Generalization of Longo-Rehren construction to subfactors of infinite depth and amenability of fusion algebras

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

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

Abstract

We extend the Longo-Rehren construction to an infinite system of bimodules (or sectors). The construction is similar to the crossed product. We also show that the Longo-Rehren construction gives an isomorphic subfactor to Popa’s symmetric enveloping algebra. We discuss the relation between the Longo-Rehren subfactors and amenability of fusion algebras in the sense of Hiai and Izumi.

1 Introduction

Since V. F. R. Jones initiated the index theory for subfactors in [11], much progress has been made not only in the field of operator algebras but also in mathematical physics, topological quantum field theory, etc.

In [13], R. Longo and K.-H. Rehren introduced the construction of an inclusion of properly infinite factors from a given inclusion of properly infinite factors of finite depth. (More precisely, they constructed a new inclusion from a finite system of sectors.) Key of their construction is the characterization of Longo’s canonical endomorphism in [12, Theorem 5.1].

In our previous paper [14], we performed the Longo-Rehren construction in the case of finite depth inclusions for factors of type II_{1} and proved that if original factors are approximately finite dimensional (AFD), then the Longo-Rehren construction gives a subfactor isomorphic to Ocneanu’s asymptotic inclusion ([17]). The asymptotic inclusion is considered as an analogue of the quantum double construction in theory of quantum groups and hence it is an important object.

A natural question is whether the Longo-Rehren construction is possible for subfactors of infinite depth. First motivation of this paper is to answer this question. In this case, we are forced to treat subfactors with infinite index. But because Longo’s characterization of canonical endomorphisms is valid only for endomorphisms with finite index, we can not generalize their method directly to subfactors of infinite depth. So we have to take a different approach. Our idea is that we construct a new inclusion like as a “crossed product”. Details of construction will be presented in the following sections. The construction is similar to that of crossed product. So we can consider the Longo-Rehren construction as a “crossed product” of M ⊗ M^{opp} by the fusion algebra arising from the inclusion N ⊂ M.

∗Supported by JSPS.

In [24], S. Popa introduced the symmetric enveloping algebras for his study of amenability in subfactor theory. Details of his theory are found in [25]. Roughly speaking, the symmetric enveloping algebra is the unique II_{1} factor generated by M, M^{opp} with [M, M^{opp}] = 0 and the Jones projection e_{N} for both inclusions N ⊂ M and N^{opp} ⊂ M^{opp}. In [25], he characterized the amenability of the standard invariant of N ⊂ M by using the symmetric enveloping algebra as follows.

The standard invariant of N ⊂ M is amenable if and only of the symmetric enveloping algebra is relatively amenable to M ∨ M^{opp} in the sense of [22].

On the other hand, F. Hiai and M. Izumi generalized the notion of the amenability to fusion algebras in [9], and investigated the relation to the amenability of subfactors. They proposed that amenability is a property for fusion algebras arising from subfactors rather than a property for subfactors themselves.

In this paper, we will prove that the Longo-Rehren subfactor is isomorphic to the symmetric enveloping algebra. Since the amenability of standard invariants is equivalent to the amenability of the fusion algebras arising from subfactors, the above results show that the fusion algebra arising from a subfactor N ⊂ M is amenable if and only if the Longo-Rehren subfactor arising from N ⊂ M is relatively amenable to M ⊗ M^{opp}. The second motivation of this paper is that we would like to take a more direct approach to the amenability of subfactors based on Hiai and Izumi’s theory.

This paper is organized as follows.

In Section 2, we extend the Longo-Rehren construction to type II_{1} subfactors in the infinite depth case. As mentioned above, the construction is similar to that of a crossed product. We also prove that when an original subfactor is of finite index, our new construction gives the same Longo-Rehren subfactor constructed as in [14].

In Section 3, we prove that the Longo-Rehren subfactor and Popa’s symmetric enveloping algebra are isomorphic. In [25], Popa proved that the symmetric enveloping algebras are similar to crossed products. Our approach shows more direct similarity between the symmetric enveloping algebras and crossed products. We also discuss the extension of automorphisms of N ⊂ M to the Longo-Rehren subfactor.

In Section 4, we investigate the relation between the Longo-Rehren subfactors and the amenability of fusion algebras. Unfortunately we can not cover the whole part of Popa’s theory, but our approach makes easier to understand the relation between them.

In Section 5, we perform the Longo-Rehren construction in the case of type III subfactors of infinite depth. Key of construction is the characterization of canonical endomorphisms due to Fidaleo and Isola [8].

Acknowledgements. The author expresses special gratitude to Professor M. Izumi for explaining the contents of [9] and much fruitful advice, especially the formula in Lemma 3.1. He is also grateful to Professor T. Isola for explaining about [8], Professor Y.

Kawahigashi for helpful suggestions and constant encouragement, and Professor S. Popa for various comments on this paper. Part of this work was done during the author’s stay at University of California, Berkeley. He thanks it for the hospitality.

2 Generalization of the Longo-Rehren construction to subfactors of infinite depth

Throughout this paper, we always assume that inclusions of factors of type II_{1} are extremal in the sense of [20].

For general theory of bimodules, see [4, Section V, Appendix 2] and [22], and for applications of bimodule theory to subfactor theory, see [5, Section 9], [16] and [26]. Also for P -Q bimodule _{P} W_{Q}, we denote the set of bounded vectors, P -valued inner product and Q-valued inner product by W^{bdd}, _{P} ⟨·|·⟩ and ⟨·|·⟩_{Q} respectively.

Let N ⊂ M be a subfactor of type II_{1} with finite index. Set

Δ := \{_{M}Z_{M}|irreducuble, _{M}Z_{M} ≺ \underbrace{_{M}M⊗_{N}M⊗_{M}M⊗_{N} ⋯ ⊗_{N}M_{M}}_{2n} for some n ∈ N\}.

Δ satisfies the following property.

(1) _{M}Z_{0M} := _{M}M_{M} ∈ Δ,
(2) _{M}Z_{iM} ∈ Δ ⇒ _{M}Z_{īM} := _{M} Z̄_{i}_{M}, ∈ Δ
(3) _{M}Z_{i}⊗_{M}Z_{jM} ≅ ⊕_{k}N_{ij}^{k}_{M}Z_{kM}, _{M}Z_{iM},_{M}Z_{jM},_{M}Z_{kM} ∈ Δ,

where N_{ij}^{k} = dim Hom (_{M}Z_{i}⊗_{M}Z_{jM},_{M}Z_{kM}).

In the following, we use the symbol Δ to denote both a bimodule system and a fusion algebra arising from a bimodule system if no confusions arises.

We can naturally get the system of M^{opp}-M^{opp} bimodules Δ^{opp} = {_{M^{opp}}Z_{iM^{opp}}^{o}} from Δ. Set B_{i} := Z_{i} ⊗ Z_{i}^{o}, where ⊗ means the usual tensor product over C. Then B_{i} is an M ⊗ M^{opp}(=: A) bimodule.

Let {V_{ijk}^{e}}_{e=1}^{N_{ij}^{k}} be an orthonormal basis of Hom (_{M}Z_{i}⊗_{M}Z_{jM},_{M}Z_{kM}) and J the natural conjugate linear map from Z_{i} to Z_{i}^{o}. (More precisely, we should use the notation J_{Z_{i}} instead of J, but we will denote it simply by J.) Set

Ṽ_{ijk} := \sqrt{\frac{d(i)d(j)}{d(k)}} ∑_{e} V_{ijk}^{e} ⊗ JV_{ijk}^{e}(J^{−1} ⊗ J^{−1}),

where d(i) is the square root of the index of _{M}Z_{iM}, i.e., d(i) := \sqrt{[End (Z_{iM}) : M]}.

Then Ṽ_{ijk} is an element of Hom (_{A}B_{i}⊗_{A}B_{jA},_{A}B_{kA}) and does not depend on the choice of an orthonormal basis {V_{ijk}^{e}}, due to the conjugate linearity of J. Especially if we put i = 0, then Ṽ_{0jk} = δ_{j,k}1_{j}, where 1_{j} means an identity map on B_{j}.

Lemma 2.1 The following equalities hold.

(1) Ṽ_{ijk_{1}} Ṽ_{ijk_{2}}^{∗} = δ_{k_{1},k_{2}}N_{ij}^{k_{1}} \frac{d(i)d(j)}{d(k_{1})},
(2) ∑_{k} Ṽ_{lmk}^{∗}Ṽ_{ijk} = ∑_{n}(Ṽ_{inl} ⊗ 1_{m})(1_{i} ⊗ Ṽ_{nmj}^{∗}) ∈ Hom (_{A}B_{i}⊗_{A}B_{jA},_{A}B_{l}⊗_{A}B_{mA}),
(3) ∑_{k} Ṽ_{knl}(Ṽ_{ijk} ⊗ 1_{n}) = ∑_{m} Ṽ_{iml}(1_{i} ⊗ Ṽ_{jnm}) ∈ Hom (_{A}B_{i}⊗_{A}B_{j}⊗_{A}B_{n},_{A}B_{lA}).

Proof of the above equalities are found in the proof of [13, Proposition 4.10]. If we put l = 0 in equality (2), we have an equality Ṽ_{ijm} = (Ṽ_{iī0} ⊗ 1_{m})(1_{i} ⊗ Ṽ_{īmj}^{∗}) and this equality means Frobenius reciprocity. Moreover if we put j = 0 and m = i, we get an equality 1 = (Ṽ_{iī0} ⊗ 1_{i})(1_{i} ⊗ Ṽ_{īi0}). In what follows, we use these equalities frequently.

Set _{A}X_{A} := ⊕_{i}_{A}B_{iA}. For ξ ∈ B_{i}^{bdd}, we will construct two linear operators λ_{i,ξ}^{l} and λ_{i,ξ}^{r} on X.

Define π^{l}(ξ) ∈ Hom (A_{A}, B_{iA}) and π^{r}(ξ) ∈ Hom (_{A}A,_{A}B_{i}) as follows,

π^{l}(ξ)â := ξ ⊗ â = ξa,
π^{r}(ξ)â := â ⊗ ξ = aξ,

where a → â is the canonical embedding map from A to L^{2}(A). Since ξ is in B_{i}^{bdd}, π^{l}(ξ) (resp. π^{r}(ξ)) is actually a bounded operator and a right (resp. left) A-linear map.

