Notes on group actions on subfactors ^{∗}^{†}

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

Abstract We construct approximately inner actions of discrete amenable groups on strongly amenable subfactors of type II_{1} with given invariants, and obtain classification results under some conditions. We also study the lifting of the relative χ group.

1 Introduction

In the theory of subfactors initiated by V. F. R. Jones in [15], analysis of automorphisms and group actions on subfactors has been done by many people. In [24], P. Loi introduced the Loi invariant for automorphisms of subfactors and obtained some results on structure of subfactors of type III_{λ}, 0 < λ < 1. In [31], S. Popa introduced the notion of proper outerness for automorphisms of subfactors (Choda and Kosaki introduced the same property in [3], and they call it strong outerness). With his classification of strongly amenable subfactors of type II_{1} in [32], he classified properly outer actions of discrete amenable groups on strongly amenable subfactors of type II_{1} by the Loi invariant, and solved the problem of classification of subfactors of type III_{λ} raised by Loi in [24]. On the other hand, related with orbifold construction, non-strongly outer automorphisms are studied in [8], [10], [11], [12], [17], [18], [19], [20], [22], [25], [35]. Also see [9, Chapter 15], in which almost all contents of the above referred papers are explained.

In [26], we classified approximately inner actions of discrete amenable groups on strongly amenable subfactors of type II_{1} by the characteristic invariant and the ν invariant under some assumptions. Among these assumptions, the most important one is the triviality of the algebraic κ invariant. (See [13] and [2] for the original definition and properties of κ invariant. For a subfactor analogue of the κ invariant, see [19].) This result is a generalization of [20, Theorem 3.1]. When the algebraic κ invariant is trivial, we can classify approximately inner actions completely. Hence we have to investigate the case when the algebraic κ invariant is not trivial. In this case, we do not know whether there exist actions with given invariants or not. Hence what we should do first is to find a systematic way to construct actions with given invariants. We remark that if the algebraic κ invariant is trivial, then our characteristic invariant is exactly the same as the original one in [14], but if the algebraic κ invariant is not trivial, then our characteristic invariant may be different from the usual one, and this makes classification more difficult.

∗2000 Mathematics Subject Classification. Primary 46L37;Secondary 46L40.

†Key Words and Phrases. Subfactors. Group actions. Characteristic invariants. κ invariant.

In this paper, we construct actions of discrete amenable groups on strongly amenable subfactors of type II_{1} with given invariants, and classify actions under an extra assumption on the ν invariant. (We emphasize that we never assume the triviality of the algebraic κ invariant.) The most essential assumption in our theory is the extendability of the ν invariant to a homomorphism from a whole group. This assumption is similar to that of [21, Theorem 20]. In [21], Kawahigashi, Sutherland and Takesaki have classified actions of a discrete abelian group G on the injective type III_{1} factor. The modular invariant ν appears as a cocycle conjugacy invariant, and this is a homomorphism from a subgroup of G to R. Essential fact in their proof is that ν can be extended to a homomorphism of G due to the divisibility of R. (Originally this idea was due to Connes. See [4, pp.466].)

In subfactor case, we can not expect such a property for the ν invariant generally. But if we assume the extendability of the ν invariant, our proof goes well as in the proof of [21, Theorem 20]. We remark that our results can be viewed as a generalization of [20, Theorem 4.1].

In appendix, we discuss liftings of χ_{a}(M, N) since we have to fix one lifting of χ_{a}(M, N) to define characteristic invariants. Acknowledgements. The author is grateful to Professor M. Izumi for informing of [30, Proposition 3.2], Professor Y. Kawahigashi and Professor S. Popa for various comments on this paper. The author is supported by Grant-in-Aid for Scientific Research, Ministry of Education (Japan).

2 Main results

Facts on automorphisms for subfactors are collected in [26, Section 2], and we will freely use notations there.

