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This ebook 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 provides the elemental conception of Hopf algebras, Van Daele's duality conception of algebraic quantum teams, and Woronowicz's compact quantum teams, staying in a in simple terms algebraic environment. half II makes a speciality of quantum teams within the atmosphere of operator algebras. Woronowicz's compact quantum teams are taken care of within the surroundings of $C^*$-algebras, and the elemental multiplicative unitaries of Baaj and Skandalis are studied intimately. an summary of Kustermans' and Vaes' complete conception of in the neighborhood compact quantum teams completes this half. half III ends up in chosen issues, resembling 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. simply 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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**Extra info for An Invitation to Quantum Groups and Duality (Ems Textbooks in Mathematics)**

**Example text**

A; f / 7! a f; turn A into a bimodule over the algebra A0 . Proof. f; a/ 7! f a turns A into a left module over A0 . id ˝ h/ ı W A ! A. g/ for all f; g 2 A0 . a; f / 7! a f turns A into a right module over A0 . 11. A; / is a Hopf algebra, then S a D Á. a// D a S for all a 2 A. 3 Properties of the antipode The antipode of a Hopf algebra satisfies several fundamental relations that are not obvious from the definition. To some extent, the antipode of a Hopf algebra behaves like the inversion of a group: the inversion of a group is antimultiplicative, and the antipode of a Hopf algebra is both antimultiplicative and anticomultiplicative.

Id ˝/ W a ˝ b 7! id ˝T2 / W a ˝ b ˝ c 7! 22 ([174]). A; / be a unital bialgebra. A; / is a Hopf algebra. Proof. A; /. Let us start with the counit. a ˝ 1A // for all a 2 A. So, consider the map E W A ! A; a 7! 3. Axiomatics of Hopf algebras 25 We show that the image of E is contained in k 1A . b/ span A ˝ A. Therefore, the calculation above shows that the image of E is contained in k 1A . Define W A ! a/ 1A for all a 2 A. We show that is a counit. Let b 2 A. b// D b. 13), we find X . b ˝ 1A / D b: It remains to show that is a homomorphism.

D! G/. We show that then ! D 0. Since the operator D! G/ is closed under left translation, it follows that D! G// D 0. We show that then also D! G// D 0, and this implies ! D 0. G/ carries a natural locally convex topology, and D! is continuous with respect to this topology [32, Chapter XVII]. G/. G/ spanned by finite-dimensional G-invariant subspaces is dense. G/. vi) Let f ¤ 0 be a representative function on G. jf / ¤ 0 for some ! g/. G/ the linear span of all right translates of f , and by W G !