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Kac algebras and duality of locally compact groups. Preface by Alain Connes, postface by Adrian Ocneanu. (English) Zbl 0805.22003
Berlin: Springer-Verlag. x, 257 p. (1992).
This book contains the theory of Kac algebras. The Kac algebras are introduced as a natural framework for duality of Pontryagin type (for locally compact abelian groups) and Tannaka-Krejn (for locally compact abelian groups). The authors’ purpose is clearly stated in the preface (written by Allan Connes): “The original motivation of M. Enock and J.- M. Schwartz can be formulated as follows: while in the Pontryagin duality theory of locally compact abelian groups a perfect symmetry exists between a group and its dual, this is no longer true in the various duality theorems of T. Tannaka, M. G. Krejn, W. F. Stinespring …dealing with nonabelian locally compact groups. The aim is then in the line proposed by G. I. Kac in 1961 and M. Takesaki in 1972, to find a good category of Hopf algebras, containing the category of locally compact groups and fulfilling a perfect duality. It is natural to look for this category as a category of Hopf-von Neumann algebras since, first, by a known result of A. Weil, a locally compact group \(G\) is fully specified by the underlying abstract group with a measure class (the class of the Haar measure), and, second, by a result of M. Takesaki, locally compact abelian groups correspond exactly to co-involutive Hopf- von Neumann algebras which are both commutative and cocommutative.”
The book under review, written by two eminent mathematicians each of whom has made major contributions to the theory of Kac algebras, includes the following chapters: Ch. I. Co-Involutive Hopf-Von Neumann Algebras (this chapter is devoted to the structure of such algebras); Ch. II. Kac Algebras (this chapter deals with technical results about Haar weights); Ch. III. Representations of a Kac Algebra; Dual Kac Algebra (this chapter deals with the construction of the dual Kac algebra); Ch. IV. Duality Theorems for Kac Algebras and Locally Compact Groups; Ch. V. The Category of Kac Algebras (in this chapter the authors put on the class of Kac algebras a structure of category, by defining convenient morphisms); Ch. VI. Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras.
The recent research shows that the close relationship between Kac algebras, Woronowicz’s work and quantum groups appears more and more often. We do hope that this book provides a window on the frontier of this active area of research.

22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
22D35 Duality theorems for locally compact groups
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16D90 Module categories in associative algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)