# 12 décembre 2017: 2 événements

### Mardi 12 décembre 13:45-15:00 - Monika Malczak - Greifswald

Lévy processes on braided *-bialgebras

Résumé : In order to use Majid’s Bosonization/Symmetrization of braided bialgebras for quantum
probability theory, we give an extension to braided *-bialgebras, which is based on a
joint work by Franz, Schott and Schürmann [1]. Furthermore we investigate a connection between quantum Lévy processes on braided *-bialgebras and their symmetrization. We present a realization as solution of quantum stochastic differential equations for Brownian motions on such braided structures, which can be considered as multi-dimensonal analogues of the Azéma martingales.
References
[1] Franz, Uwe ; Schott, René ; Schürmann, Michael : Lévy Processes and Brownian Motion on Braided Spaces. In : Franz, Uwe : The Theory of Quantum Lévy Processes, Chapter 5 of habilitation thesis.
2003. Available at https://arxiv.org/abs/math/0407488.

En savoir plus : Séminaire d’Analyse Fonctionnelle

### Mardi 12 décembre 15:00-16:00 - Philipp Varso - Greifswald

Central Limit Theorem for General Universal Products

Résumé : To model independence in quantum probability theory one uses so-called universal products, which are in general described by tensor categories of algebraic quantum probability spaces $(A,\varphi)$. In [1] Muraki has shown how to classify such products and in particular obtained that only -five universal products exist. But there are examples which do not -fit into Muraki’s framework, for instance if one wants to deal with a tuple of linear functionals $\varphi^(i)$ on the algebra $A$ like it has been done in the case of $c$-freeness by Bo\.zejko and Speicher in [2].
The case of bi-freeness by Voiculescu [3], where in particular the underlying algebra $A$ is isomorphic to the free product of two algebras $A^(1)$ and $A^(2)$, is also not covered by Muraki’s classi-cation. In [4] Schürmann and Manzel present a uni-fied approach to cumulants, which includes the above concepts of independence. This is achieved by considering a certain category of algebraic non-commutative probability spaces, denoted by $\rm algP_d,m$, which consist of an $m$-tuple of subalgebras and a $d$-tuple of linear functionals and therefore allows to investigate $(d,m)$-independence induced by a so-called u.a.u.-product.
The -first noncommutative version of a central limit theorem dates back to von Waldenfels [5]. In this talk we want to present a noncommutative version of a central limit theorem for a u.a.u.-product in $\rm algP_d,m$, where we make use of the so-called Lachs functor [6], which operates between certain tensor categories.
References
[1] Muraki, Naofumi : The -five independences as natural products. In-n. Dimens. Anal. Quantum Probab. Relat. Top. 6.3 (2003), 337-371.
[2] Bo\.zejko, Marek ; Speicher, Roland : $\psi$ -independent and symmetrized white noises. In : Quantum probability \& related topics. QP-PQ, VI, 219-236. World Sci. Publ., River Edge, NJ, 1991.
[3] Voiculescu, Dan-Virgil : Free probability for pairs of faces I. Comm. Math. Phys. 332.3 (2014), 955-980.
[4] Manzel, Sarah ; Schürmann, Michael : Non-Commutative Stochastic Independence and Cumulants. Preprint arXiv:1601.06779 (2016), 42 pages. To appear in IDAQP 20.2 (2017).
[5] von Waldenfels, W. : An algebraic central limit theorem in the anti-commuting case. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42.2 (1978), 135-140.
[6] Lachs, Stephanie : A New Family of Universal Products and Aspects of a Non-Positive Quantum Probability Theory. PhD thesis. Ernst-Moritz-Arndt-Universität Greifswald, 2015.

En savoir plus : Séminaire d’Analyse Fonctionnelle