An analogue of Connes-Haagerup approach for classification of subfactors of type III_{1}

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

Abstract

Popa proved that strongly amenable subfactors of type III_{1} with the same type II and type III graphs are completely classified by their standard invariants. We prove this classification theorem based on Connes and Haagerup’s argument on the uniqueness of the injective factor of type III_{1}, which is different from Popa’s proof.

1 Introduction

Since V. F. R. Jones introduced the theory of index for subfactors in [18], huge progress has been made by involving different area of mathematics.

One of the main problem of subfactor theory is the classification of subfactors. In [31], S. Popa introduced the notion of amenability and strong amenability for subfactors, and proved that a strongly amenable subfactor of type II_{1} has the generating property. This especially implies that the standard invariant is a complete invariant for strongly amenable subfactors. Also in [30] and [32], he gave the classification theorem from the different point of view. He introduced the notion of approximate innerness and central freeness for subfactors, and he showed that strongly amenable, approximately inner and centrally free subfactors are classified by their standard invariants in [30] for type II_{1} case, and in [32] for type III case.

One of the main applications of [32] is the classification of strongly amenable subfactors of type III_{1}. In fact, Popa proved that if an inclusion of approximately finite dimensional (AFD) type III_{1} factors has the same type II graph and type III graph, then the inclusion is approximately inner and centrally free. Hence the main theorem in [32] can be applied. More precisely, he proved the following theorem. (Notations will be explained below.)

Theorem 1.1 Let 𝒩 ⊂ ℳ be a strongly amenable subfactor of type III_{1} with the same type III graph and type II graph. Then 𝒩 ⊂ ℳ is isomorphic to 𝒩^{st} ⊗ ℛ_{∞} ⊂ ℳ^{st} ⊗ ℛ_{∞}.

The assumption on principal graphs is satisfied for finite depth subfactors by [15] for example. Hence finite depth subfactors of type III_{1} are completely classified by their standard invariants.

The main purpose of this paper is to show the above classification theorem by different approach based on [6] and [11]. In [6], A. Connes showed that an injective type III_{1} factor with trivial bicentralizer is isomorphic to the Araki-Woods factor of type III_{1} [2]. In [11], U. Haagerup proved that every injective factor of type III_{1} has trivial bicentralizer, and established the uniqueness of the injective factor of type III_{1}. Roughly speaking, the triviality of bicentralizer means the existence of a state with a large centralizer.

In [32], Popa proved the existence of a state with a large centralizer for inclusions of AFD factors of type III_{1} in a similar way as in [11]. In fact, the existence of a such state is equivalent to central freeness for subfactors in the sense of [32]. By using this result, we can formulate subfactor analogue of bicentralizer.

The main problem in [6] is approximate innerness of modular automorphisms of an injective factor of type III_{1}. Connes showed that modular automorphisms of an injective factor of type III_{1} with trivial bicentralizer is approximately inner, and that this implies the uniqueness of the injective factor of type III_{1}. The key for proof is the equivalence between the notion of semi-discreteness introduced by Effros and Lance in [8] and injectivity, which was proved by Connes in [5]. (Also see [36]). In subfactor theory, Popa introduced the notion of symmetric enveloping algebras in [34] to study various aspects of amenability for subfactors, and gave the Effros-Lance type characterization of amenability for subfactors of type II_{1}. For our purpose, we need to formulate the Effros-Lance type characterization for subfactors of type III_{1}. To do this, we have to treat the symmetric enveloping algebras for type III subfactors. Since we have a similar construction, so-called the Longo-Rehren construction [25], and we will compare these two constructions.

This paper is organized as follows. In Section 2, we collect basic facts on classification results on subfactors and their automorphisms. In Section 3, we discuss how to reduce the problem to classification of torus actions on subfactor of type III_{λ} (0 < λ < 1). We will show that we only have to prove approximate innerness of modular automorphisms. In Section 4, we introduce the relative version of bicentralizer, and discuss its fundamental properties. In Section 5, construction of symmetric enveloping algebras for type III_{1} subfactors are discussed. In Section 6, we give the Effros-Lance type characterization of amenability subfactor of type III_{1} in terms of symmetric enveloping algebras. In Section 7, we prove modular automorphisms are approximately inner by using the Effros-Lance type characterization, and finally prove that strongly amenable subfactors of type III_{1} are splitting. In appendix, we discuss the exhaustion trick, whose proof was omitted in [6].

Acknowledgements. The author would like to thank Professor Kawahigashi, Professor Izumi and Professor Ueda for helpful comments on this work. The author is supported by Grant-Aid for Scientific Research, the Ministry of Education, Culture, Sports, Science and Technology (Japan).

2 Preliminaries and notations

Let ℛ_{0}, ℛ_{λ}, ℛ_{∞} be an injective type II_{1} factor, type III_{λ} and type III_{1} factor respectively. Part of contents of this section can be found in [9]. We also refer [14] and [24] for sector theory.

2.1 Automorphism groups of subfactors

Let 𝒩 ⊂ ℳ be an inclusion of factors with [ℳ : 𝒩] < ∞, 𝒩 ⊂ ℳ(=: ℳ_{0}) ⊂ ℳ_{1} ⊂ ℳ_{2} ⊂ ⋯ the Jones tower for 𝒩 ⊂ ℳ, and e_{k} ∈ ℳ_{k} be the Jones projection. (Throughout this paper, we only deal with minimal conditional expectations in the sense of [13], and inclusions of type II factors are always extremal in the sense of [31] by assumptions or constructions.) Let Aut(ℳ, 𝒩) be the set of automorphisms of ℳ which preserve an inclusion globally. Every α ∈ Aut(ℳ, 𝒩) can be extended to Aut(ℳ_{k}) canonically by setting α(e_{k}) = e_{k}. We denote by 𝒩̃ ⊂ ℳ̃ the common continuous decomposition of a subfactor of type III 𝒩 ⊂ ℳ. For α ∈ Aut(ℳ), we denote by α̃ the canonical extension of α in the sense of [12].

Definition 2.1 ([23, Section 5]) Define Φ(α) := {α|_{ℳ′∩ℳ_{k}} }_{k=0}^{∞}. We call Φ(α) the Loi invariant for α.

Definition 2.2 ([3, Definition 1], [29, Definition 1.5.1]) For α ∈ Aut(ℳ, 𝒩), we say α is strongly outer if α satisfies the following property; if a ∈ ⋃_{k} ℳ_{k} satisfies α(x)a = ax for every x ∈ ℳ, then we have a = 0.

Strong outerness of automorphisms can be characterized by the language of sector theory as follows.

Theorem 2.3 ([3, Theorem 2], [20, Theorem 3]) Let 𝒩 ⊂ ℳ be an inclusion of factors of type III, and γ Longo’s canonical endomorphism. Then α is non-strongly outer if and only if α appears as an irreducible component of γ^{k} for some k ≥ 1.

Definition 2.4 ([37, Definition 3.1]) Let 𝒩 ⊂ ℳ be a subfactor of type III. For α ∈ Aut(ℳ, 𝒩), we say α is strongly free if α satisfies the following property; if a ∈ ⋃_{k} ℳ̃_{k} satisfies α̃(x)a = ax for every x ∈ ℳ̃, then we have a = 0.

We denote by Cnt_{o}(ℳ, 𝒩) (resp. Cnt_{f}(ℳ, 𝒩)) be the set of all non-strongly outer (resp. non-strongly free) automorphisms. Both sets are normal subgroups of Aut(ℳ, 𝒩).

If 𝒩 = ℳ, then we have Cnt_{o}(ℳ, ℳ) = Int(ℳ) and Cnt_{f}(ℳ, ℳ) = Cnt_{r}(ℳ). Every element in Cnt_{r}(ℳ) is the composition of an inner automorphism and an (extended) modular automorphism. Subfactor analogue of these facts has been obtained by Kosaki as following theorem.

Theorem 2.5 ([21, Theorem 19]) Let 𝒩 ⊂ ℳ be a subfactor of type III_{λ}, λ ≠ 0, and α a non-strongly free automorphism for 𝒩 ⊂ ℳ. Then α = β ∘ σ_{t}^{φ} for some non-strongly outer automorphism β.

2.2 Automorphism groups of strongly amenable subfactors of type II

Take a tunnel ⋯ ⊂ 𝒩_{k} ⊂ ⋯ 𝒩_{1} := 𝒩 ⊂ ℳ for 𝒩 ⊂ ℳ. We define 𝒩^{st} ⊂ ℳ^{st} := ⋁_{k}(𝒩′_{k} ∩ 𝒩) ⊂ ⋁_{k}(𝒩′_{k} ∩ ℳ), and call it the model inclusion.

Let 𝒩 ⊂ ℳ be a strongly amenable subfactor of type II_{1} in the sense of [31]. In [31], Popa proved that 𝒩 ⊂ ℳ has the generating property, i.e, there exists a choice of a tunnel ⋯ ⊂ 𝒩_{k} ⊂ ⋯ 𝒩_{1} := 𝒩 ⊂ ℳ such that 𝒩^{st} ⊂ ℳ^{st} = 𝒩 ⊂ ℳ. Especially, this means that strongly amenable subfactors of type II_{1} can be classified by their standard invariants.

In the study of automorphism groups of operator algebras, the most important classes of automorphisms are approximately inner ones and centrally trivial ones. For strongly amenable subfactors of type II, these two classes are characterized in the terms of the Loi invariant and non-strong outerness.

Theorem 2.6 ([23, Theorem 5.4]) Let 𝒩 ⊂ ℳ be a strongly amenable subfactor of type II_{1}. Then Ker Φ = \overline{Int}(ℳ, 𝒩).

Corollary 2.7 ([38, Theorem 4.6]) Let 𝒩 ⊂ ℳ be a strongly amenable subfactor of type II_{∞}. Then Ker Φ ∩ Ker mod = \overline{Int}(ℳ, 𝒩).

Theorem 2.8 ([29, Theorem 1.6]) Let 𝒩 ⊂ ℳ be a subfactor of type II_{1}. Then we have Cnt_{o}(ℳ, 𝒩) ⊂ Cnt(ℳ, 𝒩). Moreover if 𝒩 ⊂ ℳ is strongly amenable, then we also have Cnt_{o}(ℳ, 𝒩) = Cnt(ℳ, 𝒩).

Here we give a proof of Theorem 2.8 for reader’s convenience, which is different from the proof in [29]. First we prepare the following lemma.

Lemma 2.9 Let ℳ ⊃ 𝒩 ⊃ ⋯ ⊃ 𝒩_{k} ⊃ ⋯ be a tunnel, and α ∈ Aut(ℳ, 𝒩). If sup{∥α(u) − u∥_{2} | u ∈ U(𝒩_{k})} < 1 for some k, then α is non-strongly outer.

Proof. Set K := \overline{conv}\{α(u)u^{∗} | u ∈ U(𝒩_{k})\}, where closure is taken in the σ-weak topology. K is a σ-weakly compact set. By the lower semicontinuity of ∥·∥_{2} in the σ-weak topology, there exists a unique a ∈ K such that ∥a∥_{2} := min{∥b∥_{2} | b ∈ K}. Moreover by the assumption, 0 ∉ K, and hence a ≠ 0. Since ∥α(v)av^{∗}∥_{2} = ∥a∥_{2}, α(v)a = av holds for every v ∈ U(𝒩_{k}) by the uniqueness of a. This implies α is non-strongly outer. □ Proof of Theorem 2.8. First assume that σ ∈ Aut(ℳ, 𝒩) is non-strongly outer. Then we can find a non-zero a ∈ ℳ_{k} such that σ(x)a = ax holds for every x ∈ ℳ. Take {x_{n}} ∈ ℓ^{∞}(N, 𝒩) which is central in ℳ. Then ∥[x_{n}, a]∥_{2} → 0 as n goes to infinity. Then {σ(x_{n})a − x_{n}a} = {ax_{n} − x_{n}a} converges to 0 strongly, and so does {σ(x_{n})aa^{∗} − x_{n}aa^{∗}}. Here we apply the minimal conditional expectation E_{ℳ} . Since aa^{∗} ∈ ℳ′ ∩ ℳ_{k}, E_{ℳ} (aa^{∗}) is a non-zero scalar. Hence we get lim_{n→∞} σ(x_{n}) − x_{n} = 0 strongly.

Next we assume that 𝒩 ⊂ ℳ is strongly amenable. We fix a tunnel ⋯ ⊂ 𝒩_{k} ⊂ ⋯ 𝒩_{1} := 𝒩 ⊂ ℳ with the generating property.

Let σ be a strongly outer automorphism. For every k, we can choose u_{k} ∈ U(𝒩_{k}) such that ∥σ(u_{k}) − u_{k}∥_{2} ≥ 1/2 by Lemma 2.9. By the generating property, {u_{k}} is central in ℳ. Then lim inf ∥σ(u_{k}) − u_{k}∥_{2} ≥ 1/2, and this means that σ is not centrally trivial. □

Corollary 2.10 ([37, Theorem 3.4]) Let 𝒩 ⊂ ℳ be a strongly amenable subfactor of type II_{∞}. Then Cnt_{o}(ℳ, 𝒩) = Cnt(ℳ, 𝒩).

2.3 Inclusions of factors of type III_{λ}, λ ≠ 0

Let 𝒬 ⊂ 𝒫 be an inclusion of factors of type III_{λ}, λ ≠ 0. When 0 < λ < 1, we assume that 𝒬 ⊂ 𝒫 has the common discrete decomposition, i.e., there exists an inclusion of factors of type II_{∞} B ⊂ A, and an automorphism θ ∈ Aut(A, B) with mod (θ) = λ such that 𝒬 ⊂ 𝒫 is isomorphic to B ⋊_{θ} Z ⊂ A ⋊_{θ} Z. We refer to 𝒢_{𝒫,𝒬} (resp. 𝒢_{B,A}) as the type III standard invariant (resp. type II standard invariant) for 𝒬 ⊂ 𝒫. We also use the notation 𝒢̃_{𝒫,𝒬} to denote the type II standard invariant for 𝒬 ⊂ 𝒫. For subfactors of type III_{1}, we use similar notations by considering the common continuous decomposition.

The natural question is the relation between 𝒢_{𝒫,𝒬} and 𝒢̃_{𝒫,𝒬} .

Proposition 2.11 We have 𝒬′ ∩ 𝒫_{k} = (B′ ∩ A_{k})^{θ} for every k ≥ 0. Hence 𝒢_{𝒫,𝒬} = 𝒢_{A,B} holds if and only if θ acts trivially on B′ ∩ A_{k} for every k ≥ 0.

See [23, Proposition 3.1] for 0 < λ < 1 case, and [22, Corollary 6] for λ = 1 case. Remark. The type II invariant and the type III one coincide if and only if 𝒬 ⊂ 𝒫 has the same type II principal graph and type III graph.

The coincidence of the type II graph and type III graph can be characterized by Longo’s canonical endomorphism γ for 𝒬 ⊂ 𝒫.

Theorem 2.12 ([15, Theorem 3.5]) The type II graph and type III graph of 𝒫 ⊂ 𝒫_{1} coincide if and only if σ_{t}^{φ} never appears as an irreducible component of γ^{k} for any k ≥ 1 and t ∉ T (𝒫).

By combining the above theorem and Theorem 2.3, we get the following corollary.

Corollary 2.13 The type II graph and type III graph of 𝒫 ⊂ 𝒫_{1} coincide if and only σ_{t}^{φ} is strongly outer for every t ∉ T (𝒫).

By Corollary 2.7 and Corollary 2.10, we get the following classification result for strongly amenable subfactors of type III_{λ}, 0 < λ < 1.

Theorem 2.14 ([23, Theorem 6.1]) Let 𝒬 ⊂ 𝒫 be a strongly amenable subfactor of type III_{λ}, 0 < λ < 1, with 𝒢_{𝒫,𝒬} = 𝒢̃_{𝒫,𝒬} . Then 𝒬 ⊂ 𝒫 is isomorphic to 𝒬^{st} ⊗ ℛ_{λ} ⊂ 𝒫^{st} ⊗ ℛ_{λ}.

