Classification of approximately inner actions of discrete amenable groups on strongly amenable subfactors

Toshihiko MASUDA Department of Mathematics, Kochi University, 2-5-1 Akebono-cho, Kochi, 780-8520, JAPAN

Abstract

We give the classification theorem of approximately inner actions of discrete amenable groups on strongly amenable subfactor of type II_{1} by means of the characteristic invariant and ν invariant. To prove this theorem, we also give the classification theorem when the inner part and centrally trivial part of actions coincide.

1 Introduction

In the theory of operator algebras, the study of automorphisms is one of the most important subjects. In [4] and [6], A. Connes has classified automorphisms of approximately finite dimensional (AFD) semifinite factors. After Connes’ classification, V. F. R. Jones has introduced the characteristic invariant and classified actions of finite groups on the AFD type II_{1} factor in [16]. Immediately after that work, A. Ocneanu succeeded in classifying actions of discrete amenable groups on AFD semifinite factors. Based on these results, actions of discrete amenable groups on AFD factors of type III have been classified in [36], [23] and finally [18].

In subfactor theory, P. H. Loi first studied the automorphisms of subfactors in [25]. He introduced the invariant of automorphisms Φ(α), which we will call the Loi invariant, and obtained some structural results on type III_{λ} subfactors. In [34], S. Popa has introduced the proper outerness for automorphisms of subfactors, and proved that properly outer actions of discrete amenable groups on strongly amenable subfactors of type II_{1} are classified by the Loi invariant. (In [2], Choda-Kosaki have introduced the same property independently and they call it strong outerness.) Based on Popa’s result, some results on classification of actions of discrete amenable groups on strongly amenable subfactors of type III_{λ}, λ ≠ 1, have been obtained in [38], [26], [28] and [29].

So it is natural to ask if we can classify not necessary strongly outer actions of discrete amenable groups. Several people have noticed the similarity between theory of subfactors and (single) type III factors, and especially this similarity has been emphasized by Kawahigashi in [19]. Based on this similarity, Kawahigashi has introduced two outer conjugacy invariants for automorphisms, i.e., the higher obstruction and an algebraic ν invariant in [22]. The former is an analogue of the Connes obstruction (or modular obstruction) and the latter is an analogue of Sutherland-Takesaki’s modular invariant. With these two invariants, Kawahigashi has classified approximately inner automorphisms of subfactors under some extra assumptions on subfactors. After this work, Loi removed these extra assumptions and generalized Kawahigashi’s results by commuting square technique in [27].

The aim of this paper is generalization of Kawahigashi’s result to the case of actions of arbitrary discrete amenable groups. Out idea is based on [36] and [18], that is, we use duality technique. In the classification of group actions on type III factors, by taking the crossed product by modular automorphism group, the classification problem can be reduced to the case of semifinite factors. In subfactor setting, non-strongly outer automorphisms correspond to the modular automorphism, and we take the crossed product by the relative χ group and reduce classification problem to easier case. When inner part and centrally trivial part of actions coincide, we can classify actions by combining Brauer group technique, which has been developed by Connes, Jones and Ocneanu, and Popa’s classification theorem. By using duality technique, we reduce the classification problem to this case.

To classify actions, we consider two kinds of invariants. One is the characteristic invariant, and another is a ν invariant. Our characteristic invariant may or may not be different from the original one introduced by Jones, and is the generalization of Kawahigashi’s higher obstruction. Our characteristic invariant has relation with the κ invariant, which has been introduced by Jones in [15] and generalized to the subfactor case by Kawahigashi in [21]. In this paper, we consider the “algebraic” version of the κ invariant, and if this κ invariant is trivial, then our characteristic invariant coincides with the usual characteristic invariant. The second invariant ν invariant is precisely Kawahigashi’s algebraic ν invariant, and this is an analogue of Sutherland-Takesaki’s modular invariant mentioned as above.

This paper is organized as follows.

In Section 2, we recall several facts on automorphisms of subfactors which we need in this paper.

In Section 3, We prove the existence of Connes-Radon-Nikodym type cocycles. By using these cocycles, we define the extension of automorphisms onto the crossed product of a subfactor by relative Connes χ group in a similar way as in [13].

In Section 4, we discuss cocycle conjugacy invariants of actions. We introduce the characteristic invariant for actions. Mentioned as above, this characteristic invariant may or may not be different from the usual characteristic invariant. We also introduce ν invariant as the G-equivalent homomorphism from a normal subgroup of G to χ_{a}(M, N).

In Section 5, we give the classification result of actions whose inner part and centrally trivial part coincide. In this case, the Loi invariant and usual characteristic invariant are complete invariants.

In Section 6, we classify approximately inner actions of discrete amenable groups on strongly amenable subfactor of type II_{1}.

Acknowledgement. The author is grateful to Professor Y. Kawahigashi for various comments on this paper.

