Approximation properties for Kac algebras

Amenability of locally compact groups has been an important subject in the study of harmonic analysis. One of the many equivalent conditions for the amenability of such a group G is that its Fourier algebra A(G) has a bounded approximate identity. Two weaker notions of amenability, weak amenability and the approximation property, have been defined using weaker approximation properties of A(G). For discrete groups G, it is known that each of these amenability properties of G is equivalent to a corresponding approximation property of the group von Neumann algebra L(G) of G. In this paper, we extend the notions of weak amenability and the approximation property from locally compact groups to the more general setting of Kac algebras, and show that the results concerning discrete groups can be extended in a natural way to discrete Kac algebras. Operator space theory plays an important role in the paper.