asymptotically Limmentioning

confidence: 87%

The Exponential Map in Non-commutative Probability

Anshelevich

Arizmendi

2016

Int Math Res Notices

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ABSTRACT. The wrapping transformation W is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class L of measures, W also transforms the three non-commutative convolutions-free, Boolean, and monotone-to their multiplicative counterparts. Moreover, the restriction of W to L preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of numerous known, and new, results about multiplicative convolutions.

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Free evolution on algebras with two states

Cited by 11 publications

References 21 publications

Convolution powers in the operator-valued framework

Convolution powers in the operator-valued framework

Free Brownian motion and free convolution semigroups: multiplicative case

The Exponential Map in Non-commutative Probability

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