Classification of actions of discrete amenable groups on strongly amenable subfactors of type IIIλ

Using the continuous decomposition, we classify strongly free actions of discrete amenable groups on strongly amenable subfactors of type III$_\lambda, 0 < \lambda < 1$. Winsl{\o}w's fundamental homomorphism is a complete invariant. This removes the extra assumptions in the classification theorems of Loi and Winsl{\o}w and gives a complete classification up to cocycle conjugacy.