We have the following characterization of \overline{Int}(𝒫, 𝒬) and Cnt(𝒫, 𝒬).

Theorem 2.15 ([37, Theorem 3.8], [38, Theorem 4.6]) Let 𝒬 ⊂ 𝒫 be as in Theorem 2.14. Then we have \overline{Int}(𝒫, 𝒬) = Ker Φ ∩ Ker mod and Cnt(𝒫, 𝒬) = Cnt_{f}(𝒫, 𝒬).

2.4 Longo-Rehren construction

The Longo-Rehren construction was introduced in [25] originally. Here we recall the crossed product type approach of the Longo-Rehren construction based on [26]. Let Δ = {[ρ_{i}]}_{i∈I} be a set of irreducible sectors of ℳ closed under conjugation and multiplication. We assume that I is at most countable. Let j_{ℳ} be the canonical conjugate linear isomorphism from ℳ onto ℳ^{opp}. Put ρ̂:= ρ ⊗ j_{ℳ} ∘ ρ ∘ j_{ℳ}^{−1} ∈ End(ℳ ⊗ ℳ^{opp}).

Let {v(_{i,j}^{k})^{e}}_{e=1}^{N_{i,j}^{k}} ∈ (ρ_{k}, ρ_{i}ρ_{j}) be an orthonormal basis, and define the canonical intertwiner v̂_{i,j}^{k} as follows.

v̂_{i,j}^{k} := \sqrt{\frac{d(i)d(j)}{d(k)}} ∑_{e=1}^{N_{i,j}^{k}} v(_{i,j}^{k})^{e} ⊗ j_{ℳ} (v(_{i,j}^{k})^{e}).

The canonical intertwiner v̂_{i,j}^{k} does not depend on the choice of an orthonormal basis.

Lemma 2.16 The canonical intertwiners satisfy the following relations.

(1) ∑_{m} v̂_{i,j}^{m}v̂_{m,k}^{l} = ∑_{m} ρ̂_{i}(v̂_{j,k}^{m})v̂_{i,m}^{l}.

(2) ∑_{m} v̂_{i,j}^{m}v̂_{k,l}^{m∗} = ∑_{m} ρ̂_{i}(v̂_{m,l}^{j∗})v̂_{i,m}^{k}.

Set H_{i} := L^{2}(ℳ ⊗ ℳ^{opp}), i ∈ I, and define an action π of ℳ ⊗ ℳ^{opp} and an operator V_{i} on H := ⊕H_{i} as follows.

(π(x)ξ)(i) := ρ̂_{i}(x)ξ(i), x ∈ ℳ ⊗ ℳ^{opp},

(V_{i}ξ)(j) := ∑_{k} v̂_{j,i}^{k}ξ(k).

By using Lemma 2.16, we get the following relations.

Lemma 2.17 We have the following relations.

(1) V_{i}V_{j} = ∑_{k} π(v̂_{i,j}^{k})V_{k}.

(2) V_{i}π(a) = π(ρ̂_{i}(a))V_{i}, a ∈ ℳ ⊗ ℳ^{opp}.

(3) V_{i}^{∗} = π(v̂_{ī,i}^{0∗})Vī

Definition 2.18 Let B be a von Neumann algebra generated by π(ℳ⊗ℳ^{opp}) and {V_{i}}_{i∈I}. We call π(ℳ ⊗ ℳ^{opp}) ⊂ B the Longo-Rehren inclusion associated with Δ.

In fact, B is a factor. In what follows, we identify π(ℳ ⊗ ℳ^{opp}) and ℳ ⊗ ℳ^{opp}, and often omit π. When Δ is arising from a subfactor 𝒩 ⊂ ℳ, we say ℳ ⊗ ℳ^{opp} ⊂ B as the Longo-Rehren inclusion for 𝒩 ⊂ ℳ. By Lemma 2.17, {∑ π(a_{i})V_{i} | a_{i} ∈ ℳ ⊗ ℳ^{opp}, a_{i} = 0 except for finitely many i} is a dense ∗-subalgebra of B.

Let P be a projection from H onto H_{0}, and define ℰ(x) := P xP^{∗} for x ∈ B. Then ℰ is a conditional expectation from B onto ℳ ⊗ ℳ^{opp} as in the case of usual crossed product construction. Set a_{i} := ℰ(aV_{i}^{∗}), a ∈ B. Then we have a formal expansion formula a = ∑_{i∈I} π(a_{i})V_{i}.

Let ℳ ⊗ ℳ^{opp} ⊂ B be the Longo-Rehren inclusion for 𝒩 ⊂ ℳ, and ℰ a conditional expectation from B onto ℳ⊗ℳ^{opp}. We will construct the common Jones projection e_{𝒩} for 𝒩 ⊂ ℳ and 𝒩^{opp} ⊂ ℳ^{opp}. Let ι be an inclusion map 𝒩 ↪ ℳ. Let I_{0} := {i ∈ I | ρ_{i} ≺ γ}, {a_{i}^{e}} ⊂ (ι, ρ_{i}ι) an orthonormal bases, and ã_{i} := ∑_{e} a_{i}^{e} ⊗ j_{ℳ} (a_{i}^{e}). Then e_{𝒩} is expressed as e_{𝒩} = [ℳ : 𝒩]^{−1} ∑_{i∈I_{0}} \sqrt{d(i)}ã_{i}^{∗}V_{i}.

Conversely, let {w_{i}^{e}} ⊂ (ρ_{i}, γ) be a orthonormal basis, and set w̃_{i} := ∑_{e} w_{i}^{e} ⊗ j_{ℳ} (w_{i}^{e}). Then we have V_{i} = Cw̃_{i}^{∗}e_{𝒩} w̃_{0} for some 0 ≠ C ∈ C.

Next we recall the extension of automorphisms of a subfactor to the Longo-Rehren inclusion discussed in [27]. Let α ∈ Ker Φ. Then we can find a half braiding u(α, ρ) ∈ U(ℳ) in the sense of [16]. Namely u(α, ρ) satisfy the following.

(i) α ∘ ρ ∘ α^{−1} = Ad u(α, ρ)ρ, (ii) u(αβ, ρ) = α(u(β, ρ))u(α, ρ), (iii) For v ∈ (ρ_{3}, ρ_{1}ρ_{2}), u(α, ρ_{1})ρ_{1}(u(α, ρ_{2}))v = α(v)u(α, ρ_{3}).

We briefly sketch how to find u(α, ρ). Fix R ∈ (id_{ℳ} , γ) and R̄ ∈ (id_{𝒩} , γ|_{𝒩} ) such that R^{∗}R̄ = [ℳ, 𝒩]^{− \frac{1}{2}} and R̄^{∗}γ(R) = [ℳ, 𝒩]^{− \frac{1}{2}} . Next we fix u ∈ U(𝒩) such that α ∘ γ ∘ α^{−1} = Ad uγ and u^{∗}α(R) = R. Then u^{∗}α(R̄) = R̄ holds automatically. Set u_{k} := uγ(u) ⋯ γ^{k−1}(u). Then α ∘ γ^{k} ∘ α^{−1} = Ad u_{k}γ holds.

Take an isometry v ∈ (ρ, γ^{k}), and define u(α, ρ) by u(α, ρ) = α(v^{∗})u_{k}v. Then α ∈ Ker Φ assures that u(α, ρ) is well defined. We can verify u(α, ρ) satisfy the above conditions (i), (ii) and (iii) in a similar way as in [27].

We can extend α ⊗ id ∈ Aut(ℳ ⊗ ℳ^{opp}) to α ⊠ id ∈ Aut(B) by setting

α ⊠ id(V_{i}) = (u(α, ρ_{i}) ⊗ 1)V_{i}.

In fact, let U_{α} ∈ B(L^{2}(ℳ)) be the standard implementing unitary for α, and define V ∈ U(H) by

(V ξ)(i) := (u(α, ρ_{i})^{∗}U_{α} ⊗ 1)ξ(i).

Then Ad V gives an automorphism of B, and satisfies the above property.

In a similar way, we can extend α ⊗ α^{opp} ∈ Aut(ℳ ⊗ ℳ^{opp}) to α ⊠ α^{opp} ∈ Aut(B) by

α ⊠ α^{opp}(V_{i}) = (u(α, ρ_{i}) ⊗ j_{ℳ} (u(α, ρ_{i})))V_{i}.

Both extensions fix the common Jones projection e_{𝒩} .

We mainly apply the above extension to the modular automorphism σ_{t}^{φ}. In this case, u(σ_{t}^{φ}, ρ) is given by dρ^{−it}(Dφ : Dφ ∘ ϕ_{ρ})_{t}, where ϕ_{ρ} is a standard left inverse for ρ.

3 Torus actions on III_{λ} subfactors

In this section, we will discuss how to reduce the classification theorem to the classification of torus actions on strongly amenable subfactors of type III_{λ}.

Proposition 3.1 Let 𝒩 ⊂ ℳ be a subfactor of type III_{1} with 𝒢_{ℳ,𝒩} = 𝒢̃_{ℳ,𝒩} . Fix T > 0 and set θ := σ_{T}^{φ}. Then 𝒬 ⊂ 𝒫 := 𝒩 ⋊_{θ} Z ⊂ ℳ ⋊_{θ} Z is a subfactor of type III_{λ}, T = −\frac{2π}{log λ}, with 𝒢_{ℳ,𝒩} = 𝒢_{𝒫,𝒬} = 𝒢̃_{𝒫,𝒬} .

Proof. By [6, Lemma 1.1], 𝒫 and 𝒬 are type III_{λ} factors. Since 𝒢_{ℳ,𝒩} = 𝒢̃_{ℳ,𝒩} , σ_{T}^{φ} is strongly outer for any non-zero T ∈ R by Corollary 2.13. We also have σ_{T}^{φ} ∈ Ker Φ. Thus 𝒢_{ℳ,𝒩} = 𝒢_{𝒫,𝒬} holds. Next we investigate 𝒢̃_{𝒫,𝒬} . By standard argument, we can identify 𝒬̃ ⊂ 𝒫̃ with 𝒩̃ ⋊θ̃ Z ⊂ ℳ̃ ⋊θ̃ Z. In this case, θ̃ is inner. Hence

𝒬̃ ⊂ 𝒫̃ ≅ 𝒩̃ ⋊θ̃ Z ⊂ ℳ̃ ⋊θ̃ Z ≅ 𝒩̃ ⊗ L^{∞}(T) ⊂ ℳ̃ ⊗ L^{∞}(T).

This implies 𝒢_{ℳ,𝒩} = 𝒢_{𝒫,𝒬} = 𝒢̃_{𝒫,𝒬} . □

Proposition 3.2 (i) Let 𝒬 ⊂ 𝒫 be a strongly amenable subfactor of type III_{λ} with 𝒢_{𝒫,𝒬} = 𝒢̃_{𝒫,𝒬} , and θ ∈ Aut(𝒫, 𝒬) an automorphism such that 𝒬⋊_{θ} Z ⊂ 𝒫⋊_{θ} Z is a subfactor of type III_{λ} with 𝒢_{𝒫,𝒬} = 𝒢_{𝒫⋊_{θ}Z,𝒬⋊_{θ}Z} = 𝒢̃_{𝒫⋊_{θ}Z,𝒬⋊_{θ}Z} . If the dual action θ̂_{t} is strongly free for every 0 ≠ t ∈ T, then θ is outer conjugate to id_{𝒫} ⊗ σ, where σ is an aperiodic automorphism of the injective type II_{1} factor ℛ_{0}. (ii) Let 𝒬 ⊂ 𝒫 be as in (i), and θ a strongly free action of T such that 𝒬⋊_{θ} T ⊂ 𝒫⋊_{θ} T is a subfactor of type III_{λ} with 𝒢_{𝒫,𝒬} = 𝒢_{𝒫⋊_{θ}T,𝒬⋊_{θ}T} = 𝒢̃_{𝒫⋊_{θ}T,𝒬⋊_{θ}T} . Then θ is cocycle conjugate to id_{𝒫} ⊗ σ̂.

Proof. (i) If we prove θ is an approximately inner and centrally free action of Z, then we can apply [23, Theorem 4.3(2)]. By Theorem 2.15, we have \overline{Int}(𝒫, 𝒬) = Ker Φ ∩ Ker mod and Cnt_{f}(𝒫, 𝒬) = Cnt(𝒫, 𝒬). Hence we only have to prove θ is a strongly free action of Z and θ ∈ Ker Φ ∩ Ker mod . Since 𝒬 ⋊_{θ} Z and 𝒫 ⋊_{θ} Z are type III_{λ} factors, θ is an outer action of Z, and mod θ = id as the same reason in [6]. Next we compute 𝒬′ ∩ (𝒫_{n} ⋊_{θ} Z). Then by easy computation, we have

𝒬′ ∩ (𝒫_{n} ⋊_{θ} Z) = {∑_{k} a_{k}u^{k} | a_{k} ∈ 𝒫_{n}, xa_{k} = a_{k}θ^{k}(x) for every x ∈ 𝒬},

where u is an implementing unitary. Set I_{k}^{n} := {a ∈ 𝒫_{n} | xa = aθ^{k}(x) for every x ∈ 𝒬}. Then I_{k}^{n} is a finite dimensional Hilbert space, and θ acts on I_{k}^{n} as a unitary operator in a natural way. We claim that the action of θ on I_{k}^{n} is trivial. Suppose converse holds. Then we can find an eigenvalue 0 ≠ t_{0} ∈ T of θ, and an eigenvector a for t_{0}. Hence θ(a) = e^{it_{0}}a holds. We will show that θ̂_{t_{0}} is non-strongly outer. We have θ̂_{t_{0}}(x)au^{k} = xau^{k} = aθ^{k}(x)u^{k} = au^{k}x for x ∈ 𝒬. We also have θ̂_{t_{0}}(u)au^{k} = e^{−it_{0}}uau^{k} = e^{−it_{0}}θ(a)u^{k}u = au^{k}u. Hence θ̂_{t_{0}}(x)au^{k} = au^{k}x holds for every x ∈ 𝒬⋊_{θ} Z, and this implies that θ̂_{t_{0}} is non-strongly outer, which implies non-strong freeness by [21, Theorem 17]. This contradicts the assumption on θ̂. Hence θ acts on I_{k}^{n} trivially. This holds for every k and n, and hence this especially implies θ ∈ Ker Φ if we consider the case k = 0.

Next we compute (𝒬⋊_{θ}Z)′∩(𝒫_{n}⋊_{θ}Z). Since θ acts on I_{k}^{n} trivially, (𝒬⋊_{θ}Z)′∩(𝒫_{n}⋊_{θ}Z) = {∑_{k} a_{k}u^{k} | a_{k} ∈ I_{k}^{n}}. Since we assume 𝒢_{𝒫,𝒬} = 𝒢_{𝒫⋊_{θ}Z,𝒬⋊_{θ}Z} , we have I_{k}^{n} = {0} for every non-zero k ∈ Z and n ≥ 0. Hence θ is a strongly outer action of Z. In a similar way as above, we can show that θ is a strongly free action of Z.

(ii) This can be shown by using (i) and the Takesaki duality [35]. □ Remark. In Proposition 3.2(i), we assume strong freeness of θ̂_{t}. If we do not have this condition, then Proposition 3.2(i) may fail. If θ is a non-strongly outer action such that γ_{h}(θ)^{n} ≠ 1 for all non-zero n ∈ Z, where γ_{h}(θ) is the higher obstruction in [19], then the standard invariant does not change by taking the crossed product. However the author does not know whether such subfactors exist or not.

Let 𝒩 ⊂ ℳ and 𝒬 ⊂ 𝒫 be as in Proposition 3.1. Let A be a type III_{1} factor, and φ_{0} a faithful normal state of A. Set θ^{(0)} := σ_{T}^{φ_{0}} and Ā := A ⋊_{θ^{(0)}} Z. Then α_{t} := θ̂_{t} ⊗ θ̂_{−t}^{(0)} is a strongly free action of T on 𝒬 ⊗ Ā ⊂ 𝒫 ⊗ Ā with mod α_{t} = id.

Lemma 3.3 The crossed product (𝒬 ⊗ Ā) ⋊_{α} T ⊂ (𝒫 ⊗ Ā) ⋊_{α} T is a subfactor of type III_{λ}, and the standard invariant of this subfactor is equal to that of 𝒬 ⊂ 𝒫.

Proof. As in the proof of [6, Lemma I.1], we can identify (𝒬 ⊗ Ā) ⋊_{α} T ⊂ (𝒫 ⊗ Ā) ⋊_{α} T and (𝒩 ⊗ A) ⋊_{σ_{T}^{φ⊗φ_{0}}} Z ⊂ (ℳ ⊗ A) ⋊_{σ_{T}^{φ⊗φ_{0}}} Z. This is a subfactor of type III_{λ}. Moreover since 𝒩 ⊗ A ⊂ ℳ ⊗ A has the same type III standard invariant and type II standard invariant, so does (𝒩 ⊗ A) ⋊_{σ_{T}^{φ⊗φ_{0}}} Z ⊂ (ℳ ⊗ A) ⋊_{σ_{T}^{φ⊗φ_{0}}} Z. (See Proposition 3.1) □

When 𝒩 ⊂ ℳ is strongly amenable and A = ℛ_{∞} in Lemma 3.3, we can apply Proposition 3.2 since α_{t} is strongly free for t ≠ 0. Hence α is cocycle conjugate to id_{𝒫} ⊗ σ̂.

Proposition 3.4 Let 𝒩 ⊂ ℳ be a strongly amenable subfactor of type III_{1}, and 𝒬 ⊂ 𝒫 as in Proposition 3.1. If θ̂_{t} is cocycle conjugate to θ̂_{t} ⊗ id_{ℛ_{λ}} ⊗ σ̂_{t}, then 𝒩 ⊂ ℳ is isomorphic to 𝒩^{st} ⊗ ℛ_{∞} ⊂ ℳ^{st} ⊗ ℛ_{∞}.

Proof. By Theorem 2.14, 𝒬 ⊂ 𝒫 is isomorphic to 𝒩^{st} ⊗ ℛ_{λ} ⊂ ℳ^{st} ⊗ ℛ_{λ}. On one hand, id_{ℛ_{λ}} ⊗σ̂_{t} is cocycle conjugate to θ̂_{−t}^{(0)} ⊗θ̂_{t}^{(0)}. On the other hand θ̂_{t} ⊗θ̂_{−t}^{(0)} is cocycle conjugate to id_{𝒫} ⊗ σ̂_{t}, and hence cocycle conjugate to id_{ℳ^{st}} ⊗ id_{ℛ_{λ}} ⊗ σ̂_{t}. Therefore

θ̂_{t} ∼ θ̂_{t} ⊗ id_{ℛ_{λ}} ⊗ σ̂_{t}
∼ θ̂_{t} ⊗ θ̂_{−t}^{(0)} ⊗ θ̂_{t}^{(0)}
∼ id_{ℳ^{st}} ⊗ id_{ℛ_{λ}} ⊗ σ̂_{t} ⊗ θ̂_{t}^{(0)}
∼ id_{ℳ^{st}} ⊗ θ̂_{t}^{(0)}

holds. By the Takesaki duality Theorem, we get 𝒩 ⊂ ℳ ≅ 𝒩^{st} ⊗ ℛ_{∞} ⊂ ℳ^{st} ⊗ ℛ_{∞}. □

By Proposition 3.4, we only have to prove θ̂_{t} is cocycle conjugate to θ̂_{t} ⊗ id_{ℛ_{λ}} ⊗ σ̂_{t} for classification of type III_{1} subfactors. This follows from the following two conditions by the same argument in [6].

(1) 𝒩 ⊂ ℳ ≅ 𝒩 ⊗ ℛ_{λ} ⊂ ℳ ⊗ ℛ_{λ},

(2) σ_{T}^{φ} ∈ \overline{Int}(ℳ, 𝒩).

We sketch how to deduce θ̂_{t} ≅ θ̂_{t} ⊗ id_{ℛ_{λ}} ⊗ σ̂_{t} from the above conditions.

Let φ_{1} be a periodic state on ℛ_{λ}. By (1), we may assume θ = σ_{T}^{φ} ⊗ σ_{T}^{φ_{1}} = σ_{T}^{φ} ⊗ id_{ℛ_{λ}} . This implies θ̂_{t} ∼ θ̂_{t} ⊗ id_{ℛ_{λ}} . Also by (1), we have 𝒩 ⊂ ℳ ≅ 𝒩 ⊗ ℛ_{0} ⊂ ℳ ⊗ ℛ_{0}. Hence we can find a centralizing sequence {u_{n}} such that u_{n}^{2} = 0 and u_{n}^{∗}u_{n} + u_{n}u_{n}^{∗} = 1. Since u_{n} can be chosen in ℛ_{0}, we can assume σ_{t}^{φ}(u_{n}) = u_{n}. By (2), there exists {v_{n}} ⊂ U(𝒩) such that σ_{T}^{φ} = lim Ad v_{n}. Let U be a implementing unitary in (𝒩 ⊗ ℛ_{0}) ⋊_{θ} Z. Define w_{n} := u_{n}(v_{n}^{∗}⊗1)U ∈ (𝒩⊗ℛ_{0})⋊_{θ}Z. Then {w_{n}} is a centralizing sequence in (ℳ⊗ℛ_{0})⋊_{θ}Z. We have w_{n}^{2} = 1, w_{n}w_{n}^{∗} + w_{n}^{∗}w_{n} = 1 and θ̂_{t}(w_{n}) = e^{it}w_{n}. Thus we get θ̂_{t} ≅ θ̂_{t} ⊗ σ̂_{t}.

Moreover condition (1) follows from condition (2) by the “local” characterization of property L′_{λ} [1]. See [6, Theorem II.4]. The purpose of the rest of this paper is to show the approximate innerness of modular automorphisms.

4 Relative bicentralizer

Let 𝒩 ⊂ ℳ be a subfactor of type III_{1} with finite index and ℰ_{0} the minimal expectation. As an analogue of [11], we introduce the notion of the relative bicentralizer for 𝒩 ⊂ ℳ.

Definition 4.1 (i) Let φ be a faithful normal state of ℳ with φ ∘ ℰ_{0} = φ. Set C(φ) :=

{{x_{n}} ∈ l^{∞}(N, 𝒩) | lim_{n→∞} ∥[x_{n}, φ]∥ = 0}.

(ii) Define B(φ) as the set of a ∈ ℳ such that x_{n}a − ax_{n} converges to zero σ-strongly for every {x_{n}} ∈ C(φ). We call B(φ) the relative bicentralizer of φ.

It is clear that 𝒩′ ∩ ℳ ⊂ B(φ). It is also easy to see B(φ) ⊂ 𝒩′_{φ} ∩ ℳ since 𝒩_{φ} is embedded in C(φ) as constant sequences. Hence if 𝒩′ ∩ ℳ = 𝒩′_{φ} ∩ ℳ, then we have B(φ) = 𝒩′ ∩ ℳ. This condition is satisfied if 𝒩 ⊂ ℳ is a splitting inclusion of AFD type III_{1} factors.

We collect basic facts on the relative bicentralizer.

Set C_{φ}(a, δ) := \overline{conv}\{u^{∗}au | u ∈ U(𝒩), ∥[u, φ]∥ ≤ δ\}, where closure is taken in the σ-weak topology.

Lemma 4.2 For a ∈ ℳ, a ∈ B(φ) if and only if {a} = ⋂_{δ>0} C_{φ}(a, δ).

Proof. The same proof of [11, Lemma 1.2] works with appropriate changes. □

As in [11, Proposition 1.3], it is shown that B(φ) is a von Neumann subalgebra of ℳ by using the above lemma.

Proposition 4.3 Following two conditions are equivalent.

(i) B(φ) = 𝒩′ ∩ ℳ. (ii) C_{φ}(a, δ) ∩ 𝒩′ ∩ ℳ ≠ ∅ for every a ∈ ℳ and every δ > 0.

Proof. (i)⇒(ii). Put C_{φ}(a) := ⋂_{δ>0} C_{φ}(a, δ). Then C_{φ}(a) is a σ-weakly compact convex set, and C_{φ}(a) ≠ ∅ since a ∈ C_{φ}(a, δ) for every δ. Let ℌ_{φ} be a completion of ℳ with φ-norm. Then C_{φ}(a) is a norm closed convex subset of ℌ_{φ}. Hence there exists a unique element b ∈ C_{φ}(a) with ∥bξ_{φ}∥ = min{∥xξ_{φ}∥ | x ∈ C_{φ}(a)}. If a′ ∈ C_{φ}(a, δ), then C_{φ}(a′, δ) ⊂ C_{φ}(a, 2δ). Hence C_{φ}(b) = ⋂_{δ>0} C_{φ}(b, δ) ⊂ ⋂_{δ>0} C_{φ}(a, 2δ) = C_{φ}(a). If ∥[u, φ]∥ ≤ δ, then ∥u^{∗}bu∥_{φ}^{2} ≤ ∥b∥_{φ}^{2} + δ∥b∥_{φ}^{2}. By lower semicontinuity of φ-norm in the σ-weak topology, ∥x∥_{φ}^{2} ≤ ∥b∥_{φ}^{2} + δ∥b∥_{φ}^{2} for every x ∈ C_{φ}(b, δ). Hence ∥x∥_{φ}^{2} ≤ ∥b∥_{φ}^{2} holds for x ∈ C_{φ}(b). Since x ∈ C_{φ}(b) ⊂ C_{φ}(a), we get x = b, and hence C_{φ}(b) = {b} holds. By Lemma 4.2, b ∈ B(φ), and by assumption b ∈ 𝒩′ ∩ ℳ.

(ii)⇒(i). First note that 𝒩′ ∩ ℳ is finite dimensional. So the sets C_{φ}(a, δ) ∩ 𝒩′ ∩ ℳ form a family of nonempty decreasing compact sets. Thus the intersection of these sets is also non-empty. Hence there exists b ∈ C_{φ}(a) ∩ 𝒩′ ∩ ℳ. For every u ∈ U(𝒩) with ∥[u, φ]∥ ≤ δ and x ∈ 𝒩′ ∩ ℳ, we have

|φ(u^{∗}aux) − φ(ax)| = |(uφu^{∗} − φ)(ax)|
≤ ∥uφu^{∗} − φ∥∥ax∥
≤ δ∥ax∥,

and we get |φ(bx) − φ(ax)| = 0 for every x ∈ 𝒩′ ∩ ℳ. Since 0 = φ(bx) − φ(ax) = φ((b−E_{𝒩′∩ℳ}^{φ} (a))x) holds, we get b = E_{𝒩′∩ℳ}^{φ} (a), where E_{𝒩′∩ℳ}^{φ} is a φ-preserving conditional expectation on 𝒩′ ∩ ℳ.

Then a−E_{𝒩′∩ℳ}^{φ} (a) ∈ \overline{conv}\{a−u^{∗}au | u ∈ U(𝒩), ∥[u, φ]∥ ≤ δ\}. By lower semicontinuity of φ-norm in the σ-weak topology, we get ∥a − E_{𝒩′∩ℳ}^{φ} (a)∥_{φ} ≤ sup{∥a − u^{∗}au∥_{φ} | u ∈ U(𝒩), ∥[u, φ]∥ ≤ δ}.

Here if a ∈ B(φ), then the right hand side of the above inequality converges to 0 if we let δ → 0. Hence we get a = E_{𝒩′∩ℳ}^{φ} (a) and B(φ) ⊂ 𝒩′ ∩ ℳ. □

By the proof of the above proposition, B(φ) = 𝒩′∩ℳ is also equivalent to the following condition. (iii) E_{𝒩′∩ℳ}^{φ} (a) ∈ C_{φ}(a, δ) for every a ∈ ℳ and δ > 0. By duality argument, this is also equivalent to the following. (iv) ψ ∘ E_{𝒩′∩ℳ}^{φ} ∈ \overline{conv}\{uψu^{∗} | u ∈ U(𝒩), ∥[u, φ]∥ ≤ δ\} for every ψ ∈ ℳ_{∗} and δ > 0.

Corollary 4.4 If B(φ) = 𝒩′ ∩ ℳ holds for some φ with φ ∘ ℰ_{0} = φ, then B(ψ) = 𝒩′ ∩ ℳ holds for every normal faithful state ψ with ψ ∘ ℰ_{0} = ψ.

Proof. The same proof of [11, Corollary 1.5] works by using Proposition 4.3 and Connes-Størmer transitivity [7]. □

Proposition 4.5 Let 𝒩 ⊂ ℳ be a subfactor of type III_{1} with [ℳ : 𝒩] < ∞, and assume B(φ) = 𝒩′ ∩ ℳ. Then for any ε > 0, δ > 0 and any ψ_{1}, ⋯, ψ_{n} ∈ ℳ_{∗}, there exist u_{1}, ⋯, u_{m} ∈ U(𝒩) and λ_{1}, ⋯, λ_{m} > 0 such that ∑_{i=1}^{m} λ_{i} = 1, ∥[u_{i}, φ]∥ < δ and ∥ψ_{j} ∘ E_{𝒩′∩ℳ}^{φ} − ψ_{j} ∘ P ∥ < ε, j = 1, ⋯, n, where P (x) = ∑_{i} λ_{i}u_{i}^{∗}xu_{i}.

Proof. We prove by induction. When n = 1, the proposition is trivial by Proposition 4.3 (iv). Assume that the proposition holds for n − 1. Take ψ_{1}, ⋯, ψ_{n} ∈ ℳ_{∗}. By assumption of induction, for any ε′ > 0 and δ′ > 0, we can find u_{1}, ⋯, u_{m} ∈ U(𝒩) and λ_{1}, ⋯, λ_{m} > 0 satisfying the conclusion of proposition for ε′, δ′ and ψ_{1}, ⋯, ψ_{n−1}. Apply Proposition 4.3 (iv) to ψ_{n}∘P. Then we can find v_{1}, ⋯, v_{l} ∈ U(𝒩) and μ_{1}, ⋯, μ_{l} > 0 such that ∑ μ_{i} = 1, ∥[v_{i}, φ]∥ < δ′ and ∥ψ_{n} ∘ P ∘ E_{𝒩′∩ℳ}^{φ} − ψ_{n} ∘ P ∘ Q∥ < ε′, where Q(x) = ∑ μ_{j}v_{j}^{∗}xv_{j}. It is clear that P ∘ E_{𝒩′∩ℳ}^{φ} = E_{𝒩′∩ℳ}^{φ} holds, and hence we get ∥ψ_{n} ∘ E_{𝒩′∩ℳ}^{φ} − ψ_{n} ∘ P ∘ Q∥ < ε′. By the choice of {u_{i}} and {v_{j}}, ∥[u_{i}v_{j}, φ]∥ < 2δ′ holds. Next we will estimate ∥ψ_{i} ∘ E_{𝒩′∩ℳ}^{φ} − ψ_{i} ∘ P ∘ Q∥, 1 ≤ i ≤ n − 1. Since we have

∥ψ_{i} ∘ E_{𝒩′∩ℳ}^{φ} − ψ_{i} ∘ P ∘ Q∥
≤ ∥ψ_{i} ∘ (E_{𝒩′∩ℳ}^{φ} − E_{𝒩′∩ℳ}^{φ} ∘ Q)∥ + ∥ψ_{i} ∘ E_{𝒩′∩ℳ}^{φ} ∘ Q − ψ_{i} ∘ P ∘ Q∥
≤ ∥ψ_{i} ∘ (E_{𝒩′∩ℳ}^{φ} − E_{𝒩′∩ℳ}^{φ} ∘ Q)∥ + ε′,

we have to estimate ∥E_{𝒩′∩ℳ}^{φ} − E_{𝒩′∩ℳ}^{φ} ∘ Q∥. Let ∥ · ∥_{1} be a L^{1} norm on 𝒩′ ∩ ℳ for φ|_{𝒩′∩ℳ} . Note that the restriction of φ on 𝒩′ ∩ ℳ is a tracial state. Then we have

sup{|φ(yx)| | x ∈ 𝒩′ ∩ ℳ, ∥x∥ ≤ 1} = ∥y∥_{1}, y ∈ 𝒩′ ∩ ℳ.

