Amenability of Hopf von Neumann Algebras and Kac Algebras

Let (M,Γ) be a Hopf von Neumann algebra. The operator predualM*ofMis a completely contractive Banach algebra with multiplicationm=Γ*:M*⊗M*→M*. We call (M,Γ)operator amenableif the completely contractive Banach algebraM*isoperator amenable, i.e., for every operatorM*-bimoduleV, every completely bounded derivation fromM*into the dualM*-bimoduleV* is inner. There is a weaker notion of amenability introduced by D. Voiculescu. We say that a Hopf von Neumann algebra (M,Γ) isVoiculescu amenableif there exists a left invariant mean onM. We show that if a Hopf von Neumann algebra (M,Γ) is operator amenable, then it is Voiculescu amenable. c algebras, there is astrong Voiculescu amenability. We show that for discrete Kac algebras, these amenabilities are all equivalent. In fact, if we letK=(M,Γ,κ,ϕ) be a discrete Kac algebra and letK=(M,Γ,κ,ϕ) be its (compact) dual Kac algebra, then the following are equivalent: (1)Kis operator amenable; (2)Kis Voiculescu amenable; (3) The von Neumann algebraMis hyperfinite; (4)Kis strong Voiculescu amenable; (5)Kis operator amenable; (6)M*has a bounded approximate identity.