First we recall fundamental definitions and facts on automorphisms of subfactors. Let N ⊂ M be a subfactor of type II_{1} with finite index and N ⊂ M ⊂ M_{1} ⊂ M_{2} ⋯ the Jones tower. Then α ∈ Aut (M, N) can be extended to an automorphism of M_{k} inductively by setting α(e_{k}) = e_{k}, where e_{k} is the Jones projection in M_{k}.

Definition 2.1 ([24, Section 5]) For α, we put

Φ(α) := {α|_{M′∩M_{k}}}_{k=1}^{∞},

and we call Φ(α) the Loi invariant.

Definition 2.2 ([3, Definition 1], [32, Definition 1.5.1]) An automorphism α is said to be strongly outer if we have no non-zero a ∈ ⋃_{k} M_{k} satisfying σ(x)a = ax for every x ∈ M. We denote the set of non-strongly outer automorphisms by Cnt_{r} (M, N), which is a normal subgroup of Aut (M, N).

Throughout this paper, we always make the following assumptions on N ⊂ M. (A1) N ⊂ M is extremal, (A2) N ⊂ M and M ⊂ M_{1} have the trivial normalizer, (A3) Ker Φ = Aut (M, N), (A4) χ_{a}(M, N) := \frac{Ker Φ ∩ Cnt_{r} (M, N)}{Int (M, N)} is a finite group,

(A5) there exists a lifting σ of χ_{a}(M, N) to Aut (M, N).

The assumption (A3) means that every action has the trivial Loi invariant. Note that we have many classes of subfactors satisfying the above assumptions, e.g, Jones subfactors with principal graph A_{2n+1} in [15], or subfactors coming from Hecke algebras in [37]. (Also see [18] and [8].) By [12, Corollary 2.2], χ_{a}(M, N) is always abelian, and hence χ_{a}(M, N) is a finite abelian group in our setting.

By [26, Theorem 3.1], we have a Connes-Radon-Nikodym type cocycle u_{α,σ} ∈ U(N) for every α ∈ Ker Φ and σ ∈ Cnt_{r} (M, N). The algebraic κ invariant κ_{a} is defined by κ_{a}(h, k) = u_{σ_{k},σ_{h}}^{∗} for h, k ∈ χ_{a}(M, N). We can easily verify that κ_{a} is a bicharacter of χ_{a}(M, N).

Next we recall the definition of cocycle conjugacy invariants for actions considered in [26]. Let N ⊂ M be a subfactor of type II_{1} with finite index, G a discrete group, and α an action of G on N ⊂ M. Then we get cocycle conjugacy invariants in the following way. The first invariant is a normal subgroup H_{α} ⊂ G, which is the non-strongly outer part of α. Then we get a G-equivariant homomorphism ν_{α} from H_{α} to χ_{a}(M, N) by ν_{α}(h) = [α_{h}]. This ν_{α} is the second cocycle conjugacy invariant, and we call this the ν invariant. By the assumption (A5), α_{h} has the form α_{h} = Ad v_{h}σ_{ν_{α}(h)} for some unitary v_{h} ∈ U(N). Then we get two scalars λ_{α}(g, h) and μ_{α}(h, k) by the following equations for g ∈ G, h, k ∈ H_{α}.

α_{g}(v_{g^{−1}hg})u_{α_{g},σ_{να(h)}} = λ_{α}(g, h)v_{h}, v_{h}σ_{ν_{α}(h)}(v_{k}) = μ_{α}(h, k)v_{hk}.

The pair λ(g, h) and μ(h, k) satisfy the following relations for h, k, l ∈ H and g, g_{1}, g_{2} ∈ G.

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

(2) λ(g_{1}g_{2}, h) = λ(g_{1}, h)λ(g_{2}, g_{1}^{−1}hg_{1}),

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

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

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

The equation (4) shows the difference between the usual characteristic invariant and ours. This definition of λ and μ depends on the choice of v_{h}. To get rid of this dependency we have to define a suitable equivalence relation for (λ, μ). On this point see [26]. We denote the equivalence class of (λ_{α}, μ_{α}) by Λ(α) = [λ_{α}, μ_{α}], and the set of [λ, μ] by Λ(G, H|κ_{a}).

Conversely for a given normal subgroup H ⊂ G, [λ, μ] ∈ Λ(G, H|κ_{a}) and ν ∈ Hom_{G}(H, χ_{a}(M, N)), we will construct an action α with H_{α} = H, Λ(α) = [λ, μ] and ν_{α} = ν in the following proposition.

Proposition 2.3 Let N ⊂ M be a strongly amenable subfactor of type II_{1}, G a discrete amenable group. Assume that ν can be extended to a homomorphism from G. Then for every [λ, μ] ∈ Λ(G, H|κ_{a}) and ν, there exists an action α of G with H_{α} = H, Λ(α) = [λ, μ] and ν_{α} = ν.

Proof. By the assumptions, we have an extension of ν from G to χ_{a}(M, N), which we denote by ν again. Hence g → σ_{ν(g)} is an action of G on N ⊂ M. Let κ_{a} be the algebraic κ invariant for N ⊂ M, and set λ′(g, n) := κ_{a}(ν(n), ν(g))λ(g, n). Then it is easy to verify that [λ′, μ] is in Λ(G, H), that is, [λ′, μ] is a usual characteristic invariant. Let m be an action of G on the injective type II_{1} factor R_{0} with the characteristic invariant [λ′, μ]. Define an action α of G on N ⊗ R_{0} ⊂ M ⊗ R_{0} by α_{g} := σ_{ν(g)} ⊗ m_{g}. Since N ⊂ M is isomorphic to N ⊗ R_{0} ⊂ M ⊗ R_{0} by [32] (also see [1]), α can be regarded as an action of N ⊂ M. Then this α is a desired one. □

On classification of actions, we have the following result.

Theorem 2.4 Let N ⊂ M, G be as in the previous proposition. Let α and β be approximately inner actions of G. Assume ν_{α} can be extended to a homomorphism from G. Then α and β are stably conjugate if H_{α} = H_{β}, Λ(α) = Λ(β) and ν_{α} = ν_{β} hold.

Proof. Set K := χ_{a}(M, N). Let α̃ be the extension of α on N ⋊_{σ} K ⊂ M ⋊_{σ} K defined

in [26], and ˜ α̃ be the natural extension of α̃ on ˜ Ñ ⊂ ˜ M̃:= N ⋊_{σ} K ⋊σ̂ K̂ ⊂ M ⋊_{σ} K ⋊σ̂ K̂. Let w_{k} and v_{p} the usual implementing unitaries of σ, σ̂ in M ⋊_{σ} K and M ⋊_{σ} K ⋊σ̂ K̂ respectively. Then by the definition of ˜ α̃, we have ˜ α̃_{g}(x) = α_{g}(x), ˜ α̃_{g}(w_{k}) = u_{α_{g},σ_{k}}w_{k} and ˜α̃_{g}(v_{p}) = v_{p} for x ∈ M, k ∈ K and p ∈ K̂. On the other hand, the second dual action ˆ σ̂ of σ satisfies ˆ σ̂_{k} = id on M ⋊_{σ} K and ˆ σ̂_{k}(v_{p}) = \overline{⟨k, p⟩}v_{p} for p ∈ K̂.

The Takesaki duality theorem says that ˜ Ñ ⊂ ˜ M̃ is isomorphic to N ⊗ B(l^{2}(K)) ⊂ M ⊗ B(l^{2}(K)) via an isomorphism Ψ satisfying the following.

(1) (Ψ(πσ̂ ∘ π_{σ}(a))ξ)(k) = σ_{k}^{−1}(a)ξ(k),
(2) (Ψ(πσ̂(w_{l}))ξ)(k) = ξ(l^{−1}k),
(3) (Ψ(v_{p})ξ)(k) = \overline{⟨k, p⟩}ξ(k),

where π_{σ} is the embedding of M into M ⋊_{σ} K, and πσ̂ is the embedding of M ⋊_{σ} K into M ⋊_{σ} K ⋊σ̂ K̂.

