Amenability of Hopf von Neumann Algebras and Kac Algebras

✍ Scribed by Zhong-Jin Ruan

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 1008 KB
• Volume: 139
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0093

✦ Synopsis

Let (M, 1 ) be a Hopf von Neumann algebra. The operator predual M * of M is a completely contractive Banach algebra with multiplication m=1 * : M * M * Ä M * . We call (M, 1 ) operator amenable if the completely contractive Banach algebra M * is operator amenable, i.e., for every operator M * -bimodule V, every completely bounded derivation from M * into the dual M * -bimodule V* is inner. There is a weaker notion of amenability introduced by D. Voiculescu. We say that a Hopf von Neumann algebra (M, 1) is Voiculescu amenable if there exists a left invariant mean on M. We show that if a Hopf von Neumann algebra (M, 1) 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=(M, 1, }, .) be a discrete Kac algebra and let K =(M , 1 , }^, . ^) 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 M is hyperfinite; (4) K is strong Voiculescu amenable; (5) K is operator amenable; (6) M * has a bounded approximate identity. 1996 Academic Press, Inc. refered to [16], [6], [4] and [2] for details. A completely contractive Banach algebra is an associative algebra A together with an operator matrix norm such that the multiplication