An analogue of Longo’s canonical endomorphism for bimodule theory and its application to asymptotic

inclusions

Toshihiko Masuda Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153, JAPAN

Abstract We give an analogous characterization of Longo’s canonical endomorphism in the bimodule theory, and by using this, we construct an inclusion of factors of type II_{1} from a finite system of bimodules as a parallel construction to that of Longo-Rehren in a type III setting. When the original factors are approximately finite dimensional, we prove this new inclusion is isomorphic to the asymptotic inclusion in the sense of Ocneanu. This solves a conjecture of Longo-Rehren.

1 Introduction

Since V. F. R. Jones has introduced the subfactor theory, it has made a rapid progress. A. Ocneanu has introduced a paragroup in [O1] as an invariant for subfactors of type II_{1} factors. Classification of subfactors consists of two parts. One is an algebraic and combinatorial part, where the study of bimodules and their intertwiners plays an important role. The other is an analytic part, where the generating property is the main point as in [P2].

In [L1, L2], R. Longo has introduced sector theory for the study of inclusions of factors of type III and has shown equivalence between approaches based on sectors and bimodules for these factors.

In Longo’s sector theory, the analysis of canonical endomorphisms plays an important role. Let M be a properly infinite factor and ρ an endomorphism of M with dρ < +∞. In [L2, Theorem 3.1], it is proved that the canonical endomorphism for ρ(M) ⊂ M is of the form ρρ̄, where [ρ̄] is the conjugate sector of [ρ] and a characterization of an endomorphism to be canonical has been given in [L3, Proposition 4.1, Proposition 4.2, Theorem 5.1].

We further assume ρ(M) ⊂ M is of finite depth and put

Δ := {ρ_{i}|irreducible, ρ_{i} ≺ (ρρ̄)^{n}}.

The finite depth condition implies that Δ is a finite set. Then ⊕_{i}[ρ_{i} ⊗ j ∘ ρ_{i} ∘ j^{−1}] is in Sect(M ⊗ M^{opp}), where j : x → x^{∗} is the natural conjugate linear isomorphism from M to M^{opp}. In [LR], Longo and Rehren show that there exists a subfactor N ⊂ M ⊗ M^{opp} so that ⊕_{i}[ρ_{i} ⊗ j ∘ ρ_{i} ∘ j^{−1}] is the equivalent class of the canonical endomorphism for N ⊂ M ⊗ M^{opp} by using the characterization of the canonical endomorphism. They point out a similarity between N ⊂ M ⊗ M^{opp} and Ocneanu’s asymptotic inclusion M ∨ (M′ ∩ M_{∞}) ⊂ M_{∞} and they conjecture that the asymptotic inclusions are “covered” by the above construction. (Ocneanu has considered a construction of asymptotic inclusions for an analytic study of subfactors in [O1].) They also say that it is easy to construct examples for their construction which “do not describe” the asymptotic inclusions. But they consider asymptotic inclusions implicitly only for irreducible subfactors, while Ocneanu’s construction also works for reducible subfactors. Asymptotic inclusions are well understood when original factors are of type II_{1} and approximately finite dimensional (AFD) by using the graphic expression of bimodules in [O4], [EK]. So if we try to compare the above Longo-Rehren construction and the asymptotic inclusion, we have to perform the Longo-Rehren construction in the type II_{1} case.

The main purpose of this paper is to give the above construction of Longo-Rehren using bimodules instead of sectors and prove that the subfactor constructed with this method is really isomorphic to the asymptotic inclusion when the original II_{1} factors are AFD. This means that the Longo-Rehren construction and Ocneanu’s asymptotic inclusion are exactly same as methods to produce a new tensor category with quantum 6j-symbols from a given one. (Note that if we have a finite system of sectors of a type III factor, its tensor category with quantum 6j-symbols can be realized as a tensor category of bimodules over a II_{1} factor arising from a possibly reducible subfactor with the same quantum 6j-symbols.)

For this purpose, first in Section 2 we translate Longo’s characterization of a canonical endomorphism into bimodule theory, where the bimodule _{N}M_{N} corresponds to the canonical endomorphism. Though we encounter slight differences, we can characterize the bimodule _{N}M_{N} in the almost same way as the characterization of a canonical endomorphism.

In Section 3, we construct a new inclusion of Longo-Rehren type from a finite system of bimodules and compute the principal graph. To prove this inclusion is isomorphic to the asymptotic inclusion, we use Ocneanu’s graphical expression of bimodules.

In Appendix, we give the basic properties of the left inverses for bimodules which are used in this paper.

Acknowledgement. The author is grateful to Professor M. Izumi for pointing out an error in the first version of this paper, Professor Y. Kawahigashi for helpful advice and constant encouragement, Professor H. Kosaki for several comments about examples in [IK, Remark in Section 5], Professor K.-H. Rehren for comments on the first version of this paper.

2 Characterization of conjugate bimodules

The following is a “translation” of Longo’s characterization of the conjugate sector in [L2, Theorem 5.2] into bimodule theory. Also see [I, Theorem 5.3] and [FI, Theorem 5.1].

Proposition 2.1 Let N ⊂ M be an inclusion of II_{1} factors with finite index and _{N}X_{M} an N-M bimodule. The following conditions are equivalent.

(1) _{N}X_{M} ≅ _{N}M_{M},

(2) There exist coisometries T ∈ Hom (_{N}X⊗_{M}M_{N},_{N}N_{N}) and
S ∈ Hom (_{M}M⊗_{N}X_{M},_{M}M_{M}) such that
(2a) (T ⊗ 1_{_{N} X_{M}})(1_{_{N} X_{M}} ⊗ S^{∗}) = λ^{− \frac{1}{2}} ,
(1_{_{M} M_{N}} ⊗ T)(S^{∗} ⊗ 1_{_{M} M_{N}}) = λ^{− \frac{1}{2}} , λ > 0
(2b) dim (_{N}X) = dim (X_{N}).