Let T_{i} be a natural intertwiner in Hom (_{A}X_{A},_{A}B_{iA}) and set

λ_{i,ξ}^{l} := ∑_{j,k} T_{k}^{∗}Ṽ_{ijk}(π^{l}(ξ) ⊗ 1_{j})T_{j},

λ_{i,ξ}^{r} := ∑_{j,k} T_{k}^{∗}Ṽ_{jik}(1_{j} ⊗ π^{r}(ξ))T_{j}.

These two operators are well-defined on the algebraic direct sum of B_{j}’s.

Lemma 2.2 The operators λ_{i,ξ}^{l} and λ_{i,ξ}^{r} constructed above are in B(X). So λ_{i,ξ}^{l} is in End (X_{A}) and λ_{i,ξ}^{r} is in End (_{A}X).

Proof. We only prove that λ_{i,ξ}^{l} is bounded since the boundedness of λ_{i,ξ}^{r} can be proved similarly.

Since π^{l}(ξ) is bounded, there exists a constant C which satisfies ∥π^{l}(ξ) ⊗ 1_{X}∥ ≤ C. Since we have ∑_{j,k} T_{k}^{∗}Ṽ_{ijk}(π^{l}(ξ) ⊗ 1_{j})T_{j} = ∑_{j,k} T_{k}^{∗}Ṽ_{ijk}(1_{i} ⊗ T_{j})(π^{l}(ξ) ⊗ 1_{X}), we only have to prove that ∑_{j,k} T_{k}^{∗}Ṽ_{ijk}(1_{i} ⊗ T_{j}) is bounded.

Then for η in the domain, we have

∥ ∑_{j,k} T_{k}^{∗}Ṽ_{ijk}(1_{i} ⊗ T_{j})η∥ = ⟨∑_{{j_{1},k_{1} \atop j_{2},k_{2}}} T_{k_{1}}^{∗}Ṽ_{ij_{1}k_{1}}(1_{i} ⊗ T_{j_{1}})(1_{i} ⊗ T_{j_{2}}^{∗})^{∗}Ṽ_{ij_{2}k_{2}}^{∗}T_{k_{2}}η|η⟩^{\frac{1}{2}}

= ⟨ ∑_{j,k_{1},k_{2}} T_{k_{1}}^{∗}Ṽ_{ijk_{1}} Ṽ_{ijk_{2}}^{∗}T_{k_{2}}η|η⟩^{\frac{1}{2}}

= ⟨∑_{j,k} N_{ij}^{k} \frac{d(i)d(j)}{d(k)}T_{k}^{∗}T_{k}η|η⟩^{\frac{1}{2}}

= ⟨∑_{j,k} N_{īk}^{j} \frac{d(i)d(j)}{d(k)}T_{k}^{∗}T_{k}η|η⟩^{\frac{1}{2}}
= ⟨∑_{k} d(i)^{2}T_{k}^{∗}T_{k}η|η⟩^{\frac{1}{2}}
= d(i)∥η∥,

so λ_{i,ξ}^{l} is bounded and moreover we get ∥λ_{i,ξ}^{l}∥ ≤ Cd(i). □

If we take a bounded vector â from L^{2}(A), it is easy to see that λ_{0,â}^{l} (resp. λ_{0,â}^{r}) is precisely the left (resp. right) action of a on X. It is clear that the map ξ → λ_{i,ξ}^{l} is an injective linear map.

Next we define two conjugate linear operators J_{l} and J_{r} as follows.

J_{l}ξ := (π^{l}(ξ)^{∗} ⊗ 1ī)Ṽ_{iīo}^{∗}ˆ1, ξ ∈ B_{i}^{bdd},
J_{r}ξ := (1ī ⊗ π^{r}(ξ)^{∗})Ṽ_{īio}^{∗}ˆ1, ξ ∈ B_{i}^{bdd}.

Note that J_{l} and J_{r} are defined a priori only on the algebraic sum of B_{i}^{bdd} and not all the element in X. For ξ ∈ B_{i}^{bdd}, J_{l}ξ and J_{r}ξ are in Bbdd ; ī. Also note that π^{l}(J_{l}ξ) equals to (π^{l}(ξ)^{∗} ⊗ 1ī)Ṽ_{iī0}^{∗} and a similar equality holds for π^{r}(J_{r}ξ).

By making use of Lemma 2.1, we get the following relations.

Proposition 2.3 For ξ ∈ B_{i_{1}}^{bdd} and η ∈ B_{i_{2}}^{bdd}, we have the following equalities.

(1) λ_{i_{1},ξ}^{l}λ_{i_{2},η}^{l} = ∑_{i_{3}} λ_{i_{3},ζ_{i_{3}}}^{l}, where ζ_{i_{3}} = Ṽ_{i_{1}i_{2}i_{3}}(ξ ⊗ η),

(2) (λ_{i,ξ}^{l})^{∗} = λ_{ī,J_{l}ξ},

(3) λ_{i_{1},ξ}^{r}λ_{i_{2},η}^{r} = ∑_{i_{3}} λ_{i_{3},ζ_{i_{3}}}^{r}, where ζ_{i_{3}} = Ṽ_{i_{2}i_{1}i_{3}}(η ⊗ ξ),

(4) (λ_{i,ξ}^{r})^{∗} = λ_{ī,J_{r}ξ}.

Proof. We only prove equalities (1) and (2) because (3) and (4) can be verified in the same way as the proof of (1) and (2). First we verify (1).

λ_{i_{1},ξ}^{l}λ_{i_{2},η}^{l} = (∑_{j_{1},k_{1}} T_{k_{1}}^{∗}Ṽ_{i_{1}j_{1}k_{1}}(π^{l}(ξ) ⊗ 1_{j_{1}})T_{j_{1}})(∑_{j_{2},k_{2}} T_{k_{2}}^{∗}Ṽ_{i_{2}j_{2}k_{2}}(π^{l}(η) ⊗ 1_{j_{2}})T_{j_{2}})
= ∑_{j_{1},j_{2},k_{1}} T_{k_{1}}^{∗}Ṽ_{i_{1}j_{1}k_{1}}(π^{l}(ξ) ⊗ 1_{j_{1}})Ṽ_{i_{2}j_{2}j_{1}}(π^{l}(η) ⊗ 1_{j_{2}})T_{j_{2}}
= ∑_{j_{2},k_{1}} T_{k_{1}}^{∗}(∑_{j_{1}} Ṽ_{i_{1}j_{1}k_{1}}(1_{i_{1}} ⊗ Ṽ_{i_{2}j_{2}j_{1}}))(π^{l}(ξ) ⊗ 1_{i_{2}} ⊗ 1_{j_{2}})(π^{l}(η) ⊗ 1_{j_{2}})T_{j_{2}}
= ∑_{j_{2},k_{1}} T_{k_{1}}^{∗}(∑_{i_{3}} Ṽ_{i_{3}j_{2}k_{1}}(Ṽ_{i_{1}i_{2}i_{3}} ⊗ 1_{j_{2}}))((π^{l}(ξ) ⊗ 1_{i_{2}})π^{l}(η) ⊗ 1_{j_{2}})T_{j_{2}}
= ∑_{j_{2},k_{1},i_{3}} T_{k_{1}}^{∗}Ṽ_{i_{3}j_{2}k_{1}}((Ṽ_{i_{1}i_{2}i_{3}}(π^{l}(ξ) ⊗ 1_{i_{2}})π^{l}(η)) ⊗ 1_{j_{2}})T_{j_{2}}
= ∑_{i_{3}} ∑_{k,j} T_{k}^{∗}Ṽ_{i_{3}jk}(π^{l}(ζ_{i_{3}}) ⊗ 1_{j})T_{j}
= ∑_{i_{3}} λ_{i_{3},ζ_{i_{3}}}^{l}.

Next we verify (2).

(λ_{i,ξ}^{l})^{∗} = ∑_{j,k} T_{j}^{∗}(π^{l}(ξ)^{∗} ⊗ 1_{j})Ṽ_{ijk}^{∗}T_{k}
= ∑_{j,k} T_{j}^{∗}(π^{l}(ξ)^{∗} ⊗ 1_{j})(1_{i} ⊗ Ṽ_{īkj})(Ṽ_{iī0} ⊗ 1_{k})T_{k}
= ∑_{j,k} T_{j}^{∗}Ṽ_{īkj}(((π^{l}(ξ)^{∗} ⊗ 1ī)Ṽ_{iī0}) ⊗ 1_{k})T_{k}
= ∑_{j,k} T_{j}^{∗}Ṽ_{īkj}(π^{l}(J_{l}ξ) ⊗ 1_{k})T_{k}
= λ_{ī,J_{l}ξ}^{l}
□.

We investigate the properties of J_{l} and J_{r} and finally we prove that J_{l} and J_{r} are unitary involutions and J_{l} = J_{r}.

Lemma 2.4 The following equalities hold for ξ, η in B_{i}^{bdd} and a, b ∈ A.

(1) J_{l}^{2} = J_{r}^{2} = 1 on the domain.

(2) ⟨J_{l}ζ_{1}|J_{r}ζ_{2}⟩ = ⟨ζ_{2}|ζ_{1}⟩, for ζ_{1}, ζ_{2} in domain, especially J_{l}^{∗} ⊃ J_{r} and J_{r}^{∗} ⊃ J_{l} hold,

(3) J_{r}λ_{i,ξ}^{l}J_{r} = λ_{ī,J_{r}ξ}^{r}, J_{l}λ_{i,ξ}^{r}J_{l} = λ_{ī,J_{l}ξ}^{l},

(4) J_{l}(aξb) = b^{∗}(J_{l}ξ)a^{∗}, J_{r}(aξb) = b^{∗}(J_{r}ξ)a^{∗}.

Proof. Last equalities are obvious by the definitions of J_{l} and J_{r}. So we prove only equalities in (1) and (2). Before verifying the equalities, we note that 1 = (1_{i} ⊗Ṽ_{īi0})(Ṽ_{iī0}^{∗} ⊗ 1_{i}) holds for every i.

