By Thomas Timmermann
This booklet presents an creation to the idea of quantum teams with emphasis on their duality and at the atmosphere of operator algebras. half I of the textual content provides the elemental thought of Hopf algebras, Van Daele's duality conception of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a in basic terms algebraic surroundings. half II makes a speciality of quantum teams within the surroundings of operator algebras. Woronowicz's compact quantum teams are taken care of within the environment of $C^*$-algebras, and the elemental multiplicative unitaries of Baaj and Skandalis are studied intimately. an summary of Kustermans' and Vaes' complete idea of in the neighborhood compact quantum teams completes this half. half III ends up in chosen issues, comparable to coactions, Baaj-Skandalis-duality, and techniques to quantum groupoids within the atmosphere of operator algebras. The booklet is addressed to graduate scholars and non-experts from different fields. purely simple wisdom of (multi-) linear algebra is needed for the 1st half, whereas the second one and 3rd half think a few familiarity with Hilbert areas, $C^*$-algebras, and von Neumann algebras.
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