By Thomas Timmermann

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

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13. 8 i). 7]. 14. G/ ! 4. 1]. 1/ D 0g [1, p. 198]. 6. Chapter 2 Multiplier Hopf algebras and their duality A multiplier Hopf algebra is a non-unital generalization of a Hopf algebra, where the target of the comultiplication is no longer the twofold tensor product of the underlying algebra, but an enlarged multiplier algebra. 1. 4. This duality is based on left- and right-invariant linear functionals called integrals, which are analogues of the Haar measures of a group. 3. The theory of multiplier Hopf algebras was developed by Van Daele; all results presented in this chapter are taken from the articles [174], [177].

13. 8 i). 7]. 14. G/ ! 4. 1]. 1/ D 0g [1, p. 198]. 6. Chapter 2 Multiplier Hopf algebras and their duality A multiplier Hopf algebra is a non-unital generalization of a Hopf algebra, where the target of the comultiplication is no longer the twofold tensor product of the underlying algebra, but an enlarged multiplier algebra. 1. 4. This duality is based on left- and right-invariant linear functionals called integrals, which are analogues of the Haar measures of a group. 3. The theory of multiplier Hopf algebras was developed by Van Daele; all results presented in this chapter are taken from the articles [174], [177].

A0 , b 7! jb/; are isomorphisms of Hopf algebras. B; B / be Hopf -algebras and . j / W A B ! k a dual pairing of Hopf algebras. 4. b/ / for all a 2 A and b 2 B. b/ / for all a 2 A; b 2 B, and the reverse implication follows similarly. Thus, one of these conditions may be omitted in the definition of a dual pairing of Hopf -algebras. Let us consider several examples of dual pairings. 6. C// ! C/. C/ ! C/ is identified with the space of n n-matrices with vanishing trace. 14. 7. Let G be a compact connected Lie group with Lie algebra g.