Kac algebras and duality of locally compact groups

Michel Enock, Jean-Marie Schwartz ; preface by Alain Connes ; postface by Adrian Ocneanu

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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  • 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.

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書名 Kac algebras and duality of locally compact groups
著作者等 Connes, Alain
Enock, Michel
Ocneanu, Adrian
Schwartz, Jean-Marie
出版元 Springer-Verlag
刊行年月 c1992
ページ数 x, 257 p.
大きさ 25 cm
ISBN 0387547452
NCID BA19118145
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言語 英語
出版国 ドイツ