The Pontrjagin duality for locally compact Abelian groups is the best setting for a generalization of classical harmonic analysis and Fourier transforms. Dual objects to compact Abelian groups are discrete groups. To have a suitable analogue for non-commutative locally compact groups, it is necessary to find a larger category containing both groups and their duals. Such a broader scheme was recently developed under the influence of new ideas coming from physics (quantum groups). It gives an interpretation of quantum groups in the setting of C*-algebras and von Neumann algebras. The compact case (developed by Woronowicz) is included as a special case.

This book is devoted to a description of this theory. It has three parts. The first part treats quantum groups (and their duality) in a purely algebraic setting. It contains a description of the Van Daele duality of algebraic quantum groups (giving a model for further generalizations) and a discussion of the Woronowicz compact quantum groups. In the second part, quantum groups are treated in the setting of C*-algebras and von Neumann algebras. In particular, it contains a presentation of the Woronowicz of C*-algebraic compact quantum groups and a study of multiplicative unitaries. The last part contains several topics. One is the cross product construction and the duality theorem (due to Baaj and Skandalis), the other is pseudo-multiplicative unitaries on Hilbert spaces. The last chapter contains the author’s results on pseudo-multiplicative unitaries on C*-modules.

The book is nicely written and very well organized. It offers an excellent possibility for students and non-experts to learn this elegant new part of mathematics.

Reviewer:

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