## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 466-499 |

Number of pages | 34 |

Journal | Journal of Functional Analysis |

Volume | 139 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 1996 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis