Kac Algebras and Duality of Locally Compact Groups

Abstract

This is an excellent exposition of duality theory for Kacalgebras.\par The theory was originated in the early sixties byG. I. Kac [Kats] as a successful attempt at finding a self dualcategory of Hopf algebras (with multiplication, comultiplication, involution, and co involution) containing thecategory of locally compact groups. Now the solution seems tohave a final form.\par The book contains a preface (by Connes),an introduction with a historical review and six chapters: 1. Coinvolutive Hopf von Neumann algebras, 2. Kac algebras, 3.Representations of a Kac algebra; dual Kac algebra, 4. Dualitytheorems for Kac algebras and locally compact groups, 5. Thecategory of Kac algebras, and 6. Special cases: unimodular,compact, discrete and finite dimensional Kac algebras, and apostface (by Ocneanu). In particular, the theory of weights onvon Neumann algebras, the theory of Fourier Stieltjes algebrasand Eymard algebras, and various duality theorems (includingEymard's theorem, Krein's theorem, and a Tatsuuma type theorem)are carefully presented.\par Occasionally, in the course of theproofs, the book gives many nontrivial and useful calculations intensor products of von Neumann algebras. However, there are noexercises, and the reader has no alternative but to follow theauthors step by step. The reviewer thinks there are some problemsabout isomorphism and duality of concrete Kac algebras which canhelp the reader to recognize the subject as a fruitful directionin noncommutative harmonic analysis.\par The epigraph, ``Thequestion is the story itself, and whether or not it meanssomething is not for the story to tell (Paul Auster)'',illustrates the restrained viewpoint of the authors. Being quitesure that this story means enough to be told and studied, we arenot confident that it is finished. Although importantrelationships with ``quantum groups'' (mentioned by Connes) canonly be given in outline, there are topics on crossed products ofvon Neumann algebras by Kac algebras and intrinsic groups of Kacalgebras which are deeply elaborated and these seem to befruitful tools in various problems of representation theory andabstract harmonic analysis. We completely join Ocneanu in hishope ``that the authors will continue their beautiful monographwith a study of the Kac algebras in action''