First we verify (1). Take ξ ∈ B_{i}^{bdd}.

J_{l}(J_{l}ξ) = (π^{l}(J_{l}ξ)^{∗} ⊗ 1_{i})Ṽ_{īi0}^{∗}ˆ1
= ((Ṽ_{iī0}(π^{l}(ξ) ⊗ 1ī)) ⊗ 1_{i})Ṽ_{īi0}^{∗}ˆ1
= (Ṽ_{iī0} ⊗ 1_{i})(1_{i} ⊗ Ṽ_{īi0})π^{l}(ξ)ˆ1
= ξ.

We can verify J_{r}^{2} = 1 in the same way as above.

Next we verify (2). If we take ξ ∈ B_{i}^{bdd} and η ∈ B_{j}^{bdd} for i ≠ j, it is clear that the equality holds because ⟨ξ|η⟩ = ⟨J_{l}ξ|J_{r}η⟩ = 0. So we may assume ξ, η ∈ B_{i}^{bdd}. Then we have

⟨J_{l}ξ|J_{r}η⟩ = ⟨(π^{l}(ξ)^{∗} ⊗ 1ī)Ṽ_{iī0}^{∗}ˆ1|(1ī ⊗ π^{r}(η)^{∗})Ṽ_{īi0}^{∗}ˆ1⟩
= ⟨(1ī ⊗ π^{r}(η))(π^{l}(ξ)^{∗} ⊗ 1ī)Ṽ_{iī0}^{∗}ˆ1|Ṽ_{īi0}^{∗}ˆ1⟩
= ⟨(π^{l}(ξ)^{∗} ⊗ 1_{i} ⊗ 1ī)(1_{i} ⊗ 1ī ⊗ π^{r}(η))Ṽ_{iī0}^{∗}ˆ1|Ṽ_{īi0}^{∗}ˆ1⟩
= ⟨(1_{i} ⊗ Ṽ_{īi0})(Ṽ_{iī0}^{∗} ⊗ 1_{i})π^{l}(η)ˆ1|π^{r}(ξ)ˆ1⟩
= ⟨η|ξ⟩,

where we use 1 = (1_{i} ⊗ Ṽ_{īi0})(Ṽ_{iī0}^{∗} ⊗ 1_{i}) in the last equality.

Next we verify (3). Fix η ∈ B_{j}^{bdd} and we compute J_{r}λ_{i,ξ}^{l}J_{r}η. Then we have

J_{r}λ_{i,ξ}^{l}J_{r}η = J_{r}λ_{i,ξ}^{l}(1j̄ ⊗ π^{r}(η)^{∗})Ṽ_{j̄j0}ˆ1
= J_{r} ∑_{k} T_{k}^{∗}Ṽ_{ij̄k}(π^{l}(ξ) ⊗ 1j̄)(1j̄ ⊗ π^{r}(η)^{∗})Ṽ_{j̄j0}^{∗}ˆ1
= ∑_{k} T ∗ ; k̄(1k̄ ⊗ Ṽ_{j̄j0})(1k̄ ⊗ 1j̄ ⊗ π^{r}(η))(1k̄ ⊗ π^{l}(ξ)^{∗} ⊗ 1j̄)(1k̄ ⊗ Ṽ_{ij̄k}^{∗})Ṽ_{k̄k0}^{∗}ˆ1
= ∑_{k} T ∗ ; k̄(1k̄ ⊗ π^{l}(ξ)^{∗})(1k̄ ⊗ 1_{i} ⊗ Ṽ_{j̄j0})(1k̄ ⊗ Ṽ_{ij̄k}^{∗} ⊗ 1_{j})(1k̄ ⊗ 1_{k} ⊗ π^{r}(η))Ṽ_{k̄k0}^{∗}ˆ1
= ∑_{k} T ∗ ; k̄(1k̄ ⊗ π^{l}(ξ)^{∗})(1k̄ ⊗ Ṽ_{kji})(Ṽ_{k̄k0}^{∗} ⊗ 1_{j})π^{r}(η)ˆ1
= ∑_{k} T ∗ ; k̄(1k̄ ⊗ π^{l}(ξ)^{∗})Ṽ_{k̄ij}^{∗}η
= (λ_{i,ξ}^{r})^{∗}η.

In the same way as above, we can show the second equation in (3). □

Let {ξ_{e}} be a basis of _{A}B_{i}. Put ζ := ∑_{e} J_{l}ξ_{e} ⊗ ξ_{e}. We can easily verify aζ = ζa. So ζ is a nonzero central vector in _{A}Bī⊗_{A}B_{iA}, and since _{A}B_{iA} is irreducible, it is the unique central vector up to scalar multiplicity.

We compute Ṽ_{īi0}ζ. Then we have

Ṽ_{īi0}ζ = ∑_{e} Ṽ_{īi0}(J_{l}ξ_{e} ⊗ ξ_{e})
= ∑_{e} Ṽ_{īi0}((π^{l}(ξ_{e})^{∗} ⊗ 1ī)Ṽ_{iī0}^{∗}ˆ1) ⊗ ξ_{e}
= ∑_{e} Ṽ_{īi0}(π^{l}(ξ_{e})^{∗} ⊗ 1ī ⊗ 1_{i})(Ṽ_{īi0}^{∗} ⊗ 1_{i})ξ_{e}
= ∑_{e} π^{l}(ξ_{e})^{∗}(1_{i} ⊗ Ṽ_{īi0})(Ṽ_{iī0}^{∗} ⊗ 1_{i})ξ_{e}
= ∑_{e} π^{l}(ξ_{e})^{∗}ξ_{e}
= ∑_{e}⟨ξ_{e}|ξ_{e}⟩_{A}.

Since ∑_{e}⟨ξ_{e}|ξ_{e}⟩_{A} and elements of A commute, ∑_{e}⟨ξ_{e}|ξ_{e}⟩_{A} is a scalar. By the definition of the operator valued inner product,

∑_{e}⟨ξ_{e}|ξ_{e}⟩_{A} = tr_{A}(∑_{e}⟨ξ_{e}|ξ_{e}⟩_{A})
= ∑_{e}⟨ξ_{e}|ξ_{e}⟩
= tr_{A}(∑_{e}_{A}⟨ξ_{e}|ξ_{e}⟩)
= dim_{A}B_{i}

holds. Since \frac{1}{d(i)} Ṽ_{īi0} is an isometry on span\{aζ|a ∈ A\}, ∥ζ∥ = ∥\frac{1}{d(i)} Ṽ_{īi0}ζ∥ = \frac{1}{d(i)} dim_{A}B_{i} = d(i) holds.

On the other hand, ζ′ := ∑_{e} ξ̄_{e} ⊗ξ_{e} is also central, where ξ̄_{e} means a natural conjugate element in Bī, and ||ζ′|| = \sqrt{dim_{A}B_{i}} = d(i). By the uniqueness of the central vector, ζ = c_{i}ζ′ for some c_{i} in T.

So by computing (1ī ⊗ π^{r}(ξ_{e})^{∗})ζ and (1ī ⊗ π^{r}(ξ_{e})^{∗})ζ′, we get J_{l}ζ_{e} = c_{i}ξ̄_{e}. So for every bounded vector, we have J_{l}ξ = c_{i}ξ̄. Hence J_{l} preserves the inner product and can be extended onto X. Combining these with the results in Lemma 2.4, we can conclude J_{l} = J_{r}. So in the rest of this paper, we denote this J_{l} by J_{X}.

Now we can state the main theorem of this section.

Theorem 2.5 Set B^{l} := ∨{λ_{i,ξ}^{l}} and B^{r} := (∨{λ_{i,ξ}^{r}})^{opp}. Then we have the following.

(1) B^{l} and B^{r} are II_{1} factors.

(2) There is a natural bimodule isomorphism between _{A}X_{A} and _{A}L^{2}(B^{l})_{A}.

(3) J_{X}B^{l}J_{X} = B^{r} holds and the commutant of B^{l} is B^{r}.

(4) Indices of inclusions A ⊂ B^{l} and A ⊂ B^{r} are both ∑_{i} d(i)^{2}, i.e., the global index of N ⊂ M. Especially A ⊂ B^{l} is of finite index if and only if N ⊂ M is of finite depth. (5) If an original subfactor N ⊂ M is of finite depth, then A ⊂ B^{r} is the same inclusion as constructed in [14].

Proof. Let {ξ_{e}^{i}}_{e} be a basis of _{A}B_{i}. Then

B^{l(0)} = {∑_{i,e} x_{e}^{i}λ_{i,ξ_{e}^{i}}^{l}|x_{e}^{i} = 0 except for finitely many i, e}

is a dense ∗-subalgebra of B^{l(0)}.

First we define tracial states on B^{l} and B^{r}. Set

tr_{B^{l}}(x) := ⟨xˆ1|ˆ1⟩, x ∈ B^{l},
tr_{B^{r}}(x) := ⟨xˆ1|ˆ1⟩, x ∈ B^{r},

where ˆ1 ∈ L^{2}(A) ⊂ X. To prove that tr_{B^{l}} is tracial, we only have to prove tr_{B^{l}}(x^{∗}x) = tr_{B^{r}}(xx^{∗}) for x ∈ B^{l(0)}. By the definition of tr_{B^{l}}, it is not so difficult to see that tr_{B^{l}}((λ_{j,η}^{l})^{∗}λ_{i,ξ}^{l}) = tr_{B^{l}}(λ_{i,ξ}^{l}(λ_{j,η}^{l})^{∗}) = 0 holds for i ≠ j.

