On non-strongly free automorphisms of III_{0} subfactors

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

Abstract

We determine when an automorphism of a subfactor of type III_{0} with finite index is non-strongly free in the sense of C. Winsløw in the term of modular endomorphisms introduced by M. Izumi.

1 Introduction

In subfactor theory initiated by V. F. R. Jones in [13], the study of automorphisms of subfactors plays an important role. One of the important property is strong outerness for automorphisms introduced by Choda-Kosaki in [1]. (Popa also introduced the same property in [27] independently, and he calls it proper outerness.)

In [29], C. Winsløw introduced the notion of strong freeness for automorphisms of subfactors of type III as a natural generalization of strong outerness and non-pointwise innerness in the sense of [5]. This notion plays an important role in his theory of automorphisms for subfactor of type III.

In [19], H. Kosaki investigated the structure of automorphisms of type III subfactors by using sectors, and characterized non-strongly free automorphisms for subfactors of type III_{λ}, λ ≠ 0. Namely an automorphism is non-strongly free if and only if it is the composition of a non-strongly outer automorphism and a modular automorphism. (See Theorem 3.1 below.) However type III_{0} case was left unsolved. Note that the composition of a non-strongly outer automorphism and an extended modular automorphism is always non-strongly free as shown by Kosaki.

The purpose of this paper is to give a correct answer to this problem. Though one may conjecture that modular automorphisms should be replaced with extended modular automorphisms in Kosaki’s theorem, we must use modular endomorphisms introduced by M. Izumi in [10] to obtain the correct characterization. (See [11] for details of modular endomorphisms.) In general, there exist subfactors and their non-strongly free automorphisms which are not the composition of non-strongly outer automorphisms and extended modular automorphisms.

Acknowledgements. The author is grateful to Professor Izumi and Professor Kosaki for helpful comments on this work. The author is supported by Grant-in-Aid for Scientific Research, Ministry of Education (Japan).

2 Preliminaries

For basic of index theory and sector theory for subfactors, we refer to [13], [16], [7], [26], [20], [25], [23], [24], [9], [3].

Let E be the minimal conditional expectation from M onto N, φ a faithful normal semifinite weight of N, Ñ ⊂ M̃ := N ⋊_{σ^{φ}} R ⊂ M ⋊_{σ^{φ∘E}} R the continuous decomposition of N ⊂ M. Throughout this paper, we always assume that type III subfactors have the common flow of weights, that is, Z(Ñ) = Z(M̃).

First we recall the notion of strong outerness for automorphisms of subfactors.

Definition 2.1 ([1, Definition 1], [27, Definition 1.3.1]) An automorphism α ∈ Aut (M, N) is said to be strongly outer if there exist no non-zero a ∈ ⋃_{k} M_{k} such that α(x)a = ax holds for every x ∈ M.

Let α be an automorphism of N ⊂ M. Then we can define the canonical extension α̃ ∈ Aut (M̃, Ñ) of α as follows:

α̃(x) = α(x), x ∈ M,
α̃(λ(t)) = (Dφ ∘ α^{−1} : Dφ)_{t}λ(t),

where λ(t) is the implementing unitary of σ_{t}^{φ} (see [5]).

Definition 2.2 ([29, Definition 3.2]) In the above notations, α is said to be strongly free if there exist no non-zero a ∈ ⋃_{k} M̃_{k} satisfying α̃(x)a = ax for every x ∈ M̃.

By the above definition, α is non-strongly free if and only if there exist k > 0 and non-zero element a ∈ M̃_{k} which satisfies α̃(x)a = ax for every x ∈ M̃.

When N = M, α is non-strongly free if and only if α̃ is inner. In this case, by [6, Proposition 5.4], α is of the form Ad uσ_{c}^{φ} for some u ∈ U(M) and extended modular automorphism σ_{c}^{φ} (For extended modular automorphisms, see [2].)

Next we explain the canonical extension of endomorphisms introduced by Izumi in [10], [11].

Let ρ ∈ End (M) be an endomorphism of M with dρ < ∞. We denote by ϕ_{ρ} the standard left inverse of ρ. Then the canonical extension ρ̃ ∈ End (M̃) of ρ is defined as follows:

ρ̃(x) = ρ(x), x ∈ M,
ρ̃(λ(t)) = dρ^{it}(Dφ ∘ ϕ_{ρ} : Dφ)_{t}λ(t).

It is shown in [11] that the canonical extension is compatible with sector operations.

