Classification of Strongly Free Actions of Discrete Amenable Groups on Strongly Amenable Subfactors of Type Iii0

In the theory of operator algebras, classification of group actions on approximately finite dimensional (AFD) factors has been done since Connes’s work [2]. In subfactor theory, various results on classification of group actions have been obtained. The most powerful results have been obtained by Popa in [16], who classified the strongly outer actions of discrete amenable groups on strongly amenable subfactors of type II1 up to cocycle conjugacy. (Strong outerness for automorphisms are introduced by Choda-Kosaki in [1], and Popa in [16] independently. Popa use the terminology “proper outerness”.) In our previous work [13], we have classified the strongly free actions of discrete amenable groups on strongly amenable subfactors of type IIIλ, 0 < λ < 1. Our method in [13] has been based on [18] and [19]. But in [18] and [19], Sutherland and Takesaki treated factors of type IIIλ, 0 ≤ λ < 1, including the case λ = 0. So it is natural to ask if their method works for the classification of group actions on subfactors of type III0. In this paper, we classify strongly free actions of discrete amenable groups on strongly amenable subfactors of type III0. The complete invariant we use is Winsløw’s fundamental homomorphism, [22, Definition 4.2], which is an analogue of the Connes-Takesaki module ([5]) in subfactor theory. It is well known that in the single factor case, centrally free actions of discrete amenable groups on injective factors are completely classified by their Connes-Takesaki modules, [2], [14], [18], [19], [10]. And in subfactor theory, strong freeness is an analogy of centrally freeness, so the results in [13] and this paper are “subfactor-version” of these results. (In the case of strongly amenable subfactors of type II and type IIIλ, 0 < λ < 1, strong freeness is equivalent to central freeness. See [16], [21].)