Schoenberg Correspondence on Dual Groups

Schürmann, Michael; Voß, Stefan

doi:10.1007/s00220-013-1872-1

Cited by 6 publications

(8 citation statements)

References 13 publications

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“…Conversely, the recent [23] proves that, for all hermitian and conditionally positive L : U nc n → C such that L(1) = 0, there exists a free Lévy process on U n whose generator is L. We will call such a linear functional a generator, without mentioning any Lévy process. The description of the generators is made easier by the following notion of Schürmann triple.…”

Section: Generator and Schürmann Triplementioning

confidence: 99%

Haar states and Lévy processes on the unitary dual group

Cébron

Ulrich

2016

Journal of Functional Analysis

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Abstract. We study states on the universal noncommutative * -algebra generated by the coefficients of a unitary matrix, or equivalently states on the unitary dual group. Its structure of dual group in the sense of Voiculescu allows to define five natural convolutions. We prove that there exists no Haar state for those convolutions. However, we prove that there exists a weaker form of absorbing state, that we call Haar trace, for the free and the tensor convolutions. We show that the free Haar trace is the limit in distribution of the blocks of a Haar unitary matrix when the dimension tends to infinity. Finally, we study a particular class of free Lévy processes on the unitary dual group which are also the limit of the blocks of random matrices on the classical unitary group when the dimension tends to infinity.

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Section: Generator and Schürmann Triplementioning

confidence: 99%

Haar states and Lévy processes on the unitary dual group

Cébron

Ulrich

2016

Journal of Functional Analysis

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“…As the tensor product does not depend on the faces and does not mix the components, it is enough to show that it is Schoenberg for m = d = 1. This follows for example from the results of [20]. We present an alternative path.…”

Section: Positivitymentioning

confidence: 85%

“…Remark 5.7. Roughly speaking, the strategy of Schürmann and Voß [20] is to apply Lemma 2.1 (2) in order to reduce the Schoenberg correspondence on a complicated dual semigroup to the Schoenberg correspondence on a dual semigroup with primitive comultiplication; using Muraki's classification, it is known that Schoenberg correspondence holds in those cases by explicit construction of a Lévy process on a suitable Fock space via quantum stochastic calculus. Our strategy is to apply the same Lemma 2.1 (2) in order to reduce the Schoenberg correspondence for a complicated universal product to the Schoenberg correspondence for the tensor product.…”

Section: Positivitymentioning

confidence: 99%

“…As explained in [2], such a universal independence yields a convolution for linear functionals on dual semigroups. In [20], Schürmann and Voß proved Schoenberg correspondences for each universal independence, however they do not finish their proof in the axiomatic framework, but rely on Muraki's classification theorem [14,15] which states that a (positive) universal independence has to be one of the five well-studied examples: tensor, free, Boolean, monotone, and antimonotone independence.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we reinvestigate Schürmann and Voß' proof of the Schoenberg correspondence for positive universal products [20,Section 3], observe that a variation of their approach allows to furnish a proof which is independent of Muraki's classification, and simultaneously generalize the proof to the the multivariate situation.…”

Section: Introductionmentioning

confidence: 99%

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Schoenberg correspondence for multifaced independence

Gerhold¹

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The Free Central Limit Theorem and Free Cumulants

Mingo

Speicher

2017

Free Probability and Random Matrices

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We compute the fluctuation moments α m1,...,mr of a Complex Wigner Matrix X N given by the limit lim N →∞ N r−2 k r (Tr(X m1 ), . . . , Tr(X mr )). We prove the limit exists and characterize the leading order via planar graphs that result to be trees. We prove these graphs can be counted by the set of non-crossing partitioned permutations which permit us to express the moments α m1,...,mr in terms of simpler quantities κ m1,...,mr which we call the pseudo-cumulants. We prove the pseudo-cumulants coincide with the higher order free cumulants up to r = 4 which permit us to find the higher order free cumulants κ m1,...,mr associated to the moment sequence α m1,...,mr up to order 4.

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Schoenberg Correspondence on Dual Groups

Cited by 6 publications

References 13 publications

Haar states and Lévy processes on the unitary dual group

Haar states and Lévy processes on the unitary dual group

Schoenberg correspondence for multifaced independence

The Free Central Limit Theorem and Free Cumulants

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