2 Preliminaries and notations

In this section, we recall several facts on automorphisms of subfactors and fix notations.

Let N ⊂ M be a subfactor with finite index and N ⊂ M ⊂ M_{1} ⊂ M_{2} ⊂ ⋯ the Jones tower. (In this paper, we consider only minimal index, and assume that every subfactor of type II is extremal in the sense of [35].) Let α be an automorphism of N ⊂ M. Then α can be extended onto M_{k} inductively by setting α(e_{k}) = e_{k}, where e_{k} is the Jones projection for M_{k−2} ⊂ M_{k−1}.

First we recall the Loi invariant and the strong outerness for automorphisms.

Definition 2.1 ([25, Section 5]) With above notations, put

Φ(α) := {α|_{M′∩M_{k}}}_{k}.

We call this Φ the Loi invariant for α.

Definition 2.2 ([2, Definition 1], [34, Definition 1.5.1]) An automorphism α ∈ Aut (M, N) is said to be strongly outer or properly outer if we have no non-zero a ∈ ∪_{k}M_{k} satisfying α(x)a = ax for every x ∈ M.

According to the notation of [18], we use the notation Cnt_{r} (M, N) to denote the set of non-strongly outer automorphisms of N ⊂ M.

In [34], Popa proved the following important results.

Theorem 2.3 ([34, Theorem 1.6]) Non-strongly outer automorphisms are centrally trivial. Moreover if N ⊂ M is a strongly amenable subfactor of type II_{1}, then these two notions are equivalent.

Theorem 2.4 ([34, Theorem 3.1]) Let N ⊂ M be a strongly amenable subfactor of type II_{1} and G a discrete amenable group. If α is a strongly outer action of G on N ⊂ M, then α is cocycle conjugate to the action α^{st} ⊗ σ on N^{st} ⊗ R_{0} ⊂ M^{st} ⊗ R_{0}, where R_{0} is the injective factor of type II_{1} and σ is the outer action of G on R_{0}, which is unique up to cocycle conjugacy due to [31]. In other words, the Loi invariant Φ(α) is the completely cocycle conjugacy invariant for strongly outer actions of discrete amenable groups.

In [25, Theorem 5.4], Loi gave the following characterization of approximately inner automorphisms.

Theorem 2.5 If N ⊂ M is a strongly amenable subfactor of type II_{1}, then Ker Φ = \overline{Int}(M, N) holds.

Loi proved above theorem when N ⊂ M is of finite depth. But his proof works when N ⊂ M is strongly amenable.

In [19], Kawahigashi defined the relative Connes invariant χ(M, N) for a subfactor of type II_{1} N ⊂ M as an analogue of Connes’ χ group [5]. Based on this, Goto defined the algebraic χ group for subfactors in [12] as follows.

Definition 2.6 Set

χ_{a}(M, N) := \frac{Ker Φ ∩ Cnt_{r} (M, N)}{Ad N (M, N)},

where N (M, N) is the normalizer group. We call this group an algebraic χ group for N ⊂ M.

In [12, Theorem 2.1], Goto proved that χ_{a}(M, N) is an abelian group.

When N ⊂ M is strongly amenable and has the trivial normalizer, χ(M, N) = χ_{a}(M, N) holds because of Theorem 2.3 and Theorem 2.5.

Note that every θ ∈ Aut (M, N) induces an automorphism of χ(M, N) (resp. χ_{a}(M, N)) by [α] ↦ [θαθ^{−1}], which we will denote χ(θ) (resp. χ_{a}(θ)) according to the notation in [5]. We denote the image of Aut (M, N) in Aut (χ(M, N)) (resp. Aut (χ_{a}(M, N))) by χ(Aut (M, N)) (resp. χ_{a}(Aut (M, N))). But sometimes we write θ instead of χ(θ) if no confusions arise.

3 Connes-Radon-Nikodym type cocycle theorem

Throughout the rest of this paper, we always assume that N ⊂ M and M ⊂ M_{1} have the trivial normalizer. This means that α ∈ Aut (M, N) is outer on M if and only if outer on N.

In [30, Theorem 2.1], for each α ∈ Ker Φ we associated the system of a half braiding unitary {U_{α}(X)}, where X is an irreducible bimodule arising from N ⊂ M. (The notion of a half braiding has been introduced by Izumi in [14, Definition 4.2].)

If we consider the case when a bimodule X comes from a non-strongly outer automorphism of N ⊂ M, then we get the following theorem.

Theorem 3.1 For every σ ∈ Cnt_{r} (M, N) and α ∈ Ker Φ, there exists a unitary u_{α,σ} ∈ U(N) satisfying following conditions.