Definition 2.3 ([11, Definition 3.1]) An endomorphism ρ is called a modular endomorphism if ρ̃ is an inner endomorphism, that is, there exist isometries {v_{i}}_{i=1}^{n} satisfying ρ̃(x) = ∑_{i} v_{i}xv_{i}^{∗}. We denote by End (M)_{m} and Sect (M)_{m} the set of all modular endomorphisms and the image of End (M)_{m} in Sect (M) by the quotient map respectively.

In the above definition, the number n of isometries {v_{i}} is dρ. Hence the dimension of a modular endomorphism is always an integer.

Take ρ ∈ End (M)_{m}. Then we can find isometries {v_{i}}_{i=1}^{n} ⊂ M̃ with ρ̃(x) = ∑_{i} v_{i}xv_{i}^{∗}. Define c_{ij,t} := v_{i}^{∗}θ_{t}(v_{j}). Then c = (c_{ij,t}) ∈ Z_{θ}^{1}(R, U(n, Z(M̃))). If ρ_{1} and ρ_{2} are equivalent modular endomorphisms, then the associated cocycles are equivalent. Hence the map δ[ρ] := [c] ∈ H_{θ}^{1}(R, U(dρ, Z(M̃))) is well defined.

Theorem 2.4 ([11, Theorem 3.3]) The map δ gives the bijective correspondence between Sect (M)_{m} and ⨆_{n} H_{θ}^{1}(R, U(n, Z(M̃))).

We denote by ρ_{c} a modular endomorphism corresponding to a cocycle c.

3 Non-strongly free automorphisms

In [19], Kosaki obtained the following theorem, which can be considered as a subfactor analogue of [6, Proposition 5.4].

Theorem 3.1 ([19, Theorem 19]) Let N ⊂ M be a subfactor of type III_{λ}, λ ≠ 0. Then every non-strongly free automorphism is of the form ασ_{t}^{φ}, where α is a non-strongly outer automorphism.

Let γ be Longo’s canonical endomorphism of N ⊂ M. Since α ∈ Aut (M, N) is nonstrongly outer if and only if α appears in the irreducible decomposition of γ^{n}, n ≥ 1 (see [1], [17] and [19]), we can list all non-strongly free automorphisms of subfactors of type III_{λ}, 0 < λ ≤ 1, as long as we know the irreducible decomposition of γ^{n}. For example see [19, Section 6].

In the case of type III_{0} factors, as an analogue of the above theorem, one may consider that every non-strongly free automorphism is of the form ασ_{c}^{φ}, where α is a non-strongly outer automorphism and σ_{c}^{φ} is an extended modular automorphism. However this is not true in general, and we will characterize non-strongly free automorphisms by using modular endomorphisms.

Theorem 3.2 An automorphism α is non-strongly free if and only if there exists a modular endomorphism ρ_{c} such that αρ_{c} appears as an irreducible component of γ^{n} for some n.

Proof. Let (X, ν, F) be the flow of weights of M. Hence we have Z(M̃) = L^{∞}(X, ν) and θ_{t}(f)(w) = f(F_{−t}w). Then we have the common central decomposition Ñ ⊂ M̃ = ∫_{X}^{⊕}(Ñ(w) ⊂ M̃(w))dν(w).

Let α be a non-strongly free automorphism. Then there exists 0 ≠ a ∈ M̃_{k} such that α̃(x)a = ax holds for every x ∈ M̃. The argument in [19, pp.436] shows that the Connes-Takesaki module of α is trivial. Hence the above equality can be decomposed as ∫_{X}^{⊕} α_{w}(x(w))a(w)dν(w) = ∫_{X}^{⊕} a(w)x(w)dν(w) with α_{w} ∈ Aut (M(w)). By this decomposition, α_{w}(x(w))a(w) = a(w)x(w) holds for a.e. w ∈ X. Since H_{w} := {a ∈ M̃_{k}(w) | α_{w}(x)a = ax for every x ∈ M̃(w)} is a finite dimensional Hilbert space, we can take orthonormal basis {a_{j}(w)} for this Hilbert space. Note that n := dim H_{w} is constant for a.e. w ∈ X because of the ergodicity of θ_{t}.

Since θ_{t} commutes with α̃, we have α_{F_{t}w}θ_{(t,w)} = θ_{(t,w)}α_{w}. Hence θ_{(t,x)} is a unitary operator from H_{w} to H_{F_{t}w}. Then we get θ_{(t,w)}(a_{j}(w)) = ∑_{i} c_{ij}(t, w)a_{i}(F_{t}w) for some unitary matrix (c_{ij}(t, w))_{i,j}. This (c_{ij}(t, w))_{ij} satisfies the following cocycle equations.

c_{ij}(t + s, w) = ∑_{k} c_{ik}(t, F_{s}w)c_{kj}(s, w).

Define c_{t} = (c_{ij,t}) ∈ M(n; Z(M̃)) by c_{ij,t}(w) := c_{ij}(t, F_{−t}w). Then c_{t}θ_{t}(c_{s}) = c_{t+s} holds. Indeed we have the following.

∑_{k} c_{ik,t}(F_{t+s}w)θ_{t}(c_{kj,s})(F_{t+s}w) = ∑_{k} c_{ik,t}(F_{t+s}w)c_{kj,s}(F_{s}w)
= ∑_{k} c_{ik}(t, F_{s}w)c_{kj}(s, w)
= c_{ij}(t + s, w)
= c_{ij,t+s}(F_{t+s}w).

Thus we have c ∈ Z_{θ}^{1}(R, U(n, Z(Ñ))). By [10, Theorem 3.3], we have the modular endomorphism ρc̄ ∈ End (M) associate with c̄_{t}, where c̄_{ij,t} := c_{ij,t}^{∗}. Let {w_{i}}_{i=1}^{n} ⊂ Ñ be an implementing system of ρc̄. Then we have ρ̃c̄(x) = ∑_{i} w_{i}xw_{i}^{∗} and w_{i}^{∗}θ_{t}(w_{j}) = c_{ij,t}^{∗}. Set a_{j} := ∫_{X}^{⊕} a_{j}(w)dν(w). Then θ_{t}(a_{j}) = ∑_{i} c_{ij,t}a_{i} holds. It is easy to show that θ_{t}(∑_{j} w_{j}a_{j}) = ∑_{i} w_{i}a_{i}, and this means ∑_{i} w_{i}a_{i} ∈ (M̃_{k})^{θ} = M_{k}. Also we have ρc̄α(x) ∑_{i} w_{i}a_{i} = ∑_{i} w_{i}a_{i}x for every x ∈ M. In a similar way as in the proof of [1], [17] or [19], we can prove that ρc̄α appears in the irreducible decomposition of γ^{k}. □

When N ⊂ M is of type III_{λ}, λ ≠ 0, every irreducible modular endomorphism is an automorphism. Hence Theorem 3.2 covers Kosaki’s theorem.

Corollary 3.3 If either we have [M : N] < 4 or the type II graph and type III graph of N ⊂ M coincide, then every non-strongly free automorphism is the composition of a non-strongly outer automorphism and an extended modular automorphism.

Proof. Let α be a non-strongly free automorphism, Δ the set of irreducible sectors appearing in the irreducible decomposition of γ^{n}, n > 0, and Δ_{m} := Δ ∩ Sect (M)_{m}. First assume that N ⊂ M has the same type II graph and type III graph. In this case, Δ_{m} is [id_{M}] by [11, Theorem 3.5]. By Theorem 3.2, [αρ_{c}] is in Δ for some modular endomorphism ρ_{c}. Since Δ is closed under sector operations, [ρ_{c}ρc̄] = [αρ_{c}]\overline{[αρ_{c}]} can be decomposed in Δ. If dρ_{c} ≥ 2, then nontrivial modular endomorphisms appear in Δ, and this is the contradiction. Hence ρ_{c} must be an extended modular automorphism, and α can be expressed in the form βσ_{c}^{φ} for some non-strongly outer automorphism β.

Next we assume [M : N] < 4. We only have to verify the case that the type III graph of N ⊂ M is A_{4n−3} and type II graph is D_{2n}. (See [18], or [21].) In this case, we have Δ_{m} = {[id_{M}], [σ]}, where σ is an extended modular automorphism with period two. Even in this case, it is impossible that [αρ_{c}] with dρ_{c} ≥ 2 appears in Δ. □