Proof. (2)⇒(1). First we put M_{1} := End (_{N}X)^{opp}. Then we get inclusions of factors N ⊂ M ⊂ M_{1} and a projection e := T^{∗}T ∈ M_{1}. Next we define conditional expectations E_{1} : M_{1} → M and E : M → N as follows.

E_{1}(x) := S(1_{_{M} M_{N}} ⊗ x)S^{∗}, x ∈ End (_{M}M)^{opp} = M,
E_{2}(x) := T (1_{_{N} X_{M}} ⊗ x)T^{∗}, x ∈ End (_{M}M⊗_{M}X)^{opp} = M_{1}.

Then E_{1} and E are faithful, because if we assume E(x^{∗}x) = 0, then we get T (1_{_{N} X_{M}} ⊗ x^{∗}x) = 0 which implies (1_{_{N} X_{M}} ⊗ x)T^{∗} = 0 and

0 = (S ⊗ 1_{_{N} X_{M}})(1_{_{M} M_{N}} ⊗ (1_{_{N} X_{M}} ⊗ x)T^{∗})
= (S ⊗ 1_{_{N} X_{M}})(1_{_{M} M_{N}} ⊗ 1_{_{N} X_{M}} ⊗ x)(1_{_{M} M_{N}} ⊗ 1_{_{N} X_{M}} ⊗ T^{∗})
= (1_{_{M} M_{N}} ⊗ x)(1_{_{M} M_{N}} ⊗ S ⊗ 1_{_{N} X_{M}})(1_{_{M} M_{N}} ⊗ 1_{_{N} X_{M}} ⊗ T^{∗})
= (1_{_{M} M_{N}} ⊗ x)λ^{− \frac{1}{2}} .

Then λ ≠ 0 implies x = 0. The faithfulness of E_{1} can be similarly proved.

We would like to identify M_{1} with the basic extension of N ⊂ M. For this purpose, we use the next lemma.

Lemma 2.2 ([PP, Proposition 1.2], [KL, Lemma 1]) Let A ⊂ B ⊂ C be inclusions of von Neumann algebras, E [resp. F] be a faithful normal conditional expectation from B onto A [resp. C onto B] and e ∈ C be a projection with central support 1. Assume

(1) F (e) = λ^{−1}, λ > 0,
(2) λ^{−1}F (xe)e = xe, x ∈ C,
(3) exe = E(x), x ∈ B.

Then there exists an isomorphism π : C → ⟨B, e_{A}⟩, the basic extension of A ⊂ B, such that

π(e) = e_{A},
π(x) = x x ∈ B,
π ∘ F ∘ π^{−1} = Ẽ,

where Ẽ is the dual expectation of E.

We verify the conditions of Lemma 2.2 for N ⊂ M ⊂ M_{1}, E, E_{1}, and e.

The following equality means that condition (1) holds for E_{1} and e.

E_{1}(e) = S(1_{_{M} M_{N}} ⊗ T^{∗}T)S^{∗} = λ^{−1}.

To verify (2), we embed E_{1}(xe) into End (_{N}X⊗_{M}M) as 1⊗E_{1}(xe) and compute (1_{_{N} X_{M}} ⊗ E_{1}(xe))e as follows.

(1_{_{N} X_{M}} ⊗ E_{1}(xe))e
= (1_{_{N} X_{M}} ⊗ (S ⊗_{M}M_{N})(1_{_{M} M_{N}} ⊗ xT^{∗}T)(S^{∗} ⊗ 1_{_{M} M_{N}}))T^{∗}T
= (1_{_{N} X_{M}} ⊗ S ⊗ 1_{_{M} M_{N}})(1_{_{N} X_{N}} ⊗ xT^{∗})(1_{_{N} X_{N}} ⊗ T)(S^{∗} ⊗ 1_{_{M} M_{N}})T^{∗}T
= λ^{− \frac{1}{2}} (1_{_{N} X_{M}} ⊗ S ⊗ 1_{_{M} M_{N}})(1_{_{N} X_{N}} ⊗ x)(1_{_{N} X_{N}} ⊗ T^{∗})(T^{∗}T ⊗ 1_{_{N} N_{N}})
= λ^{− \frac{1}{2}} (1_{_{N} X_{M}} ⊗ S ⊗ 1_{_{M} M_{N}})(1_{_{N} X_{N}} ⊗ x)(T^{∗}T ⊗ 1_{_{N} X_{N}})(1_{_{N} X_{N}} ⊗ T^{∗})
= λ^{− \frac{1}{2}} (1_{_{N} X_{M}} ⊗ S ⊗ 1_{_{M} M_{N}})(T^{∗} ⊗ 1_{_{N} X_{N}})x(T ⊗ 1_{_{N} X_{N}})(1_{_{N} X_{N}} ⊗ T^{∗})
= λ^{−1}xT^{∗}T
= λ^{−1}xe.

Next we verify (3) as follows.

exe = T^{∗}T xT^{∗}T
= T^{∗}E(x)T
= E(x)e.

So by Lemma 2.2, there exists an isomorphism π : M_{1} → ⟨M, e_{N}⟩ satisfying the conditions in Lemma 2.2, where ⟨M, e_{N}⟩ is the basic extension of N ⊂ M and this implies d (_{N}X_{N}) = d (_{N}M_{N}), where d (_{N}X_{N}) := dim(_{N}X) dim(X_{N}). Then condition (3) of Proposition 2.1 means dim (X_{M_{1}}) = dim (M_{⟨M,e_{N} ⟩}), hence π is spatial. By the unitary implementing π, the right action of M on X is transformed to the right multiplication of M and the left action of N on X to the left action of N.

So we may identify _{N}X_{N} with _{N}(_{α}M)_{N}, where α ∈ Aut(N) and T^{∗}T = e_{N}.

There exists a coisometry T′ ∈ Hom (_{N}M_{N},_{N}N_{N}) such that T^{′∗}T′ = e_{N}. Then U := T T^{′∗} ∈ Hom (_{N_{α}}N_{N},_{N}N_{N}) is unitary, hence α is inner and we can conclude_{N}X_{N} ≅ _{N}M_{N}.

