数理解析研究所講究録977 巻 1997 年 13-22

13

JORDAN-H\"OLDER TYPE THEOREM IN NORMALINTERMEDIATE SUBFACTOR LATTICES FOR
DEPTH TWO INCLUSIONS OF AFD
FACTORS$\mathrm{I}\mathrm{I}_{1}$TAMOTSU TERUYA

照屋保)

$($

ABSTRACT. Let $N\subset M$ be a depth 2 inclusion of AFD
Jones index. Let $K$ and$L$$\mathrm{I}\mathrm{I}_{1}$factors with finite

be normal intermediate subfactors of $N\subset M$ . If

$K\cap L=N$ and $M$ is generated by $K$ and $L$ , then we can represent $M,$ $K,$ $L,$ $N$ as$M=P\otimes R,$ $K=Q\otimes R,$ $L=P\otimes S$$R\supset S$, and $N=Q\otimes S$ for some inclusins$P\supset Q$

and. Using this characterization, we shall prove Jordan-H\"older type theorem innormal intermediate subfactor lattices for depth 2 inclusions of AFD$\mathrm{I}\mathrm{I}_{1}$factors.

1. INTRODUCTION
Let $N\subset M$ be an irreducible inclusion of typefactors with finite index. In [9],$\mathrm{I}\mathrm{I}_{1}$the auther introduced the notion of normality for intermediate subfactors of $N\subset M$

as follows:
Definition 1.1. Let $K$ be an intermediate$N\subset M\subset M_{1}\subset M_{2}$.

$\mathrm{S}\mathrm{u}\mathrm{b}\mathrm{f}\mathrm{a}\mathrm{C}\mathrm{t}_{\mathrm{o}\mathrm{r}}.$of the inclusion $N\subset M$ . Let

be the Jones tower for $N\subset M$ and$K_{1}$

the basic extension

for $K\subset M$ . Then $K$ is a normal intermediate subfactor of the inclusion $N\subset M$ if$e_{K}\in Z(N’\cap M_{1})$for $K\subset M$ andand$e_{K_{1}}\in \mathcal{Z}(M’\cap M_{2})$$K_{1}\subset M_{1}$, respectively.

, where$e_{K}$

and$e_{K_{1}}$are the Jones projections

14

TAMOTSU TERUYAWith the above notation, if the depth of $N\subset M$ is 2, then $N’\cap M_{1}$ and $M’\cap M_{2}$are a dual pair of Hopf

$C^{*}$-algebras. and $K’\cap K_{1}$ is$\mathrm{a}*$-subalgebra and a left coideal

of $N’\cap M_{1}$ ( see [1]). Then $K$ is a normal intermediate subfactor of $N\subset M$ if andonly if $K’\cap K_{1}$ is a subHopf algebra and the left and right adjoint action of $N’\cap M_{1}$leave $K’\cap K_{1}$ invariant (see [3]).
$\mathrm{w}_{\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{t}}\mathrm{a}\mathrm{n}\mathrm{i}[10]$studied intermediate subfactor lattices

tween modular identity and commuting andconditions. The$\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{h}_{\mathrm{o}\mathrm{r}}[9]$$\mathrm{c}\mathrm{o}$$\mathcal{L}(N\subset M)$

and relations be--commuting (nondegenerate) square

proved if the depth of $N\subseteq M$ is 2, then the set $N(N\subset M)$

of all normal intermediate subfactors of $N\subset M$ is a sublattice of$\mathcal{L}(N\subset M)$

and amodular lattice.

Let $N\subset M$ be an irreducible, depth 2 inclusion of AFD
index. Our purpose is to show$\mathrm{J}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{a}\mathrm{n}-\mathrm{H}_{\ddot{\mathrm{O}}1\mathrm{d}\mathrm{e}}\mathrm{r}$$\mathrm{I}\mathrm{I}_{1}$factors with finite

type theorem in normal intermediatesubfactor lattices for $N\subset M$ . To be more precise, we prove that if $M=A_{0}\supset A_{1}\supset$$A_{2}\supset\cdots\supset A_{n}=N$

