Amenability of Hopf von Neumann algebras and Kac algebras

Let (Μ, Γ) be a Hopf von Neumann algebra. The operator predual Μ _* of Μ is a completely contractive Banach algebra with multiplication m = Γ _*: Μ _* ⊗̂ Μ _* → Μ _*. We call (Μ, Γ) operator amenable if the completely contractive Banach algebra Μ _* is operator amenable, i.e., for every operator Μ _*-bimodule V, every completely bounded derivation from Μ _* into the dual Μ _*-bimodule V^* is inner. There is a weaker notion of amenability introduced by D. Voiculescu. We say that a Hopf von Neumann algebra (Μ, Γ) is Voiculescu amenable if there exists a left invariant mean on Μ. We show that if a Hopf von Neumann algebra (Μ, Γ) is operator amenable, then it is Voiculescu amenable. For Kac algebras, there is a strong Voiculescu amenability. We show that for discrete Kac algebras, these amenabilities are all equivalent. In fact, if we let K = (Μ, Γ, K, φ) be a discrete Kac algebra and let K̂ = (Μ̂, Γ̂, K̂, φ̂) be its (compact) dual Kac algebra, then the following are equivalent: (1) K is operator amenable; (2) K is Voiculescu amenable; (3) The von Neumann algebra Μ̂ is hyperfinite; (4) K is strong Voiculescu amenable; (5) K̂ is operator amenable; (6) Μ̂ _* has a bounded approximate identity.