Classification of Actions of Discrete Amenable Groups on Amenable Subfactors of Type Ii

We prove a classification result for properly outer actions σ of discrete amenable groups G on strongly amenable subfactors of type II, N ⊂ M , a class of subfactors that were shown to be completely classified by their standard invariant GN,M , in ([Po7]). The result shows that the action σ is completely classified in terms of the action it induces on GN,M . As a an application of this, we obtain that inclusions of type IIIλ factors, 0 < λ < 1, having discrete decomposition and strongly amenable graph, are completely classified by their standard invariant. 0. Introduction. In ([C1]) A.Connes classified the amenable semifinite factors showing that, up to isomorphism, there is only one of type II1, the unique approximately finite dimensional II1 factor R of Murray and von Neumann, also called the hyperfinite factor, and one of type II∞ ( R ⊗ B(H) ). Then, motivated by the problem of classifying infinite amenable factors of type III, automorphisms of amenable factors of type II were classified in ([C2,5]). Further classification results were proved for actions of finite and general amenable groups on R and R ⊗ B(H) in ([J1]) and respectively ([Oc]). For inclusions of factors N ⊂ M of finite Jones index [M : N ] < ∞, the suitable notion of amenability was introduced in ([Po7]). Also, it was proved in ([Po7]) that the amenable subfactors coincide with the subfactors that can be approximated by the finite dimensional subalgebras of their higher relative commutants. In the case of a trivial inclusion N = M ⊂ M this corresponds to the uniqueness of the amenable type II1 factor. In general, this shows that amenable inclusions are completely classified by their standard invariant GN,M , the graph type combinatorial object that encodes the lattice of higher relative commutants in the Jones tower ([Po7]). GN,M consists of a pair of weighted, pointed, bipartite graphs (ΓN,M , ~s), (ΓN,M , ~ s′) called the standard (or the principal) graphs of N ⊂ M ([J2]), with some additional structure. The invariant gives rise to a canonical model N ⊂ M st and in fact the theorem in ([Po7]) states that N ⊂ M is strongly amenable, i.e., it is amenable and its standard graph is ergodic, if and only if N ⊂ M is isomorphic to its canonical model. We will prove in this paper a classification result for properly outer actions of amenable groups on strongly amenable inclusions of type II1 and II∞ factors. The result can be regarded as an equivariant version of ([Po7]). The main motivation for studying this problem is, as in the single von Neumann algebra case, the classification of inclusions of type IIIλ factors, 0 < λ < 1, for which a similar Connes type