(1)⇒(2). Because we do not use (1)⇒(2) in the following of this paper and it is easy to prove it, we omit a proof. □

Remark. To prove (2)⇒(1), we do not need the fact M is a factor and later we use this proposition in the case N is a II_{1} factor but M is a von Neumann algebra of type II_{1}.

Using the above result, we can prove the following theorem, which amounts to a characterization of a canonical endomorphism [L3, Theorem 5.1] in bimodule theory.

Theorem 2.3 Let N be a II_{1} factor and _{N}X_{N} an N-N bimodule of finite type. Suppose that T ∈ Hom (_{N}X_{N},_{N}N_{N}) and S ∈ Hom (_{N}X⊗_{N}X_{N},_{N}X_{N}) with T T^{∗} = SS^{∗} = 1 satisfy the following conditions.

(1) S(1_{_{N} X_{N}} ⊗ T^{∗}) = S(T^{∗} ⊗ 1_{_{N} X_{N}}) = λ^{− \frac{1}{2}} , λ > 0,
(2) S(1_{_{N} X_{N}} ⊗ S) = S(S ⊗ 1_{_{N} X_{N}}),
(3) S^{∗}S = (1_{_{N} X_{N}} ⊗ S)(S^{∗} ⊗ 1_{_{N} X_{N}}),
(4) dim (_{N}X) = dim (X_{N}).

Then there exists a von Neumann algebra M ⊃ N of typeII_{1} with finite index and _{N}X_{N} ≅ NM_{N}.

Moreover if dim Hom (_{N}X_{N},_{N}N_{N}) = 1, then M is a II_{1} factor and [M : N] = λ.

Proof. Put M_{1} := End (_{N}X)^{opp} and define E : M_{1} → M_{1} as follows.

E(x) := S(1_{_{N} X_{N}} ⊗ x)S^{∗}, x ∈ M_{1}.

Then

E(xE(y))
= S(1_{_{N} X_{N}} ⊗ xE(y))S^{∗}
= S(1_{_{N} X_{N}} ⊗ xS(1_{_{N} X_{N}} ⊗ y)S^{∗})S^{∗}
= S(1_{_{N} X_{N}} ⊗ x)(1_{_{N} X_{N}} ⊗ S)(1_{_{N} X_{N}} ⊗ 1_{_{N} X_{N}} ⊗ y)(1_{_{N} X_{N}} ⊗ S^{∗})S^{∗}
= S(1_{_{N} X_{N}} ⊗ x)(1_{_{N} X_{N}} ⊗ S)(1_{_{N} X_{N}} ⊗ 1_{_{N} X_{N}} ⊗ y)(S^{∗} ⊗ 1_{_{N} X_{N}})S^{∗}
= S(1_{_{N} X_{N}} ⊗ x)(1_{_{N} X_{N}} ⊗ S)(S^{∗} ⊗ 1_{_{N} X_{N}})(1_{_{N} X_{N}} ⊗ y)S^{∗}
= S(1_{_{N} X_{N}} ⊗ x)S^{∗}S(1_{_{N} X_{N}} ⊗ y)S^{∗}
= E(x)E(y).

Hence M := E(M_{1}) is a subalgebra of M_{1} and obviously weakly closed, so M is a von Neumann algebra. Moreover, since S is N-linear, N is a subfactor of M. So the isomorphism NX_{N} ≅ _{N}X⊗_{M}M_{N} holds.

If S is of the form 1_{_{N} X_{M}} ⊗ S′ ⊗ 1_{_{M} M_{N}} , S′ ∈ Hom (_{M}M⊗_{N}X_{M},_{M}M_{M}), we can apply Proposition 2.1 and we get _{N}X_{N} ≅ _{N}M_{N}. So we determine the form of S.

E(x)S = S(1_{_{N} X_{N}} ⊗ x)S^{∗}S
= S(1_{_{N} X_{N}} ⊗ x)(S ⊗ 1_{_{N} X_{N}})(1_{_{N} X_{N}} ⊗ S^{∗})
= S(S ⊗ 1_{_{N} X_{N}})(1_{_{N} X_{N}} ⊗ 1_{_{N} X_{N}} ⊗ x)(1_{_{N} X_{N}} ⊗ S^{∗})
= S(1_{_{N} X_{N}} ⊗ S)(1_{_{N} X_{N}} ⊗ 1_{_{N} X_{N}} ⊗ x)(1_{_{N} X_{N}} ⊗ S^{∗})
= S(1_{_{N} X_{N}} ⊗ E(x)).

The above equality means S ∈ Hom (_{N}X⊗_{N}X_{M},_{N}X_{M}).

Next we prove S = 1_{_{N} X_{M}} ⊗ S′. We define

E_{1} : Hom (_{N}X⊗_{N}X,_{N}X) → Hom (_{N}X⊗_{N}X,_{N}X) as
E_{1}(R) := S(1_{_{N} X_{N}} ⊗ R)(S^{∗} ⊗ 1_{_{N} X_{N}}), R ∈ Hom (_{N}X⊗_{N}X,_{N}X),
E_{2} : Hom (_{N}X,_{N}X⊗_{N}X) → Hom (_{N}X,_{N}X⊗_{N}X) as
E_{2}(R) := (S ⊗ 1_{_{N} X_{N}})(1_{_{N} X_{N}} ⊗ R)S^{∗}, R ∈ Hom (_{N}X,_{N}X⊗_{N}X).

Then the equality E_{1}(R_{1})E_{2}(R_{2}) = E(R_{1}E_{2}(R_{2})) holds.

Here we use the next lemma.

Lemma 2.4 If R ∈ Hom (_{N}X⊗_{N}X,_{N}X) and E_{1}(R) = R, then R is of the form 1_{_{N} X_{M}} ⊗ R′, R′ ∈ Hom (_{M}M⊗_{N}X,_{M}M).