By using Theorem 3.2, we can construct an example of a subfactor and a non-strongly free automorphism which does not have the form ασ_{c}^{φ} with a non-strongly outer automorphism α. Let M be a type III_{0} factor, (X, F_{t}) the flow of weights of M, c a minimal cocycle from (X, F_{t}) to a noncommutative group G of order 8 in the sense of [31]. Let π be a 2-dimensional irreducible representation of G. Let ρ be a modular endomorphism with dρ corresponding to a cocycle π ∘ c. (See [11, Section 5].) The fusion rules of sectors generated by [ρ] is the same as that of Ĝ. Hence we have [ρ^{2}] = [id] ⊕ [σ_{1}] ⊕ [σ_{2}] ⊕ [σ_{1}σ_{2}], [σ_{1}ρ] = [σ_{2}ρ] = [ρ], [σ_{1}^{2}] = [σ_{2}^{2}] = [id] and [σ_{1}σ_{2}] = [σ_{2}σ_{1}], where σ_{i} are extended modular automorphisms. Take α ∈ Ker mod which is not an extended modular automorphism, and define ρ_{1} ∈ End (M) as [ρ_{1}] = [αρ]⊕[id]. Then ρ_{1}(M) ⊂ M is a subfactor of type III_{0} with the common flow of weights. In this case, α is indeed an automorphism of ρ_{1}(M) ⊂ M up to inner perturbation because of [αρ_{1}] = [ρ_{1}α]. Easy computation shows that the set of irreducible sectors appearing in (ρ_{1}ρ̄_{1})^{n} is {[id], [αρ], [σ_{1}], [σ_{2}], [σ_{1}σ_{2}]}. By Theorem 3.2, α is non-strongly free, but never has the form βσ_{c} for some non-strongly outer automorphism β and extended modular automorphism σ_{c} because the only non-strongly outer automorphisms of ρ_{1}(M) ⊂ M are {Ad u, Ad uσ_{1}, Ad uσ_{2}, Ad uσ_{1}σ_{2} | u ∈ U(M)} by the characterization of [1] and [17].

4 The case of subfactors with the principal graph D_{2n}^{(1)}

In the end of the previous section, we construct an example of a subfactor and its nonstrongly free automorphism which does not have the form ασ_{c}^{φ} with a non-strongly outer automorphism α. However this subfactor is reducible. Hence it is natural to ask if there exist examples of irreducible subfactors. In this section, we construct subfactors with the principal graph D_{2n}^{(1)} which have the above property.

First we construct subfactors. In [4, Section 4.7], subfactors with the principal graph D_{n}^{(1)} has been obtained as A^{D_{n−2}} ⊂ (A ⊗ M_{2})^{D_{n−2}}, where A is a factor with an outer action of a dihedral group D_{n−2}. (Also see [12].)

Let a, b be generators of D_{2n} with the relations a^{2n} = b^{2} = e and bab = a^{−1}.

Fix a type III_{0} factor A, and let σ^{(0)} be an outer action of D_{2n} on A with the trivial modular invariant.

Let Rep (D_{2n}) be the category of finite dimensional representations of D_{2n}. Since σ is a dual action for some Roberts action of Rep (D_{2n}) on A^{σ}, we can take a system of isometries {v_{i}^{π}}_{i=1}^{dπ} ⊂ A with v_{i}^{π}^{∗}v_{j}^{π} = δ_{i,j} and ∑_{i} v_{i}^{π}v_{i}^{π}^{∗} = 1, and restriction of σ_{π} on ∑_{i} Cv_{i}^{π} is equivalent to π for each π ∈ Rep (D_{2n}). Then the dual action of Rep (D_{2n}) on M is given as follows,

ρ_{π}(x) = ∑_{i} v_{i}^{π}xv_{i}^{π}^{∗}, x ∈ A ⊗ M_{2}(C),

ρ_{π}(λ_{g}) = λ_{g}.

Let χ be a one-dimensional representation of D_{2n} given by χ(a) = 1 and χ(b) = −1 and χ_{1} a one dimensional representation given by χ_{1}(a) = −1, χ_{1}(b) = 1, and setχ_{2} = χχ_{1}. Let π_{ω} be a two-dimensional representation of D_{2n} given as follows.

π_{ω^{m}}(a) = ( \begin{matrix}ω^{m} & 0 \\ 0 & ω̄^{m} \end{matrix} ) , π_{ω^{m}}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,

where ω is a 2n-th primitive root of 1. If m ≠ 0, n, then ω^{m} ≠ ±1 and π_{ω^{m}} is irreducible, and we have π_{1} ≅ id ⊕ χ and π_{−1} ≅ χ_{1} ⊕ χ_{2}. Then {π_{ω}^{m}}_{m=1}^{n−1} ∪ {id, χ, χ_{1}, χ_{2}} is the set of representative of Rep (D_{2n}).

Let φ^{0} be a dominant weight of A^{σ^{0}}, and set φ := φ^{0} ∘ E, where E is the natural conditional expectation from A onto A^{σ^{0}}. We may assume that (φ, ρ_{π}) is an invariant pair in the sense of [11, Definition 2.2] for every π ∈ D̂_{2n}. Then it follows that φ is invariant under σ^{0}, and v_{i}^{π} is in A_{φ}. The former is trivial, and the latter follows from [8, Theorem 4.19].