Take λ_{i,ξ}^{l} ∈ B^{l}. Then we have

tr_{B^{l}}((λ_{i,ξ}^{l})^{∗}λ_{i,ξ}^{l}) = ⟨λ_{i,ξ}^{l}ˆ1|λ_{i,ξ}^{l}ˆ1⟩
= ⟨ξ|ξ⟩
= ⟨J_{X}ξ|J_{X}ξ⟩
= ⟨λ_{ī,J_{X}ξ}^{l}ˆ1|λ_{ī,J_{X}ξ}^{l}ˆ1⟩
= ⟨(λ_{i,ξ}^{l})^{∗}ˆ1|(λ_{i,ξ}^{l})^{∗}ˆ1⟩
= tr_{B^{l}}(λ_{i,ξ}^{l}(λ_{i,ξ}^{l})^{∗})

and hence we conclude that tr_{B^{l}} is tracial. Similarly tr_{B^{r}} is also tracial.

Next we prove that L^{2}(B^{l}, tr_{B^{l}}) is isomorphic to X as A-A bimodule. If we send λ̂_{i,ξ}^{l} ∈ L^{2}(B^{l}, tr_{B^{l}}) to ξ ∈ X, it is clear that this map gives a bimodule isomorphism between them. L^{2}(B_{l}, tr_{B^{l}}) contains only one trivial A-A bimodule and this means that there exists only one conditional expectation from B^{l} onto A. This conditional expectation E_{A} is given by E_{A}(x) = T_{0}xT_{0}∗. So B^{l} is a II_{1} factor. Similarly B^{r} is also a II_{1} factor. So we have proved (1) and (2). Also by (2), the index [B^{l} : A] is equal to ∑_{i} d(i)^{2} and we verified (4).

Next we prove (3). First statement is obvious by (3) of Lemma 2.4. When we identify AX_{A} and _{A}L^{2}(B^{l}, tr_{B^{l}})_{A} by the above isomorphism, J_{X} correspond to J_{B^{l}}, λ_{i,ξ}^{l} to left multiplication λ_{i,ξ}^{l}, and λ_{i,ξ}^{r} to right multiplication of λ_{i,ξ}^{l}. So the commutant of B^{l} is B^{r}.

Finally we prove (5). Note that (∑_{l} d(l)^{2})^{−1} ∑_{i,j,k} T_{k}^{∗}Ṽ_{ijk}(T_{i} ⊗1_{j})(1_{X} ⊗T_{j}) is precisely an intertwiner S in the proof of [14, Theorem 3.1]. In [14], the Longo-Rehren inclusion A ⊂ B is constructed by putting B := {S(1_{X} ⊗ a)S^{∗}|a ∈ End (_{A}X)}^{opp}. By easy computation, it is shown that S(1_{X} ⊗ λ_{i,ξ}^{r})S^{∗} = λ_{i,ξ}^{r}. So A ⊂ B^{r} ⊂ B holds and by comparing index, we get B = B^{r}. □

Definition 2.6 We call the above construction the generalized Longo-Rehren construction and we denote A(Δ) := B^{l}. We call the inclusion A ⊂ A(Δ) the generalized Longo-Rehren subfactor, or simply the Longo-Rehren subfactor.

The Longo-Rehren construction is similar to that of a crossed product. In fact, λ^{l} (resp. λ^{r}) is an analogue of the left (resp. right) regular representation of groups. If a bimodule _{A}B_{iA} comes from an automorphism α of A, i.e., it is isomorphic to _{A}_{α^{−1}}A_{A}, then λ_{i,ˆ1}^{l} is the implementing unitary of α. So if Δ arises from an action α of a discrete group G, A(Δ) is precisely A ⋊_{α⊗α^{opp}} G.

As in the usual crossed product case, x ∈ A(Δ) is expressed as x = ∑_{i,e} x_{i,e}λ_{i,ξ_{i}^{e}}^{l}. This summation does not converge in any usual operator topology, but converges in the L^{2} norm. Also note that though each x_{i,e} is not uniquely determined, a vector ∑_{e} x_{i,e}ξ_{i}^{e} ∈ B_{i} is uniquely determined by x.

Set B_{1} := End (X_{A}). Then an inclusion A ⊂ A(Δ) ⊂ B_{1} is the Jones tower. The relative commutant of A ⊂ B_{1} is ⨁_{k} Cp_{k} ≅ l^{∞}(Δ), where p_{k} = T_{k}^{∗}T_{k}.

Let E_{1} be the dual operator valued weight of E_{A} from B_{1} to A(Δ) such that E_{1}(e_{A}) = 1 and put tr_{B_{1}} := tr_{A(Δ)} ∘ E_{1}.

Proposition 2.7 The restriction of E_{1} to A′ ∩ B_{1} is equal to the Haar measure of Δ in the sense of [9], i.e., E_{1}(p_{k}) = d(k)^{2} holds for every k. Especially tr_{B_{1}} is semifinite on A′ ∩ B_{1}.

Proof. Let {ξ_{e}} be a basis for B_{iA}. We compute ∑_{e}(λ_{i,ξ_{e}}^{l})^{∗}p_{0}λ_{i,ξ_{e}}^{l}. It is easy to see that ∑_{e} π^{l}(ξ_{e})π^{l}(ξ_{e})^{∗} = 1 holds, so by using this equality, we can easily get

∑_{e}(λ_{i,ξ^{e}}^{l})^{∗}p_{0}λ_{i,ξ_{e}}^{l} = ∑_{e,j_{1},j_{2},k_{1},k_{2}} T_{k_{1}}^{∗}Ṽ_{j_{1}ik}(1_{j_{1}} ⊗ π^{l}(ξ_{e}))T_{j_{1}}T_{0}^{∗}T_{0}T_{j_{2}}^{∗}(1_{j_{2}} ⊗ π^{l}(ξ_{e})^{∗})Ṽ_{j_{2}ik_{2}}^{∗}T_{k_{2}}^{∗}
= ∑_{e,k_{1},k_{2}} T_{k_{1}}^{∗}Ṽ_{0ik_{1}}π^{l}(ξ_{e})π^{l}(ξ_{e})^{∗}Ṽ_{0ik_{2}}^{∗}T_{k_{2}}^{∗}
= T_{i}^{∗}T_{i}.

In the same way as above, we can show that ∑_{e}(λ_{i,ξ_{e}}^{l})^{∗}λ_{i,ξ_{e}}^{l} = d(i)^{2} holds. Hence we have the following equality.

E(T_{i}^{∗}T_{i}) = ∑_{e}(λ_{i,ξ_{e}}^{l})^{∗}E_{1}(T_{0}^{∗}T_{0})λ_{i,ξ_{e}}^{l}
= ∑_{e}(λ_{i,ξ_{e}}^{l})^{∗}λ_{i,ξ_{e}}^{l}
= d(i)^{2}.
□

3 Relation between the symmetric enveloping algebras and the Longo-Rehren subfactors

In this section, we investigate the relation between Popa’s symmetric enveloping algebras ([24], [25]) and the Longo-Rehren subfactors. For the theory of the symmetric enveloping algebras, we refer [24] and [25]. We also use the notation in [24] and [25] freely.

First in this section, we construct a projection in A(Δ) which satisfies the Jones relation for both inclusions N ⊂ M and N^{opp} ⊂ M^{opp}.

Set Δ_{0} := {_{M}Z_{iM}|irreducuble, _{M}Z_{iM} ≺ _{M}M_{1M}}. By Frobenius reciprocity, _{N}M_{M} ≺ NM⊗_{M}Z_{iM} ≅ _{N}Z_{iM} holds for every element in Δ_{0}. Let {S_{i}^{e}} ⊂ Hom (_{N}Z_{iM},_{N}M_{M}) be an orthonormal basis and put S̃_{i} := ∑_{e} S_{i}^{e} ⊗ JS_{i}^{e}J^{−1}. Then S̃_{i} is an element of Hom (_{N⊗N^{opp}}B_{A},_{N⊗N^{opp}}A_{A}).

The author learned the next formula from Professor Izumi ([10]). He is grateful to Professor Izumi for permitting him to present this formula here.

Lemma 3.1 Set e := [M : N]^{−1} ∑_{i∈Δ_{0}} \sqrt{d(i)}λ_{i,S̃_{i}^{∗}(ˆ1)}^{l}. Then e is a projection and satisfies relations E_{A}(e) = [M : N]^{−1}, exe = E_{N}(x)e for x ∈ M and exe = E_{N^{opp}}(x)e for x ∈ M^{opp}, where E_{N} (resp.E_{N^{opp}} ) is the trace preserving conditional expectation from M onto N (resp. M^{opp} onto N^{opp} ).

Proof. First we prove e = e^{∗}. Note that λ_{i,S̃_{i}^{∗}(ˆ1)}^{l} is equal to ∑_{j,k} T_{k}^{∗}Ṽ_{ijk}(S̃_{i}^{∗} ⊗ 1_{j})T_{j}. and

hence e is equal to [M : N]^{−1} ∑_{i,j,k} \sqrt{d(i)}T_{k}^{∗}Ṽ_{ijk}(S̃_{i}^{∗} ⊗ 1_{j})T_{j}.

Then

(Ṽ_{ijk}(S̃_{i}^{∗} ⊗ 1_{j}))^{∗} = (S̃_{i} ⊗ 1_{j})Ṽ_{ijk}^{∗}
= (S̃_{i} ⊗ 1_{j})(1_{i} ⊗ Ṽ_{īkj})(Ṽ_{iī0}^{∗} ⊗ 1_{k})
= Ṽ_{īkj}((S̃_{i} ⊗ 1ī)Ṽ_{iī0}^{∗} ⊗ 1_{k})

holds. Here (S̃_{i} ⊗ 1ī)Ṽ_{iī0}^{∗} is an element of Hom (_{N⊗N^{opp}}A_{A},_{N⊗N^{opp}}B_{īA}). If this intertwiner is equal to S̃∗ ; ī, we easily get e = e^{∗}. Fix an orthonormal basis {S_{i}^{e}} ⊂ Hom (_{N}Z_{iM},_{N}M_{M}) and let S̄_{i}^{e} ∈ Hom (_{N}Z_{īM},_{N}M_{M}) be a Frobenius dual intertwiner. Then {S̄_{i}^{e}} is a basis of Hom (_{N}Z_{īM},_{N}M_{N}) and S̄_{i}^{e} = \sqrt{d(i)}V_{iī0}(S_{i}^{e∗} ⊗ 1ī) holds. (See [26, Theorem 1.5].) Since S̃ī is independent from the choice of an orthonormal basis of Hom (_{N}Z_{īM},_{N}M_{M}), S̃ī = ∑_{e} S̄_{i}^{e} ⊗ JS̄_{i}^{e}J^{−1} = Ṽ_{iī0}(S̃_{i}^{∗} ⊗ 1ī) holds. Hence e is self adjoint.