On the other hand, we have the following inequality.

|φ((E_{𝒩′∩ℳ}^{φ} (y) − E_{𝒩′∩ℳ}^{φ} ∘ Q(y))x)| = |φ(yx − Q(y)x)|
= |φ(yx − ∑_{i} μ_{i}v_{i}^{∗}yxv_{i})|
≤ ∑_{i} μ_{i}|φ(yx) − v_{i}φv_{i}^{∗}(yx)|
= ∑_{i} μ_{i}|(φ − v_{i}φv_{i}^{∗})(yx)|
≤ δ′∥yx∥.

Combining these, we get ∥E_{𝒩′∩ℳ}^{φ} (y) − E_{𝒩′∩ℳ}^{φ} ∘ Q(y)∥_{1} ≤ δ′∥y∥. Since 𝒩′ ∩ ℳ is finite dimensional, there exists a constant 0 ≠ C such that ∥ · ∥_{1} ≥ C∥ · ∥. Hence we get ∥E_{𝒩′∩ℳ}^{φ} (y) − E_{𝒩′∩ℳ}^{φ} ∘ Q(y)∥ ≤ δ′/C∥y∥, and ∥E_{𝒩′∩ℳ}^{φ} − E_{𝒩′∩ℳ}^{φ} ∘ Q∥ ≤ δ′/C. Finally we get ∥ψ_{i} ∘ E_{𝒩′∩ℳ}^{φ} − ψ_{i} ∘ P ∘ Q∥ < max ∥ψ_{i}∥δ′/C + ε′. If we take ε′ and δ′ as 2δ′ < δ and max ∥ψ_{i}∥δ′/C + ε′ < ε, then {λ_{i}μ_{j}} and {u_{i}v_{j}} satisfy the conclusion of the proposition for n. □

Let 𝒩 ⊂ ℳ be an inclusion of AFD factors of type III_{1} with 𝒢_{ℳ,𝒩} = 𝒢̃_{ℳ,𝒩} , and ℰ_{k} the minimal conditional expectation from ℳ_{k} onto ℳ_{k−1}. We extend φ to ℳ_{k} by φ ∘ ℰ_{1} ∘ ⋯ ∘ ℰ_{k}, and denote by φ for simplicity.

Popa proved the following theorem in [32, Theorem 4.2].

Theorem 4.6 Let 𝒩 ⊂ ℳ be an inclusion of AFD type III_{1} factors, and φ_{0} be a dominant weight for ℳ with φ_{0} ∘ ℰ_{0} = φ_{0}. If we have 𝒩′_{φ_{0}} ∩ ℳ = 𝒩′ ∩ ℳ, then there exists a faithful normal state φ such that φ ∘ ℰ_{0} = φ_{0} and 𝒩_{φ} ∩ ℳ = 𝒩′ ∩ ℳ.

By applying the above theorem, we get 𝒩′ ∩ ℳ_{k} = B(φ) for every k.

Set ℳ_{∞} := (⋃_{k} ℳ_{k})^{−}, where closure is taken in the GNS representation with respect to φ.

Proposition 4.7 We can find {u_{n}^{k}}_{k=1}^{m_{n}} ⊂ U(𝒩) and {λ_{n}^{k}}_{k=1}^{m_{n}} ⊂ R_{+}^{∗} such that ∑_{k=1}^{m_{n}} λ_{n}^{k} = 1 for each n, sup_{k} ∥[u_{n}^{k}, φ]∥ ≤ 1/n and ψ∘P_{n} → ψ∘E_{𝒩′∩ℳ_{∞}}^{φ} , n → ∞, for every ψ ∈ (ℳ_{∞})_{∗}, where P_{n}(x) = ∑_{k=1}^{m_{n}} λ_{n}^{k}u_{n}^{k∗}xu_{n}^{k}.

Proof. Let E_{k}^{∞} be a φ preserving conditional expectation from ℳ_{∞} onto ℳ_{k}. By martingale convergence theorem [4, Lemma I.2], we have lim_{k→∞} ∥ψ − ψ ∘ E_{k}^{∞}∥ = 0. Let V_{k} be a dense countable subset of (ℳ_{k})_{∗}. We regard φ ∈ (ℳ_{k})_{∗} as the element of (ℳ_{∞})_{∗} via φ → φ ∘ E_{k+1}^{∞}. Set V := ⋃_{k} V_{k}. Then V is a dense countable subset of (ℳ_{∞})_{∗}. We index V as {ψ_{i}}_{i=1}^{∞} By the definition of V, there exists k > 0 such that ψ_{i} ∘ E_{k}^{∞} = ψ_{i}, 1 ≤ i ≤ n. We regard ψ_{i} as a normal functional on ℳ_{k}, and apply Proposition 4.5. Then we can find λ_{k}^{n} ⊂ R_{+}^{∗}, ∑_{k} λ_{k}^{n} = 1, and {u_{k}^{n}} ⊂ U(𝒩) such that ∥ψ_{i}∘P_{n}−ψ_{i}∘E_{𝒩′∩ℳ_{k}}^{φ} ∥ < 1/n and ∥[ψ_{i}, u_{k}^{n}]∥ ≤ 1/n, where P_{n} = ∑_{k} λ_{k}^{n}Ad u_{k}^{n}. Then we get ∥ψ_{i} ∘P_{n} −ψ_{i} ∘E_{𝒩′∩ℳ_{∞}}^{φ} ∥ < 1/n since E_{𝒩′∩ℳ_{k}}^{φ} ∘E_{k}^{∞} = E_{k}^{∞} ∘E_{𝒩′∩ℳ_{∞}}^{φ} and E_{k}^{∞} ∘P_{n} = P_{n} ∘E_{k}^{∞}. These {λ_{k}^{n}} and {u_{k}^{n}} are desired ones. □

By Proposition 4.7, P_{n}(x) converges to E_{𝒩′∩ℳ_{k}}^{φ} (x) σ-strongly* for x ∈ ℳ_{k}. This is the key in the following sections.

5 Symmetric enveloping algebras and Longo-Rehren inclusions

Now we discuss how to construct the symmetric enveloping algebra for a subfactor of type III_{1} 𝒩 ⊂ ℳ, since detail of construction in type III case is not presented in [34].

Let ℳ ⊃ 𝒩 =: ℳ_{−1} ⊃ ℳ_{−2} ⊃ ⋯ be a tunnel for 𝒩 ⊂ ℳ, e_{k} ∈ ℳ_{k} be the Jones projection for ℳ_{k−1} ⊃ ℳ_{k−2}, and ℰ_{k} be the minimal conditional expectation from ℳ_{k} on ℳ_{k−1}. We assume ℳ acts on L^{2}(ℳ), and identify ℳ^{opp} and ℳ′ = JℳJ via x^{opp} ↔ Jx^{∗}J. Set e_{k} := (e_{2−k})^{opp} for k ≥ 2. Then ℳ ⊂ ⋯ ⊂ ℳ_{k} := Alg⟨ℳ_{k−1}, e_{k}⟩ ⊂ ℳ_{k+1} ⊂ ⋯ is algebraically isomorphic to the Jones tower, and each ℳ_{k} is σ-weakly closed in B(L^{2}(ℳ)). Set B_{0} := ⋃_{k≥0} ℳ^{opp}ℳ_{k}ℳ^{opp} ⊂ B_{1} := C^{∗}(ℳ, e_{𝒩} , ℳ^{opp}) ⊂ B(L^{2}(ℳ)). Then B_{0} is a dense ∗-subalgebra of B_{1}. See [34].

We recall the construction of the symmetric enveloping algebra for type II_{1} case. By the relative Dixmier property [33] (also see [34, Appendix]), we can construct a conditional expectation E from B_{1} on ℳ^{opp} by xyz ∈ ℳ^{opp}ℳ_{k}ℳ^{opp} → xE_{ℳ′∩ℳ_{k}} (y)z ∈ ℳ^{opp}. Indeed for 0 = ∑_{i} x_{i}y_{i}z_{i} ∈ ℳ^{opp}ℳ_{k}ℳ^{opp}, we can choose {λ_{n}^{k}}_{k=1}^{m_{n}} ⊂ R_{+}^{∗}, ∑_{k} λ_{n}^{k} = 1, and {u_{n}^{k}}_{k=1}^{m_{k}} ⊂ U(ℳ) such that P_{n}(y_{i}) := ∑_{k=1}^{m_{n}} λ_{n}^{k}u_{n}^{k}^{∗}y_{i}u_{n}^{k} converges to E_{ℳ′∩ℳ_{k}} (y_{i}) in the norm topology for every i. (In fact, σ-strong convergence is enough in the following argument since ℳ_{k} is σ-weakly closed.) Then we have

∑_{i} x_{i}E_{ℳ′∩ℳ_{k}} (y_{i})z_{i} = lim_{n} ∑_{i} x_{i}P_{n}(y_{i})z_{i}
= lim_{n} ∑_{i} x_{i}(∑_{k} λ_{n}^{k}u_{n}^{k}^{∗}y_{i}u_{n}^{k})z_{i}
= lim_{n} ∑_{k} λ_{n}^{k}u_{n}^{k}^{∗} ∑_{i}(x_{i}y_{i}z_{i})u_{n}^{k}
= 0,

so E is well-defined. Then it is shown that tr_{ℳ^{opp}} ∘ E is a unique tracial state for B_{1}. Then by GNS construction, we get the symmetric enveloping algebra as ℳ ⊠_{e_{𝒩}} ℳ^{opp} :=

π_{tr_{ℳ^{opp}} ∘E}(B_{1})′′, which is a II_{1} factor due to the uniqueness of a tracial state.

In the type III case, it is not clear that we have such E. However in the case 𝒩 ⊂ ℳ is a strongly amenable subfactor of type III_{1} with 𝒢_{ℳ,𝒩} = 𝒢̃_{ℳ,𝒩} , we can use Proposition 4.7 instead of the relative Dixmier property. Hence xyz → xE_{ℳ′∩ℳ_{k}}^{φ} (y)z is a well defined map from B_{0} onto ℳ^{opp}. This map can be extended to a conditional expectation E from B_{1} onto ℳ^{opp}. Then we get a state ψ on B_{1} by ψ(x) := ⟨E(x)ξ_{φ}, ξ_{φ}⟩ , where ξ_{φ} ∈ L^{2}(ℳ)_{+} is the representing vector of φ. As in type II_{1} case, we get the symmetric enveloping algebra ℳ ⊠_{e_{𝒩}} ℳ^{opp} := π_{ψ}(B_{1})′′. But it is not so clear that this is a factor.

Here we already have (another) similar construction, the Longo-Rehren construction. We compare the above construction and the Longo-Rehren construction.

Let (A :=)ℳ ⊗ ℳ^{opp} ⊂ B be the Longo-Rehren inclusion arising from 𝒩 ⊂ ℳ. As explained in the preliminary, we can construct the common Jones projection e_{𝒩} in B. Once we have the common Jones projection, we can construct a Jones tower within B as in the symmetric enveloping algebra case. Then B_{0} = ⋃_{k} ℳ^{opp}ℳ_{k}ℳ^{opp} is a dense ∗-subalgebra of B.

Here φ ⊗ φ^{opp} ∘ ℰ(xyz) = φ^{opp}(xE_{ℳ′∩ℳ_{k}}^{φ} (y)z), x, z ∈ ℳ^{opp}, y ∈ ℳ_{k}, holds. (Since ℳ′ ∩ B = ℳ^{opp}, E_{ℳ′∩ℳ_{k}}^{φ} (y) ∈ ℳ^{opp}.) This is because

|φ ⊗ φ^{opp} ∘ ℰ(xyz) − φ ⊗ φ^{opp} ∘ ℰ(xP_{n}(y)z)|
= |φ ⊗ φ^{opp} ∘ ℰ(xyz) − ∑_{k} λ_{n}^{k}φ ⊗ φ^{opp} ∘ ℰ(xu_{n}^{k}^{∗}yu_{n}^{k}z)|
≤ ∑_{k} λ_{n}^{k}|φ ⊗ φ^{opp} ∘ ℰ(xyz) − u_{n}^{k}φu_{n}^{k∗} ⊗ φ^{opp} ∘ ℰ(xyz)|
= ∑_{k} λ_{n}^{k}|(φ − u_{n}^{k}φu_{n}^{k∗}) ⊗ φ^{opp} ∘ ℰ(xyz)|
≤ \frac{1}{n}∥ℰ(xyz)∥ → 0

if n → ∞.

Since B_{0} is dense in B, π_{φ⊗φ^{opp}∘ℰ}(B_{0})′′ = π_{φ⊗φ^{opp}∘ℰ}(B)′′ = B. On B_{0}, φ ⊗ φ^{opp} ∘ ℰ = φ^{opp} ∘ E = ψ, and hence we get π_{φ⊗φ^{opp}∘ℰ}(B_{0})′′ = π_{ψ}(B_{0})′′ = π_{ψ}(B_{1})′′.

We summarize the above argument as follows.

Theorem 5.1 Let ψ be a state on C^{∗}(ℳ, e_{𝒩} , JℳJ) defined as above. Let π_{ψ} be the GNS representation of C^{∗}(ℳ, e_{𝒩} , JℳJ) via ψ. Then π_{ψ}(C^{∗}(ℳ, e_{𝒩} , JℳJ))′′ is isomorphic to the Longo-Rehren inclusion for 𝒩 ⊂ ℳ.

Here we make a remark. Let W_{j} be the canonical implementing isometry for ρ_{j} defined as in [10, Appendix A]. Then {d(j)W_{j}}_{j∈Δ} satisfy the same relation of V_{j}’s in the Longo-Rehren construction by [10, Proposition A.4]. We also have the formula e_{𝒩} = [ℳ : 𝒩]^{−1} ∑_{i∈I_{0}} \sqrt{d(i)}ã_{i}^{∗}d(i)W_{i} with the same notations in Section 2. (Here we have to replace j_{ℳ} (x) by JxJ). We can construct a conditional expectation from C^{∗}(ℳ, e_{𝒩} , JℳJ) onto C^{∗}(ℳ, JℳJ) as follows.

First we show the following lemma. (In the following, we use notations in Section 2.4.)

Lemma 5.2 For every x ∈ M and 0 ≠ j ∈ Δ_{0}, we have E_{ℳ′∩ℳ_{1}}^{φ} (xw_{j}^{e∗}e_{𝒩} w_{0}^{f}) = 0.

Proof. We verify φ ∘ ℰ_{1}(yxw_{j}^{e∗}e_{𝒩} w_{0}^{f}) = 0 for every y ∈ ℳ′ ∩ ℳ_{1}, which implies the conclusion. First note that there exists a unique z ∈ 𝒩′ ∩ ℳ such that ze_{𝒩} = ye_{𝒩} by the push-down lemma. Then we have

φ ∘ ℰ_{1}(yxw_{j}^{e∗}e_{𝒩} w_{0}^{f}) = φ ∘ ℰ_{1}(xw_{j}^{e∗}ye_{𝒩} w_{0}^{f})
= φ ∘ ℰ_{1}(xw_{j}^{e∗}ze_{𝒩} w_{0}^{f})
= [ℳ : 𝒩]^{−1}φ ∘ ℰ_{1}(xw_{j}^{e∗}zw_{0}^{f}).

It is easy to see w_{j}^{e∗}zw_{0}^{f} ∈ (id, ρ_{j}). Hence w_{j}^{e∗}zw_{0}^{e} = 0 if j ≠ 0. □ We compute ψ(xJyJW_{j}) for x, y ∈ ℳ. As explained in Section 2.4, we have

W_{j} = Cw̃_{j}^{∗}e_{𝒩} w̃_{0} = C ∑_{e,f} w_{j}^{e∗}Jw_{j}^{e∗}Je_{𝒩} w_{0}^{f}Jw_{0}^{f}J

for some nonzero constant C.

Hence for 0 ≠ j ∈ Δ_{0}, we have

ψ(xJyJW_{j}) = Cψ(xJyJ ∑_{e,f} w_{j}^{e∗}Jw_{j}^{e∗}Je_{𝒩} w_{0}^{f}Jw_{0}^{f}J)
= C ∑_{e,f} ψ(Jyw_{j}^{e∗}Jxw_{j}^{e∗}e_{𝒩} w_{0}^{f}Jw_{0}^{f}J)
= C ∑_{e,f} φ^{opp}(Jyw_{j}^{e∗}JE_{ℳ′∩ℳ_{1}}^{φ} (xw_{j}^{e∗}e_{𝒩} w_{0}^{f})Jw_{0}^{f}J)
= 0.

In a similar way, we can show ψ(xJyJW_{j}) = 0 for all 0 ≠ j ∈ Δ. (Consider ℳ_{−k} ⊂ ℳ ⊂ ℳ_{k} instead of 𝒩 ⊂ ℳ ⊂ ℳ_{1}.) Note that ψ(xJyJ) = φ(x)φ^{opp}(JyJ) = φ ⊗_{alg} φ^{opp}(x ⊗ JyJ).

Take a ∈ Alg(ℳ, JℳJ, {W_{j}}_{j∈Δ}) and assume a = 0. Then a has the form a = ∑_{i} x_{i}W_{i}, x_{i} ∈ Alg(ℳ, JℳJ). Hence a^{∗}a = ∑_{i,j,k} v̂_{ī,i}^{0∗}ρī(x_{i}^{∗}x_{j})v̂_{ī,j}^{k}W_{k} holds. By the above argument, 0 = ψ(a^{∗}a) = φ ⊗_{alg} φ^{opp}(∑_{i} v̂_{ī,i}^{0∗}ρ̂ī(x_{i}^{∗}x_{i})v̂_{ī,i}^{0}). Thus we get ρ̂ī(x_{i})v̂_{ī,i}^{0} = 0 for every i, and we get x_{i} = x_{i}v̂_{i,ī}^{0∗}ρ̂_{i}(v̂_{ī,i}^{0}) = v̂_{i,ī}^{0∗}ρ̂_{i}(ρ̂ī(x_{i})v̂_{ī,i}^{0}) = 0. Hence a map ℰ

ℰ : Alg(ℳ, e_{𝒩} , JℳJ) ∋ a = ∑_{i} a_{i}W_{i} → a_{0} ∈ Alg(ℳ, JℳJ)

is well-defined.

This also shows ψ = φ ⊗_{alg} φ^{opp} ∘ ℰ on Alg(ℳ, e_{𝒩} , JℳJ) = Alg(ℳ, JℳJ, {W_{j}}_{j∈Δ}).

6 Effros-Lance type characterization of amenability for subfactors.

In [34, Theorem 8.1], Popa gave a following subfactor analogue of the Effros-Lance characterization of amenability for subfactors.

Theorem 6.1 Let 𝒩 ⊂ ℳ be an inclusion of AFD factors of type II_{1}. Then the inclusion is amenable if and only if C^{∗}(ℳ, e_{𝒩} , JℳJ) ≅ C_{min}^{∗}(ℳ, e_{𝒩} , ℳ^{opp}). Here C_{min}^{∗}(ℳ, e_{𝒩} , ℳ^{opp}) means a C^{∗}-subalgebra of ℳ ⊠_{e_{𝒩}} ℳ^{opp} generated by ℳ, e_{𝒩} and ℳ^{opp}.

In this section, we prove the following theorem.

Theorem 6.2 Let 𝒩 ⊂ ℳ be a strongly amenable inclusion of AFD factors of type III_{1} with 𝒢_{ℳ,𝒩} = 𝒢̃_{ℳ,𝒩} . Then we have C^{∗}(ℳ, e_{𝒩} , JℳJ) ≅ C_{min}^{∗}(ℳ, e_{𝒩} , ℳ^{opp}).

We need some preparations for proof.

Proposition 6.3 Let 𝒩̃ ⊂ ℳ̃ be a strongly amenable subfactor of type II_{∞}. Then we have C^{∗}(ℳ̃, e𝒩̃, Jℳ̃J) ≅ C_{min}^{∗}(ℳ̃, e𝒩̃, ℳ̃^{opp}).

Proof. Let A ⊂ B a type II_{1} part of 𝒩̃ ⊂ ℳ̃. Hence we have the decomposition 𝒩̃ ⊂ ℳ̃ = A ⊗ B(l^{2}) ⊂ B ⊗ B(l^{2}). Then by [31], A ⊂ B has the generating property. Hence there exists a choice of a tunnel B ⊃ A ⊃ A_{1} ⊃ A_{2} ⊃ ⋯ such that B = ∨_{k}(A′_{k} ∩B) and A = ∨_{k}(A′_{k} ∩ A). Fix a = ∑_{i} x_{i}y_{i}z_{i} ∈ C^{∗}(ℳ̃, e𝒩̃, Jℳ̃J), x_{i}, z_{i} ∈ Jℳ̃J, y_{i} ∈ ℳ̃_{n}. Let E_{k} be the trace preserving conditional expectation from B_{n} onto A′_{k} ∩ B_{n}. Then lim_{k} E_{k}(x) = x holds in the strong topology. Then a_{k} := ∑_{i} x_{i}E_{k} ⊗ id(y_{i})z_{i} converges to a in the strong topology. Note ∥a_{k}∥_{min} ≤ ∥a∥_{min}, because xyz ∈ A^{opp}B_{n}A^{opp} → xE_{k}(y)z is a conditional expectation. See the argument in the previous section. Here E_{k} ⊗id(y_{i}) ∈ (A′_{k} ∩ B) ⊗ B(l^{2}) ⊂ (JB_{k}J ⊗ C1) ∨ (C1 ⊗ B(l^{2})). Hence a_{k} ∈ (C1 ⊗ B(l^{2})) ∨ Jℳ̃_{k}J. Since B(l^{2}) is semidiscrete, ∥a_{k}∥_{B(L^{2}(ℳ̃))} = ∥a_{k}∥_{min} ≤ ∥a∥_{min}. By lower semicontinuity of ∥ · ∥_{B(L^{2}(ℳ̃))} in the strong topology, ∥a∥_{B(L^{2}(ℳ̃))} ≤ ∥a∥_{min} holds. □

The minimal tensor product ⊗_{min} has the following property. If we have two inclusions of C^{∗}-algebras A_{1} ⊂ A_{2} and B_{1} ⊂ B_{2}, we have an inclusion A_{1} ⊗_{min} B_{1} ⊂ A_{2} ⊗_{min} B_{2}. The next lemma can be considered as an analogue of the above fact for symmetric enveloping algebras. (Also see [34, Proposition 2.10].)

Lemma 6.4 Let 𝒩 ⊂ ℳ be as in Theorem 6.2, and 𝒩̃ ⊂ ℳ̃ := 𝒩 ⋊_{σ^{φ}} R ⊂ ℳ ⋊_{σ^{φ}} R the continuous decomposition. Then we have an inclusion ℳ ⊠_{e_{𝒩}}ℳ^{opp} ⊂ ℳ̃ ⊠_{e_{𝒩}} ℳ̃^{opp}. Especially

we have an inclusion C_{min}^{∗}(ℳ, e_{𝒩} , ℳ^{opp}) ⊂ C_{min}^{∗}(ℳ̃, e_{𝒩} , ℳ̃^{opp}).

Proof. We will prove that the Longo-Rehren subfactor arising from 𝒩 ⊂ ℳ can be embedded into the Longo-Rehren subfactor arising from 𝒩̃ ⊂ ℳ̃.

Let Δ (resp. Δ̃) be the system of ℳ-ℳ sectors (resp. ℳ̃-ℳ̃ sectors) arising from 𝒩 ⊂ ℳ (resp. 𝒩̃ ⊂ ℳ̃). For ρ ∈ End(ℳ), we denote by ρ̃ the canonical extension of ρ in the sense of [17]. Since 𝒢_{ℳ,𝒩} = 𝒢̃_{ℳ,𝒩} , ρ̃ is an irreducible endomorphism by [15, Theorem 3.5], and [ρ̃] ∈ Δ̃ for [ρ] ∈ Δ. Hence we have Δ̃ = {[ρ̃] | [ρ] ∈ Δ}. Moreover we have (ρ, σπ) = (ρ̃, σ̃π̃) for [ρ], [σ], [π] ∈ Δ. From these facts, we have the desired embedding. □

To construct the Jones tower in the symmetric enveloping algebra, we have to fix a tunnel for 𝒩 ⊂ ℳ. For the compatibility of choice of a tunnel and the continuous crossed product we need the following lemma due to Popa.

Lemma 6.5 ([32, Lemma 2.10]) Let 𝒩 ⊂ ℳ be a subfactor of type III_{1} with finite index, and φ a faithful normal state such that φ ∘ ℰ_{0} = φ. Then for any ε > 0, we can choose a tunnel for 𝒩 ⊂ ℳ such that φ ∘ ℰ_{−j} ∘ ⋯ ∘ ℰ_{−1} ∘ ℰ_{0} converges to φ_{0} and ∥φ − φ_{0}∥ < ε.

Remark. The state φ_{0} in Lemma 6.5 is faithful. See [32, p.34].

Let ⋯ ⊂ ℳ_{i} ⊂ ⋯ ⊂ ℳ_{−1} := 𝒩 ⊂ ℳ_{0} := ℳ be a tunnel in the above lemma. We take a continuous crossed product with respect to φ_{0} in the above lemma. Let λ_{t} ∈ ℳ̃ be an implementing unitary. Set ℳ̃_{i} := ℳ_{i} ∨ {λ_{t}}. Then ⋯ ⊂ ℳ̃_{i} ⊂ ⋯ ⊂ 𝒩̃ ⊂ ℳ̃ is a tunnel for 𝒩̃ ⊂ ℳ̃, and we have a natural embedding ℳ_{k} ⊂ ℳ̃_{k} in the symmetric enveloping algebra. Proof of Theorem 6.2. The argument is similar to [36].

By the definition of the crossed product, ℳ̃ is generated by π(ℳ) and {λ_{t}}, where π is defined by (π(a)ξ)(t) = σ_{−t}^{φ_{0}}(a)ξ(t), and (λ_{t}ξ)(s) = ξ(s − t) for ξ ∈ L^{2}(R, L^{2}(ℳ)). Let π′ be an embedding of ℳ′ into ℳ̃′ given by π′(a) = a⊗id. Here note that π can be extended to ℳ_{k} as (π(a)ξ)(t) = σ_{−t}^{φ_{0}}(a)ξ(t) due to Lemma 6.5. Fix ∑_{i} x_{i}y_{i}z_{i} ∈ ℳ^{opp}ℳ_{n}ℳ^{opp} ⊂ C^{∗}(ℳ, e_{𝒩} , JℳJ). By Proposition 6.3 and Lemma 6.4, we have

∥∑_{i} π′(x_{i})π(y_{i})π′(z_{i})∥_{B(L^{2}(ℳ̃))} ≤ ∥∑_{i} π′(x_{i})π(y_{i})π(z_{i})∥_{min} = ∥∑_{i} x_{i}y_{i}z_{i}∥_{min} .

Take ξ ∈ L^{2}(ℳ), and f(t) ∈ C_{0}(R). We have

∫_{R} ∥∑_{i} x_{i}σ_{−t}^{φ_{0}}(y_{i})z_{i}ξ∥_{L^{2}(ℳ)}^{2} |f(t)|^{2}dt = ∥∑_{i} π′(x_{i})π(y_{i})π′(z_{i})(ξ ⊗ f)∥_{L^{2}(ℳ̃)}^{2}

≤ ∥∑_{i} π′(x_{i})π(y_{i})π′(z_{i})∥_{B(L^{2}(ℳ̃))}^{2} ∥ξ∥_{L^{2}(ℳ)}^{2} ∥f∥_{L^{2}(R)}^{2}

= ∥∑_{i} x_{i}y_{i}z_{i}∥_{min}^{2} ∥ξ∥_{L^{2}(ℳ)}^{2} ∥f∥_{L^{2}(R)}^{2}.

Since ∥ ∑_{i} x_{i}σ_{−t}^{φ_{0}}(y_{i})z_{i}ξ∥_{L^{2}(ℳ)} is continuous for t, we get

∥∑_{i} x_{i}σ_{−t}^{φ_{0}}(y_{i})z_{i}ξ∥ ≤ ∥∑_{i} x_{i}y_{i}z_{i}∥_{min} ∥ξ∥_{L^{2}(ℳ)} .

If we put t = 0, then we get

∥∑_{i} x_{i}y_{i}z_{i}ξ∥_{L^{2}(ℳ)} ≤ ∥∑_{i} x_{i}y_{i}z_{i}∥_{min} ∥ξ∥.

□

In the end of this section, we discuss the extension of automorphisms to the symmetric enveloping algebra. Let 𝒩 ⊂ ℳ be as in Theorem 6.2. Take α ∈ Ker Φ. Then we can extend α ⊗ id ∈ Aut(ℳ ⊗ ℳ^{opp}) to α ⊠ id ∈ Aut(ℳ ⊠_{e_{𝒩}}ℳ^{opp}) such that α ⊠ id(e_{𝒩} ) = e_{𝒩} as

explained in section 2. It is clear that α ⊠ id is an automorphism of C_{min}^{∗}(ℳ, e_{𝒩} , ℳ^{opp}). Hence by Theorem 6.2, we have α̃ ∈ Aut(C^{∗}(ℳ, e_{𝒩} , JℳJ)) such that α̃ = α on ℳ, α̃ = id on ℳ′ and α̃(e_{𝒩} ) = e_{𝒩} . Especially modular automorphisms have the trivial Loi invariant, and hence we can apply the above argument for modular automorphisms. Here we remark that α̃(e_{k}) = e_{k} for k ≥ 1 since α̃ is trivial on JℳJ. Hence α̃ preserves ℳ_{k}, and this coincides with the usual extension of α to ℳ_{k}. So we denote α̃|_{ℳ_{k}} by α.

7 Modular automorphisms

As an analogue of [6, Theorem III.1], approximately inner automorphisms can be characterized as follows. (Also see appendix.)

Theorem 7.1 For θ ∈ Aut(ℳ, 𝒩), θ ∈ \overline{Int}(ℳ, 𝒩) if and only if for any φ_{1}, ⋯, φ_{n} ∈ (ℳ_{∗})_{+} and ε > 0, we can find 0 ≠ x ∈ 𝒩 such that

∥xξ_{φ_{j}} − θ(ξ_{φ_{j}})x∥^{2} ≤ ε^{2} ∑_{j} φ_{j}(x^{∗}x),

where ξ_{φ_{j}} ∈ L^{2}(ℳ)_{+} is a representing vector for φ_{j}.

By using Theorem 7.1, we will prove the following theorem.

Theorem 7.2 For any 0 ≠ T ∈ R and any faithful normal state ψ with ψ ∘ ℰ_{0} = ψ, σ_{T}^{ψ} is approximately inner.

As remarked in the end of Section 3, this implies Theorem 1.1. Proof. Fix ε > 0 and φ_{1}, ⋯, φ_{n} ∈ (ℳ_{∗})_{+}. We may assume φ_{1} is faithful. Set φ := [ℳ : 𝒩] ∑_{j} φ_{j} ∘ ℰ_{0}. Then φ ∘ ℰ_{0} = φ and φ ≥ φ_{j} hold. For the last inequality, we used the Pimsner-Popa inequality ℰ_{0}(x) ≥ [ℳ : 𝒩]^{−1}x [28]. For simplicity, we denote ξ_{φ_{j}} by ξ_{j}. Put α = σ_{T}^{φ}. Define T_{j} as T_{j}xξ_{φ} = xξ_{j}. Then ∥T_{j}∥ ≤ 1, and T_{j} belongs to ℳ′. Set b_{j} := JT_{j}^{∗}J ∈ ℳ. Then b_{j}^{∗}ξ_{φ} = ξ_{φ}b_{j} = ξ_{j} holds by definition. We have ∥b_{j}∥ ≤ 1. Set A := C^{∗}(ℳ, e_{𝒩} , JℳJ). Define X := e_{𝒩} ∑_{j} |Jb_{j}^{∗}J − b_{j}^{∗}|^{2}e_{𝒩} ∈ A.

Fix f(x) ∈ C_{c}^{∞}(R_{+}^{∗}) as in [6, p.205]. Then 0 ≤ 1−e_{𝒩} +X +e_{𝒩} |f(Δ_{φ})−1|^{2}e_{𝒩} ≤ 4n+2 holds. Here note that e_{𝒩} ξ_{φ} = ξ_{φ}, Xξ_{φ} = 0 and e_{𝒩} |f(Δ_{φ}) − 1|^{2}e_{𝒩} ξ_{φ} = 0. Hence we get ∥4n + 2 − (1 − e_{𝒩} ) − X − e_{𝒩} |f(Δ_{φ}) − 1|^{2}e_{𝒩} ∥ = 4n + 2.

Let α̃ be an automorphism of A as in the end of the last section. We assume ∥4n+2− (1 − e_{𝒩} ) − α̃(X) − e_{𝒩} |f(Δ_{φ}) − 1|^{2}e_{𝒩} ∥ = 4n + 2. Then we can find a unit vector η ∈ L^{2}(ℳ) such that

(i) ∥(Jb_{j}^{∗}J − α(b_{j})^{∗})e_{𝒩} η∥ ≤ ε, (ii) ∥(f(Δ_{φ}) − 1)e_{𝒩} η∥ ≤ ε, (iii) ∥(1 − e_{𝒩} )η∥ ≤ ε.

Set η_{1} := e_{𝒩} η. Then η_{1} ∈ e_{𝒩} L^{2}(ℳ) = L^{2}(𝒩). By (iii), 1 ≥ ∥η_{1}∥ ≥ 1 − ε. Next set η_{2} := f(Δ_{φ})η_{1}. Then η_{2} ∈ 𝒟(Δφ^{\frac{1}{2}}) ∩ L^{2}(𝒩). By (ii), we have ∥η_{2} − η_{1}∥ ≤ ε, and hence we get 1 ≥ ∥η_{2}∥ ≥ 1 − 2ε. By (i), we get ∥(Jb_{j}^{∗}J − α(b_{j}^{∗}))η_{2}∥ ≤ 3ε. By the choice of f(x),

∥(Δφ^{\frac{1}{2}} − 1)η_{2}∥ ≤ ε∥η_{2}∥ ≤ ε. Since 𝒩ξ_{φ} is dense in 𝒟(Δφ^{\frac{1}{2}}) ∩ L^{2}(𝒩) in the graph norm,

we can choose xξ_{φ} ∈ 𝒩ξ_{φ} such that ∥η_{2} − xξ_{φ}∥ ≤ ε and ∥Δφ^{\frac{1}{2}}η_{2} − Δφ^{\frac{1}{2}}xξ_{φ}∥ ≤ ε. Then

∥xξ_{φ}∥ ≥ ∥η_{2}∥ − ∥η_{2} − xξ_{φ}∥ ≥ 1 − 3ε holds. Thus we get

∥(Jb_{j}^{∗}J − α(b_{j}^{∗}))xξ_{φ}∥ ≤ ∥(Jb_{j}^{∗}J − α(b_{j}^{∗}))η_{2}∥ + ∥(Jb_{j}^{∗}J − α(b_{j}))(η_{2} − xξ_{φ})∥
≤ 3ε + 2ε
= 5ε
≤ \frac{5ε}{1 − 3ε}∥xξ_{φ}∥

and

∥xξ_{φ} − ξ_{φ}x∥ = ∥xξ_{φ} − Δφ^{\frac{1}{2}}xξ_{φ}∥
≤ ∥xξ_{φ} − η_{2}∥ + ∥η_{2} − Δφ^{\frac{1}{2}}η_{2}∥ + ∥Δφ^{\frac{1}{2}}η_{2} − Δφ^{\frac{1}{2}}xξ_{φ}∥
≤ 3ε
= \frac{3ε}{1 − 3ε}∥xξ_{φ}∥.

Hence if we take sufficiently small ε, we can assume ∥xξ_{φ} − ξ_{φ}x∥ ≤ ε/2∥xξ_{φ}∥ and ∥(Jb_{j}^{∗}J − α(b_{j}^{∗}))xξ_{φ}∥ ≤ ε/2∥xξ_{φ}∥.

Finally we have

∥xξ_{j} − α(ξ_{j})x∥ = ∥xξ_{φ}b_{j} − α(b_{j}^{∗}ξ_{φ})x∥
= ∥Jb_{j}^{∗}Jxξ_{φ} − α(b_{j}^{∗})ξ_{φ}x∥
≤ ∥Jb_{j}^{∗}Jxξ_{φ} − α(b_{j}^{∗})xξ_{φ}∥ + ∥α(b_{j}^{∗})(xξ_{φ} − ξ_{φ}x)∥
≤ ε∥xξ_{φ}∥.

Hence we have ∥xξ_{j} − α(ξ_{j})x∥^{2} ≤ ε^{2}∥xξ_{φ}∥^{2} = ε^{2}φ(x^{∗}x) = ε^{2}[ℳ : 𝒩] ∑_{j} φ_{j} ∘ ℰ_{0}(x^{∗}x) = ε^{2}[ℳ : 𝒩] ∑_{j} φ_{j}(x^{∗}x). By Theorem 7.1, α is approximately inner. □

In the above proof, we assumed ∥4n+2−(1−e_{𝒩} )−α̃(X)−e_{𝒩} |f(Δ_{φ})−1|^{2}e_{𝒩} ∥ = 4n+2. Hence we have to verify this. To do this, we use the symmetric enveloping algebra for 𝒩 ⊂ ℳ, so we fix a tunnel for 𝒩 ⊂ ℳ and φ_{0} ∈ (ℳ_{∗})_{+} as in Lemma 6.5.

Lemma 7.3 Let K be a compact metric space, μ a probability measure on K, a(k), c(k) bounded σ-strongly* continuous maps from K to ℳ′, and b(k) a bounded σ-strongly* continuous map from K to ℳ_{n}, and consider ∫_{K} a(k)b(k)c(k)dμ(k) ∈ B(L^{2}(ℳ)).

(i) The set B ⊂ B(L^{2}(ℳ)) of the elements of the above form, changing K, μ and n, is a ∗-subalgebra of B(L^{2}(ℳ)). (ii) B is invariant under Ad Δ_{φ}^{it}.

Proof. (i) It is easy to see that B is closed under summation and ∗-operation. We verify that B is closed under multiplication. Let a_{i}(k_{i}), c_{i}(k_{i}) be bounded σ-strongly* continuous maps from K_{i} to ℳ^{opp}, and b_{i}(k_{i}) a bounded σ-strongly* continuous map from K_{i} to ℳ_{n}, i = 1, 2, and consider ∫_{K_{i}} a_{i}(k)b_{i}(k_{i})c_{i}(k_{i})dμ_{i}(k_{i}), i = 1, 2. Let f_{n} ∈ ℳ^{opp} be the Jones projection for ℳ_{−2n}^{opp} ⊂ ℳ_{−n}^{opp} ⊂ ℳ^{opp}, and choose v ∈ ℳ^{opp} such that v^{∗}f_{n}v = 1. Note ℳ_{n} ∨ {f_{n}} = ℳ_{2n}. Then

c_{1}(k_{1})a_{2}(k_{2}) = c_{1}(k_{1})a_{2}(k_{2})v^{∗}f_{n}v
= [ℳ : 𝒩]^{2}ℰ_{ℳ_{−n}^{opp}} (c_{1}(k_{1})a_{2}(k_{2})v^{∗}f_{n})f_{n}ℰ_{ℳ_{−n}^{opp}} (f_{n}v)

holds by the push-down lemma. Set d(k_{1}, k_{2}) := [ℳ : 𝒩]^{2}ℰ_{ℳ_{−n}^{opp}} (c_{1}(k_{1})a_{2}(k_{2})v^{∗}f_{n}), and v′ := ℰ_{ℳ_{−n}^{opp}} (f_{n}v). Then d(k_{1}, k_{2}) is a σ-strongly* continuous map from K_{1} × K_{2} to ℳ_{−n}^{opp}.

Then we have

(∫_{K_{1}} a_{1}(k_{1})b_{1}(k_{1})c_{1}(k_{1})dμ_{1}(k_{1})) (∫_{K_{2}} a_{2}(k_{2})b_{2}(k_{2})c_{2}(k_{2})dμ_{2}(k_{2}))
= ∫_{K_{1}×K_{2}} a_{1}(k_{1})b_{1}(k_{1})c_{1}(k_{1})a_{2}(k_{2})b_{2}(k_{2})c_{2}(k_{2})d(μ_{1} × μ_{2})(k_{1}, k_{2})
= ∫_{K_{1}×K_{2}} a_{1}(k_{1})b_{1}(k_{1})d(k_{1}, k_{2})f_{n}v′b_{2}(k_{2})c_{2}(k_{2})d(μ_{1} × μ_{2})(k_{1}, k_{2})
= ∫_{K_{1}×K_{2}} a_{1}(k_{1})d(k_{1}, k_{2})b_{1}(k_{1})f_{n}b_{2}(k_{2})v′c_{2}(k_{2})d(μ_{1} × μ_{2})(k_{1}, k_{2}).

Here a_{1}(k_{1})d(k_{1}, k_{2}) and v′c_{2}(k_{2}) are σ-strongly* continuous maps from K_{1} × K_{2} to ℳ^{opp}, and b_{1}(k_{1})f_{n}b_{2}(k_{2}) is a σ-strongly* continuous map from K_{1} × K_{2} to ℳ_{2n}. Hence the above operator is in B.

(ii) By the choice of φ_{0} and a tunnel for 𝒩 ⊂ ℳ, we have Ad Δ_{φ_{0}}^{it}ℳ_{n} = ℳ_{n} for every n ≥ 0. We have Δ_{φ}^{it} = u_{t}Ju_{t}JΔ_{φ_{0}}^{it}, where u_{t} := (Dφ : Dφ_{0})_{t}. Let b(k) be a σ-strongly* continuous map from K to ℳ_{n}. Then we have Ad Δ_{φ}^{it}(b(k)) = Ju_{t}Ju_{t}Ad Δ_{φ_{0}}^{it}(b(k))u_{t}^{∗}Ju_{t}^{∗}J. Here Ad u_{t}Δ_{φ_{0}}^{it}(b(k)) is a σ-strongly∗ continuous map from K to ℳ_{n}. Now it is easy to see that Ad Δ_{φ}^{it} preserves B. □

Lemma 7.4 Define 𝒜 := {T ∈ B | t → Δ_{φ}^{it}T Δ_{φ}^{−it} is norm continuous.}^{−∥·∥}.

(i) 𝒜 is a C^{∗}-algebra. (ii) θ_{t} := Ad Δ_{φ}^{it} is a pointwise norm continuous action of R on 𝒜. (iii) For a ∈ C^{∗}(ℳ, e_{𝒩} , ℳ^{opp}) and f ∈ C_{c}(R), ∫_{R} f(t)Δ_{φ}^{it}aΔ_{φ}^{−it}dt is in 𝒜.

Proofs of (i) and (ii) are same as [6, Lemma IV.3]. To prove (iii), we only have to prove (iii) holds for abc, a, c ∈ ℳ^{opp} and b ∈ ℳ_{n}. We have

Ad Δ_{φ}^{it}(abc) = Ad Δ_{φ}^{it}(a)Ad Δ_{φ}^{it}(b)Ad Δ_{φ}^{it}(c)
= Ad Δ_{φ}^{it}(a)Ju_{t}Ju_{t}Ad Δ_{φ_{0}}^{it}(b)u_{t}^{∗}Ju_{t}JAd Δ_{φ}^{it}(c).

Here Ad Δ_{φ}^{it}(a)Ju_{t}J, Ju_{t}^{∗}JAd Δ_{φ}^{it}(c) ∈ ℳ^{opp} and Ad u_{t}Δ_{φ_{0}}^{it}(b) ∈ ℳ_{n} are σ-strongly* continuous maps. Hence ∫_{R} f(t)Ad Δ_{φ}^{it}abcdt ∈ B. It is clear that this operator is in 𝒜. □

Let ℌ be the standard Hilbert space for ℳ ⊠_{e_{𝒩}} ℳ^{opp}. Note that L^{2}(ℳ ⊗ ℳ^{opp}) =

L^{2}(ℳ) ⊗ L^{2}(ℳ^{opp}) is a subspace of ℌ. and φ ⊗ φ^{opp} ∘ ℰ is given by a vector state for ξ_{φ} ⊗ ξ_{φ^{opp}}. As explained in Section 2, we have extensions σ_{t}^{φ} ⊠ id and σ_{t}^{φ} ⊠ σ_{−t}^{φ^{opp}}. Let Δ_{φ}^{it} ⊠id and Δ_{φ}^{it} ⊠Δ_{φ^{opp}}^{−it} be the standard implementing unitaries for σ_{t}^{φ} ⊠id and σ_{t}^{φ} ⊠σ_{−t}^{φ^{opp}} respectively.

Lemma 7.5 (A) There exists a sequence {ξ_{ν}} ⊂ ℌ such that (1) ∥Δ_{φ}^{it}⊠Δ_{φ^{opp}}^{−it}ξ_{ν} −ξ_{ν}∥ → 0 for any t ∈ R and (2) ⟨abcξ_{ν}, ξ_{ν}⟩ → ⟨abcξ_{φ}, ξ_{φ}⟩ for any a, c ∈ ℳ^{opp} and b ∈ ℳ_{k}. (B) There exists a sequence {Ψ_{ν}} of normal states on B(L^{2}(ℳ)) such that (1) Ψ_{ν}(Δ_{φ}^{it}) → 1 for any t ∈ R and (2) Ψ_{ν}(abc) → ⟨abcξ_{φ}⊗ξ_{φ^{opp}}, ξ_{φ}⊗ξ_{φ^{opp}}⟩ for any a, c ∈ ℳ^{opp} and b ∈ ℳ_{k}.

Proof. (A) Since C^{∗}(ℳ, e_{𝒩} , JℳJ) ≅ C_{min}^{∗}(ℳ, e_{𝒩} , ℳ^{opp}), Ψ(abc) := ⟨abcξ_{φ}, ξ_{φ}⟩ can be viewed as a state of C_{min}^{∗}(ℳ, e_{𝒩} , ℳ^{opp}), which we extend to a state on B(ℌ) by the Hahn-Banach extension Theorem. We choose a sequence {Ψ_{n}} of normal states on B(ℌ) such that Ψ_{n}(abc) → Ψ(abc) as n → ∞ as follows. Let {x_{n}^{k}} ⊂ ℳ_{k} be a strongly dense countable subset of a unit ball of ℳ_{k}. We reindex {(x_{0}^{k})^{opp}} ⊂ ℳ^{opp} and {x_{n}^{k}}_{n,k} ⊂ ⋃_{k} ℳ_{k} as {x_{i}} and {y_{i}} respectively. Then there exists a normal state on B(ℌ) such that

|(x_{i}y_{j}z_{k}ξ_{φ}, ξ_{φ}) − Ψ_{n}(x_{i}y_{j}z_{k})| < 1/n, 1 ≤ i, j, k ≤ n.

This {Ψ_{n}} satisfies a required condition.