Define an action σ of D_{2n} on A ⊗ M_{2}(C) as in [12, Theorem 5.5]. Then a subfactor N ⊂ M := A ⋊_{σ} D_{2n} ⊂ (A ⊗ M_{2}(C)) ⋊_{σ} D_{2n} has the principal graph D_{2n+2}^{(1)}. Moreover this subfactor has the common flow of weights and its flow is given by (Z(Ã)^{σ}, θ_{t}|_{Z(Ã)^{σ}}). See [15], [28]. The continuous decomposition Ñ ⊂ M̃ can be identify with Ã ⋊σ̃ D_{2n} ⊂ (Ã ⊗ M_{2}(C)) ⋊σ̃ D_{2n}. Here the action σ̃ of D_{2n} on Ã ⊗ M_{2} is given by the canonical extension of σ. Set H := Ker mod σ^{(0)}, and assume that H = {e, a^{n}}. Hence D_{2n}/H acts on Z(Ã) faithfully. The flow {Z(Ã), θ_{t}|_{Z(Ã)}} can be expressed as the skew product of Z(Ñ) and its minimal cocycle to D_{2n}/H. (See [31], or [11, Appendix A.].) Hence we may assume Z(Ã) = Z(Ñ) ⊗ l^{∞}(D_{2n}/H). We denote by ẋ an equivalence class of x ∈ D_{2n} in D_{2n}/H, and by δẋ the characteristic function of {ẋ} in l^{∞}(D_{2n}/H).

Now we will prove ρ̃_{π_{ω}}(x) = w_{1}α̃(x)w_{1}^{∗} + w_{2}α̃(x)w_{2}^{∗} for some α̃ ∈ Aut (M̃, Ñ) commuting with θ_{t}. If we can find such α̃, define α as the restriction of α̃ on M = M̃^{θ}. Since we have τM̃ ρ̃_{π_{ω}} = 2τM̃ by [11, Proposition 2.5], it is easy to see τα̃ = τ. By [14, Lemma 1.1], α̃ is the canonical extension of α. We will show that α is a non-strongly free automorphism. In what follows, we denote ρ_{π_{ω}} by ρ, and v_{i}^{π_{ω}} by v_{i} for simplicity. We have σ_{a}(v_{0}) = ωv_{0}, σ_{a}(v_{1}) = ω̄v_{1}, σ_{b}(v_{0}) = v_{1} and σ_{b}(v_{1}) = v_{0}. The canonical extension ρ̃ of ρ is given as follows:

ρ̃(x) = v_{0}xv_{0}^{∗} + v_{1}xv_{1}^{∗}, x ∈ Ã,
ρ̃(λ_{g}) = λ_{g}.

Lemma 4.1 Define w_{1} and w_{2} as follows.

w_{0} := v_{0} ∑_{i=0}^{n−1} ω^{i}δ_{}_{ȧ^{i}} + v_{1} ∑_{i=0}^{n−1} ω̄^{i}δ_{ȧ_{i}ḃ},

w_{1} := v_{1} ∑_{i=0}^{n−1} ω̄^{i}δ_{}_{ȧ^{i}} + v_{0} ∑_{i=0}^{n−1} ω^{i}δ_{ȧ_{i}ḃ}.

Then w_{0} and w_{1} are mutually orthogonal isometries with support 1.

Proof. First we compute w_{0}^{∗}w_{0}. Then

w_{0}^{∗}w_{0} = (∑_{i=0}^{n−1} v_{0}^{∗}ω̄^{i}δ_{}_{ȧ^{i}} + v_{1}^{∗}ω^{i}δ_{ȧ_{i}ḃ})(∑_{j=0}^{n−1} v_{0}ω^{j}δ_{}_{ȧ^{j}} + v_{1}ω̄^{j}δ_{ȧ_{j} ḃ})

= ∑_{i=0}^{n−1} δ_{}_{ȧ^{i}} + δ_{ȧ_{i}ḃ}
= 1

In a similar way, we can show w_{1}^{∗}w_{1} = 1 and w_{0}^{∗}w_{1} = 0.