Next we verify exe = E_{N}(x)e for x ∈ M. First note that since S̃_{i} is not left M ⊗ M^{opp} linear, x and S̃_{i} do not commute.

exe = [M : N]^{−2} ∑_{{i_{1},j_{1},k_{1}, \atop i_{2},j_{2},k_{2}}} \sqrt{d(i_{1})d(i_{2})}T_{i_{1}}^{∗}Ṽ_{i_{1}j_{2}k_{3}}(S̃_{i_{1}}^{∗} ⊗ 1_{j_{1}})T_{j_{1}}xT_{k_{2}}^{∗}Ṽ_{i_{2}j_{2}k_{2}}(S̃_{i_{2}}^{∗} ⊗ 1_{j_{2}})T_{j_{2}}

= [M : N]^{−2} ∑_{{i_{1},j_{1},k, \atop i_{2},j_{2}}} \sqrt{d(i_{1})d(i_{2})}T_{k}^{∗}Ṽ_{i_{1}j_{1}k}(S̃_{i_{1}}^{∗} ⊗ 1_{j_{1}})xṼ_{i_{2}j_{2}j_{1}}(S̃_{i_{2}}^{∗} ⊗ 1_{j_{2}})T_{j_{2}}

= [M : N]^{−2} ∑_{{i_{1},i_{2}, \atop k,j_{2}}} \sqrt{d(i_{1})d(i_{2})}T_{k}^{∗}(∑_{j_{1}} Ṽ_{i_{1}j_{1}k}(1_{i_{1}} ⊗ Ṽ_{i_{2}j_{2}j_{1}}))((S̃_{i_{1}}^{∗} ⊗ 1_{i_{2}})xS̃_{i_{2}} ⊗ 1_{j_{2}})T_{j_{2}}

= [M : N]^{−2} ∑_{{i_{1},i_{2}, \atop k,j_{2}}} \sqrt{d(i_{1})d(i_{2})}T_{k}^{∗}(∑_{j_{1}} Ṽ_{j_{1}j_{2}k}(Ṽ_{i_{1}i_{2}j_{1}} ⊗ 1_{j_{2}}))((S̃_{i_{1}}^{∗} ⊗ 1_{i_{2}})xS̃_{i_{2}} ⊗ 1_{j_{2}})T_{j_{2}}

= [M : N]^{−2} ∑_{{i_{1},j_{1},k, \atop i_{2},j_{2}}} \sqrt{d(i_{1})d(i_{2})}T_{k}^{∗}Ṽ_{j_{1}j_{2}k}(Ṽ_{i_{1}i_{2}j_{1}}(S̃_{i_{1}}^{∗} ⊗ 1_{i_{2}})xS̃_{i_{2}}^{∗} ⊗ 1_{j_{2}})T_{j_{2}}.

Here we compute ∑_{i_{1},i_{2}} \sqrt{d(i_{1})d(i_{2})}Ṽ_{i_{1}i_{2}j}(S̃_{i_{1}}^{∗} ⊗ 1_{i_{2}})xS̃_{i_{2}}^{∗}. Let \{V_{i_{1}i_{2}j}^{e}\}_{e=1}^{N_{i_{1}i_{2}}^{j}} be an orthonormal basis of Hom (_{M}Z_{i_{1}}⊗_{M}Z_{i_{2}M},_{M}Z_{jM}). Then we have

∑_{i_{1},i_{2}} \sqrt{d(i_{1})d(i_{2})}Ṽ_{i_{1}i_{2}j}(S̃_{i_{1}}^{∗} ⊗ 1_{i_{2}})xS̃_{i_{2}}^{∗}

= ∑_{{i_{1},i_{2}, \atop e_{1},e_{2},e_{3}}} d(i_{1})d(i_{2})d(j)^{− \frac{1}{2}} (V_{i_{1}i_{2}j}^{e_{1}}(S_{i_{1}}^{e_{2}∗} ⊗ 1_{i_{2}})xS_{i_{2}}^{e_{3}∗}) ⊗ JV_{i_{1}i_{2}j}^{e_{1}}(S_{i_{1}}^{e_{2}∗} ⊗ 1_{i_{2}})S_{i_{2}}^{e_{3}∗}J^{−1}.

Then V_{i_{1}i_{2}j}^{e_{1}}(S_{i_{1}}^{e_{2}∗} ⊗ 1_{i_{2}})S_{i_{2}}^{e_{3}∗} is in Hom (_{N}M_{M},_{N}Z_{jM}), so

V_{i_{1}i_{2}j}^{e_{1}}(S_{i_{1}}^{e_{2}∗} ⊗ 1_{i_{2}})S_{i_{2}}^{e_{3}∗} = ∑_{e_{4}} \begin{matrix}i_{1} & →^{e_{2}} & 0 \\ ↓_{e_{1}} & & ↓_{e_{3}} \\ j & →^{e_{4}} & 0 \end{matrix} S_{j}^{e_{4}∗}

holds, here we use the notation of connection in [5].

Let R̄ ∈ Hom (_{N}M_{N},_{N}N_{N}) and R ∈ Hom (_{M}M⊗_{N}M_{M},_{M}M_{M}) be coisometries which satisfy R(1_{_{M} M_{N}} ⊗ R̄^{∗}) = [M : N]^{− \frac{1}{2}} and (R̄ ⊗ 1_{_{N} M_{M}})R^{∗} = [M : N]^{− \frac{1}{2}} . Then R_{i}^{e} := \sqrt{d(i)}(R ⊗ 1_{i})(1_{_{M} M_{N}} ⊗ S_{i}^{e∗}) ∈ Hom (_{M}M⊗_{N}M_{M},_{M}Z_{iM}) is a Frobenius dual intertwiner of S_{i}^{e} and S_{i}^{e} = [M : N]^{\frac{1}{2}} d(i)^{− \frac{1}{2}} (R̄ ⊗ 1_{_{N} M_{M}})R_{i}^{e∗} holds.

By renormalization rule,

\begin{matrix}i_{1} & →^{e_{2}} & 0 \\ ↓_{e_{1}} & & ↓_{e_{3}} \\ j & →^{e_{4}} & 0 \end{matrix} = \sqrt{\frac{d(j)}{d(i_{1})}} \overline{\begin{matrix}0 & →^{e_{2}} & i_{1} \\ ↓_{e_{3}} & & ↓_{e_{1}} \\ 0 & →^{e_{4}} & j \end{matrix}} = \sqrt{\frac{d(j)}{d(i_{1})}}V_{i_{1}i_{2}j}^{e_{1}}(R_{i_{1}}^{e_{2}} ⊗ 1_{i_{2}})(1_{_{M} M_{N}} ⊗ S_{i_{2}}^{e_{3}∗})R_{j}^{e_{4}∗}

holds.

So we get an equality

∑_{i_{1},i_{2}} \sqrt{d(i_{1})d(i_{2})}Ṽ_{i_{1}i_{2}j}(S̃_{i_{1}}^{∗} ⊗ 1_{i_{2}})xS̃_{i_{2}}^{∗}

= [M : N]^{\frac{1}{2}} ∑_{{i_{1},i_{2},e_{1}, \atop e_{2},e_{3},e_{4}}} \begin{matrix}0 & →^{e_{2}} & i_{1} \\ ↓_{e_{3}} & & ↓_{e_{1}} \\ 0 & →^{e_{2}} & j \end{matrix} d(i_{2})V_{i_{1}i_{2}j}^{e_{1}}(R_{i_{1}}^{e_{2}} ⊗ 1_{i_{2}})(R̄^{∗} ⊗ 1_{_{N} M_{M}} ⊗ 1_{i_{2}})xS_{i_{2}}^{e_{3}∗}

⊗JS_{j}^{e_{4}∗}J^{−1} =: (∗).

Here we have the following equality.

∑_{i_{1},e_{1},e_{2}} \begin{matrix}0 & →^{e_{2}} & i_{1} \\ ↓_{e_{3}} & & ↓_{e_{1}} \\ 0 & →^{e_{4}} & j \end{matrix} V_{i_{1}i_{2}j}^{e_{1}}(R_{i_{1}}^{e_{2}} ⊗ 1_{i_{2}}) = R_{j}^{e_{4}}(1_{_{M} M_{N}} ⊗ S_{i_{2}}^{e_{3}}).

Hence we have the following.

(∗) = [M : N]^{\frac{1}{2}} ∑_{i_{2},e_{3},e_{4}} d(i_{2})R_{j}^{e_{4}}(1_{_{M} M_{N}} ⊗ S_{i_{2}}^{e_{3}})(R̄^{∗} ⊗ 1_{_{N} M_{M}} ⊗ 1_{i_{2}})xS_{i_{2}}^{e_{3}∗} ⊗ J_{Z_{j}}S_{j}^{e_{4}∗}J_{L^{2}(M)}^{−1}
= [M : N]^{\frac{1}{2}} ∑_{i_{2},e_{3},e_{4}} d(i_{2})R_{j}^{e_{4}}(R̄^{∗} ⊗ 1_{_{N} M_{M}})S_{i_{2}}^{e_{3}}xS_{i_{2}}^{e_{3}∗} ⊗ J_{Z_{j}}S_{j}^{e_{4}∗}J_{L^{2}(M)}^{−1}
= ∑_{i_{2},e_{3},e_{4}} d(i_{2})d(j)^{\frac{1}{2}} S_{j}^{e_{4}∗}S_{i_{2}}^{e_{3}}xS_{i_{2}}^{e_{3}∗} ⊗ J_{Z_{j}}S_{j}^{e_{4}∗}J_{L^{2}(M)}^{−1}
= [M : N] ∑_{i_{2},e_{3},e_{4}} d(j)^{\frac{1}{2}} S_{j}^{e_{4}∗}(R̄ ⊗ 1_{_{N} M_{M}})R_{i_{2}}^{e_{3}∗}xR_{i_{2}}^{e_{3}}(R̄^{∗} ⊗ 1_{_{N} M_{M}}) ⊗ J_{Z_{j}}S_{j}^{e_{4}∗}J_{L^{2}(M)}^{−1}
= [M : N]\sqrt{d(j)} ∑_{e_{4}} S_{j}^{e_{4}∗}(R̄ ⊗ 1_{_{N} M_{N}})x ∑_{i_{2},e_{3}} R_{i_{2}}^{e_{3}∗}R_{i_{2}}^{e_{3}}(R̄^{∗} ⊗ 1_{_{N} M_{M}}) ⊗ J_{Z_{j}}S_{j}^{e_{4}∗}J_{L^{2}(M)}^{−1}
= [M : N]\sqrt{d(j)} ∑_{e_{4}} S_{j}^{e_{4}∗}(R̄ ⊗ 1_{_{N} M_{M}})x(R̄^{∗} ⊗ 1_{_{N} M_{M}}) ⊗ J_{Z_{j}}S_{j}^{e_{4}∗}J_{L^{2}(M)}^{−1}
= [M : N]\sqrt{d(j)} ∑_{e_{4}} S_{j}^{e_{4}∗}E_{N}(x) ⊗ J_{Z_{j}}S_{j}^{e_{4}∗}J_{L^{2}(M)}^{−1}
= [M : N]\sqrt{d(j)}E_{N}(x)S̃_{j}.