Replacing Ψ_{n} by

∫_{−N_{n}}^{N_{n}}(Δ_{φ}^{it} ⊠ Δ_{φ^{opp}}^{−it})Ψ_{n}(Δ_{φ}^{−it} ⊠ Δ_{φ^{opp}}^{it})dt,

we may assume that ∥Ψ_{n} − (Δ_{φ}^{it} ⊠ Δ_{φ^{opp}}^{−it})Ψ_{n}(Δ_{φ}^{−it} ⊠ Δ_{φ^{opp}}^{it})∥ → 0. Let ξ_{n} ∈ ℌ be the representing vector of Ψ_{n}|_{ℳ ⊠_{e_{𝒩}} ℳ^{opp}} . Then {ξ_{n}} satisfies (1) and (2).

(B) By Proposition 4.7, we can find {λ_{n}^{k}}_{k=1}^{m_{n}} ⊂ R_{+}^{∗} and {u_{n}^{k}}_{k=1}^{m_{n}} ⊂ U(ℳ) such that ∑_{k} λ_{n}^{k} = 1, ∥[u_{n}^{k}, φ]∥ < 1/n and ∑_{k=1}^{m_{n}} λ_{n}^{k}u_{n}^{k∗}xu_{n}^{k} → E_{ℳ′∩ℳ_{k}}^{φ} (x) σ-strongly* for x ∈ ℳ_{k}. Define a state Ψ_{n} on B(L^{2}(ℳ)) by Ψ_{n}(x) := ∑_{k} λ_{n}^{k}⟨xu_{n}^{k}ξ_{φ}, u_{n}^{k}ξ_{φ}⟩. Since ∥[u_{n}^{k}, φ]∥ < 1/n, Ψ_{n}(Δ_{φ}^{it}) → 1 for any t ∈ R. For a, c ∈ ℳ^{opp} and b ∈ ℳ_{k}, Ψ_{n}(abc) → ⟨aE_{ℳ′∩ℳ_{k}}^{φ} (b)cξ_{φ}, ξ_{φ}⟩ = ⟨abcξ_{φ} ⊗ ξ_{φ^{opp}}, ξ_{φ} ⊗ ξ_{φ^{opp}}⟩. (See Section 5 for the last equation.) □

By Lemma 7.4, (𝒜, θ) is a C^{∗}-dynamical system and (𝒜, Δ_{φ}^{it}) is a C^{∗}-covariant representation for (𝒜, θ), say π. Set ℬ := π(𝒜 ⋊_{θ} R). Then any element of ℬ is a norm limit of operators of the form ∫ a(s)Δ_{φ}^{is}ds, where a(s) is a norm continuous map from R to 𝒜 with a compact support.

Lemma 7.6 (1) With the notations in Lemma 7.3, there exists an automorphism α̃ ∈ Aut(𝒜) such that

α̃ (∫_{K} a(s)b(s)c(s)dμ(s)) = ∫_{K} a(s)α(b(s))c(s)dμ(s).

(2) There exists an automorphism β ∈ Aut(ℬ) such that

β (∫ a(s)Δ_{φ}^{is}ds) = ∫ α̃(a(s))Δ_{φ}^{is}ds.

Proof. (1) Let 𝒜_{1} be a subalgebra of B(ℌ) formed by the operators ∫_{K} a(k)b(k)c(k)dμ(k) with the notations in Lemma 7.3. To distinguish elements in B and 𝒜_{1}, we use the notations ∫_{K,B} a(k)b(k)c(k)dμ(k) and ∫_{K,𝒜_{1}} a(k)b(k)c(k)dμ(k). Define ϱ by

ϱ (∫_{K,𝒜_{1}} a(k)b(k)c(k)dμ(k)) = ∫_{K,B} a(k)b(k)c(k)dμ(k).

By Lemma 7.5(A) and the Lebesgue convergence theorem,

⟨∫_{K,B} a(k)b(k)c(k)dμ(k)ξ_{φ}, ξ_{φ}⟩ = lim_{ν→∞} ⟨∫_{K,𝒜_{1}} a(k)b(k)c(k)dμ(k)ξ_{ν}, ξ_{ν}⟩ .

Hence |⟨ϱ(T)ξ_{φ}, ξ_{φ}⟩| ≤ ∥T ∥ holds for T ∈ 𝒜_{1}. Since ξ_{φ} is cyclic for B, ϱ is a well-defined. By Lemma 7.5(B) and the Lebesgue convergence theorem, we have

⟨∫_{K,𝒜_{1}} a(k)b(k)c(k)dμ(k)ξ_{φ⊗φ^{opp}}, ξ_{φ⊗φ^{opp}}⟩ = lim_{ν→∞} Ψ_{ν} (∫_{K,B} a(k)b(k)c(k)dμ(k)) .

Hence we get ⟨T ξ_{φ⊗φ^{opp}}, ξ_{φ⊗φ^{opp}}⟩ ≤ ∥ϱ(T)∥ for T ∈ 𝒜_{1}. Since ξ_{φ} ⊗ ξ_{φ^{opp}} is cyclic for 𝒜_{1}, ϱ is an isometry.

Then

∥∫_{K,B} a(k)α(b(k))c(k)dμ(k)∥ = ∥ϱ (∫_{K,𝒜_{1}} a(k)α(b(k))c(k)dμ(k))∥
= ∥∫_{K,𝒜_{1}} a(k)α(b(k))c(k)dμ(k)∥
= ∥(Δ_{φ}^{iT} ⊠ id) ∫_{K,𝒜_{1}} a(k)b(k)c(k)dμ(k)(Δ_{φ}^{−iT} ⊠ id)∥
= ∥∫_{K,𝒜_{1}} a(k)b(k)c(k)dμ(k)∥
= ∥∫_{K,B} a(k)b(k)c(k)dμ(k)∥ .

Since α̃ and θ_{t} commutes, α̃ ∈ Aut(𝒜).

(2) By (1), there exists an isomorphism η from 𝒜 into B(ℌ) such that

η (∫_{K,B} a(k)b(k)c(k)dμ(k)) = ∫_{K,𝒜_{1}} a(k)b(k)c(k)dμ(k).

Then η(Δ_{φ}^{it}XΔ_{φ}^{−it}) = Δ_{φ}^{it} ⊠ Δ_{φ^{opp}}^{−it}η(X)Δ_{φ}^{−it} ⊠ Δ_{φ^{opp}}^{it} holds. Let ℬ_{1} ⊂ B(L^{2}(ℳ)) be a ∗-algebra of elements of the form T = ∫_{R} a(s)Δ_{φ}^{is}ds, a(s) ∈ C_{c}(R, 𝒜). Define η_{1} by

η_{1} (∫_{R} a(s)Δ_{φ}^{is}ds) = ∫_{R} η(a(s))Δ_{φ}^{is} ⊠ Δ_{φ^{opp}}^{−is}ds.

Then by Lemma 7.5(B), lim Ψ_{ν}(T) = ⟨η_{1}(T)ξ_{φ} ⊗ ξ_{φ^{opp}}, ξ_{φ} ⊗ ξ_{φ^{opp}}⟩. Indeed we can verify this as follows. First note that lim Ψ_{ν}(a) = ⟨η(a)ξ_{φ} ⊗ ξ_{φ^{opp}}, ξ_{φ} ⊗ ξ_{φ^{opp}}⟩ for a ∈ 𝒜. We have

Ψ_{ν}(T) = Ψ_{ν} (∫_{R} Y (s)Δ_{φ}^{is}ds)
= Ψ_{ν} (∫_{R} Y (s)(Δ_{φ}^{is} − 1)ds) + Ψ_{ν} (∫_{R} Y (s)ds)
= ∫_{R} Ψ_{ν} (Y (s)(Δ_{φ}^{is} − 1)) ds + ∫_{R} Ψ_{ν} (Y (s)) ds.

Here

|Ψ_{ν}(Y (s)(Δ_{φ}^{is} − 1))| ≤ Ψ_{ν}(Y (s)Y (s)^{∗})Ψ_{ν} ((Δ_{φ}^{is} − 1)^{∗}(Δ_{φ}^{is} − 1))
= Ψ_{ν}(Y (s)Y (s)^{∗})Ψ_{ν}(2 − Δ_{φ}^{is} − Δ_{φ}^{−is})
→ 0

as ν goes to infinity. Hence

lim_{ν→∞} Ψ_{ν}(T) = lim_{ν→∞} ∫_{R} Ψ_{ν}(Y (s))ds
= ∫_{R}⟨η(Y (s))ξ_{φ} ⊗ ξ_{φ^{opp}}, ξ_{φ} ⊗ ξ_{φ^{opp}}⟩ds
= ⟨∫_{R} η(Y (s))Δ_{φ}^{is} ⊠ Δ_{φ^{opp}}^{−is}ξ_{φ} ⊗ ξ_{φ^{opp}}, ξ_{φ} ⊗ ξ_{φ^{opp}}⟩

holds.

Hence we get

⟨η_{1}(T)ξ_{φ⊗φ^{opp}}, ξ_{φ⊗φ^{opp}}⟩ ≤ ∥T ∥.

Since ξ_{φ} ⊗ ξ_{φ^{opp}} is cyclic for η_{1}(ℬ_{1}), η_{1} is a well-defined homomorphism. In a similar way, we have

⟨∫_{R} a(s)Δ_{φ}^{is}dsξ_{φ}, ξ_{φ}⟩ = lim_{ν→∞} ⟨∫_{R} η(a(s))Δ_{φ}^{is} ⊠ Δ_{φ^{opp}}^{−is}dsξ_{ν}, ξ_{ν}⟩

by using Lemma 7.5(A). So ⟨T ξ_{φ}, ξ_{φ}⟩ ≤ ∥η_{1}(T)∥ holds. Since ξ_{φ} is cyclic for ℬ_{1}, η_{1} is an isometry. Then

∥∫_{R} α̃(a(s))Δ_{φ}^{is}ds∥ = ∥η_{1} (∫_{R} α̃(a(s))Δ_{φ}^{is}ds)∥
= ∥∫_{R} η(α̃(a(s)))Δ_{φ}^{is} ⊠ Δ_{φ^{opp}}^{−is}ds∥
= ∥∫_{R}(Δ_{φ}^{iT} ⊠ id)η(a(s))(Δ_{φ}^{−iT} ⊠ id)Δ_{φ}^{is} ⊠ Δ_{φ^{opp}}^{−is}ds∥
= ∥(Δ_{φ}^{iT} ⊠ id) (∫_{R} η(a(s))Δ_{φ}^{is} ⊠ Δ_{φ^{opp}}^{−is}ds) (Δ_{φ}^{−iT} ⊠ id)∥
= ∥∫_{R} η(a(s))Δ_{φ}^{is} ⊠ Δ_{φ^{opp}}^{−is}ds∥
= ∥∫_{R} a(s)Δ_{φ}^{is}ds∥ .

Hence we get a desired isomorphism. □

Lemma 7.7 With the notations in the proof of Theorem 7.2, we have

∥4n + 2 − (1 − e_{𝒩} ) − α̃(X) − e_{𝒩} |f(Δ_{φ}) − 1|^{2}e_{𝒩} ∥ = 4n + 2.

Proof. Let h(t) ∈ C_{c}^{∞}(R) be a positive function with ∫_{R} h(t)dt = 1. Then we have ∫_{R} Δ_{φ}^{it}XΔ_{φ}^{−it}h(t)dt ∈ 𝒜. Also

Z := (∫_{R} Δ_{φ}^{it}(4n + 2 − 1 + e_{𝒩} + X + e_{𝒩} |f(Δ_{φ}) − 1|^{2}e_{𝒩} )Δ_{φ}^{−it}h(t)dt) ∫_{R} Δ_{φ}^{it}h(t)dt ∈ ℬ.

By Lemma 7.6,

β(Z) = (∫_{R} Δ_{φ}^{it}(4n + 2 − 1 + e_{𝒩} + α̃(X) + e_{𝒩} |f(Δ_{φ}) − 1|^{2}e_{𝒩} )Δ_{φ}^{−it}h(t)dt) ∫_{R} Δ_{φ}^{it}h(t)dt, and ∥β(Z)∥ = ∥Z∥. Here ∫_{R} h(t)Δ_{φ}^{it}dt = ĥ(Δ_{φ}), and ∥ĥ(Δ_{φ})∥ ≤ 1. Hence we have

∥β(Z)∥ ≤ ∥4n + 2 − (1 − e_{𝒩} ) + α̃(X) + e_{𝒩} |f(Δ_{φ}) − 1|^{2}e_{𝒩} ∥.

Since ξ_{φ} is an eigenvector of Z for an eigenvalue 4n + 2, ∥Z∥ ≥ 4n + 2. Hence we get ∥4n + 2 − (1 − e_{𝒩} ) + α̃(X) + e_{𝒩} |f(Δ_{φ}) − 1|^{2}e_{𝒩} ∥ = 4n + 2. □

A Exhaustion trick

In this appendix we give a proof of a subfactor-version of [6, Lemma III.4], since the proof was omitted in [6]. This lemma is the key for the proof of Theorem 7.1. As mentioned in [6], idea of proof is similar to that of [6, Theorem II.2].

Lemma A.1 Let 𝒩 ⊂ ℳ, ε, φ_{j} and θ be as in Theorem 7.1, and ξ_{j} ∈ L^{2}(ℳ)_{+} be a representing vector of φ_{j}. Let c be as in [6, III.Lemma 2]. Then there exists a projection E ∈ 𝒩 and y ∈ 𝒩 such that,

(1) ∥y∥ ≤ 1, y = θ(E)yE,

(2) ∑ ∥yξ_{j}∥^{2} ≥ 2^{−6}c^{−1} ∑ ∥Eξ_{j}∥^{2},

(3) ∥[E, ξ_{j}]∥^{2} ≤ ε^{2} ∑ ∥Eξ_{j}∥^{2},

(4) ∥yξ_{j} − θ(ξ_{j})y∥^{2} ≤ ε^{2} ∑ ∥yξ_{j}∥^{2}.

Proof of this lemma is same as the proof of [6, III.Lemma 3]. Here note that since x ∈ 𝒩 in Theorem 7.1, E and y can be chosen in 𝒩 by construction.

Lemma A.2 Let 𝒩 ⊂ ℳ, and θ be as in Theorem 7.1. Then there exists a bounded sequence {y_{n}} ⊂ 𝒩 such that {y_{n}} does not converge to 0 strongly and

∥y_{n}φ − θ(φ)y_{n}∥ → 0, φ ∈ ℳ_{∗}.

Proof. Fix φ_{1}, ⋯, φ_{n} ∈ ℳ_{∗}^{+}. Let ε > 0 be such that 2^{−6}c^{−1} − nε^{2} ≥ 2^{−7}c^{−1}, where c is a constant defined in [6, Lemma III.3]. Let ξ_{j} ∈ L^{2}(ℳ)_{+} be a representing vector of φ_{j}, i.e., φ_{j}(x) = ⟨xξ_{j}, ξ_{j}⟩. Let R be a set of r = (E, x, α_{1}, ⋯, α_{n}) ∈ Proj (𝒩) × 𝒩 × L^{2}(ℳ)^{n} satisfying the following conditions.

(i) θ(E)xE = x, ∥x∥ ≤ 1. (ii) Eα_{j} = α_{j}, η_{j} := ξ_{j} − α_{j} − Jα_{j} ∈ L^{2}(ℳ)_{+} and [E, η_{j}] = 0. (iii) ∥α_{j}∥^{2} ≤ ε^{2} ∑_{j} ∥Eξ_{j}∥^{2}. (iv) ∑_{j} ∥xη_{j}∥^{2} ≥ 2^{−7}c^{−1} ∑_{j} ∥Eη_{j}∥^{2}.

(v) ∥xη_{j} − θ(η_{j})x∥^{2} ≤ ε^{2} ∑_{j} ∥xξ_{j}∥^{2}.