Define a unitary c_{g} ∈ N ⊗ B(l^{2}(K)) by (c_{g}ξ)(k) := u_{α_{g},σ_{k}^{−1}}^{∗}ξ(k). Since c_{g} commutes with elements in N′ ⊗ C1, c_{g} is indeed in N ⊗ B(l^{2}(K)). Moreover since we have

(c_{g}α_{g} ⊗ id(c_{h})ξ)(k) = u_{α_{g},σ_{k}^{−1}}^{∗}α_{g}(u_{α_{h},σ_{k}^{−1}}^{∗})ξ(k)
= u_{α_{gh},σ_{k}^{−1}}^{∗}ξ(k)
= (c_{gh}ξ)(k),

c_{g} is an α ⊗ id cocycle. Then as in the argument in [23, Section 5], it is shown that Ψ ∘ ˜ α̃_{g} ∘ Ψ^{−1} = Ad c_{g}(α_{g} ⊗ id) holds.

On the other hand we have Ψ ∘ ˆ σ̂_{k} ∘ Ψ^{−1} = σ_{k} ⊗ Ad ρ_{k}^{−1}, where ρ is the left regular representation of K.

Here we consider the Connes-Radon-Nikodym type cocycle for Ad c_{g}α_{g} ⊗ id and σ_{k} ⊗ Ad ρ_{k}^{−1}. Take 0 ≠ a ∈ M_{n} with σ_{k}(x)a = ax for every x ∈ M. By [26, Theorem 3.1], α_{g}(a) = u_{α_{g},σ_{k}}a holds. It is obvious that σ_{k} ⊗ Ad ρ_{k}^{−1}(x)(a ⊗ ρ_{k}^{−1}) = (a ⊗ ρ_{k}^{−1})x holds for every M ⊗ B(l^{2}(K)). Here we have the following.

(Ad c_{g}(α_{g} ⊗ id)(a ⊗ ρ_{k}^{−1})ξ)(l) = (c_{g}(α_{g}(a) ⊗ ρ_{k}^{−1})c_{g}^{∗}ξ)(l)
= u_{α_{g},σ_{l}^{−1}}^{∗}α_{g}(a)(c_{g}^{∗}ξ)(kl)
= u_{α_{g},σ_{l}^{−1}}^{∗}u_{α_{g},σ_{k}}au_{α_{g},σ_{kl}^{−1}}ξ(kl)
= u_{α_{g},σ_{l}^{−1}}^{∗}u_{α_{g},σ_{k}}σ_{k}(u_{α_{g},σ_{kl}^{−1}})aξ(kl)
= u_{α_{g},σ_{l}^{−1}}^{∗}u_{α_{g},σ_{l}^{−1}}aξ(kl)
= (a ⊗ ρ_{k}^{−1}ξ)(l).

By [26, Theorem 3.1], the above equality implies u_{Ad c_{g}(α⊗id),σ_{k}⊗Ad ρ_{k}^{−1}} = 1 for every g ∈ G and k ∈ K. Hence by replacing α and β if necessary, we may assume that u_{α_{g},σ_{k}} = 1 and u_{β_{g},σ_{k}} = 1 hold for every g ∈ G and k ∈ K. This especially implies α_{g}σ_{k} = σ_{k}α_{g} and β_{g}σ_{k} = σ_{k}β_{g}.

Define two new actions ᾱ and β̄ by ᾱ_{g} := α_{g}σ_{ν(g)}^{−1} and β̄_{g} := β_{g}σ_{ν(g)}^{−1}. Since α and β commute with σ, ᾱ and β̄ are indeed actions of G. By the construction of ᾱ and β̄, it is easy to see H_{α} = ᾱ^{−1}(Cnt (M, N)) = ᾱ^{−1}(Int (M, N)) = β̄^{−1}(Cnt (M, N)) = β̄^{−1}(Int (M, N)).

