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

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]).