So

exe = [M : N]^{−2} ∑_{j_{1},j_{2},k} T_{k}^{∗}Ṽ_{j_{1}j_{2}k}([M : N]\sqrt{d(j_{1})}E_{N}(x)S̃_{j_{1}} ⊗ 1_{j_{2}})T_{j_{2}}
= [M : N]^{−1}E_{N}(x) ∑_{j_{1},j_{2},k} \sqrt{d(j_{1})}T_{k}^{∗}Ṽ_{j_{1}j_{2}k}(S̃_{j_{1}} ⊗ 1_{j_{2}})T_{j_{2}}
= E_{N}(x)e.

holds. By similar computations, it is shown that exe = E_{N^{opp}}(x)e holds for x ∈ M^{opp}. If we put x = 1, then we get e^{2} = e. Hence e is a projection. Finally since T_{0}(∑_{j,k} T_{k}^{∗}Ṽ_{ijk}(S̃_{i}^{∗} ⊗ 1_{j})T_{j})T_{0}^{∗} = δ_{i,0} holds, we get T_{0}eT_{0}^{∗} = [M : N]^{−1}. □

Lemma 3.2 The algebra A(Δ) is generated by M, M^{opp} and e.

Proof. Let X^{0} be an A-A bimodule generated by e and A. Then it is obvious that X^{0} is isomorphic to ⊕_{i∈Δ_{0}}B_{i} as an A-A bimodule. So X^{0} contains an element of the form λ_{i,ξ}^{l}, i ∈ Δ_{0} and ξ ≠ 0. Since Δ_{0} generates Δ, it is easy to see that such elements generate A(Δ). □

Let M_{1} (resp. M_{1}^{opp}) ⊂ A(Δ) be a von Neumann algebra generated by e and M (resp. M^{opp}).

Lemma 3.3 We have M′ ∩ A(Δ) = M^{opp}, M^{opp}′ ∩ A(Δ) = M, N′ ∩ A(Δ) = M_{1}^{opp} and M′_{1} ∩ A(Δ) = N^{opp}.

Proof. First we prove that if ξ ∈ B_{i} (B_{i} ≠ L^{2}(A)) satisfies xξ = ξx for every x ∈ M, then ξ = 0.

Let {ζ_{k}} be a CONS of Z_{i}^{o} and we express ξ = ∑_{k} ξ_{k} ⊗ ζ_{k}. By assumption, ∑_{k} xξ_{k} ⊗ ζ_{k} = ∑_{k} ξ_{k}x ⊗ ζ_{k} holds. This means that xξ_{k} = ξ_{k}x holds for every k. So for every k, ξ_{k} is a central vector in _{M}Z_{iM}. But since we assume that _{M}Z_{iM} ≠ _{M}M_{M} and irreducibility of _{M}Z_{iM}, ξ_{k} must be zero. So ξ is a zero vector.

Take x ∈ A(Δ) and express x = ∑_{i∈Δ} λ_{i,ξ_{i}}^{l}. Then ax = xa implies aξ_{i} = ξ_{i}a for every i. By the above result, ξ_{i} = 0 for every i ≠ 0 and we get x ∈ M^{opp}. In a similar way as above, we can show M^{opp} ∩ A(Δ) and N′ ∩ A(Δ) = M_{1}^{opp}.

It is clear that N^{opp} ⊂ M′_{1} ∩ A(Δ) holds. Conversely if x ∈ M′_{1} ∩ A(Δ), first we get x ∈ M^{opp}. Moreover ex = xe implies E_{N^{opp}}(x)e = xe, so we get T_{0}E_{N^{opp}}(x)eT_{0}^{∗} = T_{0}xeT_{0}^{∗} and by Lemma 3.1, we get E_{N^{opp}}(x) = x. □

By Lemma 3.3, M_{1} is a II_{1} factor. By the characterization of the basic construction in [21, Proposition 1.2], N ⊂ M ⊂ M_{1} and N^{opp} ⊂ M^{opp} ⊂ M_{1}^{opp} are standard.

By [25, Corollary 2.3] the symmetric enveloping algebras are characterized by the above conditions, hence we get the following theorem.

Theorem 3.4 The Longo-Rehren subfactor A ⊂ A(Δ) is isomorphic to M ∨ M^{opp} ⊂

M ⊠_{e_{N}} M^{opp}.

At the end of this section, we discuss an extension of automorphisms of a subfactor N ⊂ M to A(Δ). Let α ∈ Aut (M, N). Then as mentioned in [24], we get an automorphism α ⊠ α^{opp} on A(Δ) by defining α ⊠ α^{opp}|_{A} = α ⊗ α^{opp} and α ⊠ α^{opp}(e) = e. On the other hand, if a bimodule _{Mα^{−1}}M_{M} is in Δ, then there is a unitary U in A(Δ) implementing α ⊗ α^{opp} as remarked after Definition 2.6. This U satisfies UeU^{∗} = e, because we have M_{α}M⊗_{N}M_{αM} ≅ _{M}M⊗_{N}M_{M}, and if _{M}Z_{iM} is in Δ_{0}, then _{M}_{α}(Z_{i})_{αM} is in Δ_{0}. So Ad U is coincide with α ⊠ α^{opp}. Since a bimodule _{Mα^{−1}}M_{M} is in Δ if and only if α is not strongly outer (See [1, Theorem 2] and [6, Theorem 2.2]), so we get the following.

Proposition 3.5 Let α be an automorphism in Aut (M, N). If α is not strongly outer, then the extension α ⊠ α^{opp} of α to A(Δ) is an inner automorphism.

Conversely, let us assume that the extension α ⊠ α^{opp} of α is an inner automorphism. Let U be a unitary such that Ad U = α ⊠ α^{opp}. Set B_{U} be the ∥ · ∥_{2} norm closure of {aU|a ∈ A} in L^{2}(A(Δ)). Then B_{U} is an irreducible A-A submodule of_{A}A(Δ)_{A}, hence the irreducible component of _{A}A(Δ)_{A}. Also it is isomorphic to _{Aα⊠α^{opp}^{−1}}A_{A}, so we conclude that _{Mα^{−1}}M_{M} is in Δ. This implies α is not strongly outer as remarked above. So we can summarize these arguments as follows.

Theorem 3.6 For α ∈ Aut (M, N), α is not strongly outer if and only if the extension α ⊠ α^{opp} of α to A(Δ) is inner.

More properties of the above extension of automorphisms will be studied in [15].

4 Amenability of fusion algebras

By combining Theorem 3.4 and [25, Theorem 5.3], A(Δ) is relatively amenable to A if and only if the standard invariant of N ⊂ M is amenable.

In this section, we give another approach to the above result based on the amenability of fusion algebras.

First recall the definition of the amenability of fusion algebras in [9].

Definition 4.1 ([9, Definition 4.3]) Let Δ be a fusion algebra, σ the Haar measure on Δ and ρ_{i}^{l} a left convolution operator by i ∈ Δ.

(1) A fusion algebra Δ is said to be weakly amenable if there exists an invariant mean on Δ, i.e., there exists a state ϕ ∈ l^{∞}(Δ)^{∗} such that ϕ(ρ_{i}^{l}(f)) = ϕ(f) for every i ∈ Δ and every f ∈ l^{∞}(Δ).

(2) A fusion algebra Δ is said to be amenable if there exists an almost invariant l^{2}-sequence in l^{2}(Δ, σ), i.e., there exists a sequence {f_{n}}_{n=1}^{∞} ⊂ l^{2}(Δ, σ) such that lim_{n→∞} ∥ρ_{i}^{l}(f_{n}) − f_{n}∥_{2,σ} = 0 for every i ∈ Δ.

Amenability of Δ is equivalent to the following Følner type condition ([9, Theorem 4.6]).

(FC) For any ε > 0 and any i_{1}, i_{2}, ⋯ i_{n} ∈ Δ, there exists a finite set ∅ ≠ F ⊂ Δ such that

∥ρ_{i_{k}}^{l}(χ_{F}) − χ_{F} ∥_{1,σ} < ε

holds for 1 ≤ k ≤ n, where χ_{F} means a characteristic function of F. Of course we can replace ρ_{i}^{l} by the right convolution operator ρ_{i}^{r} in Definition 4.1 and (FC).

An important point is that weak amenability and amenability are not necessary equivalent in general fusion algebras, though in the group case they are equivalent as proved in [7]. (See [9, Example 6.10].) This fact makes situations more difficult than the group cases.

Let A ⊂ A(Δ) ⊂ B_{1} be the basic construction. By using the above Følner condition, we can prove the following proposition.

Proposition 4.2 If Δ is amenable, there exists a norm one projection from B_{1} onto A(Δ)

Proof. We number the elements of Δ as i_{1}, i_{2}, i_{3}, . . ..