Next we compute Λ(ᾱ). For m ∈ H, take v_{m} ∈ U(N) with α_{m} = Ad v_{m}σ_{ν(m)}. In this case, we have Ad ᾱ_{m} = Ad v_{m} for m ∈ H. Moreover since

1 = u_{α_{h},σ_{k}}
= u_{Ad v_{m}σ_{ν(m)},σ_{k}}
= Ad v_{m}(u_{σ_{ν(m)},σ_{k}})u_{Ad v_{m},σ_{k}}
= \overline{κ_{a}(k, ν(m))}v_{m}σ_{k}(v_{m}^{∗})

holds, we have σ_{k}(v_{m}) = \overline{κ_{a}(k, ν(m))}v_{m}.

First we compute λᾱ. Since we have u_{α_{g},σ_{k}} = 1, α_{g}(v_{g^{−1}ng}) = λ_{α}(g, n)v_{n} holds by the definition of λ_{α}. Then we get

ᾱ_{g}(v_{g^{−1}ng}) = α_{g}σ_{ν(g)}^{−1}(v_{g^{−1}ng})
= \overline{κ_{a}(ν(g)^{−1}, ν(n))}α_{g}(v_{g^{−1}ng})
= κ_{a}(ν(g), ν(n))λ_{α}(g, n)v_{n},

and λᾱ(g, n) = κ_{a}(ν(g), ν(n))λ_{α}(g, n) holds. Next we compute μᾱ. By the definition of μ_{α}, we have v_{m}σ_{ν(m)}(v_{n}) = μ_{α}(m, n)v_{mn}, m, n ∈ H. Hence we get v_{m}v_{n} = κ_{a}(ν(m), ν(n))μ_{α}(m, n)v_{mn} and consequently μᾱ(m, n) = κ_{a}(ν(m), ν(n))μ_{α}(m, n).

Similar computation is valid for β̄, and by the assumption Λ(α) = Λ(β), we get Λ(ᾱ) = Λ(β̄). Hence ᾱ and β̄ are cocycle conjugate by [26, Theorem 5.1]. Then ᾱ and β̄ are stably conjugate, and hence there exists an automorphism θ ∈ Aut (M ⊗B(l^{2}(G)), N ⊗B(l^{2}(G))) with θ ∘ (β̄_{g} ⊗ Ad ϱ_{g}) ∘ θ^{−1} = ᾱ_{g}, where ϱ is the right regular representation of G.

To prove the main theorem, we need the following proposition where N ⊂ M can be an arbitrary subfactor of finite index.

Proposition 2.5 Let σ be a non-strongly outer automorphism. Take 0 ≠ a ∈ M_{n} such that σ(x)a = ax holds for every x ∈ M. Then v ∈ M is in M^{σ} if and only if va = av holds.

Proof. First assume that v ∈ M^{σ}. Then we have va = σ(v)a = av. Conversely assume that va = av holds. Then σ(v)a = av = va holds. Hence we have σ(v)aa^{∗} = vaa^{∗}. Here aa^{∗} is in M′ ∩ M_{n}. Let E be the minimal conditional expectation from M_{n} onto M. Then we get σ(v)E(aa^{∗}) = E(σ(v)aa^{∗}) = E(vaa^{∗}) = vE(aa^{∗}), and E(aa^{∗}) ∈ M ∩ M′ = C. Since a is not zero, E(aa^{∗}) is a non-zero scalar. Hence v is in M^{σ}. □ Remark. The above proposition can be regarded as a subfactor-analogue of the characterization of the centralizer of type III factors. Namely let M be a type III factor, ϕ a faithful normal state of M. Then a ∈ M is in M_{ϕ} if and only if [ϕ, a] = 0.

