Kac Algebras and Duality of Locally Compact Groups

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Overview

The theory of Kac lagebras and their duality, elaborated independently in the seventies by Kac and Vainermann and by the authors of this book, has nowreached a state of maturity which justifies the publication of a comprehensive and authoritative account in bookform. Further, the topic of
"quantum groups" has recently become very fashionable and attracted the attention of more and more mathematicians and theoretical physicists. However a good characterization of quantum groups among Hopf algebras in analogy to the characterization of Lie groups among locally compact groups is still missing. It is thus very valuable to develop the generaltheory as does this book, with emphasis on the analytical aspects of the subject instead of the purely algebraic ones.
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 Tannaka, Krein, Stinespring and others dealing with non-abelian locally compact groups. Kac (1961) and Takesaki (1972) formulated the objective of finding a good category of Hopf algebras, containing the category of locally compact groups and fulfilling a perfect duality.
The category of Kac algebras developed in this book fully answers the original duality problem, while not yet sufficiently non-unimodular to include quantum groups.
This self-contained account of thetheory will be of interest to all researchers working in quantum groups,
particularly those interested in the approach by Lie groups and Lie algebras or by non-commutative geometry, and more generally also to those working in C* algebras or theoretical physics.

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Editorial Reviews

Booknews
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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Product Details

  • ISBN-13: 9780387547459
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 11/28/1992
  • Pages: 257

Table of Contents

Introduction 1
Ch. 1 Co-Involutive Hopf-Von Neumann Algebras 7
1.1 Von Neumann Algebras and Locally Compact Groups 8
1.2 Co-Involutive Hopf-Von Neumann Algebras 13
1.3 Positive Definite Elements in a Co-Involutive Hopf-Von Neumann Algebra 19
1.4 Kronecker Product of Representations 23
1.5 Representations with Generator 30
1.6 Fourier-Stieltjes Algebra 36
Ch. 2 Kac Algebras 44
2.1 An Overview of Weight Theory 45
2.2 Definitions 55
2.3 Towards the Fourier Representation 58
2.4 The Fundamental Operator W 60
2.5 Haar Weights Are Left-Invariant 66
2.6 The Fundamental Operator W Is Unitary 71
2.7 Unicity of the Haar Weight 76
Ch. 3 Representations of a Kac Algebra; Dual Kac Algebra 83
3.1 The Generator of a Representation 84
3.2 The Essential Property of the Representation [lambda] 89
3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra 92
3.4 Eymard Algebra 97
3.5 Construction of the Dual Weight 101
3.6 Connection Relations and Consequences 104
3.7 The Dual Kac Algebra 111
Ch. 4 Duality Theorems for Kac Algebras and Locally Compact Groups 124
4.1 Duality of Kac Algebras 125
4.2 Takesaki's Theorem on Symmetric Kac Algebras 130
4.3 Eymard's Duality Theorem for Locally Compact Groups 136
4.4 The Kac Algebra K[subscript 3](G) 140
4.5 Characterisation of the Representations and Wendel's Theorem 144
4.6 Heisenberg's Pairing Operator 152
4.7 A Tatsuuma Type Theorem for Kac Algebra 158
Ch. 5 The Category of Kac Algebras 161
5.1 Kac Algebra Morphisms 162
5.2 H-Morphisms of Kac Algebras 166
5.3 Strict H-Morphisms 172
5.4 Preliminaries About Jordan Homomorphisms 174
5.5 Isometries of the Preduals of Kac Algebras 176
5.6 Isometries of Fourier-Stieltjes Algebras 184
Ch. 6 Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras 192
6.1 Unimodular Kac Algebras 193
6.2 Compact Type Kac Algebras 197
6.3 Discrete Type Kac Algebras 208
6.4 Krein's Duality Theorem 213
6.5 Characterisation of Compact Type Kac Algebras 219
6.6 Finite Dimensional Kac Algebras 232
Postface 243
Bibliography 245
Index 255
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