Fix n ∈ N and let F_{n} be a Følner set for i_{1}, i_{2}, ⋯, i_{n} and \frac{1}{n}. (Here we consider the right convolution operator in (FC).)

Define a map P_{n} on B_{1} as follows.

P_{n}(x) := ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{i∈F_{n}} ∑_{e} λ_{i,ξ_{i}^{e}}^{r}x(λ_{i,ξ_{i}^{e}}^{r})^{∗},

where {ξ_{i}^{e}}_{e} is a basis for _{A}B_{i}.

Fix a free ultrafilter ω over N and set P (x) := lim_{n→ω} P_{n}(x). We show that P is a norm one projection from B_{1} onto A(Δ). But it is easy to see that P is of norm one and P (x) = x holds for every x ∈ A(Δ). So we only have to prove that the range of P is indeed A(Δ).

To prove this, we only have to verify that P (x)λ_{i,ξ}^{r} = λ_{i,ξ}^{r}P (x) for every i ∈ Δ and ξ ∈ B_{i}^{bdd}. Then by (3) of Theorem 2.5, we get the conclusion.

Let i be one of i_{1}, ⋯ i_{n} and take ζ ∈ B_{i}^{bdd}. Then ∥ρ_{i}^{r}(χ_{F_{n}}) − χ_{F_{n}}∥_{1,σ} ≤ \frac{1}{n} holds. The left hand of inequality is equal to

∑_{k∈F_{n}} ∑_{m∈F_{n}i\F_{n}} N_{ki}^{m} \frac{d(k)d(m)}{d(i)} + ∑_{k∈F_{n}} ∑_{m∈F_{n}ī\F_{n}} N_{kī}^{m} \frac{d(k)d(m)}{d(i)}.

(See the proof of [9, Theorem 4.6].)

Here we estimate ∥λ_{i,ζ}^{r}P_{n}(x) − P_{n}(x)λ_{i,ζ}^{r}∥. So first we compute λ_{i,ζ}^{r}P_{n}(x).

In the following computations, we denote the right action of a ∈ A by the symbol π^{r}(a) to avoid confusions. Then we have

λ_{i,ζ}^{r}P_{n}(x) = ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{e} λ_{i,ζ}^{r}λ_{k,ξ_{k}^{e}}^{r}x(λ_{k,ξ_{k}^{e}}^{r})^{∗}
= ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{e} ∑_{m∈F_{n}i} λ_{Ṽ_{kim}(ξ_{k}^{e}⊗ζ)}^{r}x(λ_{k,ξ_{k}^{e}}^{r})^{∗}
= ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{e} ∑_{m∈F_{n}i} ∑_{f} λ_{m,ξ_{m}^{f}}^{r}π^{r}(_{A}⟨Ṽ_{kim}(ξ_{k}^{e} ⊗ ζ)|ξ_{m}^{f}⟩)x(λ_{k,ξ_{k}^{e}}^{r})^{∗}
= ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{e} ∑_{m∈F_{n}i} ∑_{f} λ_{m,ξ_{m}^{f}}^{r}x π^{r}(_{A}⟨Ṽ_{kim}(ξ_{k}^{e} ⊗ ζ)|ξ_{m}^{f}⟩)(λ_{k,ξ_{k}^{f}})^{∗}.

Here

A⟨Ṽ_{kim}(ξ_{k}^{e} ⊗ ζ)|ξ_{m}^{f}⟩ = _{A}⟨ξ_{k}^{e}|(1_{k} ⊗ π^{r}(ζ)^{∗})Ṽ_{kim}^{∗}ξ_{m}^{f}⟩
= _{A}⟨ξ_{k}^{e}|(1_{k} ⊗ π^{r}(ζ)^{∗})(Ṽ_{mīk} ⊗ 1_{i})(1_{m} ⊗ Ṽ_{īi0}^{∗})ξ_{m}^{f}⟩
= _{A}⟨ξ_{k}^{e}|Ṽ_{mīk}(1_{m} ⊗ (1ī ⊗ π^{r}(ζ)^{∗})Ṽ_{īi0}^{∗})ξ_{m}^{f}⟩
= _{A}⟨ξ_{k}^{e}|Ṽ_{mīk}(ξ_{m}^{f} ⊗ J_{X}ζ)⟩

holds, so

λ_{i,ζ}^{r}P_{n}(x) = ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{e} ∑_{m∈F_{n}i} ∑_{f} λ_{ξ_{m}^{f}}^{r}x(λ_{k,ξ_{k}^{e}}_{A}⟨Ṽ_{mīk}(ξ_{m}^{f} ⊗ J_{X}ζ)|ξ_{k}^{e}⟩)^{∗}
= ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{m∈F_{n}i} ∑_{f} λ_{ξ_{m}^{f}}^{r}x(λ_{k,Ṽ_{mīk}(ξ_{m}^{f}⊗J_{X}ζ)}^{r})^{∗}

holds.

By similar computations, we also have the following equality.

P_{n}(x)λ_{i,ζ}^{r} = ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{e} ∑_{m∈F_{n}ī}λ_{k,ξ_{k}^{e}}x(λ_{m,Ṽ_{kīm}(ξ_{k}^{e}⊗J_{X}ζ)})^{∗}.

Hence we get the following.

λ_{i,ζ}^{r}P_{n}(x) − P_{n}(x)λ_{i,ζ}^{r}

= ∥χ_{F_{n}}∥_{1,σ}^{−1}( ∑_{k∈F_{n}} ∑_{m∈F_{n}i\F_{n}} ∑_{f} λ_{m,ξ_{m}^{f}}^{r}x(λ_{k,Ṽ_{mīk}(ξ_{m}^{f}⊗J_{X}ζ)}^{r})^{∗}

− ∑_{k∈F_{n}} ∑_{e} ∑_{m∈F_{n}ī\F_{n}} λ_{k,ξ_{k}^{e}}^{r}x(λ_{m,Ṽ_{kīm}(ξ_{k}^{e}⊗J_{X}ζ)}^{r})^{∗}).

So we have to estimate ∥ ∑_{f} λ_{m,ξ_{m}^{f}}^{r}x(λ_{k,Ṽ_{mīk}(ξ_{m}^{f}⊗J_{X}ζ)}^{r})^{∗}∥. Then by putting η^{f} :=

Ṽ_{mīk}(ξ_{m}^{f} ⊗ J_{X}ζ), we have

∑_{f} λ_{m,ξ_{m}^{f}}^{r}x(λ_{k,η^{f}}^{r} )^{∗}

= ∑_{f} ∑_{{j_{1},j_{2}, \atop n_{1},n_{2}}} T_{n_{1}}^{∗}Ṽ_{j_{1}mn_{1}}(1_{j_{1}} ⊗ π^{r}(ξ_{m}^{f}))T_{j_{1}}xT_{j_{2}}^{∗}(1_{j_{2}} ⊗ π^{r}(η^{f}))Ṽ_{j_{2}kn_{2}}^{∗}T_{n_{2}}

= ∑_{f} ∑_{{j_{1},j_{2}, \atop n_{1},n_{2}}} T_{n_{1}}^{∗}Ṽ_{j_{1}mn_{1}}(T_{j_{1}} ⊗ 1_{m})x(1_{X} ⊗ π^{r}(ξ_{m}^{f})π^{r}(η^{f})^{∗})(T_{j_{2}}^{∗} ⊗ 1_{k})Ṽ_{j_{2}kn_{2}}^{∗}T_{n_{2}}.

Then as in the proof of Lemma 2.2, we have ∥ ∑_{j_{1},n_{1}} T_{n_{1}}^{∗}Ṽ_{j_{1}mn_{1}}(T_{j_{1}} ⊗ 1_{m})∥ = d(m) and for η ∈ B_{k},

∑_{f} π^{r}(ξ_{m}^{f})π^{r}(η^{f})^{∗}η = ∑_{f} π^{r}(ξ_{m}^{f})π^{r}(Ṽ_{mīk}(ξ_{m}^{f} ⊗ J_{X}ζ))η
= ∑_{f} π^{r}(ξ_{m}^{f})_{A}⟨η |Ṽ_{mīk}(ξ_{m}^{f} ⊗ J_{X}ζ)⟩
= ∑_{f}_{A}⟨(1_{m} ⊗ π^{r}(J_{X}ζ)^{∗})Ṽ_{mīk}^{∗}η |ξ_{m}^{f}⟩ξ_{m}^{f}
= (1_{m} ⊗ π^{r}(J_{X}ζ)^{∗})Ṽ_{mīk}^{∗}η

holds. Hence we have

∥λ_{i,ζ}^{r}P_{n}(x) − P_{n}(x)λ_{i,ζ}^{r}∥ ≤ ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{m∈F_{n}i\F_{n}} d(m)∥(1_{m} ⊗ π^{r}(J_{X}ζ)^{∗})Ṽ_{mīk}∥d(k)

+ ∑_{k∈F_{n}} ∑_{m∈F_{n}ī\F_{n}} d(k)∥(1_{k} ⊗ π^{r}(J_{X}ζ)^{∗})Ṽ_{kīm}∥d(m)

≤ ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{m∈F_{n}i\F_{n}} d(k)d(m)\sqrt{\frac{d(m)d(i)}{d(k)}N_{mī}^{k}}∥π^{r}(J_{X}ζ)∥

+ ∑_{k∈F_{n}} ∑_{m∈F_{n}ī\F_{n}} d(k)d(m)\sqrt{\frac{d(k)d(i)}{d(m)}N_{kī}^{m}}∥π^{r}(J_{X}ζ)∥

= ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{m∈F_{n}i\F_{n}} \sqrt{d(k)d(m)d(i)N_{mī}^{k}}d(m)∥π^{r}(J_{X}ζ)∥

+ ∑_{k∈F_{n}} ∑_{m∈F_{n}ī\F_{n}} \sqrt{d(m)d(k)d(i)N_{kī}^{m}}d(k)∥π^{r}(J_{X}ζ)∥.

Here note that if N_{mī}^{k} = N_{ki}^{m} > 0, then d(m) ≤ d(k)d(i) holds. So we get the following inequality.