(1) Ad u_{α,σ}σ = ασα^{−1},

(2) u_{α,σ_{1}σ_{2}} = u_{α,σ_{1}}σ_{1}(u_{α,σ_{2}}),

(3) u_{αβ,σ} = α(u_{β,σ})u_{α,σ},

(4) u_{α,Ad v} = α(v)v^{∗}, u_{Ad v,σ} = uσ(u^{∗}), v ∈ U(N),

(5) Take nonzero a ∈ M_{k} satisfying σ(x)a = ax. Then u_{α,σ} satisfies α(a) = u_{α,σ}a. Moreover u_{α.σ} is characterized by conditions (1) and (5).

Proof. In the following, we identify an automorphism α ∈ Aut (M, N) with a bimodule M_{α^{−1}}M_{M}. First set u_{α,σ} := σ(U_{α}(σ)ˆ1) ∈ M. Here we have

U_{α}(σ) ∈ Hom (_{M}_{σ^{−1}α^{−1}}M_{M},_{M}_{σ^{−1}}M_{αM}).

Then on one hand, we have

U_{α}(σ)(x̂) = U_{α}(σ)(ˆ1 · x)
= U_{α}(σ)(ˆ1) · x
= σ^{−1}(u_{α,σ})α(x).

On the other hand, we have

U_{α}(σ)(x̂) = U_{α}(σ)(ασ(x) · ˆ1)
= ασ(x) · U_{α}(σ)(ˆ1)
= σ^{−1}ασ(x)σ^{−1}(u_{α,σ}).

Hence we get ασα^{−1}(x)u_{α,σ} = u_{α,σ}σ(x). Since U_{α}(_{M}_{σ^{−1}}M_{M}) is a unitary, u_{α,σ} is a unitary and falls in U(N).

Since we have U_{α}(σ_{1}σ_{2}) = (1_{σ_{1}} ⊗_{M} U_{α}(σ_{2}))(U_{α}(σ_{1}) ⊗_{M} 1_{σ_{2}}) and U_{αβ}(σ) = (U_{β}(σ) ⊗_{M} 1_{α})(1_{β} ⊗_{M} U_{α}(σ)), we get (2) and (3).

Next we will verify (4). Take u ∈ U(N). First of all U_{α}(id) = id holds. An intertwiner T from _{M}M_{M} to _{M}_{Ad u^{∗}}M_{M} is given by T x̂ = û^{∗}x. By using the braiding fusion equation, we get u_{α,Ad u} = α(u)u^{∗}.

Since U_{Ad u}(X) is given by U_{Ad u}(X)ξ = uξu^{∗}, we get u_{Ad u,σ} = uσ(u^{∗}).

To prove (5) first set I := {a ∈ M_{k} | σ(x)a = ax for all x ∈ M}. We have a natural Hilbert space structure on I. Since we have ασα^{−1} = Ad u_{α,σ}σ, we can define an operator U_{α} on I by setting U_{α}(a) = u_{α,σ}^{∗}α(a). It is clear that U_{α} is a unitary. Then by the same argument as in the proof of [22, Theorem 2.2], we can show that U_{α} is a scalar. Then we can apply the same argument in the proof of [30, Proposition 3.3] and verify U_{α} = 1.

Let assume v_{α,σ} ∈ U(N) satisfies (1) and (5). Then from condition (1), we have v_{α,σ} = cu_{α,σ} for some c ∈ T. But condition (5) implies c = 1. Hence u_{α.σ} is characterized by conditions (1) and (5). □

Remark. When α itself is non-strongly outer and has the trivial Loi invariant, u_{α,α} is equal to the higher obstruction γ_{h}(α). So in general, u_{α,α} fails to be 1. Examples of such automorphisms are non-strongly outer automorphisms of subfactors with Dynkin diagram A_{4n−1}. See [20], [40] and [10].

We compare the above theorem and Connes-Radon-Nikodym cocycle theorem [3, Theorem 1.2.1] and [7, Theorem 4.2.4]. Let M be a type III factor, F(M) the flow of weights of M and ϕ a dominant weight of M. (See [7] for these notations.) For each c ∈ Z^{1}(F(M)), we can associate an extended modular automorphism σ_{c}^{ϕ} ∈ Aut (M). Take α ∈ M. Then we have ασ_{c}^{ϕ}α^{−1} = σ_{mod (α)(c)}^{ϕ∘α^{−1}} = Ad [Dϕ ∘ α^{−1} : Dϕ]_{c}σ_{mod (α)(c)}^{ϕ}. Especially if mod (α) is trivial, then α and σ_{c}^{ϕ} commute in Aut (M)/Int(M). Here we have the correspondence as in Table 1.

Table 1:

Subfactor case  α ∈ Ker Φ  σ ∈ Cnt_{r} (M, N)  u_{α,σ}
Type III factor case α ∈ Ker mod σ_{c}^{ϕ}, c ∈ Z^{1}(F(M)) [Dϕ ∘ α^{−1} : Dϕ]_{c}

Next we define the algebraic κ invariant. Here we take α, β ∈ Ker Φ ∩ Cnt_{r} (M, N) and assume that α and β commute. In this case u_{α,β} is a scalar. We set κ_{a}(α, β) := u_{β,α}^{∗} and call this scalar the algebraic κ invariant, or the κ invariant simply. The origin of this notation is Jones’ κ invariant in [15] and when N ⊂ M is strongly amenable, it can be proved that this coincides with the relative Jones κ invariant defined in [21, Section 2] in the same way as in the proof of [22, Theorem 4.1]. (See [1] for details of κ invariant.)

Let K be a discrete group and σ be an outer action of K on N ⊂ M such that σ_{k} ∈ Cnt_{r} (M, N) for every k ∈ K. By using Theorem 3.1, we can extend α ∈ Ker Φ onto the crossed product subfactor N ⋊_{σ} K ⊂ M ⋊_{σ} K in the same way as [13].

Proposition 3.2 Let w_{g} be an implementing unitary in N ⋊_{σ} K ⊂ M ⋊_{σ} K. For each α ∈ Ker Φ, there exists a unique automorphism α̃ ∈ Aut (M ⋊_{σ} K, N ⋊_{σ} K) satisfying α̃(x) = α(x) for x ∈ M and α̃(w_{g}) = u_{α,σ_{g}}w_{g}.

Moreover we have the following.

(1) For α, β ∈ Ker Φ, α̃β̃ = α̃β and α̃^{−1} = α̃^{−1} hold.

(2) For α = Ad u, u ∈ U(N), α̃ = Ad u holds.

(3) If K is abelian, then α̃ commutes with the dual action of K̂.

Proof. Let U_{α} be the canonical implementing unitary of α on L^{2}(M). Define a unitary W_{α} on L^{2}(M) ⊗ l^{2}(K) by (W_{α}ξ)(h) = U_{α}u_{α^{−1},σ_{h}^{−1}}ξ(h). By using Theorem 3.1, we get W_{αβ} = W_{α}W_{β} and W_{α}^{∗} = W_{α^{−1}}.

The crossed product algebra M ⋊_{σ} K is generated by π(M) and {w_{g}}_{g∈K}, where (π(x)ξ)(h) = σ_{h}^{−1}(x)ξ(h) and (w_{g}ξ)(h) = ξ(g^{−1}h). Then direct computations show that Ad W_{α}π(a) = π(α(x)) and Ad W_{α}(w_{g}) = π(u_{α,σ_{g}})w_{g} hold. From these facts, it is clear that Ad W_{α} induces an automorphism of M ⋊_{σ} K. Since π(M) and w_{g} generate M ⋊ K, the extension α̃ of α is characterized by these two conditions. If one uses the property of W_{α} mentioned above, it is easy to see that the condition (1) holds.

Since we have u_{Ad u,σ_{g}} = uσ_{g}(u^{∗}), we get (2).

Assume that K is abelian. Then the dual action σ̂ of K̂ is defined by σ̂_{p}(x) = x for x ∈ M and σ̂_{p}(w_{g}) = \overline{⟨p, g⟩}w_{g}. Then it is obvious that the dual action σ̂ and α̃ commute. □

4 Cocycle conjugacy invariants

Let G be a discrete group and α be an action of G with trivial Loi invariant. We consider the cocycle conjugacy invariants for α.

Before giving the definition of invariants, we make three assumptions. Firstly we assume that N ⊂ M and M ⊂ M_{1} have the trivial normalizer. Secondly we assume that there exists the lifting from χ_{a}(M, N) to Aut (M, N). Thirdly we assume that χ_{a}(M, N) is a finite cyclic group. It is rather technical assumption that χ_{a}(M, N) is a cyclic group, and the reason of this assumption will be clarified in the following arguments. But here we remark that we have many examples of subfactors satisfying these assumptions, e.g., SU(n)_{k} subfactors.

Set H := {n ∈ G | α_{n} ∈ Cnt_{r} (M, N)}. Obviously H is a normal subgroup of G and a cocycle conjugacy invariant. Define a homomorphism ν from H to χ_{a}(M, N) by ν(n) = [α_{n}] ∈ χ_{a}(M, N). Since we assume that α has the trivial Loi invariant, this ν is well defined. This ν satisfies ν(gng^{−1}) = ν(n) for every g ∈ G and n ∈ H. In [22], Kawahigashi introduced an algebraic ν invariant ν_{alg} and an analytic ν invariant ν_{ana} as an analogue of Sutherland-Takesaki’s modular invariant in [36, Definition 5.8]. Obviously our ν is a generalization of an algebraic ν invariant ν_{alg}.

By assumptions, we have the lifting σ from χ_{a}(M, N) to Aut (M, N). We fix this lifting. Note that fixing of the lifting is independent from actions. We can regard this σ as an action of χ_{a}(M, N). Hence for n ∈ H we have α_{n} = Ad v_{n}σ_{ν(n)} for some unitary v_{n} ∈ U(N).

From equations α_{m}α_{n} = α_{mn} and α_{g}α_{g^{−1}ng}α_{g}^{−1} = α_{n}, m, n ∈ H and g ∈ G, we get scalars λ(g, n) and μ(m, n) by the following equations in a similar way as in the single factor case.

v_{m}σ_{ν(m)}(v_{n}) = μ(m, n)v_{mn},

α_{g}(v_{g^{−1}ng})u_{α_{g},σ_{ν(n)}} = λ(g, n)v_{n}.

Proposition 4.1 For h, k, l ∈ H and g, g_{1}, g_{2} ∈ G, we have the following equalities.

(1) μ(h, k)μ(hk, l) = μ(k, l)μ(h, kl),

(2) λ(g_{1}g_{2}, n) = λ(g_{1}, n)λ(g_{2}, g_{1}^{−1}ng_{1}),

(3) λ(h, k) = μ(h, h^{−1}kh)\overline{μ(k, h)κ_{a}(ν(k), ν(h))},

(4) λ(g, hk)\overline{λ(g, h)}\overline{λ(g, k)} = μ(h, k)\overline{μ(g^{−1}hg, g^{−1}kg)},

(5) λ(g, 1) = λ(1, h) = μ(h, 1) = μ(1, h) = 1.

Proof. We verify only the equality (3), because other equalities can be verified in the same way as in the single factor case and the equality (3) shows the difference between the subfactor case and the single factor case.

Then by the definition, we have

λ(h, k) = α_{h}(v_{h^{−1}kh})u_{h,ν(k)}v_{k}^{∗}
= v_{h}σ_{ν(h)}(v_{h^{−1}kh})v_{h}^{∗}u_{h,ν(k)}v_{k}^{∗}
= μ(h, h^{−1}kh)v_{kh}v_{h}^{∗}u_{Ad v_{h}σ_{ν(h)},ν(k)}v_{k}^{∗}
= μ(h, h^{−1}kh)\overline{μ(k, h)}v_{k}σ_{ν(k)}(v_{h})v_{h}^{∗}Ad v_{h}(u_{ν(h),ν(k)})u_{Ad v_{h},ν(k)}v_{k}^{∗}
= μ(h, h^{−1}kh)\overline{μ(k, h)}v_{k}σ_{ν(k)}(v_{h})\overline{κ_{a}(ν(k), ν(h))}v_{h}^{∗}v_{h}σ_{ν(k)}(v_{h}^{∗})v_{k}^{∗}
= μ(h, h^{−1}kh)\overline{μ(k, h)κ_{a}(ν(k), ν(h))}.

Hence equality (3) holds. □

By Z(G, H|κ_{a}), we denote the set of (λ, μ) satisfying the above conditions. The above definition of λ and μ are depend on the choice of unitaries v_{n}. To get rid of this dependence, we define the equivalence relation in Z(G, H|κ_{a}) as follows. Two elements (λ_{1}, μ_{1}) and (λ_{2}, μ_{2}) are equivalent if and only if there exists a map c from H to T with c_{e} = 1 such that λ_{1}(g, n) = \overline{c_{g^{−1}ng}}λ_{2}(g, n)c_{n} and μ_{1}(m, n) = \overline{c_{m}}\overline{c_{n}}μ_{2}(m, n)c_{mn} hold. We set Λ(G, H|κ_{a}) = Z(G, H|κ_{a})/∼, where ∼ denotes the above equivalence relation. If the κ-invariant is trivial, then this is a usual characteristic invariant Λ(G, N). Here we remark that Λ(G, H|κ_{a}) does not have a natural group structure, but has a natural Λ(G, H) space structure.

For a given action α of G, we get [λ, μ] ∈ Λ(G, H|κ_{a}) and ν ∈ Hom_{G} (H, χ_{a}(M, N)). When we have to specify an action α, we also use notations H_{α}, Λ(α) = [λ_{α}, μ_{α}] and ν_{α}.

Proposition 4.2 Let α be an action of G with trivial Loi invariant. Then the triplet (H_{α}, Λ(α), ν_{α}) is a cocycle conjugacy invariant in the following sense. If β is an another action of G which is cocycle conjugate to α, then we have H_{α} = H_{β}, Λ(α) = Λ(β) and ν_{α} = χ_{a}(θ)(ν_{β}) for some χ(θ) ∈ χ_{a}(Aut (M, N)).

Proof. It is obvious that H_{α} is a cocycle conjugacy invariant. So we will verify that Λ(α) and ν_{α} are cocycle conjugacy invariants. First we compare α and θαθ^{−1} for some θ ∈ Aut (M, N). Here for a ∈ χ_{a}(M, N), we can take a unitary w_{a} ∈ U(N) such that θσ_{a}θ^{−1} = Ad w_{a}σ_{θ(a)} holds. Since we assume that χ_{a}(M, N) is a finite cyclic group and this means that H^{2}(χ_{a}(M, N), T) is trivial, we can take that w_{a}’s satisfy the cocycle identity, i.e., w_{a}σ_{θ(a)}(w_{b}) = w_{ab}. (This is the reason why we assume χ_{a}(M, N) is a finite cyclic group.)

It is clear that ν_{θαθ^{−1}} = θ(ν_{α}). Then we have θα_{n}θ^{−1} = Ad θ(v_{n})w_{ν(n)}σ_{θ(ν(n))} for n ∈ H. So we replace v_{n} to θ(v_{n})w_{ν(n)} in the definition of λ and μ and simple computations show that λ_{α} = λ_{θαθ^{−1}} and μ_{α} = μ_{θαθ^{−1}} hold. For example, we have

μ_{θαθ^{−1}}(m, n) = θ(v_{m})w_{m}σ_{θ(ν(m))}(θ(v_{n})w_{m})w_{mn}^{∗}θ(v_{mn}^{∗})
= θ(v_{m}σ_{ν(m)}(v_{n}))w_{m}σ_{θ(ν(m))}(w_{n})w_{mn}^{∗}θ(v_{mn}^{∗})
= θ(v_{m}σ_{ν(m)}(v_{n})v_{mn}^{∗})
= μ_{α}(m, n).

We use the cocycle identity w_{m}σ_{θ(m)}(w_{n}) = w_{mn} in the third equality. In the same way, we can verify λ_{α}(g, n) = λ_{θαθ^{−1}}(g, n).

Hence Λ(α) and ν_{α} are conjugacy invariants.

Next we compare α and its cocycle perturbation Ad w_{g}α_{g}. Here we have Ad w_{n}α_{n} = Ad w_{n}v_{n}σ_{ν(n)} for n ∈ H_{α}. In the same way as above, we can verify that these two actions have the same invariants. □ Remark. In the above lemma, we assumed that χ_{a}(M, N) is a finite cyclic group. But note that the above argument is valid if the range of ν is a finite cyclic group. Also note that when Aut (M, N) = Ker Φ holds, we do not have to make this assumption because of Theorem 3.1.

Example 4.3 We describe these invariants in the case G = Z. Fix α ∈ Ker Φ. Let p := p_{s}(α) be the strongly outer period of α. So we have H_{α} = p_{s}(α)Z. Next we compute λ_{α}. We only have to compute λ_{α}(1, p). Since σ_{ν(p)} is not strongly outer, there exist l > 0 and non-zero a ∈ M_{l} such that σ_{ν(p)}(x)a = ax holds for every x ∈ M. Hence α^{p}(x)v_{p}a = v_{p}ax holds. By (5) of Theorem 3.1, α(a) = u_{α,σ_{ν(p)}}a holds. Also we have α(v_{p}) = λ(1, p)v_{p}u_{α,σ_{ν(n)}}^{∗} by the definition of λ. Hence we have α(v_{p}a) = λ(1, p)v_{p}u_{α,σ_{ν(p)}}^{∗}u_{α,σ_{ν(p)}}a = λ(1, p)v_{p}a. Here recall the definition of the higher obstruction γ_{h}(α) in [22, Section 2]. Then we get λ(1, p) = γ_{h}(α). Since H^{2}(Z, T) is trivial, we may choose μ(m, n) = 1. So summarizing the above computations, we have H_{α} = p_{s}(α)Z, λ(m, p_{s}(α)n) = γ_{h}(α)^{mn}, μ ≡ 1 and ν(p_{s}(α)n) = nν(p_{s}(α)).

5 First classification theorem

In this section, we treat actions whose inner part and centrally trivial part coincide. (We do not assume the triviality of the Loi invariant in this section.) In this case, we can define characteristic invariant Λ(α) ∈ Λ(G, H) in the same way as in the single factor case.

The purpose of this section is to prove the following theorem.

Theorem 5.1 Let N ⊂ M be a strongly amenable subfactor of type II_{1}, G a discrete amenable group. Let α and β be actions of G on N ⊂ M such that H := α^{−1}(Int(M, N)) = α^{−1}(Cnt (M, N)) = β^{−1}(Int(M, N)) = β^{−1}(Cnt (M, N)). Then α and β are cocycle conjugate if and only if (Φ(α), Λ(α)) = (Φ(β), Λ(β)) holds.

To prove Theorem 5.1, we need some preparations. The following lemma can be verified in the same way as in the proof of [31, Lemma 2.3].

Lemma 5.2 Let N ⊂ M, G and α be as in Theorem 5.1. Assume Λ(α) is trivial. Then we can choose α-cocycle w_{g} such that Ad w_{h}α_{h} = id for h ∈ H.

Lemma 5.3 Let N ⊂ M, G, α and β be as in Theorem 5.1 and assume that the characteristic invariants of α and β are trivial. Then α and β are cocycle conjugate.

Proof. By the previous lemma, we can choose an α-cocycle w_{g}^{1} and a β-cocycle w_{g}^{2} such that Ad w_{h}^{1}α_{h} = Ad w_{h}^{2}β_{h} = id for h ∈ H. So we can regard these action as the centrally free actions of G/H. Since they have the same Loi invariant, these two actions are cocycle conjugate by [34, Theorem 3.1]. Since this means that they are cocycle conjugate as actions of G, original actions α and β are cocycle conjugate. □

Lemma 5.4 Let N ⊂ M, G and α be as in Theorem 5.1. Then α is cocycle conjugate to the action α ⊗ σ^{(0)}, where σ^{(0)} is the model action of G/H on the AFD type II_{1} factor R_{0} constructed by Ocneanu in [31], and we regard σ^{(0)} as the action of G in the natural way.

Proof. From α, we get a G/H-kernel naturally. Then this G/H-kernel is centrally free. Hence in a similar argument in the proof of [31, Theorem 1.6], this is conjugate to α⊗σ^{(0)} as G/H-kernels. This means that there exist unitaries v_{g}, g ∈ G such that Ad v_{g}α_{g} is conjugate to α ⊗ σ^{(0)}. Then the same argument in the proof of [31, Lemma 2.4] works in this case and we can conclude that α and α ⊗ σ^{(0)} are cocycle conjugate. □

Proof of Theorem 5.1. Let σ be an action of G on R_{0} whose characteristic invariant is an inverse of Λ(α) and σ̄ an action of G on R_{0} with the characteristic invariant Λ(α). Then the characteristic invariants of α ⊗ σ and β ⊗ σ are trivial, so they are cocycle conjugate by Lemma 5.3. Then we have

α ∼ α ⊗ σ^{(0)} ∼ α ⊗ σ ⊗ σ̄
∼ β ⊗ σ ⊗ σ̄ ∼ β ⊗ σ^{(0)}
∼ β.

Hence α and β are cocycle conjugate. □

Corollary 5.5 Let N ⊂ M be a strongly amenable subfactor of type II_{∞} and G, α, β as in Theorem 5.1. Then α and β are cocycle conjugate if and only if (Φ(α), Λ(α), mod (α)) = (Φ(β), Λ(β), mod (β)) holds.

Proof. In a similar way as in the proof of Theorem 5.1, we can reduce the problem to the case when α and β acts freely on N ⊂ M. Then we can apply [38, Theorem 4.3]. □

6 Second classification theorem

In this section we prove the main theorem in this paper. Until the end of the proof of Theorem 6.1, we always assume that N ⊂ M is a strongly amenable subfactor of type II_{1}, has the trivial normalizer, the lifting from χ(M, N) to Aut (M, N) and trivial Jones κ invariant. We also assume that χ(M, N) is a finite group. Examples of such subfactors are SU(n)_{k} subfactors, where n is a odd number, or n is even number and 2n divides k. See [8], [40] and [11]. Fix the lifting σ_{g} of χ(M, N) to Aut (M, N).

Theorem 6.1 Let N ⊂ M be as above and G a discrete amenable group. Let α and β be approximately inner actions of G, whose range of ν are cyclic groups. Then α and β are cocycle conjugate if and only if (H_{α}, Λ(α), ν_{α}) = (H_{β}, Λ(β), θ(ν_{β})) hold for some θ ∈ χ_{a}(Aut χ(M, N)).

Set K := χ(M, N). Then K is an abelian group. We consider the crossed product P ⊂ Q := N ⋊_{σ} K ⊂ M ⋊_{σ} K. Let w_{h} be an implementing unitary. We may also assume ν_{α} = ν_{β}.

By Proposition 3.2, we have an action of G × K̂ by (g, p) ↦ α̃_{g}σ̂_{p}, which we denote α̃_{g,p} for simplicity.

Lemma 6.2 Let Q ⊂ P ⊂ P_{1} ⊂ P_{2} ⊂ ⋯ be the Jones tower for Q ⊂ P. Then we have

P′ ∩ P_{n} = {∑_{k∈K} a_{k}w_{k}|a_{k} ∈ M_{n}, xa_{k} = a_{k}σ_{k}(x) for every x ∈ M}.

We have a similar expression for Q′ ∩ P_{n}.

Proof. Take a = ∑_{k∈K} a_{k}w_{k} ∈ Q_{n} and assume that a commutes with every x ∈ M first. Then we have x ∑_{k∈K} a_{k}w_{k} = ∑_{k∈K} a_{k}w_{k}x. Hence we get xa_{k} = a_{k}σ_{k}(x). Next assume that a commutes with w_{h}, h ∈ K. Then σ_{h}(a_{k}) = a_{k} must hold. By (5) of Theorem 3.1, we have σ_{h}(a_{k}) = a_{k}u_{σ_{h},σ_{k}}^{∗} = κ(k, h)a_{k}. But one of our assumptions is the triviality of κ invariant. Hence σ_{h}(a_{k}) = a_{k} holds. □.

Lemma 6.3 The dual action σ̂ of K̂ is strongly outer.

Proof. Fix e ≠ p ∈ K̂. Suppose that a = ∑_{k∈K} a_{k}w_{k} ∈ P_{n} satisfies σ̂_{p}(x)a = ax for every x ∈ P. First we take x from M. Then by the definition of the dual action, we have σ̂_{p}(x) = x. Then in the same way in as the proof of Lemma 6.2, we get xa_{k} = a_{k}σ_{k}(x). Assume a_{k} ≠ 0. This means that the strong outerness of σ_{k} breaks at M_{n}. Next put x = w_{h}, h ∈ K. Then we get σ_{h}(a_{k}) = ⟨p, h⟩a_{k}. Since we assume the triviality of κ invariant, we have σ_{h}(a_{k}) = a_{k}. Hence ⟨p, h⟩ = 1 holds for every h ∈ K and this contradicts the fact p ≠ e. □

Lemma 6.4 Actions α̃ and β̃ of G × K̂ on P ⊂ Q satisfy the assumption of Theorem 5.1.

Proof. First we compute Φ(α̃). By Lemma 6.2, we know the relative commutant P′ ∩P_{n}. Fix ∑_{k∈K} a_{k}w_{k} ∈ P′ ∩ P_{n}. Then by (5) of Theorem 3.1 and the definition of α̃,

α̃_{g}(∑_{k∈K} a_{k}w_{k}) = ∑_{k∈K} α̃_{g}(a_{k})α̃(w_{k})
= a_{k}u_{g,k}^{∗}u_{g,k}w_{k}
= ∑_{k∈K} a_{k}w_{k}

holds for every g ∈ G. Hence Φ(α̃_{g}) is trivial for every g ∈ G. The same results holds for β̃_{g}. On the other hand, it is clear that Φ(α̃_{p}) = Φ(β̃_{p}) holds for every p ∈ K̂. Hence we get Φ(α̃) = Φ(β̃).

Next we consider α̃_{h}, h ∈ H. Since we assume the triviality of κ invariant, we have σ̃_{k} = Ad w_{k}. Hence α̃_{h} = Ad v_{h}w_{ν(h)} holds for h ∈ H and α̃_{h} is inner.

Next suppose that α̃_{g}(x) ∑_{k∈K} a_{k}w_{k} = ∑_{k∈K} a_{k}w_{k}x for g ∈ G \ H. holds. If x ∈ M, then we have α_{g}(x)a_{k} = a_{k}σ_{k}(x) and hence we get α_{g}σ_{k}^{−1}(x)a_{k} = a_{k}x. Since Cnt_{r} (M, N) is a subgroup of Aut (M, N), α_{g}σ_{k}^{−1} is strongly outer. So a_{k} is equal to zero for every k ∈ K. Hence α̃_{g} is strongly outer. Similar results hold for β̃, so H = α̃^{−1}(Int(P, Q)) = α̃^{−1}(Cnt (P, Q)) = β̃^{−1}(Int(P, Q)) = β̃^{−1}(Cnt (P, Q)) holds. Let [λα̃, μα̃] be the characteristic invariant of α̃. By easy computations, we can verify that λα̃((g, k), n) = λ_{α}(g, n)\overline{⟨k, ν(n)⟩} and μα̃(m, n) = μ_{α}(m, n) hold. Then by the assumption Λ(α) = Λ(β), we can conclude that Λ(α̃) = Λ(β̃) holds. □

Proof of Theorem 6.1. By [39, Theorem 6.1], Q ⊂ P is strongly amenable. By Lemma 6.4 and Theorem 5.1, α̃ and β̃ are cocycle conjugate. Then we take the partial crossed product Q ⋊σ̂ K̂ ⊂ P ⋊σ̂ K̂, extend the actions α̃ and β̃ of G canonically and we denote them ˜ α̃ and ˜ β̃ respectively. Then by [36, Proposition 1.1], ˜ α̃ and ˜ β̃ are cocycle conjugate. With Takesaki’s duality theorem, we can conclude α and β are cocycle conjugate. □

By using Corollary 5.5, we can prove the following in the same way as above.

Corollary 6.5 Let N ⊂ M be a strongly amenable subfactor of type II_{∞} and A ⊂ B be a subfactor of type II_{1} associated with N ⊂ M, which satisfies the assumption in Theorem 6.1. Let G be a discrete amenable group, α and β be actions of G with trivial Loi invariant whose range of ν invariant is cyclic. Then α and β are cocycle conjugate if and only if N_{α} = N_{β}, Λ(α) = Λ(β), ν_{α} = χ(θ)(ν_{β}) and mod (α) = mod (β) hold for some χ_{a}(θ) ∈ χ_{a}(Aut (M, N)).

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