Next we compute w_{0}w_{0}^{∗}

w_{0}w_{0}^{∗} = (∑_{j=0}^{n−1} v_{0}ω^{j}δ_{}_{ȧ^{j}} + v_{1}ω̄^{j}δ_{ȧ_{j} ḃ})(∑_{i=0}^{n−1} v_{0}^{∗}ω̄^{i}δ_{}_{ȧ^{i}} + v_{1}^{∗}ω^{i}δ_{ȧ_{i}ḃ})

= v_{0}v_{0}^{∗} ∑_{i=0}^{n−1} δ_{}_{ȧ^{i}} + v_{1}v_{1}^{∗} ∑_{i=0}^{n−1} δ_{ȧ_{i}ḃ}

In a similar way, we get w_{1}w_{1}^{∗} = v_{1}v_{1}^{∗} ∑_{i=0}^{n−1} δ_{}_{ȧ^{i}} + v_{0}v_{0}^{∗} ∑_{i=0}^{n−1} δ_{ȧ_{i}ḃ}. Now w_{0}w_{0}^{∗} + w_{1}w_{1}^{∗} = 1 is clear. □

Lemma 4.2 w_{i}w_{j}^{∗} ⊂ (ρ̃, ρ̃) and w_{1}^{∗}ρ̃(·)w_{1} = w_{2}^{∗}ρ̃(·)w_{2} =: β is an automorphism.

Proof. In the proof of the above lemma, we already know w_{0}w_{0}^{∗} = v_{0}v_{0}^{∗}e_{0} + v_{1}v_{1}^{∗}e_{1}, where e_{0} = ∑_{i=0}^{n−1} δ_{}_{ȧ^{i}} and e_{1} = ∑_{i=0}^{n−1} δ_{ȧ_{i}ḃ}. Since e_{0} and e_{1} are in Z(Ã) and ρ̃(x) = v_{0}xv_{0}^{∗} + v_{1}xv_{1}^{∗} for x ∈ Ã, it is shown that w_{0}w_{0}^{∗}ρ̃(x) = ρ̃(x)w_{0}w_{0}^{∗} holds for x ∈ Ã. We have Ad λ_{a}v_{i}v_{i}^{∗} = v_{i}v_{i}^{∗} and Ad λ_{a}e_{i} = e_{i}. Hence we get w_{0}w_{0}^{∗}ρ̃(λ_{a}) = ρ̃(λ_{a})w_{0}w_{0}^{∗}. Similarly w_{0}w_{0}^{∗}ρ̃(λ_{b}) = ρ̃(λ_{b})w_{0}w_{0}^{∗} holds. Hence w_{0}^{∗}ρ̃(·)w_{0} is an endomorphism of M̃. If x ∈ Ã, then

w_{0}^{∗}ρ̃(x)w_{0} = (∑_{i=0}^{n−1} v_{0}^{∗}ω̄^{i}δ_{}_{ȧ^{i}} + v_{1}^{∗}ω^{i}δ_{ȧ_{i}ḃ})(v_{0}xv_{0}^{∗} + v_{1}xv_{1}^{∗})(∑_{j=0}^{n−1} v_{0}ω^{j}δ_{}_{ȧ^{j}} + v_{1}ω̄^{j}δ_{ȧ_{j} ḃ})
= e_{0}x + e_{1}x
= x

holds.

On the other hand, we have

w_{0}^{∗}ρ̃(λ_{a})w_{0} = (∑_{i=0}^{n−1} v_{0}^{∗}ω̄^{i}δ_{}_{ȧ^{i}} + v_{1}^{∗}ω^{i}δ_{ȧ_{i}ḃ})λ_{a}(∑_{j=0}^{n−1} v_{0}ω^{j}δ_{}_{ȧ^{j}} + v_{1}ω̄^{j}δ_{ȧ_{j} ḃ})

= (∑_{i=0}^{n−1} v_{0}^{∗}ω̄^{i}δ_{}_{ȧ^{i}} + v_{1}^{∗}ω^{i}δ_{ȧ_{i}ḃ})(∑_{j=0}^{n−1} v_{0}ω^{j+1}δ_{}_{ȧ^{j+1}} + v_{1}ω̄^{j+1}δ_{ȧ_{j+1}ḃ})λ_{a}

= ((∑_{i=0}^{n−1} ω̄^{i}δ_{}_{ȧ^{i}})(∑_{j=0}^{n−1} ω^{j+1}δ_{}_{ȧ^{j+1}}) + (∑_{i=0}^{n−1} ω^{i}δ_{ȧ_{i}ḃ})(∑_{j=0}^{n−1} ω^{j+1}δ_{ȧ_{j+1}ḃ})) λ_{a}

= (−δė − δḃ + ∑_{i=1}^{n−1} δ_{}_{ȧ^{i}} + δ_{ȧ_{i}ḃ})λ_{a}.