Proof of Lemma 2.4. Let {ξ_{i}}_{i∈I} be a basis of _{N}X. We define maps V_{i}, i ∈ I, from L^{2}(N) to X ≅ N⊗_{N}X as follows.

V_{i}(n̂) = n̂ ⊗ ξ_{i}, n̂ ∈ L^{2}(N).

Then V_{i} ∈ Hom (_{N}N,_{N}X) and ∑_{i} V_{i}V_{i}^{∗} = 1.

Clearly E_{2}(1_{_{N} X_{N}} ⊗ V_{i}) = 1_{_{N} X_{N}} ⊗ V_{i}. So if E_{1}(R) = R, then the following identity holds.

R(1_{_{N} X_{N}} ⊗ V_{i}) = E_{1}(R)E_{2}(1_{_{N} X_{N}} ⊗ V_{i})
= E(RE_{2}(1_{_{N} X_{N}} ⊗ V_{i}))
= E(R(1_{_{N} X_{N}} ⊗ V_{i})).

By the definition of M, the above equality implies that R(1_{_{N} X_{N}} ⊗ V_{i}) is of the form 1_{_{N} X_{M}} ⊗ m_{i}, m_{i} ∈ M.

Then the following equality holds.

R = ∑_{i} R(1_{_{N} X_{N}} ⊗ V_{i}V_{i}^{∗})
= ∑_{i}(1_{_{N} X_{M}} ⊗ m_{i})(1_{_{N} X_{N}} ⊗ V_{i}^{∗})
= 1_{_{N} X_{M}} ⊗ (∑_{i} m_{i}(1_{_{M} M_{N}} ⊗ V_{i}^{∗})).

Hence R is of the form 1_{_{N} X_{M}} ⊗ R′. □

We continue the proof of Theorem 2.3. The following equality means that S is of the form 1_{_{N} X_{M}} ⊗ S′ by the above lemma.

E_{1}(S) = S(1_{_{N} X_{N}} ⊗ S)(S^{∗} ⊗ 1_{_{N} X_{N}}) = S(S ⊗ 1_{_{N} X_{N}})(S^{∗} ⊗ 1_{_{N} X_{N}}) = S.

Then we can conclude that S is of the form 1_{_{N} X_{M}} ⊗ S′ ⊗ 1_{_{M} M_{N}}. Hence we can apply Proposition 2.1 and get _{N}X_{N} ≅ _{N}M_{N}.

When dim Hom (_{N}X_{N},_{N}N_{N}) = 1 holds, the conditional expectation from M to N is uniquely determined. Because N is a type II_{1} factor, this means that the trace of M is unique. Hence M is a II_{1} factor and [M : N] = λ. □

Remark 1. In Proposition 2.1 and Theorem 2.3, we assume dim (_{N}X) = dim (X_{N}). This condition is used to prove the isomorphism π appearing in the proof of Proposition 2.1 is spatial. In sector theory, because factors of type III are considered, an isomorphism is automatically spatial. But in bimodule theory, we deal with factors of type II_{1} and an isomorphism is not necessary spatial. So the condition dim (_{N}X) = dim (X_{N}) is necessary.

Remark 2. In Theorem 2.3, the construction of M depends on a choice of S, and in general S satisfying the condition of Theorem 2.3 is not determined uniquely. Indeed, there exist non-isomorphic subfactors N ⊂ M and N ⊂ M̃ with _{N}M_{N} ≅ _{N} M̃_{N}. See the remark in [IK, Section 5].

3 Asymptotic inclusions

Let M be a II_{1} factor and Δ = {_{M}Z_{iM}}_{i∈I} be a finite set of irreducible M-M bimodules of finite type satisfying the following conditions.

MZ_{iM} ∈ Δ ⇒ _{M} Z̄_{iM} ∈ Δ,

MZ_{i}⊗_{M}Z_{jM} ≅ ⨁_{k} N_{ij}^{k}_{M}Z_{kM}, _{M}Z_{kM} ∈ Δ,

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

Obviously _{M}Z_{0M} := _{M}M_{M} ∈ Δ.

For an M-M bimodule _{M}X_{M}, we naturally get an M^{opp}-M^{opp} bimodule _{M^{opp}}X_{M^{opp}}^{o}. As a Hilbert space, X^{o} = X^{∗} and we identify the left action of M on X^{o} with the right action of M^{opp}. Similarly the right action of M is identified with the left action of M^{opp}.

Let J_{X} be the natural conjugate linear map from X to X^{o}. For T ∈ Hom (_{M}X_{M},_{M}Y_{M}), an intertwiner J_{Y} T J_{X}^{−1} is an element of Hom (_{M^{opp}}X_{M^{opp}}^{o},_{M^{opp}}Y_{M^{opp}}^{o}). In the following, we omit the index of J_{X} and write simply J.

We construct a new inclusion of II_{1} factors from Δ as a parallel construction to that of Longo-Rehren in [LR, Section 4].

We put

AX_{A} := ⨁_{i}_{A}(Z_{i}⊗_{c}Z_{i}^{o})_{A},

where A := M ⊗ M^{opp} and ⊗_{c} denotes the usual tensor product of Hilbert spaces over C. The following Theorem is a direct translation of [LR, Proposition 4.10].

Theorem 3.1 There exists a II_{1} factor B ⊃ A such that _{A}B_{A} ≅ _{A}X_{A} with [B : A] = λ =: ∑_{i} d(i)^{2},where d(i) = \sqrt{d(_{M}Z_{iM})}.

Proof. It is clear that the equality dim (_{A}X) = dim (X_{A}) holds. So by Theorem 2.3, we only have to find coisometries which satisfy conditions of Theorem 2.3.

We put

MY_{M} := ⨁_{i}_{M}Z_{iM}, _{A}W_{A} := _{A}(Y ⊗_{c}Y^{o})_{A}.

Clearly _{A}X_{A} ≺ _{A}W_{A}. We fix coisometries

U_{i} ∈ Hom (_{M}Y_{M},_{M}Z_{iM}), R ∈ Hom (_{A}W_{A},_{A}X_{A}).

