Kac Algebras and Duality of Locally Compact Groups / Edition 1

by Michel Enock, A. Ocneanu, A. Connes, Jean-Marie Schwartz

Overview

This book deals with the theory of Kac algebras and their dual­ ity, elaborated independently by M. Enock and J . -M. Schwartz, and by G. !. Kac and L. !. Vajnermann in the seventies. The sub­ ject has now reached a state of maturity which fully justifies the publication of this book. Also, in recent times, the topic of "quantum groups" has become very fashionable and attracted the attention of more and more mathematicians and theoret­ ical physicists. One is still missing a good characterization of quantum groups among Hopf algebras, similar to the character­ ization of Lie groups among locally compact groups. It is thus extremely valuable to develop the general theory, as this book does, with emphasis on the analytical aspects of the subject instead of the purely algebraic ones. The original motivation of M. Enock and J. -M. Schwartz can be formulated as follows: while in the Pontrjagin 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. Krein, W. F. Stinespring . . . dealing with non abelian locally compact groups. The aim is then, in the line proposed by G. !. 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.

Product Details

ISBN-13:	9783642081286
Publisher:	Springer Berlin Heidelberg
Publication date:	12/06/2010
Edition description:	Softcover reprint of hardcover 1st ed. 1992
Pages:	257
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

1. Co-Involutive Hopf-Von Neumann Algebras.- 2. Kac Algebras.- 3. Representations of a Kac Algebra; Dual Kac Algebra.- 4. Duality Theorems for Kac Algebras and Locally Compact Groups.- 5. The Category of Kac Algebras.- 6. Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras.- Postface.

Editorial Reviews

A comprehensive account of Kac algebras and their duality, developed during the 1970s, for researchers working on quantum groups, particularly those interested in the approach by Lie groups and Lie algebras, or by non-commutative geometry. Discusses co-involutive Hopf-Von Neumann algebras, duality theorems for Kac algebras and locally compact groups, and the special cases of unimodular, compact, discrete, and finite- dimension Kac algebras. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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