Put u = (−δė−δḃ+∑_{i=1}^{n−1} δ_{}_{ȧ^{i}}+δ_{ȧ_{i}ḃ}). By similar computation, we can verify w_{0}^{∗}ρ̃(λ_{b})w_{0} = −uλ_{b}. Due to u^{2} = 1, β := w_{0}^{∗}ρ̃(·)w_{0} is an automorphism with period 2.

In the same way as above, w_{1}^{∗}ρ̃(x)w_{1} = β(x) can be shown. □. By the above lemma, {w_{i}w_{j}^{∗}}_{0≤i,j≤1} are matrix units in (ρ̃, ρ̃).

Lemma 4.3 We have θ_{t}βθ_{−t} = Ad u_{t}^{∗}β for some θ_{t}-cocycle u_{t}.

Proof. Since θ_{t} commutes with ρ̃, we have ρ̃(x)θ_{t}(w_{0}) = θ_{t}(w_{0})θ_{t}βθ_{−t}(x). Since θ_{t}(w_{0}w_{0}^{∗}) is equivalent to w_{0}w_{0}^{∗} in (ρ̃, ρ̃), β and θ_{t}βθ_{−t} are unitary equivalent. Hence there exists a unitary u_{t} with Ad u_{t}^{∗}β = θ_{t}βθ_{−t}. Since θ_{t+s}βθ_{−(t+s)} = θ_{t}θ_{s}βθ_{−s}θ_{−t}, u_{t+s} and u_{t}θ_{t}(u_{s}) differ by a center valued 2-cocycle. However it is always a coboundary (for example see [2, Appendix]), and hence we can choose u_{t} as a θ-cocycle. □

By stability of θ_{t}, there exists a unitary u with u^{∗}θ_{t}(u) = u_{t}. Then Ad uβ commutes with θ_{t}. Let α be the restriction of Ad uβ on M. As remarked in the beginning of this section, Ad uβ is the canonical extension of α.

Proposition 4.4 An automorphism α is a non-strongly free automorphism.

Proof. Since ρ is an irreducible component of γ, and ρα is a modular endomorphism, the conclusion follows by Theorem 3.2. □.

Theorem 4.5 Let N ⊂ M be A ⋊_{σ} D_{4n} ⊂ (A ⊗ M_{2}(C)) ⋊_{σ} D_{4n} and α an automorphism constructed above. Then α is not the composition of a non-strongly outer automorphism and an extended modular automorphism.

Proof. By examining the fusion rule of Δ, we can show Δ_{m} = {[id], [ρ_{χ}], [ρ_{χ_{1}}], [ρ_{χ_{2}}]} ∪ {[ρ_{π_{ω^{2m}}}]}_{m=1}^{n−1}. For example, we have [ρ_{π_{ω}}ρ_{π_{ω}}] ≅ [id] ⊕ [ρ_{χ}] ⊕ [ρ_{π_{ω^{2}}}]. Since an outer period of α̃ is two, ρ_{π_{ω}}ρ_{π_{ω}} is a modular endomorphism. Hence its irreducible components are also modular endomorphisms. Non-strongly outer automorphisms of N ⊂ M are inner perturbation of id, ρ_{χ}, ρ_{χ_{1}} and ρ_{χ_{2}}, and all of them are extended modular automorphisms. On the other hand, α is not an extended modular automorphism. Hence α is not the composition of a non-strongly outer automorphism and an extended modular automorphism. □

Remark 1. If N ⊂ M is a subfactor constructed in this section by using D_{4n−2}, then Δ_{m} associated with this subfactor is {[id], [σ_{χ}]} ∪ {ρ_{π_{ω^{2m}}}}_{m=1}^{n−1}. Hence ρ_{χ_{1}} and ρ_{χ_{2}} are nonstrongly outer automorphism and neither of them is an extended modular automorphism. In this case, all non-strongly free automorphism are the composition of ρ_{χ_{1}} and modular extended modular automorphisms. In fact, it is easy to see [ρ̃_{χ_{1}}] = [ρ̃_{χ_{2}}] = [α̃].

2. We have a cocycle θ from (X, F) to the Loi part of Ñ(w) ⊂ M̃(w). In a similar way as in [19], we can prove that ρ_{c} appears in γ^{n} if and only if there exist {v_{i}} ∈ M̃′ ∩ M̃_{k} satisfying θ_{t}(v_{j}) = ∑_{i} c_{ij,t}v_{i}, and this is a representation of a cocycle θ. There exists a 1 to 1 correspondence between the representation of a cocycle and the representation of the minimal group associated with a cocycle θ. (See [31].) Then the minimal group associated with this cocycle is D_{2n}/{e, a^{n}}. (Note that the Loi part of subfactors with index 4 is completely determined by Loi in [22].) Moreover this minimal cocycle determines the relative flow of weights for N ⊂ M, which plays an essential role in the classification of type III_{0} subfactors in [30].