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 put

Ṽ_{ijk}^{e} := U_{k}^{∗}V_{ijk}^{e}(U_{i} ⊗ 1)(1 ⊗ U_{j}).

Then Ṽ_{ijk}^{e} ∈ Hom (_{M}Y ⊗_{M}Y_{M},_{M}Y_{M}). Define T̃ and S̃ as following.

T̃ := U_{0} ⊗ JU_{0}J^{−1},

S̃ := λ^{− \frac{1}{2}} ∑_{i,j,k,e} \sqrt{\frac{d(i)d(j)}{d(k)}}Ṽ_{ijk}^{e} ⊗ JṼ_{ijk}^{e}J^{−1}.

We put T := T̃R^{∗} and S := RS̃(1_{W} ⊗ R^{∗})(R^{∗} ⊗ 1_{X}). Then it is verified that T and S satisfy the conditions of Theorem 2.3 in the same way as in the proof of [LR, Proposition 4.10]. □

Let N ⊂ M be an inclusion of II_{1} factors with finite index and finite depth. Put

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

Then Δ satisfies the conditions at the beginning of this section. It is also easy to see any such Δ arises from such a possibly reducible subfactor N ⊂ M. By Theorem 3.1, we can construct B ⊃ A = M ⊗ M^{opp} such that

AB_{A} ≅ _{A}(⨁_{i} Z_{i} ⊗ Z_{i}^{o})_{A},

and [B : A] = ∑_{i} d(_{M}Z_{iM}) is the global index (for definition of global index, see [O1, pp. 137]) of N ⊂ M. Then we know the principal graph of A ⊂ B by the following Proposition.

Proposition 3.2 Put _{A}B_{kB} := _{A}((Z_{k}⊗_{M}M^{o})⊗_{A}B)_{B}. Then the following hold. (1) _{A}B_{kB} is irreducible,

(2) _{A}B_{k} ⊗_{B} B_{A} ≅ ⨁_{i,j} N_{ij̄}^{k}_{A}(Z_{i}⊗_{c}Z_{j}^{o})_{A},

(3) _{A}(Z_{i}⊗_{c}Z_{j}^{o})⊗_{A}B_{B} ≅ ⨁_{k} N_{ij̄}^{k}_{A}B_{kB}.

Proof. First we verify (1).

AB_{k} ⊗_{B} B̄_{kA} ≅ _{A}((Z_{k}⊗_{c}M^{o})⊗_{A}B⊗_{A}(Z̄_{k}⊗_{c}M^{o}))_{A}
≅ _{A}((⨁_{i} Z_{k}⊗_{M}Z_{i}⊗_{M} Z̄_{k})⊗_{c}Z_{i}^{o})_{A}.

So if _{M}Z_{iM} ≅ \ _{M}M_{M}, then _{A}A_{A} does not appear in _{A}((Z_{k}⊗_{M}Z_{i}⊗_{M} Z̄_{k})⊗_{c}Z_{i}^{o})_{A}. When MZ_{iM} = _{M}M_{M}, the bimodule _{A}A_{A} is included in _{A}((Z_{k}⊗_{M} Z̄_{k})⊗_{c}M^{o})_{A} with multiplicity one because _{M}Z_{kM} is irreducible. Hence _{A}B_{kA} is irreducible and (1) is verified.

Next we verify (2).

AB_{kB} ≅ _{A}(Z_{k}⊗_{c}M^{o})⊗_{A}B_{B}
≅ _{A}(⨁_{j}(Z_{k}⊗_{M}Z_{j})⊗_{c}Z_{j}^{o})_{B}
≅ _{A}(⨁_{i,j} N_{ik}^{j}Z_{i}⊗_{c}Z_{j}^{o})_{B}
≅ _{A}(⨁_{i,j} N_{ij̄}^{k}Z_{i}⊗_{c}Z_{j}^{o})_{B}.

So

AB_{kA} ≅ ⨁_{i,j} N_{ij̄}^{k}_{A}(Z_{i}⊗_{c}Z_{j}^{o})_{A}

and (2) is proved.

Finally we verify (3).

A(Z_{i}⊗_{c}Z_{j}^{o})⊗_{A}B_{B} ≅ _{A}(⨁_{k}(Z_{i}⊗_{M}Z_{k})⊗_{c}(Z_{j}⊗_{M}Z_{k})^{o})_{B}
≅ _{A}(⨁_{l,m} ∑_{k} N_{ik}^{l}N_{jk}^{m}Z_{l}⊗_{c}Z_{m}^{o})_{B}.

Here the equality

∑_{k} N_{ik}^{l}N_{jk}^{m} = ∑_{k} N_{ik}^{l}N_{j̄m}^{k} = ∑_{k} N_{km}^{l}N_{ij̄}^{k} = ∑_{k} N_{ij̄}^{k}N_{lm̄}^{k}

holds, where the first and third equalities come from Frobenius reciprocity and the second equality comes from associativity of the relative tensor product. Then

A(Z_{i}⊗_{c}Z_{j}^{o})⊗_{A}B_{B} ≅ _{A}(⨁_{l,m} ∑_{k} N_{ij̄}^{k}N_{lm̄}^{k}Z_{l}⊗_{c}Z_{m}^{o})_{B}
≅ _{A}(⨁_{i,j} N_{ij̄}^{k}B_{k})_{B}
≅ ⨁_{i,j}^{k} N_{ij̄}^{k}_{A}B_{kB}.

So (3) is proved. □

By the above lemma, the principal graph of A ⊂ B is the connected component containing (_{M}M_{M},_{M}M_{M}) of the fusion graph (for definition of the fusion graph, see [O2, Section III. 1]) of the M-M bimodules arising from the inclusion N ⊂ M.

In the rest of this paper, we assume N and M are AFD. Here we recall the definition of Ocneanu’s asymptotic inclusion.

