Algèbres de Kac moyennables

Abstract

We generalize the classical theory of amenable locally compact groups to Kac algebras. Most of the equivalent definitions of amenability are translated into the formalism of Kac algebras and still remain equivalent. Among others, we see that every "group dual" (i.e. symmetric Kac algebra) is amenable.Results on actions of amenable groups on von Neumann algebras are extended as well; this allows us to obtain new properties on group co-actions (i.e. actions of "group duals"). Dans cet article, nous montrons que la theorie des groupes moyennables, et en particulier les deux resultats cites plus haut de Hulanicki et Leptin s'etendent facilement au cadre des algebres de Kac; cela peut paraitre surprenant dans la mesure oύ la moyennabilitέ n'est pas reliee a Introduction

Amenable Representations and Reiter's Property for Kac Algebras

“…The equivalences of (I)-(IV) were proved in [3] and their equivalence with (V) can be found in [11, 3.6(a)]. …”

“…In [11,3.1], we defined a more general notion of amenable Hopf C g -algebras which can be shown to be equivalent to the above (see [11, 3.5 and 3.6]). In particular, the above theorem is also true in the case of locally compact quantum groups (as defined in [9]).…”

Amenability of Hopf von Neumann Algebras and Kac Algebras

“…In [18] and [9], Voiculescu, and Enock and Schwartz studied various amenability conditions on Kac algebras. We list some of equivalence conditions in the following theorem.…”

“…But it is still an open question for general Kac algebras. M. Enock and J.-M. Schwartz have tried to prove this in [9]. However, there is a gap in their proof (see [9, 2.6.8, page 237]).…”

An example of amenable Kac systems

“…The study of amenability of Kac algebras began with [6]. In [2], amenability of regular multiplicative unitaries was also defined.…”