Kac Algebras and Duality of Locally Compact Groups

267 Pages·1992·8.62 MB·English

by  Michel Enock

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Anthropomorphic carving
representing Duality
(Totonac culture)
Michel Enock
Jean-Marie Schwartz
Kac Algebras and
Duality
of Locally Compact
Groups
Preface
by Alain Connes
Postface
by Adrian Ocneanu
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Michel Enock
Jean-Marie Schwartz
CNRS, Laboratoire de Mathematiques Fondamentales
Universite Pierre et Marie Curie
4 place Jussieu
F-75252 Paris Cedex 05, France
The sculpture reproduced on cover and frontispiece is exhibited at the
Museo de antropologia de la Universidad Veracruzana, Jalapa, E. U. de
Mexico
Mathematics Subject Classification (1980):
22 D 25, 22 D 35, 43 A 30, 43 A 65
ISBN 3-540-54745-2 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-54745-2 Springer-Verlag New York Berlin Heidelberg
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To Professor Jacques Dixmier
"What's the matter?"
Macbeth (11,2)
The question is the story itself, and
whether or not it means something is
not for the story to tell.
Paul Auster (City of glass)

Preface
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.I. Kac and L.I. 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.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 Haax 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.
A co-involutive Hopf-von Neumann algebra is given by a
morphism 1' : M --p M 0 M of a von Neumann algebra M
VIII Preface
in its tensor square M ® M and a co-involution n which to-
gether turn the predual M* into an involutive Banach algebra.
A Kac algebra is a co-involutive Hopf-von Neumann algebra
with a Haar weight, i.e. a semi-finite faithful normal weight on
M which is left-invariant in a suitable way. In this book, the
theory of Kac algebras and their duality is brought to a quite
mature state, relying a lot on the modular theory of weights de-
velopped also in the seventies. The resulting category of Kac al-
gebras fully answers the original duality problem, but is not yet
sufficiently non-unimodular to include quantum groups. This
of course opens a very interesting direction of research, under-
taken recently by S. Baaj and G. Skandalis.
Paris Alain Connes
Table of Contents
Introduction
. . . . . . . . . . . . . . . . 1
Chapter 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
Chapter 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
Chapter 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 A 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
X Table of Contents
Chapter 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 Ks (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
Chapter 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
Chapter 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
Pos tf ace . . . . . . . . . . . . . . . . . . 243
Bibliography . . . . . . . . . . . . . . . . 245
Index . . . . . . . . . . . . . . . . . . . 255