and $M=B_{0}\supset B_{1}\supset B_{2}\supset\cdots\supset B_{m}=N$ are maximal chainsof $N(N\subset M)$ , then $m=n$ and the inclusions
inclusions$B_{j-1}\supset B_{j}$$A_{i-1}\supset A_{i}$are isomorphic to thein some order. To show this , we characterize tensor productsof depth 2 inclusions of AFD$\mathrm{I}\mathrm{I}_{1}$factors with finite index as follows: Let $N\subset M$ bean irreducible, depth 2 inclusion of AFD$\mathrm{I}\mathrm{I}_{1}$factors with finite index. Let $K$ and$L$

be normal intermediate subfactors for $N\subset M$ . If $K\cap L=N$ and $M$ is generated by
$K$

and

$L$

, then we can represent $M,$ $K,$ $L,$ $N$ as $M=P\otimes R,$ $K=Q\otimes R,$ $L=P\otimes S$

and $N=Q\otimes S$ for some inclusions $P\supset Q$ and $R\supset S$ .

15$\mathrm{J}\mathrm{O}\mathrm{R}\mathrm{D}\mathrm{A}\mathrm{N}-\mathrm{H}\check{\mathrm{o}}$LDER TYPE THEOREM

2. A CHARACTERIZATION OF TENSOR PRODUCTS OF DEPTH 2 INCLUSIONS
Let $N\subset M$ be an irreducible, depth 2 inclusion of$\mathrm{I}\mathrm{I}_{1}$factors with $[M : N]<\infty$

and $N(N\subset M)$ the all normal intermediate subfactors of $N\subset M$ . Suppose that
$K,$ $L\in N(N\subset M)$

and $M$ is generated by $K$ and$K$$\subset$$N$$\langle K, e_{K}^{M}\rangle$$e_{L}^{M}$

and$\mathrm{c}\mathrm{o}$, and $N=K\cap L$ . Then

$M$$\cup$$\cup$

is commuting and$L$$\subset$$L$-commuting (nondegenerate) square (see [6, 8]). Let $K_{1}=$$L_{1}=\langle L, e_{L}^{M}\rangle$

be the basic extension with the Jones projections$e_{K}^{M}$

andfor $K\subset M$ and $L\subset M$ , respectively. Then it is well known that$M$$\subset$and$\cup$$\cup$$L$$M$$K_{1}$$\subset$$\cup$$\cup$$K$$\langle L, e_{K}^{M}\rangle$$L_{1}$$\subset$$\subset$$\langle K,e_{L}^{M}\rangle$

are also nondegenerate commuting squares.

Lemma 2.1. With the above notation,

inclusions. Moreover, $M$ andand, $M$ and$\langle K, e_{L}^{M}\rangle$$\langle L, e_{K}^{M}\rangle$$L\subset K_{1}$and$K\subset L_{1}$are irreducible, depth 2

are normal intermediate subfactors ofare normal intermediate subfactors of$K\subset L_{1}$$L\subset K_{1}$16

TAMOTSU TERUYAProof. Since$L\subset M$

andis 2 by [7]. Similarly,
andare depth 2 inclusion by [9], the depth of$L\subset K_{1}$is a depth 2 inclusion. It is easy to see that

$L\subset K_{1}$$M\subset K_{1}$$K\subset L_{1}$are irreducible inclusions.

$K\subset L_{1}$$\square$Lemma 2.2. With the above notation, we have$K’\cap K_{1}=\langle K, eLM\rangle’\cap M1=N’\cap\langle L, e^{M}\rangle K$$L’\cap L_{1}=\langle L, e_{K}^{M}\rangle’\cap M1=N’\cap\langle K, e^{M}\rangle L$.