Definition 3.3 ([O2, Section III.1.]) Let N ⊂ M be an inclusion of factors of type II_{1} with finite index and

N ⊂ M ⊂ M_{1} ⊂ M_{2} ⋯

the Jones tower. Put M_{∞} := ∨_{i}M_{i}. We call the inclusion M ∨ (M′ ∩ M_{∞}) ⊂ M_{∞} the asymptotic inclusion of N ⊂ M.

In [O2, Section III], Ocneanu has claimed that if N ⊂ M has finite index and finite depth then the index [M_{∞} : M ∨ (M′ ∩ M_{∞})] is equal to ∑_{i} d(_{M}Z_{iM}) and the principal graph of the asymptotic inclusion is the connected component containing (∗, ∗) of the fusion graph. (See [EK].)

In [LR], a similarity of asymptotic inclusions to A ⊂ B constructed above has been pointed out. In fact, if the original subfactor N ⊂ M is AFD, then the following theorem holds.

Theorem 3.4 If N ⊂ M is AFD, then A ⊂ B constructed above is conjugate to the asymptotic inclusion M ∨ (M′ ∩ M_{∞}) ⊂ M_{∞}.

Proof. We use the notation in [O4] freely. (Also see [EK].) By [P1, Theorem 4.7],

N ⊂ M ≅ ∨_{i}(N_{i} ∩ N) ⊂ ∨_{i}(N_{i} ∩ M),

where

⋯ ⊂ N_{3} ⊂ N_{2} ⊂ N_{1} = N ⊂ M

is a tunnel for N ⊂ M with the generating property. Hence we can naturally identify M′ ∩ M_{∞} with M^{opp} and M ∨ (M′ ∩ M_{∞}) with A = M⊗_{c}M^{opp}.

The bimodules _{M}Z_{iM}, M and B_{Z_{i},Z_{i}} can be expressed graphically as in Figures 3.1, 3.2 and 3.3 respectively.

[figure: M; M; N; M; N; M; N; M; N; _{M}Z_{iM}; _{N}M_{M}; _{M}M_{N}; _{N}M_{M}; _{M}M_{N}; _{N}M_{M}; _{M}M_{N}; _{N}M_{M}; _{M}M_{N}; Figure 3.1]

[figure: Figure 3.2]

[figure: _{M}Z_{iM}; _{M}Z_{iM}; Figure 3.3]

Figure 3.1 represents an element of Hom (_{M}M^{n}⊗_{M}Z_{iM},_{M}M_{M}^{n}), where

MM^{n}_{M} := \underbrace{_{M}M⊗_{N}M⊗_{M} ⋯ ⊗_{N}M_{M}}_{2n} .

An element of _{M^{opp}}Z_{iM^{opp}}^{o} is identified with an element of Hom (_{M}M_{M}^{n},_{M}Z_{i}⊗_{M}M_{M}^{n}). Using the above identification, a ⊗ b with a ∈ Hom (_{M}M^{n}⊗_{M}Z_{iM},_{M}M_{M}^{n}) and b ∈

Hom (_{M}M_{M}^{n},_{M}Z_{i}⊗_{M}M_{M}^{n}) is embedded in End (_{M}M^{n}⊗_{M}M_{M}^{n}) as (a ⊗ 1)(1 ⊗ b).

Since the inner products on End (_{M}M^{n}⊗_{M}M_{M}^{n}) come from the Markov trace, so for x, y ∈ End (_{M}M^{n}⊗_{M}M_{M}^{n}), we have

⟨x, y⟩ = Φ_{_{M} M^{n}⊗_{M} M_{M}^{n}} (y^{∗}x),

where Φ_{_{M} M^{n}⊗_{M} M_{M}^{n}} is the minimal left inverse for _{M}M^{n}⊗_{M}M_{M}^{n}. (See the appendix for left inverses.)

Next on Hom (_{M}M^{n}⊗_{M}Z_{iM},_{M}M_{M}^{n}) we consider the1 inner product as follows.

⟨a, b⟩ = Φ_{_{M} M^{n}⊗_{M} Z_{iM}} (b^{∗}a), a, b ∈ Hom (_{M}M^{n}⊗_{M}Z_{iM},_{M}M_{M}^{n}),

and on Hom (_{M}M_{M}^{n},_{M}Z_{i}⊗_{M}M_{M}^{n}) we consider the inner product

⟨a, b⟩ = Φ_{_{M} M}_{M}^{n} (b^{∗}a), a, b ∈ Hom (_{M}M_{M}^{n},_{M}Z_{i}⊗_{M}M_{M}^{n}).

Then the above three inner products are compatible with the embedding a⊗b → (a⊗1)(1⊗ b). Indeed, for a_{1}, a_{2} ∈ Hom (_{M}M^{n}⊗_{M}Z_{iM},_{M}M_{M}^{n}) and b_{1}, b_{2} ∈ Hom (_{M}M_{M}^{n},_{M}Z_{i}⊗_{M}M_{M}^{n}), the following equality holds.

⟨(a_{1} ⊗ 1)(1 ⊗ b_{1}), (a_{2} ⊗ 1)(1 ⊗ b_{2})⟩ = Φ_{_{M} M^{n}⊗_{M} M_{M}^{n}} ((1 ⊗ b_{2}^{∗})(a_{2}^{∗}a_{1} ⊗ 1)(1 ⊗ b_{1}))
= Φ_{_{M} M}_{M}^{n} (b_{2}^{∗}Φ_{_{M} M}_{M}^{n} (a_{2}^{∗}a_{1} ⊗ 1)b_{1})
= Φ_{_{M} M}_{M}^{n} (a_{2}^{∗}a_{1})Φ_{_{M} M}_{M}^{n} (b_{2}^{∗}b_{1})
= Φ_{_{M} M^{n}⊗_{M} Z_{iM}} (a_{2}^{∗}a_{1})Φ_{_{M} M}_{M}^{n} (b_{2}^{∗}b_{1})
= ⟨a_{1}, a_{2}⟩ ⟨b_{1}, b_{2}⟩.

