数理解析研究所講究録
第 839 巻 1993 年 8-118On Type II and Type III Principal Graphs
Masaki Izumi
Research Institute for Mathematical Sciences
Kyoto University, Kyoto 606, Japanizumi@kurims. kyoto-u. ac. jp\S 0. Introduction
Let us consider a pair of type

$III_{1}$factors with finite index. Then it is known that there

exists the simultaneous continuous crossed product decomposition of the pair. Therefore wehave two principal graphs, so-called the type II principal graph and the type III principalgraph [KL]. In this note we shall characterize the condition that above two graphs aredifferent, in terms of Longo’s sectors.
In the case of type $III_{\lambda}(0<\lambda<1)$ factors, we can show the same type of characteri-zation when the pair has simultaneous discrete crossed product decomposition.

\S 1. The type
Let $M\supset N$ be a pair of type

$III_{1}$$III_{1}$case

factors with finite index. Then we can construct

the simultaneous continuous crossed product decomposition as follows. Let$\omega$

be a faithfulnormal state on $N$ and $E:Marrow N$ the minimal conditonal expectation of Hiai [H]. We

define a pair of type

$II_{\infty}$

factors $M\supset N$ as follows.

$\overline{M}=Mx_{E}R\supset\tilde{N}=N\sigma^{\omega}\sigma^{\lambda_{\omega}}$

Let$\tau$be the trace on

action of$\sigma^{\omega}$$\overline{M}$restricted toand$\overline{N}$$\theta$

the dual action of$\sigma^{\omega\cdot E}$R.

, which coincides with the dual

. Thanks to Takesaki duality we can identify $M\supset N$ with

9$\overline{M}x_{\theta}R\supset\tilde{N}\rangle\triangleleft\theta$R. Let$M_{n}\supset M_{n-1}\supset\cdots\supset M\supset N$$\overline{M}_{n}\supset\overline{M}_{n-1}\supset\cdots\supset\overline{M}\supset\overline{N}$be the towers associated with $M\supset N$ andidentified with–
$M_{n}\rangle\triangleleft\theta$

R. (Note that

Defintion 1.1. We call the princip$\theta$$al$$\overline{M}\supset\overline{N}$. In the same way as above

is canonically extended to$M_{n}.$$M_{n}$can be)graph of $M\supset N$ the type II principal graph andthat of $M\supset N$ th $etypeIII$ prin cipal graph.
It is known that the following relation holds [KL].
$(\overline{M}_{n}\cap\overline{N}’)_{\theta}=M_{n}\cap N’$.

So in general two principal graphs do not coincide. Actually there exist examples having

different principal graphs [S]. We show the necessary and sufficient condition that thisphenomenon happens in terms of Longo’s sectors. (For the definition of the sectors, see[L2, I].)

Theorem 1.2. Let $M\supset N$ be a pair of typ $eIII_{1}$ factors with finite index and$\gamma$: $Marrow N$

the canonical endomorph $ism[L3]$ . The type II and th type III prin cipal graphs of $M_{1}\supset M$$e$

do not coincide if and only if th ere $app$ ears the modular au tomorphism$M[\gamma^{n}]_{M}$$M[\sigma_{t}^{\hat{\tau}}]_{M}(t\neq 0)$

infor some $n\in N$ .

As a corollary we have the following.

Corollary 1.3. Moreover if the depth of $M\supset N$ is finite the $typeII$ and $type\Pi I$ principalgraphs oincide.
$c$

It is easy to show the sufficiency. So we sketch the proof of the necessity. Assume

that the two graphs are different. Since a one-parameter action can not move any central

10projections in the higher relative commutant algebras, we can see that there exist twodifferent ( $M-M$ or $M-N$ ) sectorsthat there exists $t\neq 0$ and$[\rho_{1}],$$[\rho_{2}]=[\sigma_{t}^{\overline{\tau}}\cdot p_{1}]$$[\rho_{2}]$which coincide restricted to$\overline{M}$. This means. Therefore$[\rho_{2}]\overline{[p_{1}]}=[\sigma_{t}^{\hat{\tau}}][\rho_{1}\overline{p_{1}}]$

contains$[\sigma_{t}^{\hat{\tau}}]$. Corollary 2.3 follows from the fact that no aperiodic automorphism canappear in the decsendent $M-M$ sectors when the depth is finite.

\S 2. The type $III_{\lambda}(0<\lambda<1)$ caseIn the case of type $III_{\lambda}(0<\lambda<1)$ factors, we have to assume that there sxist a pairof type

$II_{\infty}$factors$\overline{M}\supset\tilde{N}$and $\theta\in Aut(M)$ scaling the trace and globally preserving $N$

such that$M=\overline{M}*\theta Z\supset N=\tilde{N}\lambda\theta$Z.

Note that the assumption is satified if $M$ is isomorphic to $N$ and the center of $M\cap N’$ istrivial [Li].
We define the two principal graphs in the same way as before using the discrete crossed
product decompositon. Then the same type of theorem holds.

Theorem 1.2. Let $M\supset N$ be a pair of type $\Pi I_{\lambda}(0<\lambda<1)$ factors with finite in $dex$ and$\gamma$: $Marrow N$ the canonical en domorphi$sm$ . We assume that the pair as the $sim$ ultaneous$h$discrete crossed product decomposition.

Then The type II and the type III princip$ai$

graphs of $M_{1}\supset M$ do not coincide if and only if there appears the modular automorphism$M[\sigma_{t}^{\hat{\tau}}]_{M}(t\not\in T(M))$

in$M[\gamma^{n}]_{M}$In contrast to the typefor some $n\in N$ .

$III_{1}$case the discrete action can move central projections in

the higher relative commutant algebras. So we have to treat an additional case. Assume

11
that

$\theta$moves some central projections. Then we can show that there exists a decsendant( $M-M$ or $M-N$ ) sector$[\rho]$satisfying$[\sigma_{t}^{\hat{\tau}}\rho]=[p](t\not\in T(1\mathfrak{l}/I))$. Hence,$[\rho\overline{\rho}]=[\sigma_{t}^{\hat{\tau}}p\overline{\rho}]$contains$[\sigma_{t}^{\hat{\tau}}]$.

References.
[GHJ] F. Goodman, P. de la Harpe, V. Jones, Coxeter graphs and towers of algebras,

MSRI Publications 14, Springer Verlag, 1989.
[H] F. Hiai, Minimizing indices of conditional expectations on a subfactor, Publ.

RIMS, Kyoto Univ. 24 (1988), 673-678.
[I] M. Izumi, Goldman’s type theorem for index 3, to appear in Publ. RIMS, Kyoto

Univ..
[J] V. Jones, Index for subfactors, Invent. Maht. 72 (1983), 1-25.

[K] H. Kosaki, Extension of Jones theory on index to arbitrary factors, J. Funct. Anal.

66 (1986), 123-140.
[KL] H. Kosaki, R. Longo, A remark on the minimal index of subfactors, preprint.
[L1] R. Longo, Index of subfactors and statistics of quantum fields I, Commun. Math.

Phys. 126 (1989), 217-247.
[L2] R. Longo, Index of subfactors and statistics of quantum fields$\Pi$, Commun. Math.

Phys. 130 (1990), 285-309.
[L3] R. Longo, Simple injective subfactors, Adv. Math. 63 (1987), 152-171.
[L4] R. Longo, Minimal index and braided subfactors, preprint.

[Li] P. Loi, On automorphisms of subfactors, preprint.
[S] Y. Sekine, An inclusion of type

preorint.

$III_{1}$factors without splitting into type

$II_{1}$inclusion,