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

In: Kac Algebras and Duality of Locally Compact Groups

Author

Listed:
  • Michel Enock

    (Université Pierre et Marie Curie, CNRS, Laboratoire de Mathématiques Fondamentales)

  • Jean-Marie Schwartz

    (Université Pierre et Marie Curie, CNRS, Laboratoire de Mathématiques Fondamentales)

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]).

Suggested Citation

  • Michel Enock & Jean-Marie Schwartz, 1992. "Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras," Springer Books, in: Kac Algebras and Duality of Locally Compact Groups, chapter 0, pages 192-241, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-02813-1_7
    DOI: 10.1007/978-3-662-02813-1_7
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