∥λ_{i,ζ}^{r}P_{n}(x) − P_{n}(x)λ_{i,ζ}^{r}∥ ≤ ∥χ_{F_{n}}∥_{1,σ}^{−1} ∑_{k∈F_{n}} ∑_{m∈F_{n}i\F_{n}} d(k)d(i)d(m)N_{ki}^{m}∥π^{r}(J_{X}ζ)∥

+ ∑_{k∈F_{n}} ∑_{m∈F_{n}ī\F_{n}} d(k)d(i)d(m)N_{kī}^{m}∥π^{r}(J_{X}ζ)∥

= ∥χ_{F_{n}}∥_{1,σ}^{−1}∥ρ(χ_{F_{n}}) − χ_{F_{n}}∥_{1,σ}d(i)^{2}∥π^{r}(J_{X}ζ)∥
≤ εd(i)^{2}∥π^{r}(J_{X}ζ)∥.

So for every ϕ ∈ (B_{1})_{∗}, lim_{n→ω} ϕ(λ_{i,ζ}^{r}P_{n}(x) − P_{n}(x)λ_{i,ζ}^{r}) = 0 holds, and we get λ_{i,ζ}^{r}P (x) = P (x)λ_{i,ζ}^{r} for every i ∈ Δ and ζ ∈ B_{i}^{bdd}. So P is a norm one projection from B_{1} onto A(Δ). □

The natural question is that whether the inverse implication of the above proposition holds or not. Of course, if there exists a norm one projection P from B_{1} onto A(Δ), the restriction of P to A′ ∩ B_{1} is an invariant mean, but as remarked after Definition 4.1, this fact does not imply the amenability of Δ.

Proposition 4.3 Assume that there exists a state ϕ ∈ B_{1}^{∗} whose centralizer includes A(Δ). Moreover assume that ϕ ∘ E_{A′∩B_{1}} = ϕ. Then Δ is amenable, where E_{A′∩B_{1}} means the unique trace preserving conditional expectation from B_{1} onto A′ ∩ B_{1} whose existence is guaranteed by Proposition 2.7.

Before proof, we remark that the condition in the above proposition corresponds to the condition (3) of [25, Theorem 5.3]. Proof. First note that an easy computation shows that

d(i)^{−2} ∑_{e} λ_{i,ξ_{e}}^{l}p_{j}(λ_{i,ξ_{e}}^{l})^{∗} = ∑_{k} \frac{d(j)}{d(i)d(k)}N_{ij}^{k}p_{k}

holds, so for f ∈ A′ ∩ B_{1}, d(i)^{−2} ∑_{e} λ_{i,ξ_{e}}^{l}f(λ_{i,ξ}^{l})^{∗} coincide with ρ_{i}^{l}(f).

The idea of proof is the same as [2]. Fix unitaries u_{1}, u_{2}, …, u_{n} of A(Δ). By assumption, ϕ can be approximated by the net of normal state {ψ_{i}} in σ(B_{1}^{∗}, B_{1}) topology, where each ψ_{i} satisfies ψ_{i} ∘ E_{A′∩B_{1}} = ψ_{i}.

So the norm closure of a set

{(ψ ∘ Ad u_{1} − ψ, ψ ∘ Ad u_{2} − ψ, …, ψ ∘ Ad u_{n} − ψ)|ψ ∘ E_{A′∩B_{1}} = ψ}^{−∥·∥}

contains (0, 0, ⋯, 0). A Radon-Nikodym derivative \frac{dψ∘E_{A′∩B_{1}}}{dtr_{B_{1}}} is in A′ ∩ B_{1}. So for any ε > 0 there exists f ∈ A′ ∩ B_{1} with ∥f∥_{1,tr_{B_{1}}} = 1 which satisfies ∥u_{i}fu_{i}^{∗} − f∥_{1,tr_{B_{1}}} < ε for 1 ≤ i ≤ n. So by Powers-Størmer inequality, there exists f ∈ A′ ∩ B_{1} with ∥f∥_{2,tr_{B_{1}}} = 1 which satisfies ∥u_{i}fu_{i}^{∗} − f∥_{2,tr_{B_{1}}} < ε for 1 ≤ i ≤ n.

So for any ε > 0 and i_{1}, i_{2}, ⋯, i_{n} ∈ Δ, there exists f ∈ A′ ∩ B_{1} with ∥f∥_{2,tr_{B_{1}}} = 1 satisfying ∥λ_{i_{k},ξ_{i}^{e}}^{l}f − fλ_{i_{k},ξ_{i}^{e}}^{l}∥_{2,tr_{B_{1}}} < ε.

Hence

∥ρ_{i_{k}}^{l}(f) − f∥_{2,tr_{B_{1}}} = d(i)^{−2}∥ ∑_{e} λ_{i_{k},ξ_{i_{k}}^{e}}^{l} x(λ_{i_{k},ξ_{i_{k}}^{e}}^{l})^{∗} − f ∑_{e} λ_{i_{k},ξ_{i_{k}}^{e}}^{l} (λ_{i_{k},ξ_{i_{k}}^{e}}^{l})∥_{2,tr_{B_{1}}}
≤ ∑_{e} ∥λ_{i_{k},ξ_{i_{k}}^{e}}^{l} f − fλ_{i_{k},ξ_{i_{k}}^{e}}^{l} ∥_{2,tr_{B_{1}}}
< Cε

holds where C is a constant depending on ξ_{i}^{e}’s, and we get an almost invariant l^{2} sequence. So Δ is amenable. □ Remark. With more effort, one can show that for a norm one projection P constructed in Proposition 4.2, tr_{A(Δ)} ∘ P ∘ E_{A′∩B_{1}} = tr_{A(Δ)} ∘ P holds.

5 Longo-Rehren construction in the type III subfactor case

Originally Longo and Rehren considered their construction in the type III subfactor case by using the characterization of Longo’s canonical endomorphism in [12, Therem 5.1].

So in this section, we consider the generalized Longo-Rehren construction in the type III case by using the result in [8].

In this section, we do not present the detail of construction, because almost arguments are similar to type II_{1} case.

Throughout this section, we always assume that von Neumann algebras are properly infinite.

First we cite the characterization of canonical endomorphisms in [8]

Theorem 5.1 ([8, Theorem 4.1]) Let M be a von Neumann algebra and λ be an endomorphism of M. Assume that there exists an isometry v ∈ (id, λ) and {m_{i}} ⊂ M such that ∑_{i} m_{i}^{∗}vv^{∗}m_{i} = 1 and setting N := ⟨λ(M), {m_{i}}⟩, one has x ∈ N, xv = 0 → x = 0.

Then λ is a canonical endomorphism for N ⊂ M.

Let N ⊂ M be a subfactor with finite index and Δ be a set of irreducible component of the powers of the canonical endomorphism γ for N ⊂ M.

Let j_{M} : M → M^{opp} be a natural conjugate linear isomorphism and for ρ ∈ End(M), set ρ^{opp} := j_{M} ∘ ρ ∘ j_{M}^{−1}. Fix a set of isometries {w_{i}}_{i∈Δ} ⊂ M ⊗ M^{opp} which satisfies ∑_{i} w_{i}w_{i}^{∗} = 1 and put λ(x) := ∑_{i} w_{i}ρ_{i} ⊗ ρ_{i}^{opp}(x)w_{i}^{∗}.

Let \{v_{ijk}^{e}\}_{e} be an orthonormal basis of (ρ_{k}, ρ_{i}ρ_{j}) and set ṽ_{ijk} := ∑_{e} \sqrt{\frac{d(i)d(j)}{d(k)}}v_{ijk}^{e} ⊗ j_{M}(v_{ijk}^{e}). Then these ṽ_{ijk}’s satisfy similar conditions in Lemma 2.1.

Set V_{j} := ∑_{i,k} w_{i}ṽ_{ijk}w_{k}^{∗}.

Lemma 5.2 We have the following.

(1) V_{j}λ(x) = λ ∘ ρ ⊗ ρ^{opp}(x)V_{j}.

(2) V_{j}^{∗} = λ(ṽ_{j̄j0}^{∗})Vj̄.

(3) V_{i}V_{j} = ∑_{k} λ(ṽ_{ijk})V_{k}.

Proof. Easy computations. □

By the above lemma, B^{(0)} := span{λ(x_{j})V_{j}|j ∈ Δ} is a ∗-subalgebra of A. Let B be a σ-weak closure of B^{(0)}. We verify that B, λ, w_{0} and V_{j} satisfy the condition of Theorem 5.1.

Since V_{j}^{∗}w_{0} = ∑_{k} w_{k}ṽ_{0jk}^{∗} = w_{j},

∑_{j} V_{j}^{∗}w_{0}w_{0}^{∗}V_{j} = ∑_{j} w_{j}w_{j}^{∗}
= 1

holds and the first condition is verified.

Next we verify the second condition, i.e., for x ∈ B, xw_{0} = 0 implies x = 0. As in type II_{1} case, x is expressed as a formal sum ∑_{j∈Δ} λ(x_{j})V_{j}. Take x = ∑_{j} λ(x_{j})V_{j} ∈ B and assume xw_{0} = 0. Then xw_{0} = ∑_{j} λ(x_{j})w_{j} = ∑_{j} w_{j}ρ ⊗ ρ_{j}^{opp}(x_{j}). So 0 = w_{j}^{∗}xw_{0} = ρ_{j} ⊗ ρ_{j}^{opp}(x_{j}) holds and we get x_{j} = 0. Hence x = 0 and the second condition holds.

So we have verified the conditions in Theorem 5.1, thus we can conclude that λ is a canonical endomorphism for A ⊃ B. Since dim(id, λ) = 1, B is indeed a factor. We call A ⊃ B the Longo-Rehren subfactor.

As in the II_{1} case, we can show that when the original subfactors N ⊂ M is of finite depth, this construction gives the same subfactor constructed in [13, Proposition 4.10].

Finally we remark that {M ⊗ M^{opp}, λ, w_{0}, {V_{j}}_{j}} is an irreducible semidiscrete Qsystem in the sense of [8, Definition 5.1, 5.4].

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