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

Radon 

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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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