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We will study two kinds of actions of a discrete amenable Kac algebra. Thefirst one is an action whose modular part is normal. We will construct a newinvariant which generalizes a characteristic invariant for a discrete groupaction, and we will present a complete classification. The second is acentrally free action. By constructing a Rohlin tower in an asymptoticcentralizer, we will show that the Connes-Takesaki module is a completeinvariant.

Classification of actions of discrete Kac algebras on injective factors