We continue the proof of Theorem 2.4. Since an outer action of a finite group is stable, we can find a unitary w ∈ N ⊗ B(l^{2}(G)) such that w^{∗}σ_{k} ⊗ id(w) = u_{θ,σ_{k}⊗id}. Hence θ ∘ σ_{k} ⊗ id ∘ θ^{−1} = Ad u_{θ,σ_{k}⊗id} ∘ σ_{k} ⊗ id = Ad w^{∗} ∘ σ_{k} ⊗ id ∘ Ad w holds. If we can prove that wᾱ_{g} ⊗Ad ϱ_{g}(w^{∗}) is in (M ⊗B(l^{2}(G)))^{K}, then wᾱ_{g} ⊗Ad ϱ_{g}(w^{∗}) is an α⊗Ad ϱ = ᾱσ ⊗Ad ϱ cocycle and

ᾱ_{g} ⊗ Ad ϱ_{g} ∘ θ ∘ σ_{ν(g)} ⊗ id ∘ θ^{−1} = ᾱ_{g} ⊗ Ad ϱ_{g} ∘ Ad w^{∗} ∘ σ_{ν(g)} ⊗ id ∘ Ad w
= Ad (ᾱ ⊗ Ad ϱ_{g}(w^{∗}))ᾱ_{g}σ_{ν(g)} ⊗ Ad ϱ_{g} ∘ Ad w
= Ad w^{∗} ∘ Ad (wᾱ_{g} ⊗ Ad ϱ_{g}(w^{∗}))α_{g} ⊗ Ad ϱ_{g} ∘ Ad w

holds and we have the following.

α ⊗ Ad ϱ = ᾱ ⊗ Ad ϱ ∘ σ ⊗ id
∼ ᾱ ⊗ Ad ϱ ∘ θ ∘ σ ⊗ id ∘ θ^{−1}
= θ ∘ β̄ ⊗ Ad ϱ ∘ θ^{−1}θ ∘ σ ⊗ id ∘ θ^{−1}
= θ ∘ β̄σ ⊗ Ad ϱ ∘ θ
= θ ∘ β ⊗ Ad ϱ ∘ θ^{−1}.

Hence α and β are stably conjugate. So we only have to prove that wᾱ_{g} ⊗ Ad ϱ_{g}(w^{∗}) is in (M ⊗ B(l^{2}(G)))^{K}.

It is easy to see u_{α_{g}⊗Ad ϱ_{g},σ_{k}⊗id} = 1, so that u_{ᾱ_{g}⊗Ad ϱ_{g},σ_{k}⊗id} = u_{α_{g}σ_{ν(g)}^{−1}⊗Ad ϱ_{g},σ_{k}⊗id} = κ_{a}(k, ν(g)) holds. In the same way, we can see u_{β̄_{g}⊗Ad ϱ_{g},σ_{k}⊗id} = κ_{a}(k, ν(g)). Hence

u_{ᾱ_{g}⊗Ad ϱ_{g},Ad w^{∗}∘σ_{k}⊗id∘Ad w} = u_{θ∘β̄_{g}⊗Ad ϱ_{g}∘θ^{−1},θ∘σ_{k}⊗id∘θ^{−1}}
= θ(u_{β̄_{g}⊗Ad ϱ_{g},σ_{k}⊗id})
= κ_{a}(k, ν(g))

holds. Take 0 ≠ a ∈ M_{n} ⊗ B(l^{2}(G)) such that σ_{k} ⊗ id(x)a = ax holds for every x ∈ M ⊗ B(l^{2}(G)). Then ᾱ_{g} ⊗ Ad ϱ_{g}(a) = κ_{a}(k, ν(g))a holds by [26, Theorem 3.1]. Since Ad w^{∗}∘σ_{k}⊗id∘Ad w(x)w^{∗}aw = w^{∗}awx, we also have ᾱ_{g}⊗Ad ϱ_{g}(w^{∗}aw) = κ_{a}(k, ν(g))w^{∗}aw. From these two equalities, we get ᾱ_{g} ⊗ Ad ϱ_{g}(w^{∗})aᾱ_{g} ⊗ Ad ϱ_{g}(w) = w^{∗}aw. Hence wᾱ_{g} ⊗ Ad ϱ_{g}(w^{∗}) satisfies the condition in Proposition 2.5, and wᾱ_{g} ⊗ Ad ϱ_{g}(w^{∗}) is in the fixed point algebra (M ⊗ B(l^{2}(G)))^{K}. □

Corollary 2.6 If G is a finite group in Theorem 2.4, then α and β are cocycle conjugate if and only if H_{α} = H_{β}, Λ(α) = Λ(β) and ν_{α} = ν_{β} hold.