Using these expressions, we write down S : Hom (_{A}M_{∞}⊗_{A}M_{∞A},_{A}M_{∞A}) concretely. In general, for a subfactor N ⊂ M, S : M⊗_{N}M → M is given by

x⊗_{N}y → [M : N]^{− \frac{1}{2}} xy.

Then S : _{A}M_{∞}⊗_{A}M_{∞A} → _{A}M_{∞A} is expressed as in Figure 3.4, where λ = ∑_{i} d(_{M}Z_{iM}) = [M_{∞} : M ∨ (M′ ∩ M_{∞})].

[figure: i; i; ⊗_{A}; j; j; →; λ^{− \frac{1}{2}}; i; j; i; j; Figure 3.4]

It is clear that the equality in Figure 3.5 holds.

[figure: i; j; i; j; =; ∑_{k,e}; k; e; k; e; Figure 3.5]

Then we have to investigate the map expressed as in Figures 3.6 and 3.7. We fix i, j and k, and we denote these maps by R_{1} and R_{2}.

[figure: i; j; →; k; e; Figure 3.6]

[figure: i; j; →; k; e; Figure 3.7]

Figures 3.6 and 3.7 mean that

R_{1}a = a(1 ⊗ V_{ijk}^{∗}), a ∈ Hom (_{M}M^{n}⊗_{M}Z_{i}⊗_{M}Z_{jM},_{M}M_{M}^{n}) and
R_{2}b = (V_{ijk} ⊗ 1)b, b ∈ Hom (_{M}M_{M}^{n},_{M}Z_{i}⊗_{M}Z_{j}⊗_{M}M_{M}^{n}).

We claim R_{1}R_{1}^{∗} = \frac{d(i)d(j)}{d(k)} and R_{2}R_{2}^{∗} = 1. For a_{1} ∈ Hom (_{M}M^{n}⊗_{M}Z_{i}⊗_{M}Z_{jM},_{M}M_{M}^{n}) and a_{2} ∈ Hom (_{M}M^{n}⊗_{M}Z_{kM},_{M}M_{M}^{n}),

⟨R_{1}a_{1}, a_{2}⟩ = Φ_{_{M} M^{n}⊗_{M} Z_{kM}} (a_{2}^{∗}a_{1}(1 ⊗ V_{ijk}^{∗}))
= Φ_{_{M} Z_{kM}} (Φ_{_{M} M}_{M}^{n} (a_{2}^{∗}a_{1})V_{ijk}^{∗})
= \frac{d(i)d(j)}{d(k)}Φ_{_{M} Z_{i}⊗_{M} Z_{jM}} (V_{ijk}^{∗}Φ_{_{M} M}_{M}^{n} (a_{2}^{∗}a_{1}))
= \frac{d(i)d(j)}{d(k)}Φ_{_{M} M^{n}⊗_{M} Z_{i}⊗_{M} Z_{jM}} ((1 ⊗ V_{ijk}^{∗})a_{2}^{∗}a_{1})
= \frac{d(i)d(j)}{d(k)}⟨a_{1}, a_{2}(1 ⊗ V_{ijk})⟩

holds. Hence R_{1}^{∗}a = \frac{d(i)d(j)}{d(k)}a(1⊗V_{ijk}) and we can easily verify the equality R_{1}R_{1}^{∗} = \frac{d(i)d(j)}{d(k)}. Next for b_{1} ∈ Hom (_{M}M_{M}^{n},_{M}Z_{i}⊗_{M}Z_{j}⊗_{M}M_{M}^{n}) and b_{2} ∈ Hom (_{M}M_{M}^{n},_{M}Z_{k}⊗_{M}M_{M}^{n}),

⟨R_{2}b_{1}, b_{2}⟩ = Φ_{_{M} M}_{M}^{n} (b_{2}^{∗}(1 ⊗ V_{ijk})b_{1})
= ⟨b_{1}, (1 ⊗ V_{ijk}^{∗})b_{2}⟩

holds. Hence R_{2}^{∗}b = (1 ⊗ V_{ijk}^{∗})b, and the equality R_{2}R_{2}^{∗} = 1 is easily verified.

Then S is expressed as in Figure 3.8 and from the above computation, the right-upper part and the right-lower part of Figure 3.8 represent coisometries. So this has the same form as S constructed by Longo-Rehren. Hence we get A ⊂ B ≅ A ⊂ M_{∞}. □

[figure: i; i; ⊗_{A}; j; j; →; λ^{− \frac{1}{2}}; ∑_{k,e}; \sqrt{\frac{d(i)d(j)}{d(k)}}; \sqrt{\frac{d(k)}{d(i)d(j)}}; k; e; k; e; Figure 3.8]

A Left inverses for bimodules

In this appendix, we treat the left inverses for bimodules. This appendix is the translation of theory of left inverses for sectors into bimodule theory. In this section, N and M are arbitrary II_{1} factors and we do not assume N ⊂ M.

Definition A.1 For an N-M bimodule _{N}X_{M}, fix a coisometric intertwiner T ∈ Hom (_{M} X̄⊗_{N}X_{M},_{M}M_{M}). For any left M-modules _{M}Y ,_{M}Z, define

Φ_{X}(R) := (T ⊗ 1_{Z})(1X̄ ⊗ R)(T^{∗} ⊗ 1_{Y}), R ∈ Hom (_{N}X⊗_{M}Y,_{M}X⊗_{M}Z),

then Φ_{X} is a linear map from Hom (_{M}X⊗_{M}Y,_{M}X⊗_{M}X) to Hom (_{M}Y ,_{M}Z). We call Φ_{X} the left inverse for _{M}X_{M}.

Proposition A.2 For Φ_{X}, the following hold.

(1) Φ_{X}((1_{X} ⊗ R_{3})R_{2}(1_{X} ⊗ R_{1})) = R_{3}Φ_{X}(R_{2})R_{1}, R_{1} ∈ Hom (_{M}Y_{1},_{M}Y_{2}), R_{2} ∈ Hom (_{N}X⊗_{M}Y_{2},_{N}X⊗_{M}Y_{3}), R_{3} ∈ Hom (_{M}Y_{3},_{M}Y_{4}).