Proof. By Lemma 2.1 and [8], we have $[M:K]=[L:N]=[L_{1} : \langle K, e_{L}^{M}\rangle]$ . Therefore
we have$\dim_{\mathbb{C}(K’}\cap K_{1})=\dim_{\mathbb{C}}(\langle K, e^{M}L\rangle’\cap M_{1})=\dim_{\mathbb{C}(N’}\cap\langle L, e_{K}^{M}\rangle)$

Let

$x$

be an element of $K’\cap K_{1}$ . Since$K’\cap K_{1}\subset N’\cap M_{1},$

have

$x$

and

$K’\cap K_{1}\subset\langle K, e_{L}^{M}\rangle’\cap M_{1}$$K’\cap K_{1}=\langle K, e_{L}^{M}\rangle’\cap M_{1}$

Since
have

$e_{L}^{M}$$M_{1}$

if is an element of$x$$N’\cap\langle L, e_{K}^{M}\rangle$. And

Similarly, we haveis an element of the center of $N’\cap M_{1}$ andare commutative and hence. By$\dim_{\mathbb{C}(K’}\cap K_{1}$)$x\in\langle K, e_{L}^{M}\rangle’\cap M_{1}$$=\dim_{\mathbb{C}}(\langle K, e^{M}\rangle’L\cap M_{1})$. So we, we have.

is the basic extension of$\langle L, e_{K}^{M}\rangle=\{e_{L}^{M}\}’\cap K1$$e_{L}^{M}$.

. Since$K’\cap K_{1}$, then$e_{L}^{M}$$K_{1}$

by$\langle L, e_{K}^{M}\rangle$

with the Jones projection

is an element of the center of$x\in\{e_{L}^{M}\}’\cap K1=N’\cap\langle L, e_{K}\rangle M$$K’\cap K_{1}=N’\cap\langle L, e_{K}^{M}\rangle$

by$\dim_{\mathbb{C}(K’}\cap K_{1}$$L’\cap L_{1}=\langle L, e_{K}^{M}\rangle’\cap M_{1}=N’\cap\langle K, e_{L}^{M}\rangle$.

)$e_{L}^{M}$, we$N’\cap M_{1}(\supset K’\cap K_{1})$. And hence$K’\cap K_{1}\subset$$=\dim_{\mathbb{C}}(N’\cap\langle L, e_{K}^{M}\rangle)$$\square$

,

.

17

$\mathrm{J}\mathrm{O}\mathrm{R}\mathrm{D}\mathrm{A}\mathrm{N}-\mathrm{H}\tilde{\mathrm{O}}$LDER TYPE THEOREM

Theorem 2.3. Let $N\subset M$ be an irreducible, depth 2 inclusion ofwith $[M : N]<\infty$ . If $K$ and$L$$AFDII_{1}fact_{\mathit{0}}r\mathit{8}$are normal intermediate subfactors of $N\subset M$ suchthat $K\cap L=N$ and $M$ is generated by $K$ and , then we can represent $M,$ $N,$ $K,$
$L$

$L$

as $M=P\otimes R,$ $N=Q\otimes S,$ $K=Q\otimes R$ and $L=P\otimes S$

Proof. $N\subset M$ has the generating property, i.e., there exists a tunnel $M=N_{0}\supset$$N=N_{1}\supset N_{2}\supset\cdots\supset N_{i}\supset\cdots$such that$\overline{\infty}eak$$\overline{\infty}eak$$\supset N=\cup(N\mathrm{n}i=1N’i)$$M= \bigcup_{i=1}(M\cap N_{i’})$(see for example [4, 5]). Let$A_{00}\supset A_{01}\supset A_{02}\supset\cdots$$\cup$$\cup$$\cup$$A_{10\supset}A11\supset A12\supset\cdots$$\cup$$\cup$$\cup$$A_{20\supset}A21\supset A22\supset\cdots$$\cup$$\cup$$\cup$

18

TAMOTSU TERUYA

be the commuting and$\mathrm{c}\mathrm{o}$-commuting squares such that the initial commuting square

is
$M$

$L$

$\supset$$\cup$$\cup$$K$

and $A_{ii}=N_{i}$ for $i=1,2,$$\ldots$

as in [8]. Note that for the square$A_{kl}$$\supset$$\cup$$A_{k+1,l}$$A_{kl}\supset A_{k+1,l1}+$$N$$\supset$$A_{k,l+1}$$\cup$$\supset$$A_{k+1,l1}+$

is again irreducible, depth 2 and,mediate subfactors of$A_{kl}\supset A_{k+1,l+1}$’$A_{k,l+1}$

and$A_{k+1,l}$are normal inter-. We put$\overline{\infty}eak$$P=\overline{\bigcup_{i=1}^{\infty}(A_{0}0^{\cap}A\prime)i0}eak$$\supset Q=\cup(A_{1}0\cap A_{i}’0)i=1$$\overline{\infty}eak$$R=\cup(A_{00}i=1\mathrm{n}A_{0i};)$$\overline{\infty}$weak$\supset S=\bigcup_{i=1}(A_{0}1^{\cap}A_{0i}/)$

