Quantum Lévy processes on dual groups

Abstract

We present a theory of quantum (non-commutative) Lévy processes on dual groups which generalizes the theory of Lévy processes on bialgebras. It follows from a result of N. Muraki that there exist exactly 5 notions of non-commutative 'positive' stochastic independence. We show that one can associate a commutative bialgebra with each pair consisting of a dual group and one of the 5 notions of independence. This construction is related to a construction of U. Franz. Our construction has the advantage that the important case of free independence is included. We show that Lévy processes are given by their generators which are precisely the conditonally positive linear functionals on the dual group.

Haar states and Lévy processes on the unitary dual group

“…In [3], it is proved that L is well-defined and determines completely the family of law (φ•j t ) t≥0 . The generator satisfies L(1) = 0, is hermitian and is conditionally positive, in the sense that…”

“…One other direction in the understanding of the unitary dual group is the study of free Lévy processes on it, in the sense of [1]. Quantum Lévy processes on quantum groups, bialgebras or dual groups have been intensively studied by Ben Ghorbal, Franz, Schürmann and Voss (see [2,3,11,22,23,31,30]). Very recently, the second author of the present article enlightened a deep link between free Lévy processes on U n and random matrices in [26].…”

“…The case k r(q) = k r(1) , k r(2) = k r(3) , · · · , k r(q−2) = k r(q−1) in B(the continuous line represents σ while the dashed line represents K(σ)) Those computations shows that the quantity (h * φ)•∆(m) expressed as (3) does not depend on the choice of φ, and in particular, we can replace φ by δ and obtain (h * φ)•∆(m) = (h * δ)•∆(m). Since m is arbitrary, we have (h * φ)•∆ = (h * δ)•∆.…”

Schoenberg correspondence for multifaced independence

“…It is a remarkable feature of noncommutative probability that there are other independences which can be used to define Lévy processes. A particularly nice class of independences are the universal independences; noncommutative random variables are called independent if their noncommutative joint distribution (modelled on the free product of algebras) coincides with the product distribution with respect to a universal product in the sense of [2]. As explained in [2], such a universal independence yields a convolution for linear functionals on dual semigroups.…”

“…A particularly nice class of independences are the universal independences; noncommutative random variables are called independent if their noncommutative joint distribution (modelled on the free product of algebras) coincides with the product distribution with respect to a universal product in the sense of [2]. 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.…”