References

[1] Choda, M. and Kosaki, H., Strongly outer actions for an inclusion of factors, J. Funct. Anal. 122 (1994), 315–332.

[2] Connes, A. and Takesaki, M., The flow of weights on factors of type III, Tohoku J. Math. 29 (1977), 473–555.

[3] Evans, D.E. and Kawahigashi, Y., Quantum symmetries on operator algebras, Oxford Press, 1998.

[4] Goodman, F.M., de la Harpe, P., and Jones, V.F.R., Coxeter graphs and towers of algebras, Springer-Verlag, 1989.

[5] Haagerup, U. and Størmer, E., Equivalence of normal states on von neumann algebras and the flow of weights, Adv. Math. 83 (1990), 180–262.

[6] Haagerup, U. and Størmer, E., Pointwise inner automorphisms on von neumann algebras, J. Funct. Anal. 92 (1990), 177–201.

[7] Hiai, F., Minimalizing indices of conditional expectations onto a subfactor, Publ. Res. Inst. Math. Soc. 24 (1988), 673–678.

[8] Isola, T., Modular structure of the crossed product by a compact group dual, J. Operator Theory 33 (1995), 3–32.

[9] Izumi, M., Application of fusion rules to classification of subfactors, Publ. Res. Inst. Math. Soc. 27 (1991), 953–994.

[10] Izumi, M., Canonical extension of endomorphisms on factors, Subfactors (Araki, H. et al., eds.), World Scientific, 1994, pp. 129–138.

[11] Izumi, M., Canonical extension of endomorphisms of type III factors, preprint, 2001.

[12] Izumi, M. and Kawahigashi, Y., Classification of subfactors with principal graph D_{n}^{(1)}, J. Funct. Anal (1993), 257–286.

[13] Jones, V. F. R., Index for subfactors, Invent. Math. 72 (1983), 1–25.

[14] Katayama, Y., Sutherland, C. E., and Takesaki, M., The characteristic square of a factor and the cocycle conjugacy of discrete group actions on factors, Invent. Math. 132 (1998), 331–380.

[15] Kawahigashi, Y. and Takesaki, M., Compact abelian group actions on injective factors, J. Funct. Anal. 105 (1992), 112–128.

[16] Kosaki, H., Extension of Jones’ theory on index to arbitrary factors, J. Funct. Anal. 66 (1986), 123–140.

[17] Kosaki, H., Automorphisms in the irreducible decomposition of sectors, Quantum and Non-commutative Analysis (Araki., H. et al., eds.), Kluwer Academic Publishers, 1993, pp. 305–316.

[18] Kosaki, H., AFD factor of type III_{0} with many isomorphic index 3 subfactors, J. Operator Theory 32 (1994), 17–29.

[19] Kosaki, H., Sector theory and automorphisms for factor-subfactor pairs, J. Math. Soc. Japan 48 (1996), 427–454.

[20] Kosaki, H. and Longo, R., A remark on the mminimal index of subfactors, J. Funct. Anal. (107), 458–470.

[21] Kosaki, H. and Sano, T., Non-splitting inclusions of factors of type III_{0}, Pacific J. Math. 178 (1997), 95–125.

[22] Loi, P., On the derived towers of certain inclusions of type III_{λ} factors of index 4, Pacific J. Math. 165 (1994), 321–345.

[23] Longo, R., Index of subfactors and statistics of quantum fields I, Comm. Math. Phys. 126 (1989), 217–247.

[24] Longo, R., Index of subfactors and statistics of quantum fields II, Comm. Math. Phys. 130 (1990), 285–309.

[25] Longo, R., Minimal index and braided subfactors, J. Funct. Anal. 109 (1992), 98– 112.

[26] Pimsner, M. and Popa, S., Entropy and index for subfactors, Ann. Sci. Ecole. Norm. 19 (1986), 57–106.

[27] Popa, S., Classification actions of discrete amenable groups on subfactors of type II, preprint, 1992.

[28] Sekine, Y., Flow of weights of the crossed products of type III factors by discrete groups, Publ. Res. Inst. Math. Sci. 26 (1990), 655–666.

[29] Winløw, C., Strongly free actions on subfactors, Internat. J. Math 4 (1993), 675–688.

[30] Winsløw, C., Classification of strongly amenable subfactors of type III_{0}, J. Funct. Anal 148 (1997), 296–313.

[31] Zimmer, R., Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373–409.