Then we can see $M=P\otimes R,$ $N=Q\otimes S,$ $K=Q\otimes R$ and $L=P\otimes S$ by Lemma 2.2

and [2].

$\square$3.

$\mathrm{J}\mathrm{o}\mathrm{R}\mathrm{D}\mathrm{A}\mathrm{N}$-H\"OLDER TYPE THEOREM

In this section, we shall prove Jordan-H\"older type theorem for depth 2 inclusionsof AFD$\mathrm{I}\mathrm{I}_{1}$factors.

19

LDER TYPE THEOREM$\mathrm{J}\mathrm{O}\mathrm{R}\mathrm{D}\mathrm{A}\mathrm{N}-\mathrm{H}\vec{\mathrm{o}}$Theorem 3.1. Let $N\subset M$ be an irreducible, depth 2 inclusion of $AFDII_{1}$ factor.

If

and

$K$

are normal intermediate subfactors of $N\subset M$ , then $K\subset K\vee L$ and$L$$K\cap L\subset L$

are conjugate.

Proof. Since the set $N(N\subset M)$ of all normal intermediate subfactors ofa sublattice of$\mathcal{L}(N\subset M),$

$K\vee L$

isand $K\cap L$ are elements of $N(N\subset M)$ . Therefore$N\subset KL$ and $N\subset K\cap L$ are depth 2 inclusion by [9, Theorem 4.6].
$K\cap L$$N\subset M$

Moreoveris a normal intermediate subfactor of $N\subset K\vee L$ by [9, Proposition 3.7]. So wehave $K\cap L\subset K\vee L$ is depth 2 inclusion by [9, Theorem 4.6]. By theorem 2.3, there

exist inclusions $P\supset Q$ and$R\supset S$

such that $K\vee L=P\otimes R,$ $K=P\otimes S,$ $L=Q\otimes R$

and $K\cap L=Q\otimes S$ . So we can see both $KL\subset K$ and $L\supset K\cap L$ are conjugate
to $R\subset S$ .

$\square$

Theorem 3.2. Let $N\subset M$ be an irreducible, depth 2 inclusion of $AFDII_{1}$ factors

with $[M : N]<\infty$ . Let$K\supset\tilde{K}$and$\tilde{L}(\tilde{K}\cap L)$

Proof. Since
and$L\supset\tilde{L}$$K,\tilde{K},$$L,\tilde{L}$be normal intermediate subfactors of $N\subset M$ with. Then the pairs$\tilde{K}(K\cap L)\supset\tilde{K}(K\cap\tilde{L})$and $\tilde{L}(K\cap L)\supset$are conjugate.
$\tilde{K}\vee(K\cap L)=(\tilde{K}\vee(K\mathrm{n}\tilde{L}))\vee(K\cap L)$$K\cap L\supset(K\cap L)\cap(\tilde{K}\vee(K\mathrm{n}\tilde{L}))$, the pairs$\tilde{K}\vee(K\cap L)\supset\tilde{K}\vee(K\cap\tilde{L})$are conjugate by the previous theorem. Similarly,

the pair $\tilde{L}(K\cap L)\supset\tilde{L}(\tilde{K}\cap L)$ and $K\cap L\supset(K\cap L)\cap(\tilde{L}\vee(\tilde{K}\cap L))$ . areconjugate. Since $N(N\subset M)$ is a modular lattice by [9], we hve $(K\cap L)\cap(\tilde{K}$$(K\cap\tilde{L}))=((K\cap L)\cap\tilde{L})(K\cap\tilde{L}))=(K\cap\tilde{L})(K\cap\tilde{L})$. Similarly, we hvave

20

TAMOTSU TERUYA$(K\cap L)\cap(\tilde{L}\vee(\tilde{K}\cap L))=(K\cap\tilde{L})(K\cap\tilde{L})$In a lattice
and

$L$

, a finite chain$x_{i}\supseteq a\supseteq x_{i+1}$. We have thus proved the theorem.

