Kac Algebras

Abstract

This chapter deals with technical results about Haar weights, as they have been studied by the authors in [36] and [136], and, independently, by Vaĭnerman and Kac ([180]). On a co-involutive Hopf-von Neumann algebra (M, Γ, k), a Haar weight is a faithful, semi-finite, normal weight on M ⁺, which is left-invariant with respect to Γ, i.e. such that:

$$(i \otimes \varphi )\Gamma (x) = \varphi (x)1$$

for all x in M ⁺ (in 2.5, we show, after Kirchberg, that this axiom may be weakened), and, roughly speaking, satisfies two other axioms involving k. The quadruple (M, Γ, k, φ) is then called a Kac algebra.