Locally compact quantum groups and applications in noncommutative ergodic theory

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The concept of group duality is fundamental in the analysis of locally compact abelian groups.
The theory of (analytic) quantum groups was developed in order to provide a framework for duality of general locally compact groups.
The simple set of axioms describing "locally compact quantum groups" (LCQGs) introduced in '00 by Kustermans and Vaes is built on preceding, deep works of Kac and Vainerman, Enock and Schwartz, Woronowicz, Baaj and Skandalis, Masuda and Nakagami and many others.
LCQGs have an intriguing structure theory, and numerous results on locally compact groups have already been generalized to LCQGs.
In this talk we will motivate and introduce the definition of LCQGs, explain and exemplify how they are constructed and mention some of their applications.
Afterwards, we shall describe a generalization of recent work on aspects of ergodic theory of semigroup actions on von Neumann algebras to the context of quantum semigroups.
These results give a Jacobs-de Leeuw-Glicksberg splitting at the von Neumann algebra level.