(2) For an M-N bimodule _{M}W_{N}, the map Φ_{X} ∘ Φ_{W} is the left inverse for MW ⊗_{N}X_{M}.

Proof. Easy computations. □

Note that if _{M}Y = _{M}Z, then the map Φ_{X} is a conditional expectation from End (_{N}X⊗_{M}Y) to End (_{M}Y).

In general, the left inverse depends on the choice of intertwiners. But if the bimodule is irreducible, it is uniquely determined. By the above Proposition and remark, Φ_{X} ∘ Φ_{W} is a conditional expectation from End (_{M}W ⊗_{N}X) to End (_{M}M). The maps Φ_{W} and Φ_{X} are uniquely determined, hence they are minimal. So Φ_{X} ∘ Φ_{W} is the minimal conditional expectation by the multiplicativity of the minimal expectations ([PP], [KL], [L4]).

In the following, we assume that N = M and bimodules are irreducible. Then we get the following proposition.

Proposition A.3 Let _{M}W ⊗_{M}X_{M} ≅ ⨁_{i} _{M}Z_{iM} the irreducible decomposition and R_{i} ∈ Hom (_{M}W ⊗_{M}X_{M},_{M}Z_{iM}) an intertwiner with R_{i}R_{i}^{∗} = 1. Then the map Φ_{W} ∘Φ_{X} is tracial on End (_{M}W ⊗_{M}X_{M}) and

Φ_{X} ∘ Φ_{W} (R_{i}^{∗}R_{i}) = \sqrt{\frac{d(_{M}Z_{iM})}{d(_{M}W ⊗_{M}X_{M})}}

holds.

Corollary A.4 Let _{M}X_{M}, _{M}W_{M} and _{M}Z_{iM} be as in the above proposition. For V_{1}, V_{2} ∈ Hom (_{M}W ⊗_{M}X_{M},_{M}Z_{iM}),

V_{1}V_{2}^{∗} = \sqrt{\frac{d(_{M}X_{M})d(_{M}W_{M})}{d(_{M}Z_{iM})}}Φ_{X} ∘ Φ_{W} (V_{2}^{∗}V_{1})

holds.

Proof. Let {R}_{i=1}^{N_{W,X}^{Z_{i}}} be an orthonormal basis of Hom (_{M}W ⊗_{M}X_{M},_{M}Z_{iM}). Then R_{i}R_{j}^{∗} = δ_{i,j}. We only have to prove the above equality for R_{i}. Then

Φ_{X} ∘ Φ_{W} (R_{i}^{∗}R_{j}) = Φ_{X} ∘ Φ_{W} (R_{i}^{∗}R_{i}R_{i}^{∗}R_{j})
= Φ_{X} ∘ Φ_{W} (R_{i}^{∗}R_{j}R_{i}^{∗}R_{i}) (Φ_{W} Φ_{X} is tracial.)
= δi, jΦ_{X} ∘ Φ_{W} (R_{i}^{∗}R_{i})
= δi, j\sqrt{\frac{d(_{M}Z_{iM})}{d(_{M}X_{M})d(_{M}W_{M})}} ,

and we get the desired equality. □

References

[EK] Evans, D. E. and Kawahigashi, Y., On Ocneanu’s theory of asymptotic inclusions for subfactors, topological quantum field theories and quantum doubles, Internat. J. Math. 6 (1995), 205–208.

[FI] Fidaleo, F. and Isola, T., On conjugate endomorphism for discrete and compact inclusions, preprint (1993).

[I] Isola, T., Modular structure of the crossed product by a compact group dual, J. Operator Theory 33 (1995), 3–31.

[IK] Izumi, M. and Kosaki, H., Finite dimensional Kac algebras arising from certain group actions on a factor, preprint (1995).

[J] Jones, V. F. R., Index for subfactors, Invent. Math. 72 (1983), 1–15.

[KL] Kosaki, H. and Longo, R., A remark on the minimal expectation of subfactors, J. Funct. Anal. 107 (1992), 458–470.

[L1] Longo, R., Index for subfactors and statistics of quantum fields I, Comm. Math. Phys. 126 (1989), 217–267.

[L2] Longo, R., Index for subfactors and statistics of quantum fields II, Comm. Math. Phys. 130 (1990), 285–309.

[L3] Longo, R., Duality for Hopf algebras and for subfactors, Comm. Math. Phys. 159 (1994), 133–150.

[L4] Longo, R., Minimal index and braided subfactors, J. Funct. Anal. 109 (1992), 98–112.

[LR] Longo, R., Rehren, K.-H., Nets for subfactors, Rev. Math. Phys. 7 (1995), 567–597.

[O1] Ocneanu, A., Quantized group string algebras and Galois theory for algebras, in “Operator algebras and applications, Vol. 2 (Warwick, 1987),” London Math. Soc. Lect. Note Series Vol. 136, Cambridge University Press, 1988, pp. 119–172.

[O2] Ocneanu, A., Quantum symmetry, differential geometry of finite graphs and classification of subfactors, University of Tokyo Seminary Notes 45, (Notes recorded by Y. Kawahigashi.) (1991).

[O3] Ocneanu, A., An invariant coupling between 3-manifolds and subfactors, with connections to topological and conformal quantum field theory, manuscript 1991.

[O4] Ocneanu, A., Seminar talk at University of California, Berkeley, June 1993.

[PP] Pimsner, M. and Popa, S., Iterating the basic construction, Trans. Amer. Math. Soc. 310 (1988), 127–133.

[P1] Popa, S., Classification of subfactors : reduction to commuting squares, Invent. Math. 101 (1990), 19–43.

[P2] Popa, S., Classification of amenable subfactors of type II_{1}, Acta. Math. 172 (1994), 163–255.

[Y] Yamagami, S., A note on Ocneanu’s approach to Jones index theory, Internat. J. Math. 4 (1993), 859–871.