We define a partial ordering r = (E, x, α_{1}, ⋯, α_{n}) ≤ r′ = (E′, x′, α′_{1}, ⋯, α′_{n}) as follows. (α) E ≤ E′. (β) θ(E)x′E = x. (γ) E(α′_{j} − α_{j}) = 0. (δ) ∥α′_{j} − α_{j}∥^{2} ≤ ε^{2} ∑_{j} ∥(E′ − E)ξ_{j}∥^{2}.

We claim that this is indeed an order.

(1) It is trivial that r ≤ r.

(2) Assume r ≤ r′ and r′ ≤ r. By (α), E = E′. By (β) and (i), x = θ(E)x′E = θ(E′)x′E′ = x′. By (δ), α_{j} = α′_{j}. Hence we get r = r′.

(3) Assume r ≤ r′ and r′ ≤ r′′. It is clear that E ≤ E′′. Next θ(E)x′′E = θ(E)θ(E′)x′′E′E = θ(E)x′E = x, so (β) holds. E(α′′_{j} − α_{j}) = E(α′′_{j} − α′_{j}) + E(α′_{j} − α_{j}) = E′(α′′_{j} − α′_{j}) = 0. Hence (γ) holds. To prove (δ), first note ⟨α′′_{j} − α′_{j}, α′_{j} − α_{j}⟩ = ⟨E′(α′′_{j} − α′_{j}), α′_{j} − α_{j}⟩ = 0. Then ∥α′′_{j} − α_{j}∥^{2} = ∥α′′_{j} − α′_{j}∥^{2} + ∥α′_{j} − α_{j}∥^{2} ≤ ε^{2} ∑_{j} ∥(E′′ − E′)ξ_{j}∥^{2} + ε^{2} ∑_{j} ∥(E′ − E)ξ_{j}∥^{2} = ε^{2} ∑_{j} ∥(E′′ − E)ξ_{j}∥^{2}.

Next we prove that R is an inductive ordered set. Let {r_{i}}_{i∈I} be a totally ordered subset. Then there exists a cofinal sequence {r_{i_{k}}}. So we may assume that {r_{k}} is an increasing sequence. Set r_{k} = (E_{k}, x_{k}, α_{1}^{k}, ⋯, α_{n}^{k}). Then {E_{k}} is an increasing sequence of projections, and hence E = lim_{k} E_{k} exists in a strong topology. By a similar reason, x = lim_{k} x_{k} also exists. By (δ), ∥α_{j}^{k} − α_{j}^{l}∥^{2} ≤ ε^{2} ∑_{j} ∥(E_{k} − E_{l})ξ_{j}∥^{2}. By letting k, l → ∞, we know that {α_{j}^{k}} is a Cauchy sequence. Hence a limit α_{j} = lim_{k} α_{j}^{k} exists. Then r = (E, x, α_{1}, ⋯, α_{n}) is in R by continuity. By construction r is an upper bound of {r_{k}}. Hence R is inductive.

By Zorn’s lemma, there exists a maximal element r = (E, x, α_{1}, ⋯, α_{n}) ∈ R. We claim E = 1.

Suppose E ≠ 1. Let u ∈ 𝒩 be a unitary such that uθ(1 − E)u^{∗} = 1 − E, and consider θ̄:= Ad uθ|_{ℳ_{(1−E)}} . Since Eα_{j} = α_{j}, we have (1 − E)α_{j} = 0 and (Jα_{j})(1 − E) = J(1 − E)α_{j} = 0, so that (1 − E)ξ_{j}(1 − E) = (1 − E)η_{j}(1 − E)(=: ζ_{j}). By applying the above lemma to ℳ_{(1−E)}, θ̄ and {ζ_{j}}, we can find 0 ≠ y ∈ 𝒩_{(1−E)} and F ∈ Proj (𝒩_{(1−E)}) such that

(1) θ̄(F)yF = y, ∥y∥ ≤ 1,

(2) ∑_{j} ∥yζ_{j}∥^{2} ≥ 2^{−6}c^{−1} ∑_{j} ∥F ζ_{j}∥^{2},

(3) ∥[F, ζ_{j}]∥^{2} ≤ ε^{2} ∑_{j} ∥F ζ_{j}∥^{2},

(4) ∥yζ_{j} − θ̄(ζ_{j})y∥^{2} ≤ ε^{2} ∑_{j} ∥yζ_{j}∥^{2}.

Set x′ = x + u^{∗}y, E′ = E + F, α′_{j} = α_{j} + F η_{j}(1 − F). We claim that r′ = (E′, x′, α′_{1}, ⋯, α′_{n}) ∈ R and r′ majorizes r. Note that F η_{j} = F ζ_{j} and η_{j}F = ζ_{j}F since 1 − E ≥ F and (1 − E) commutes with η_{j}.

(i) First note that θ̄(F)y = y implies θ(F)u^{∗}y = u^{∗}y. Then ∥x′∥ ≤ 1 is clear from the facts θ(E)xE = x, θ(F)u^{∗}yF = u^{∗}y.

(ii) E′α′_{j} = (E +F)(α_{j} +F η_{j}(1−F)) = Eα_{j} +F η_{j}(1−F) = α′_{j}. Set η′_{j} = ξ_{j} −α′_{j} −Jα′_{j}. Since

η′_{j} = ξ_{j} − α′_{j} − Jα′_{j}
= ξ_{j} − α_{j} − F η_{j}(1 − F) − Jα_{j} − (1 − F)η_{j}F
= η_{j} − F η_{j}(1 − F) − (1 − F)η_{j}F
= F η_{j}F + (1 − F)η_{j}(1 − F),

we have η′_{j} ∈ L^{2}(ℳ)_{+}.

Next we verify [η′, E′] = 0. Note that Eη_{j} = η_{j}E = Eη_{j}E holds due to [E, η_{j}] = 0.

Then we get the following.

[E′, η′_{j}]

= (E + F)(F η_{j}F + (1 − F)η_{j}(1 − F)) − (F η_{j}F + (1 − F)η_{j}(1 − F))(E + F)
= F η_{j}F + Eη_{j}(1 − F) − F η_{j}F − (1 − F)η_{j}E
= Eη_{j}(1 − F) − (1 − F)η_{j}E
= Eη_{j}E(1 − F) − (1 − F)Eη_{j}E
= 0.

(iii) Since Eα_{j} = α_{j}, α_{j} and F η_{j}(1 − F) are orthogonal. Then

∥α′_{j}∥^{2} = ∥α_{j}∥^{2} + ∥F η_{j}(1 − F)∥^{2}
≤ ε^{2} ∑_{j} ∥Eξ_{j}∥^{2} + ∥F ζ_{j}(1 − F)∥^{2}
= ε^{2} ∑_{j} ∥Eξ_{j}∥^{2} + ∥F [F, ζ_{j}]∥^{2}
≤ ε^{2} ∑_{j} ∥Eξ_{j}∥^{2} + ε^{2} ∑_{j} ∥F ζ_{j}∥^{2}
= ε^{2} ∑_{j} ∥Eξ_{j}∥^{2} + ε^{2} ∑_{j} ∥F ξ_{j}(1 − E)∥^{2}
≤ ε^{2} ∑_{j} ∥Eξ_{j}∥^{2} + ε^{2} ∑_{j} ∥F ξ_{j}∥^{2}
= ε^{2} ∑_{j} ∥(E + F)ξ_{j}∥^{2}
= ε^{2} ∑_{j} ∥E′ξ_{j}∥^{2}.

(iv) First we compute x′η′_{j}.

x′η′_{j} = (x + u^{∗}y)(F η_{j}F + (1 − F)η_{j}(1 − F))
= xη_{j}(1 − F) + u^{∗}yη_{j}F
= xη_{j} + u^{∗}yζ_{j}F.

Hence ∥x′η′_{j}∥^{2} = ∥xη_{j}∥^{2} + ∥u^{∗}yζ_{j}F ∥^{2} holds. We also have

∥yζ_{j}∥^{2} = ∥yζ_{j}F ∥^{2} + ∥yζ_{j}(1 − F)∥^{2}
≤ ∥yζ_{j}F ∥^{2} + ∥F ζ_{j}(1 − F)∥^{2}
= ∥yζ_{j}F ∥^{2} + ∥F [F, ζ_{j}]∥^{2}
≤ ∥yζ_{j}F ∥^{2} + ε^{2} ∑_{j} ∥F ζ_{j}∥^{2}.

Hence ∑_{j} ∥yζ_{j}F ∥ ≥ ∑_{j} ∥yζ_{j}∥^{2} − ε^{2}n ∑_{j} ∥F ζ_{j}∥^{2} holds.

We will estimate ∑_{j} ∥x′η′_{j}∥.

∑_{j=1}^{n} ∥x′η′_{j}∥^{2} ≥ ∑_{j} ∥xη_{j}∥^{2} + ∑_{j} ∥yζ_{j}∥^{2} − ε^{2}n ∑_{j} ∥F ζ_{j}∥^{2}
≥ 2^{−7}c^{−1} ∑_{j} ∥Eη_{j}∥^{2} + (2^{−6}c^{−1} − nε^{2}) ∑_{j} ∥F ζ_{j}∥^{2}
≥ 2^{−7}c^{−1} ∑_{j} ∥Eη_{j}∥^{2} + 2^{−7}c^{−1} ∑_{j} ∥F ζ_{j}∥^{2}
≥ 2^{−7}c^{−1} ∑_{j} ∥E(1 − F)η_{j}(1 − F)∥^{2} + 2^{−7}c^{−1} ∑_{j} ∥F ζ_{j}F ∥^{2}
= 2^{−7}c^{−1} ∑_{j} ∥E(1 − F)η_{j}(1 − F) + F ζ_{j}F ∥^{2}
= 2^{−7}c^{−1} ∑_{j} ∥(E + F)((1 − F)η_{j}(1 − F) + F η_{j}F)∥^{2}
= 2^{−7}c^{−1} ∑_{j} ∥E′η′_{j}∥^{2}.

(v) We compute θ(η′_{j})x′.

θ(η′_{j})x′ = θ(F η_{j}F + (1 − F)η_{j}(1 − F))(x + u^{∗}y)
= θ(F η_{j})u^{∗}y + θ((1 − F)η_{j})x
= u^{∗}θ̄(F ζ_{j})y + θ((1 − F)η_{j})x.

Then we have the following estimate.

∥x′η′_{j} − θ(η′_{j})x′∥^{2} = ∥xη_{j} + u^{∗}yζ_{j}F − θ((1 − F)η_{j})x − u^{∗}θ̄(F ζ_{j})y∥^{2}
= ∥xη_{j} − θ(η_{j})x + u^{∗}yζ_{j}F − u^{∗}θ̄(F)θ̄(ζ_{j})y∥^{2}
= ∥xη_{j} − θ(η_{j})x∥^{2} + ∥u^{∗}yζ_{j}F − u^{∗}θ̄(F)θ̄(ζ_{j})y∥^{2}
≤ ε^{2} ∑_{j} ∥xξ_{j}∥^{2} + ∥yζ_{j}F − θ̄(ζ_{j})y∥^{2}
≤ ε^{2} ∑_{j} ∥xξ_{j}∥^{2} + ∥yζ_{j} − θ̄(ζ_{j})y∥^{2}
≤ ε^{2} ∑_{j} ∥xξ_{j}∥^{2} + ε^{2} ∑_{j} ∥yζ_{j}∥^{2}
= ε^{2} ∑_{j} ∥xξ_{j}∥^{2} + ε^{2} ∑_{j} ∥y(1 − E)ξ_{j}(1 − E)∥^{2}
≤ ε^{2} ∑_{j} ∥xξ_{j}∥^{2} + ε^{2} ∑_{j} ∥u^{∗}yξ_{j}∥^{2}
= ε^{2} ∑_{j} ∥(x + u^{∗}y)ξ_{j}∥^{2}
= ε^{2} ∑_{j} ∥x′ξ_{j}∥^{2}

Hence r′ is in R. Next we verify r′ > r. (α) is obvious. Since θ(E)(x + u^{∗}y)E = θ(E)xE + θ(E)θ(F)u^{∗}yF E = x, so (β) holds. Next we verify (γ). Then E(α′_{j} − α_{j}) = E(F η_{j}(1−F) = 0). Finally ∥α′_{j} −α_{j}∥^{2} = ∥F η_{j}(1−F)∥^{2} = ∥F ([F, η_{j}])∥^{2} ≤ ε^{2} ∑_{j} ∥F ζ∥^{2} = ε^{2} ∑_{j} ∥F ξ(1 − E)∥^{2} ≤ ∥(E′ − E)ξ∥^{2}, so (δ) holds. This contradicts the maximality of r, so we get E = 1.

Hence we get x ∈ 𝒩, α_{j} ∈ L^{2}(ℳ) satisfying the following conditions.

(1) ∥x∥ ≤ 1.

(2) ∥α_{j}∥^{2} ≤ ε^{2} ∑_{j} ∥ξ_{j}∥^{2}.

(3) With η_{j} = ξ_{j} − α_{j} − Jα_{j}, ∑_{j} ∥xη_{j}∥ ≥ 2^{−7}c^{−1} ∑_{j} ∥η_{j}∥^{2}.

(4) ∥xη_{j} − θ(η_{j})x∥^{2} ≤ ε^{2} ∑_{j} ∥ξ_{j}∥^{2}.

Set a := \sqrt{∑_{j} ∥ξ_{j}∥^{2}}. Then ∥α_{j}∥ ≤ εa holds. By (4), we get

∥xξ_{j} − θ(ξ_{j})x∥ = ∥xη_{j} − θ(η_{j})x + x(α_{j} + Jα_{j}) + θ(α_{j} + Jα_{j})x∥
≤ 5εa,

and

∥xη_{j}∥ = ∥x(ξ_{j} − α_{j} − Jα_{j})∥
≤ ∥xξ_{j}∥ + 2∥α_{j}∥
≤ ∥xξ_{j}∥ + 2εa.

Hence we have the following.

∑_{j} ∥xη_{j}∥^{2} ≤ ∑_{j}(∥xξ_{j}∥ + 2εa)^{2}
= ∑_{j}(∥xξ_{j}∥^{2} + 4εa∥xξ_{j}∥ + 4ε^{2}a^{2})
≤ ∑_{j} ∥xξ_{j}∥^{2} + 4εn^{\frac{1}{2}} (∑_{j} ∥xξ_{j}∥^{2})^{\frac{1}{2}} a + 4ε^{2}na^{2}
= ((∑_{j} ∥xξ_{j}∥^{2})^{\frac{1}{2}} + 2εan^{\frac{1}{2}} )^{2}.

On the other hand, we have the following.

∑_{j} ∥η_{j}∥^{2} = ∑_{j} ∥ξ_{j} − α_{j} − Jα_{j}∥^{2}
≥ ∑_{j}(∥ξ_{j}∥ − 2εa)^{2}
= ∑_{j}(∥ξ_{j}∥^{2} − 4εa∥ξ_{j}∥ + 4ε^{2}a^{2})
≥ a^{2} − 4εn^{\frac{1}{2}} a^{2} + 4ε^{2}a^{2}n
= a^{2}(1 − 2εn^{\frac{1}{2}} )^{2}.

So by (3), ((∑_{j} ∥xξ_{j}∥^{2})^{\frac{1}{2}} + 2εan^{\frac{1}{2}} )^{2} ≥ 2^{−7}c^{−1}a^{2}(1−2εn^{\frac{1}{2}} )^{2} holds. If we take sufficient small ε, then we get x ∈ 𝒩, ∥x∥ ≤ 1 such that ∥xξ_{j} − θ(ξ_{j})x∥ ≤ εa and ∑_{j} ∥xξ_{j}∥^{2} ≥ d′a^{2} for some constant 0 ≠ d′.

Since ℌ is separable, we can construct a bounded sequence {y_{n}} ⊂ 𝒩 such that {y_{n}} does not converge to 0 strongly and lim_{n→∞} ∥y_{n}φ − θ(φ)y_{n}∥ = 0 for every φ ∈ ℳ_{∗}.

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