Proof. On one hand, α ⊗ Ad ϱ and β ⊗ Ad ϱ are conjugate by Theorem 2.4. On the other hand, in the same way as in the proof of [16, Lemma 6.5], we can prove that α is cocycle conjugate to α ⊗ m, where m is an outer action of G/H on the injective type II_{1} factor R_{0} and we regard m as an action of G in the natural way. Hence α and β are cocycle conjugate since m and m ⊗ Ad ϱ are cocycle conjugate. □

In the rest of this section, we treat examples which satisfy the assumption in Theorem 2.4. The first example is taken from [20, Theorem 4.1].

Example 2.7 We consider the case G = Z. Take α ∈ Aut (M, N). Let p be the strongly outer period of α. Set σ := α^{p}. Then ν_{α} is given by ν_{α}(pm) = [σ^{m}]. Let n be the outer period of σ. Here assume (p, n) = 1. Then we can find k, l ∈ Z such that pk + nl = 1. Set ν(g) := [σ^{gk}]. Then we have ν(p) = [σ^{pk}] = [σ^{−nl+1}] = [σ], and hence ν_{α} can be extended to a homomorphism from Z.

Example 2.8 Assume that G is of the form G = H_{α} ⋊ K. For (h, k) ∈ G = H_{α} ⋊ K, define ν(h, k) := ν_{α}(h). Then by using the fact ν_{α}(knk^{−1}) = ν_{α}(n), we get

ν((h_{1}, k_{1})(h_{2}, k_{2})) = ν(h_{1}k_{1}h_{2}k_{1}^{−1}, k_{1}k_{2})
= ν_{α}(h_{1}k_{1}h_{2}k_{1}^{−1})
= ν_{α}(h_{1})ν_{α}(k_{1}h_{2}k_{1}^{−1})
= ν(h_{1}, k_{1})ν(h_{2}, k_{2}).

Hence we can extend ν_{α} to a homomorphism from G, and we can apply the main theorem.

A On liftings of the relative χ group

In [26] and this paper, we fixed a lifting of χ_{a}(M, N) to Aut (M, N) for the definition of the characteristic invariants and classification of group actions on subfactors. However, in general, a different choice of a lifting produces a 2-cocycle, and this 2-cocycle may change the characteristic invariants of actions. Hence the definition of characteristic invariants depends on the choice of a lifting. In this appendix, we show that we can choose a unique lifting up to cocycle perturbation by using the algebraic κ invariant.

Take a lifting σ. Then the algebraic κ invariant κ_{a}(h, k) is defined as κ_{a}(h, k) := u_{σ_{k},σ_{h}}^{∗}. To specify σ, we denote this κ_{a} by κ_{a}^{σ}(h, k).

Fix σ and take another lifting σ̃. Then we can find a unitary u_{h} ∈ U(N) with Ad u_{h}σ_{h} = σ̃_{h}. Since σ̃ is a lifting, there exists a 2-cocycle μ(h, k) ∈ Z^{2}(χ_{a}(M, N), T) with u_{h}σ_{h}(u_{k}) = μ(h, k)u_{hk}. To compare κ_{a}^{σ} and κσ̃ ; a, we compute κσ̃ ; a. Then