$x=x_{0}\supseteq x_{1}\supseteq\cdots\supseteq x_{d}=y$implies $x=a$ or $x_{i+1}=a$ for $i=1,2,$$\ldots,$

is maximal if$\square$$x_{i}\wedge\supset x_{i+1}$$d-1$ .

Theorem 3.3. Let $N\subset M$ be an irreducible, depth 2 inclusion of$AFDII_{1}faCt_{\mathit{0}}r\mathit{8}$with $[M:N]<\infty$ . If $M=A_{0}\supset A_{1}\supset\cdots\supset A_{n}=N$ and $M=B_{0}\supset B_{1}\supset\cdots\supset B_{m}$are two maximal chain of $N(N\subset M)$ , then $m=n$ and the$inClu\mathit{8}ionSA_{i1}-\supset A_{i}$areisomorphic to the $inclu\mathit{8}i_{onS}B_{j1}-\supset Bj$ in some order.

Proof. Put$A_{ij}=A_{i}\vee(Ai-1\cap Bj)$

and
$B_{ji}=B_{j}\mathrm{v}(Ai\cap Bj-1)$

Then$A_{i,j-1}\supset A_{ij}$$..\supset A_{s}$is isomorphic to$B_{j,i-1}\supset B_{ji}$.

by Theorem 3.2. Since

is maximal chain, for any $i(i=1,2, \ldots , s)$ , there uniquely exists$A_{i-1}=A_{i,j-1}\supset A_{ij}=A_{i}$isomorphic to$B_{j-1}\supset B_{j}$Example 3.4. Let$G$. Then.

$B_{j-1}=B_{j,i-1}arrow\supset B_{ji}=B_{j}$. And hence$\gamma$be a semi direct groupis an outer action of(see for example [7]). Let$j$such that$A_{i-1}\supset A_{i}$is$\square$$B\rangle\triangleleft A$of finite groups$A$$M=P\rangle\triangleleft_{\gamma}B\supset N=P^{()}A,\gamma=\{x\in P|\gamma_{a}(x)=x,\forall a\in A\}$

where

$A_{0}\supset A_{1}\supset$

$G$

on

$\mathrm{I}\mathrm{I}_{1}$and $B$ . Let,

factor $P$ . Then the depth of $N\subset M$ is 2

$A_{0}=A\wedge\supset A_{1}arrow\supset\cdots\wedge\supset A_{r}=\{e\}$and$B_{0}=B\wedge\supset B_{1}\supset\wedge$

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JORDAN-HOLDER TYPE THEOREM

... 2
of

$G$

$B_{s}=\{e\}$with

be normal subgroups of$A_{i-1}\supseteq H\supset A_{i}$

or

$G$

such that if $H$ is a normal subgroup$B_{j-1}\supseteq H\supset B_{j}$, then $H=A_{i}$ or $H=B_{j}$ . Then

$M=P\lambda_{\gamma}B_{0}\supset P\rangle\triangleleft_{\gamma}B_{1}\supset:\cdot\cdot P=P^{(A_{r},\gamma)}\supset P^{(A_{\mathit{7}-}}1,\gamma)\supset\cdots P^{(A0,\gamma)}=N$

is amaximal chain of $N(N\subset M)$ by [9]. Therefore if $M=C_{0}\supset C_{1}\supset\cdots C_{n}=N$a maximal chain of $N(N\subset M)$ , then $n=r+s$ and the inclusins
isomorphic to$R\rangle\triangleleft F\supset P$

or$R\supset R^{F}$

for some$\mathrm{I}\mathrm{I}_{1}$$C_{k-1}\supset C_{k}$arefactor and some finite group $F$ .

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6. S. Popa. Classification of amenable subfactors of type II. Acta Math., 172:163-255, 1994.

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TAMOTSU TERUYA
10. Y. Watatani. Lattices of intermediate subfactors. J. Funct. Anal., to appear.