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.

Kac Algebras and Duality of Locally Compact Groups PDF

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