\overline{κσ̃ ; a(h, k)} = u_{σ̃_{k},σ̃_{h}}
= u_{Ad u_{k}σ_{k},Ad u_{h}σ_{h}}
= u_{Ad u_{k}σ_{k},Ad u_{h}}Ad u_{h}(u_{Ad u_{k}σ_{k},σ_{h}})
= u_{k}σ_{k}(u_{h})u_{k}^{∗}u_{h}^{∗}Ad u_{h}(Ad u_{k}(u_{σ_{k},σ_{h}})u_{Ad u_{k},σ_{h}})
= u_{k}σ_{k}(u_{h})u_{k}^{∗}u_{σ_{k},σ_{h}}u_{k}σ_{h}(u_{k}^{∗})u_{h}^{∗}
= \overline{κ_{a}^{σ}(h, k)}\overline{μ(h, k)}μ(k, h)

holds, so we get κσ̃ ; a(h, k) = κ_{a}^{σ}(h, k)μ(h, k)\overline{μ(k, h)}.

Therefore if κσ̃ ; a = κ_{a}^{σ}, then we must have μ(h, k) = μ(k, h). By [30, Proposition 3.2], μ is a coboundary, so we can choose u_{h} as a σ-cocycle. Hence we have shown the following proposition.

Proposition A.1 Let σ and σ̃ be liftings of χ_{a}(M, N) to Aut (M, N). If κ_{a}^{σ} = κσ̃ ; a holds, then σ̃ is a cocycle perturbation of σ.

By Proposition A.1, we can find a unique lifting σ up to cocycle perturbation once we fix the algebraic κ invariant.

In the next proposition, we do not assume Ker Φ = Aut (M, N). Every θ ∈ Aut (M, N) induces an automorphism χ_{a}(θ) of χ_{a}(M, N) by χ_{a}(θ)([σ]) := [θ ∘ σ ∘ θ^{−1}].

Proposition A.2 Let σ be a lifting of χ_{a}(M, N) to Aut (M, N). Assume that κ_{a}^{σ}(h, k) = κ_{a}^{σ}(χ_{a}(θ)(h), χ_{a}(θ)(k)) holds for every θ ∈ Aut (M, N). Then there exists a σ_{χ_{a}(θ)(·)}-cocycle w_{h} such that θ ∘ σ_{h} ∘ θ^{−1} = Ad w_{h}σ_{χ_{a}^{σ}(θ)(h)} holds.

Proof. Take a unitary w_{h} with Ad w_{h}σ_{χ_{a}(θ)(h)} = θ ∘σ_{h} ∘θ^{−1}. Then there exists a 2-cocycle μ(h, k) satisfying w_{h}σ_{χ_{a}(θ)(h)}(w_{k}) = μ(h, k)w_{hk}. On one hand, we have u_{θ∘σ_{h}∘θ^{−1},θ∘σ_{k}∘θ^{−1}}^{∗} = θ(u_{σ_{h},σ_{k}})^{∗} = κ_{a}^{σ}(k, h). On the other hand, we have

u_{θ∘σ_{h}∘θ^{−1},θ∘σ_{k}∘θ^{−1}}^{∗} = u_{Ad w_{h}σ_{χa(θ)(h)},Ad w_{k}σ_{χa(θ)(k)}}^{∗}
= κ_{a}^{σ}(χ_{a}(θ)(k), χ_{a}(θ)(h))μ(k, h)\overline{μ(h, k)}.

By the assumption on κ_{a}^{σ}, we can choose w_{h} as a cocycle as in the proof of Proposition A.1. □

The assumption on κ_{a} in the above proposition is satisfied when either κ_{a} is trivial or χ_{a}(M, N) is cyclic. The former is trivial so that we will see the latter. It is easy to see κ_{a}^{σ}(h, h) = κ_{a}^{σ}(χ_{a}(θ)(h), χ_{a}(θ)(h)) holds from the above computation. If χ_{a}(M, N) is cyclic and g is a generator of χ_{a}(M, N), then

κ_{a}^{σ}(g^{m}, g^{n}) = κ_{a}^{σ}(g, g)^{mn}
= κ_{a}^{σ}(χ_{a}(θ)(g), χ_{a}(θ)(g))^{mn}
= κ_{a}^{σ}(χ_{a}(θ)(g^{m}), χ_{a}(θ)(g^{n}))

holds.

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