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Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras

  • Michel Enock
  • Jean-Marie Schwartz
Chapter

Abstract

Let \(\Bbbk = (M,\Gamma ,k,\varphi )\) be a Kac algebra, \(\hat \Bbbk = (\hat M,\hat \Gamma ,\hat k,\hat \varphi )\) the dual Kac algebra. We have seen that the modular operator \(\hat \Delta = {\Delta _{\hat \varphi }}\) is the RadonNikodym derivative of the weight \(\varphi \) with respect to the weight \(\varphi ok\) (3.6.7).So, it is just a straightforward remark to notice that so is invariant under is if and only if \(\hat \varphi \) is a trace. Moreover, the class of Kac algebras whose Haar weight is a trace invariant under k is closed under duality (6.1..4). These Kac algebras are called “unimodular” because, for any locally compact group G,the Kac algebra \({\Bbbk _a}\left( G \right)\) is unimodular if and only if the group G is unimodular. Unimodular Kac algebras are the objects studied by Kac in 1961 ([66], [70]).

Keywords

Irreducible Representation Hopf Algebra Compact Group Banach Algebra Fourier Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Michel Enock
    • 1
  • Jean-Marie Schwartz
    • 1
  1. 1.CNRS, Laboratoire de Mathématiques FondamentalesUniversité Pierre et Marie CurieParis Cedex 05France

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