TY - JOUR

T1 - Amenability of Hopf von Neumann algebras and Kac algebras

AU - Ruan, Zhong Jin

N1 - Funding Information:
* This research was partially supported by the National Science Foundation.

PY - 1996/8/1

Y1 - 1996/8/1

N2 - 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.

AB - 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.

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U2 - 10.1006/jfan.1996.0093

DO - 10.1006/jfan.1996.0093

M3 - Article

AN - SCOPUS:0030211228

VL - 139

SP - 466

EP - 499

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -