Classification of strongly free actions of discrete amenable groups on strongly amenable subfactors of type III_{0}

Toshihiko Masuda ^{∗} Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153, JAPAN

Abstract We classify the strongly free actions of discrete amenable groups on strongly amenable subfactors of type III_{0}. Winsløw’s fundamental homomorphism is a complete invariant.

1 Introduction

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 II_{1} 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 III_{λ}, 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 III_{λ}, 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 III_{0}. In this paper, we classify strongly free actions of discrete amenable groups on strongly amenable subfactors of type III_{0}. 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 III_{λ}, 0 < λ < 1, strong freeness is equivalent to central freeness. See [16], [21].)

∗e-mail address : masuda@ms.u-tokyo.ac.jp

To classify actions on subfactors of type III_{0}, we must consider actions of groupoids on subfactors of type II_{∞}. Groupoid actions on semifinite factors are studied in [9] first, and developed in [18] and [19]. A basic idea to classify groupoid actions is the followings. First we split a groupoid into an isotropy part and a principal part, second apply the classification results of group actions to the isotropy part, and the cohomology lemma to the principal part, and finally combine them. Our idea to classify groupoid actions on strongly amenable subfactor of type II_{∞} owes much to this idea.

Note that in the type III_{λ} case, 0 < λ < 1, we need only classification results on group actions and do not have to use the cohomology lemma because the form of the flow of weights of type III_{λ} factor is simple and the principal parts become trivial.

This paper is organized as follows.

In section 2, we collect facts on group actions and strongly amenable subfactor of type III_{0} which we need in this paper.

In section 3, we explain how to reduce the classification of group actions to that of groupoid actions. We also discuss relation between invariants of group actions and those of groupoid actions.

In section 4, we classify groupoid actions on strongly amenable subfactor of II_{∞}. We split the groupoid into the isotropy part and the principal part. To classify the isotropy part, we use Popa’s classification result, [16, Theorem 3.1] and to classify the principal part, we use the cohomology lemma, [17, Theorem 5.5]. Then we get outer conjugacy, but it may happen that the actions are not cocycle conjugate. To obtain cocycle conjugacy, we use model actions constructed by Sutherland and Takesaki in [18] and we complete the classification. Acknowledgement. The author is grateful to Prof. M. Izumi and Prof. Y. Kawahigashi for helpful advice and constant encouragement. He also thanks Professor C. Winsløw for pointing out his misunderstanding in Theorem 2.7

2 Preliminaries and notations

In this section, we recall several facts about group actions on subfactors and strongly amenable subfactors of type III_{0}.

Definition 2.1 ([1, Definition 1], [16, Definition 1.5.1]) Let N ⊂ M be a subfactor with finite index and N ⊂ M ⊂ M_{1} ⊂ ⋯ be the Jones tower. For α ∈ Aut (M, N), α is said to be strongly outer if there are no nonzero a ∈ M_{k} such that α(x)a = ax holds for every x ∈ M. An action α of G is said to be strongly outer if α_{g} is strongly outer for g ∈ G \ {e}.

Definition 2.2 ([12, Section 5]) Set Φ(α) := {α|_{M′∩M_{k}}}_{k}. We call Φ(α) the Loi invariant.

In the case of subfactors in type II, the following classification results are obtained by Popa and Winsløw.

Theorem 2.3 ([16, Theorem 2.1, Theorem 3.1], [21, Theorem 4.3]) Let N ⊂ M be a strongly amenable subfactor of type II_{1} or II_{∞}. Let α and β be actions of a discrete amenable group G. Then α and β are cocycle conjugate if and only if Φ(α) = Φ(β) and mod (α) = mod (β). (In the type II_{1} case, the last condition is not necessary.)

Let φ be a faithful normal state of N and E the minimal conditional expectation from M onto N. Set (Ñ ⊂ M̃) := (N ⋊_{σ^{φ}} R ⊂ M ⋊_{σ^{φ∘E}} R). Let α ∈ Aut (M, N) and α̃ the canonical extension of α to Ñ ⊂ M̃. (See [7]).

Definition 2.4 ([21, Definition 3.2], [22, Definition 4.2]) Let N ⊂ M be a subfactor type III with finite index and Ñ ⊂ M̃ ⊂ M̃_{1} ⊂ ⋯ be the Jones tower for Ñ ⊂ M̃. For α ∈ Aut (M, N), α is said to be strongly free if there are no nonzero a ∈ M̃_{k} such that α̃(x)a = ax holds for every x ∈ M̃. An action α of G is said to be strongly free if α_{g} is strongly free for g ∈ G \ {e}.

Definition 2.5 ([22, Definition 4.2]) Set Υ(α) := {α̃|_{M̃′∩M̃_{k}}}_{k}. We call Υ(α) the fundamental homomorphism.

In the case of subfactors of type II and type III_{λ}, strong amenability has been introduced by Popa, [15] and [16]. Based on strong amenability of subfactor of type II, Winsløw introduced strong amenability for subfactors of type III_{0}.

Definition 2.6 ([25, Definition 3.5]) Let N ⊂ M be a subfactor of type III_{0} with the common flow of weights and Ñ ⊂ M̃ := N ⋊_{σ^{φ}} R ⊂ M ⋊_{σ^{φ∘E}} R. N ⊂ M is said to be strongly amenable if Z(Ñ) = Z(M̃) and (Ñ ⊂ M̃) ≅ Z(Ñ) ⊗ (P ⊂ Q), where P ⊂ Q is a strongly amenable subfactor of type II_{∞}.

On the classification of subfactors of type III_{0}, the following result has been obtained by Winsløw.

Theorem 2.7 ([25, Theorem 4.2]) Let N ⊂ M be a strongly amenable subfactor of type III_{0}. Then N ⊂ M is classified by its relative flow of weights,

F^{M⊃N} := {Ñ′ ∩ M̃_{k} ⊂ M̃′ ∩ M̃_{k}, θ_{t}|_{M̃′∩M̃_{k}}}_{k}.

3 Reduction to groupoid actions

In the rest of this paper N ⊂ M is always a strongly amenable subfactor of type III_{0} and G denotes a discrete amenable group. We also use the notations in the previous section.

Our main theorem of this paper is the following.

Theorem 3.1 Let α and β be strongly free actions of G on N ⊂ M. Then α and β are cocycle conjugate if and only if there exists σ ∈ Aut (F^{M⊃N}) such that

Υ(α) = σ ∘ Υ(β) ∘ σ^{−1}.

We give the proof of above theorem in this section and next section.

In this section, we reduce the classification of actions of G to that of actions of groupoids.

First we consider only one action α. For the same reason as at the beginning of Section 3 in [13], we only have to classify the actions of G̃ := G × R on Ñ ⊂ M̃ defined by (g, t) → α̃θ_{t}. We also denote this action of G̃ by α̃.

Let (X, μ, F_{t}) be the flow of weight of M. Then L^{∞}(X, μ) = Z(Ñ) = Z(M̃). Then by assumption,

(Ñ ⊂ M̃ ⊂ M̃_{1} ⊂ ⋯) ≅ L^{∞}(X, μ) ⊗ (P ⊂ Q ⊂ Q_{1} ⊂ ⋯)

holds.

Let G̃ ⋉ X be the auxiliary groupoid. Then the source map s and the range map r from G̃ ⋉ X to X are the following.

s(g, x) := x, r(g, x) := gx, (g, x) ∈ G̃ ⋉ X.

We consider the groupoid action of G̃ ⋉ X on P ⊂ Q by the following equation.

α̃_{g}(a) = ∫_{X}^{⊕} α̃_{(g,g^{−1}x)}(a(g^{−1}x))dμ(x).

We express (X, μ, F_{t}) as the flow built under a ceiling function on the base transformation T on the measure space (Y, ν). So X is of the form {(y, s) ∈ Y ×R|0 ≤ s ≤ f(y)}. We define maps π : X → Y and h :→ R by the equality x = (π(x), h(x)). Set G := {γ ∈ G̃ ⋉ X|s(γ), r(γ) ∈ Y }. The groupoid G is an orbitally discrete groupoid. (See [6].) For x ∈ X, put H_{x} := {g ∈ G̃|gx = x}. Then H_{x} is a discrete amenable group. Since R is in the center of G̃, H_{x} = H_{tx} holds for t ∈ R. Since R acts on X ergodicaly, H_{x} = H for a.e x. Then we have a semidirect product of G as G = H ⋊ K, where K is a principal groupoid derived from G. In this case, K is an AF ergodic groupoid by [4], [26].

Here as in [13, Proposition 3.2], the following holds.

Proposition 3.2 If α is a strongly free action on N ⊂ M, the action of H_{x} on P ⊂ Q is strongly outer for a.e x ∈ Y.

Proof. Assume that there exists Z ⊂ Y such that ν(Z) > 0 and actions of H_{z} are not strongly outer for any z ∈ Z. Then for every z ∈ Z there exists k ∈ N, 0 ≠ a(z) ∈ P_{k} and e ≠ h ∈ H_{z} such that α̃_{h}(x)a(z) = a(z)x holds for every x ∈ P. It is possible that k and g depend on z. But since H_{z} is countable, we may assume k and g are independent from z ∈ Z. Note that since τ_{Q}α̃_{h} = τ, g is in G. Set

ã := ∫_{X}^{⊕} α̃_{(h(x),π(x))}(a(π(x)))dμ(x).

Then an easy computation shows that this element breaks strong freeness and α is not strongly free. □

By the next lemma, we know the relation between cocycle conjugacy of actions of G̃ ⋉ X and that of G.

Lemma 3.3 Let α̃ and β̃ be actions of G̃⋉X on P ⊂ Q. If α̃ and β̃ are cocycle conjugate when we restrict actions on G, then actions of G̃ ⋉ X are also cocycle conjugate.

Proof. The proof of [19, Theorem 4.2] works in the same way. But we present the proof here for the reader’s convenience.

First we define a map from G̃ ⋉ X to G as follows

p(g, x) := (h(gx)^{−1}gh(x), π(x)).

By assumption, there exist automorphisms {θ_{y}}_{y∈Y} ⊂ Aut (Q, P) and a cocycle v_{γ} ∈ Z1 ; β̃(G, U(P)) such that

θ_{r(γ)} ∘ α̃_{γ} ∘ θ_{s(γ)}^{−1} = Ad v_{γ} β̃_{γ}, γ ∈ G.

Define θ′_{x}, x ∈ X and v′_{(g,t)}, (g, t) ∈ G̃ ⋉ X as follows.

θ_{x} := β̃_{(h(x),π(x))} ∘ θ_{π}(x) ∘ α̃_{(h(x),π(x))}^{−1},
v_{(g,x)} := β̃_{(h(gx),π(gx))}(v_{p(g,x)}).

Note that (g, x) = (h(gx), π(gx))p(g, x)(h(x), π(x))^{−1} holds. Then using the above equations, we can show that

θ′_{gx} ∘ α̃_{(g,x)} ∘ θ_{x}^{′−1} = Ad v′_{(g,x)} β̃_{(g,x)}.

holds and this shows that actions of G̃ ⋉ X are cocycle conjugate. □ In the end of this section, we discuss invariants of actions of G on P ⊂ Q.

Lemma 3.4 Let α and β be actions of G on N ⊂ M such that Υ(α) = Υ(β). Let α̃ and β̃ be actions of G induced from α and β respectively by the above procedure. Then we have Φ(α̃) = Φ(β̃) and mod (α̃) = mod (β̃).

Proof. First we compare mod . Let τ be the trace of M̃ such that τθ_{t} = e^{−t}τ, where θ_{t} is the dual action. For (g, t) ∈ G̃, set ψ(g, t) := e^{−t}. Then τα̃_{h} = ψ(h) = τβ̃_{h} holds for h ∈ G̃. Express τ by direct integral, τ = ∫_{X}^{⊕} τ_{x}dμ(x). Then

τα̃_{h}(a) = ∫_{X}^{⊕} τ_{hx}(α̃_{(h,x)}(a(x)))dμ(gx)
= ∫_{X}^{⊕} τ_{hx}(α̃_{(h,x)}(a(x)))\frac{dμ ∘ g}{dμ} (x)dμ(x)

holds and we get an equality dμ ∘ g/dμ(x)τ_{hx} α̃_{(h,x)} = e^{−t}τ_{x}. A similar equality holds for β̃. From these equations, we can easily get mod (α̃_{(h,x)}) = mod (β̃_{(h,x)}).

Next we compare Φ. By assumption, α̃_{h}|_{M′∩M̃_{k}} = β̃_{h}|_{M′∩M̃_{k}} holds for every k. Note that an action α̃ of G̃ ⋉ X can be extended to Q_{k} such that the equation

α̃_{h}(a) = ∫_{X}^{⊕} α̃_{(h,h^{−1}x)}(a(h^{−1}x))dμ(x)

holds for every a ∈ M̃_{k}. (See the proof of [25, Lemma 4.3].) Then we get an equality Φ(α̃) = Φ(β̃). □

4 Classification results

Let α and β be strongly free actions of G on N ⊂ M. By the results of previous section, we get two actions α̃ and β̃ of a groupoid G such that α̃_{h} and β̃_{h} act strongly freely on P ⊂ Q and satisfy Φ(α̃) = Φ(β̃) and mod (α̃) = mod (β̃).

In this section, we classify actions of G on P ⊂ Q and complete the classification of actions of G on N ⊂ M.

Theorem 4.1 Two actions α̃ and β̃ of G are cocycle conjugate, i.e. there exists {θ_{y}}_{y∈Y} ⊂ Aut (Q, P) and a cocycle u_{γ} ∈ Z1 ; β̃(G, U(P)) such that

θ_{r(γ)} ∘ α̃_{γ} ∘ θ_{s(γ)}^{−1} = Ad u_{γ} β̃_{γ}, γ ∈ G.

In the following, we give a proof of Theorem 4.1.

We express G as the semidirect product H ⋊ K. We use Theorem 2.3 to classify the H part and the cohomology lemma to classify the K part.

First we compare the H part. By the above assumption, we have Φ(α̃_{h}) = Φ(β̃_{h}) and mod (α̃_{h}) = mod (β̃_{h}) for h ∈ H_{x}. So by Theorem 2.3, there exist automorphisms {θ_{x}}_{x∈Y} and a cocycle u_{h} ∈ Z1 ; β̃(H_{x}, U(P)) such that

θ_{x} ∘ α̃ ∘ θ_{x}^{−1} = Ad u_{h} β̃_{h}.

So we replace α̃_{γ} by θ_{r(γ)} ∘ α̃ ∘ θ_{s(γ)}^{−1}, we may assume α̃_{h} = Ad u_{h} β̃_{h} for h ∈ H_{x}.

Next we compare the K part by applying the cohomology lemma, [17, Theorem 5.5]. To use the cohomology lemma, we need several preparations.

Define two groups N_{x}^{0} and N_{x}^{1} as follows.

N_{x}^{1} := {(θ, v_{h}) | θ ∈ Aut (Q, P), τ_{Q}θ = τ_{Q}, Φ(θ) = id,

v_{h} ∈ Z1 ; β̃(H_{x}, U(P)), θ ∘ β̃_{h} ∘ θ^{−1} = Ad v_{h} β̃_{h}, for h ∈ H_{x}},

N_{x}^{0} := {(Ad u, uβ̃_{h}(u^{∗})) | u ∈ U(P), h ∈ H_{x}}.

Then obviously N_{x}^{0} ⊂ N_{x}^{1}.

Lemma 4.2 N_{x}^{0} is dense in N_{x}^{1}.

Proof. Take (θ, v_{h}) ∈ N_{x}^{1}. Since Φ(θ) = id and τ_{Q}θ = τ, θ is in \overline{Int}(Q, P). (See [22].) So there exist {u_{n}} ∈ U(P) such that lim_{n→∞} Ad u_{n} = θ. By assumption, θ ∘ β̃_{h} ∘ θ^{−1} = Ad v_{h} β̃_{h} holds, so lim_{n→∞} Ad u_{n} β̃_{h}(u_{n}^{∗})β̃_{h} = Ad v_{h} β̃_{h} holds and {v_{h}^{∗}u_{n}β̃_{h}(u_{n}^{∗})}_{n} is a central sequence.

Fix a free ultrafilter ω over N. Let C_{ω}(Q, P) be the central sequence algebra for P ⊂ Q and β̃^{(ω)} the induced action on C_{ω}(Q, P) from β̃. Then strong outerness of the action β̃ on P ⊂ Q means that β̃^{(ω)} acts freely on C_{ω}(Q, P). (See [21, Proposition 3.4] and [2, Proposition 2.1.2].)

Here we need the next proposition, which is a subfactor version of the 1-cohomology vanishing theorem of [14, Proposition 7.2].

Proposition 4.3 Every 1-cocycle w_{h} ∈ Z1 ; β̃^{(ω)}(H_{x}, U(C_{ω}(Q, P))) is a coboundary, i.e.

there exists a unitary w′ in C_{ω}(Q, P) such that w′β̃_{h}^{(ω)}(w′) = w_{h}.

The proof in [14] works in the same way. When the group is Z, the above result has been obtained by Loi in [12, Proposition 4.2]. We can easily verify that {v_{h}^{∗}u_{n}β̃_{h}(u_{n}^{∗})}_{n} is a β̃^{ω} cocycle. So by Proposition 4.3, we can find a unitary w = {w_{n}}_{n} ∈ C_{ω}(Q, P) such that {w_{n}^{∗}β̃_{h}(w_{n})}_{n} = {v_{h}^{∗}u_{n}β̃_{h}(u_{n}^{∗})}_{n} in C_{ω}(Q, P). Set y_{n} := w_{n}u_{n}. Then lim_{n→ω} Ad y_{n} = θ holds, and

lim_{n→ω} y_{n}β̃_{h}(y_{n}^{∗}) = lim_{n→ω} w_{n}u_{n}β̃_{h}(u_{n}^{∗}w_{n}^{∗})
= lim_{n→ω} v_{h}v_{h}^{∗}w_{n}u_{n}β̃_{h}(u_{n}^{∗}w_{n}^{∗})
= lim_{n→ω} v_{h}w_{n}v_{h}^{∗}u_{n}β̃_{h}(u_{n}^{∗})β̃_{h}(w_{n}^{∗})
= lim_{m→ω} v_{h}w_{n}w_{n}^{∗}β̃_{h}(w_{n})β̃_{h}(w_{n}^{∗})
= v_{h}

So by choosing a subsequence, we get unitaries {y_{n}}_{n} such that

lim_{n→∞}(Ad y_{n}, y_{n}β̃_{h}(y_{n}^{∗})) = (θ, v_{h}).

This shows that N_{x}^{0} is dense in N_{x}^{1} and the proof of Lemma 4.2 is complete. □.

For k = (y, x) ∈ K, put γ_{k} := α̃_{k}β̃_{k}^{−1}. Then it is clear that Φ(γ_{k}) = id and mod (γ_{k}) = 1. Moreover

γ_{k}β̃_{h}γ_{k}^{−1} = α̃_{k}β̃_{k^{−1}hk}α̃_{k}^{−1}
= α̃_{k}Ad u_{k^{−1}hk}^{∗} α̃_{k^{−1}hk}α̃_{k}^{−1}
= Ad α̃_{k}(u_{k^{−1}hk}^{∗}) α̃_{h}
= Ad α̃_{k}(u_{k^{−1}hk}^{∗})u_{h} β̃_{h}

holds and α̃_{k}(u_{k^{−1}hk}^{∗})u_{h} is a β̃ cocycle. So γ̃_{k} := (γ_{k}, α̃_{k}(u_{k^{−1}hk}^{∗}u_{h})) is an element of N_{x}^{1}. Next, for k = (y, x) ∈ K, we define a Borel map N_{k}^{1} from N_{x}^{1} to N_{y}^{1} as follows.

N_{k}^{1}(θ, v_{h}) := (β̃_{k} ∘ θ ∘ β̃_{k}^{−1}, β̃(v_{k^{−1}hk})).

Then (N_{x}^{1}, N_{k}^{1}) is a Borel functor in the sense of [17, Definition 4.1] and γ̃ is a N_{k}^{1} cocycle, i.e. γ̃_{k_{1}k_{2}} = γ̃_{k_{1}}N_{k_{1}}^{1}(γ̃_{k_{2}}). Here we apply the cohomology lemma, [17, Theorem 5.5], for γ̃ and id.

Then we get Borel maps θ : Y → Aut (Q, P) and a : K → U(P) and a cocycle v_{h} ∈ Z1 ; β̃(H_{x}, U(P)) which satisfy equalities

θ_{y} ∘ γ_{k} ∘ θ_{x}^{−1} = Ad a_{k}, k = (y, x) ∈ K,
θ_{x} ∘ β̃_{h} ∘ θ_{x}^{−1} = Ad v_{h} β̃_{h}.

Since K is AF, we may consider that a_{k} is a cocycle for β̃_{k}. By definition of γ_{k}, we obtain the equality

θ_{y} ∘ α̃_{k} ∘ θ_{x}^{−1} = Ad a_{k} β̃_{k}.

Then for h ∈ H_{y} and k = (y, x) ∈ K

θ_{y} ∘ α̃_{hk} ∘ θ_{x}^{−1} = θ_{y} ∘ α̃_{h} ∘ θ_{y}^{−1} ∘ θ_{y} ∘ α̃_{k} ∘ θ_{x}^{−1}
= θ_{y} ∘ Ad u_{h} ∘ β̃_{h} ∘ θ_{y}^{−1} ∘ Ad a_{k} ∘ β̃_{k}
= Ad θ_{y}(u_{h})v_{h} ∘ β̃_{h} ∘ Ad a_{k} ∘ β̃_{k}
= Ad θ_{y}(u_{h})v_{h}β̃_{h}(a_{k}) β̃_{hk}

Set a_{h} := θ_{x}(u_{h})v_{h} for h ∈ H_{x} and a_{hk} := a_{h}β̃_{h}(a_{k}), we get

θ_{y} ∘ α̃_{hk} ∘ θ_{x} = Ad a_{hk} β̃_{hk}.

Here a_{h} and a_{k} are cocycles for β̃_{h} and β̃_{k}, but a_{hk} is not necessary a cocycle for β̃_{hk}. But by comparing α̃_{h_{1}k_{1}} α̃_{h_{2}k_{2}} with α̃_{h_{1}k_{1}h_{2}k_{2}}, we get the following equality.

a_{h_{1}k_{1}} β̃_{h_{1}k_{1}}(a_{h_{2}k_{2}}) = φ(k_{1}, h_{2})a_{h_{1}k_{1}h_{2}k_{2}},

where φ(k_{1}, h_{2}) is the element of Z^{1}(K, Ĥ) and does not depend on h_{1} and k_{2}, where Ĥ is the set of 1 dimensional representations of H.

To remove φ, we use the model actions constructed by Sutherland and Takesaki.

Theorem 4.4 ([18, Theorem 3.1]) Let R_{0,1} be the AFD factor of type II_{∞}. Then there exists an action m of G satisfy the following conditions.

(1) For every y ∈ Y, H_{y} acts outer on R_{0,1}.

(2) mod (m) = mod (α̃)

(3) For every φ ∈ Z^{1}(K, Ĥ), there exist {θ_{y}}_{y∈Y} ⊂ Aut (R_{0,1}) and u_{hk} ∈ U(R_{0,1}) such that

(a) u_{h} and u_{k} are cocycle for H_{y} and K,
(b) θ_{y} ∘ m_{hk} ∘ θ_{x} = Ad u_{hk} m_{hk},
(c) u_{h_{1}k_{1}}m_{h_{1}k_{1}}(u_{h_{2}k_{2}}) = \overline{φ(k_{1}, h_{2})} u_{h_{1}k_{1}h_{2}k_{1}}.

We define the model action m̄ of G on (P ⊂ Q) ≅ (∨_{k}(P′_{k} ∩ P) ⊂ ∨_{k}(P′_{k} ∩ Q)) ⊗ R_{0,1} as follows.

m̄_{γ} := α̃_{γ}^{st} ⊗ m_{γ}.

We complete the proof of Theorem 4.1 by comparing α̃ with the model action m̄. The action m̄ has the same invariant of α̃ and m̄_{h}, h ∈ H_{x}, acts strongly outer on

P ⊂ Q by the construction of m̄. Then the above discussion is valid for α̃ and m̄, so we may assume the following equalities.

α̃_{hk} = Ad a_{hk} ∘ m̄_{hk},
a_{h_{1}k_{1}} m̄_{h_{1}k_{1}}(a_{h_{2}k_{2}}) = φ(k_{1}, h_{2})a_{h_{1}k_{1}h_{2}k_{2}}

for some φ. Take {θ_{y}}_{y∈Y} ⊂ Aut (R_{0,1}) and u_{hk} satisfying the conditions of Theorem 4.4. Then

(id ⊗ θ_{r(k)}) ∘ α̃_{hk} ∘ (id ⊗ θ_{s(k)})^{−1} = (id ⊗ θ_{r(k)}) ∘ Ad a_{hk} ∘ m̄_{hk} ∘ (id ⊗ θ_{s(k)})^{−1}
= Ad (id ⊗ θ_{r(k)})(a_{hk})α̃_{hk}^{st} ⊗ (θ_{r(k)} ∘ m_{hk} ∘ θ_{s(k)})
= Ad (id ⊗ θ_{r(k)})(a_{hk})α̃_{hk}^{st} ⊗ (Ad u_{hk}m_{hk})
= Ad (id ⊗ θ_{r(k)})(a_{hk})(1 ⊗ u_{hk})m̄_{hk}

and

(id ⊗ θ_{r(k_{1})})(a_{h_{1}k_{1}})(1 ⊗ u_{h_{1}k_{1}})m̄_{h_{1}k_{1}}((id ⊗ θ_{r(k_{2})})(a_{h_{2}k_{2}})(1 ⊗ u_{hk}))
= (id ⊗ θ_{r(k_{1})})(a_{h_{1}k_{1}}) (α̃_{h_{1}k_{1}}^{st} ⊗ θ_{r(k_{1})}m_{h_{1}k_{1}})(a_{h_{2}k_{2}}) (1 ⊗ u_{h_{1}k_{1}}) (1 ⊗ m_{h_{1}k_{1}}(u_{h_{2}k_{2}}))
= (id ⊗ θ_{r(k_{1})})(a_{h_{1}k_{1}}m̄_{h_{1}k_{1}}^{st}(a_{h_{2}k_{2}})) (1 ⊗ u_{h_{1}k_{1}}m_{h_{1}k_{1}}(u_{h_{2}k_{2}}))
= (id ⊗ θ_{r(k_{1})})(φ(k_{1}, h_{2})a_{h_{1}k_{1}h_{2}k_{2}}) (1 ⊗ \overline{φ(k_{1}, h_{2})}u_{h_{1}k_{1}h_{2}k_{2}})
= (id ⊗ θ_{r(k_{1})})(a_{h_{1}k_{1}h_{2}k_{2}})u_{h_{1}k_{1}h_{2}k_{2}}

hold, so (id ⊗ θ_{r(k)})(a_{hk})(1 ⊗ u_{hk}) is a cocycle for m̄_{hk} and finally we can conclude two actions α̃ and m̄ are cocycle conjugate and proof of Theorem 4.1 is complete.

Proof of Theorem 3.1. Let α and β be strongly free actions of G on N ⊂ M such that Υ(α) = Υ(β). Then by the results of Section 3 and Section 4, two actions α̃ and β̃ of G on P ⊂ Q are cocycle conjugate. Then by Theorem 3.3, two actions α̃ and β̃ are cocycle conjugate, i.e. there exist automorphisms {θ_{x}}_{x∈X} ⊂ Aut (Q, P) and a cocycle u_{γ} ∈ Zβ̃(G̃ ⋉ X, U(P)) such that

θ_{r(γ)} ∘ α̃_{γ} ∘ θ_{s(γ)}^{−1} = Ad u_{γ} β̃_{γ}.

We put θ̃:= ∫_{X}^{⊕} θ_{x} dμ(x) and u_{g} := ∫ ⊕_{X} u_{(g,x)} dμ(x), then we get

θ̃ ∘ α̃_{g} ∘ θ̃^{−1} = Ad u_{g} β̃_{g}

for g ∈ G̃, so actions α̃ and β̃ of G̃ = G × R are cocycle conjugate and we conclude α and β are cocycle conjugate as explained at the beginning of Section 3. □

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