It is not only the ‘book of involutions’, but also the ‘book of the classical groups’ ... a very welcome addition to the literature. The topics treated here have been the objects of very intensive research. The specialists felt the need of a reference book, and the beginners of a good introduction. Both of these needs are fulfilled by the ‘book of involutions’ ... a very useful reference source ... these results are not yet published elsewhere ... very well-written ... In addition to being an excellent exposition of many basic results concerning algebras with involution and the classical groups, the book also contains many new ideas and new results, often due to the authors themselves. The topic is a very beautiful and vital one, object of intensive current research. This research is now made easier thanks to the impressive work of the four authors.

This volume is a compendious study of algebras with involution, a subject with many facets which becomes particularly interesting for central simple algebras. The book is excellently written, and the chapters on algebraic groups and Galois cohomology alone would make the book an ideal read for aspiring postgraduate students of an algebraic persuasion. In addition, there is plenty of material to enlighten even those of us who already know something about the subject. All in all, this book recommends itself to anyone who wants a thorough reference source, complete with an ample selection of enlightening exercises and historical notes, which deals with exceptional Jordan algebras, Clifford algebras and modules, Tits' algebras and algebraic groups in a modern manner.

The book under review is an important work which records many of the significant advances in the theory of algebras with involution that have taken place in recent years. Much of the material has not previously appeared in a book before; indeed, there is a substantial amount that has not appeared anywhere before. There is an interesting selection of exercises for each chapter which cover many ancillary results not included in the body of the text. Additionally, there are carefully prepared and highly informative notes at the end of each chapter which give historical commentary and sources for the topics covered. Overall, this book is an outstanding achievement. It will be an indispensable reference for the specialist, and a challenging but highly rewarding introduction for the novice.