Sugaku Expositions
Volume 4, Number 2, December 1991

INDEX THEORY FOR OPERATOR ALGEBRASHIDEKI KOSAKI
An operator algebra is a natural generalization of the matrix algebra MC).
aWhen one operator algebra is sitting in a larger one, by their index we mean
copies
many
how
count
to
like
would
“ratio” between them. In other words, weof the smaller one are in the larger one. For example, M,(C) = M,(C) ® M,(C)contains the subalgebra M,(C) ® 1 isomorphic to M,(C).
A
0i ; Aem©}.
01={[
02m)

linearIn this case, the index should be the ratio between the dimensions (asspaces) of the algebras

dim M,(C)/ dim M,(C) = 4°/2* = 4.
so thatHowever, all the interesting operator algebras are of infinite dimension
would
the above simple-minded method gives us oo/oo and makes no sense. We
index
meaningful
developtoandway
like to define the index in a reasonable
theory.
§2)
In the epoch-making article [32] V. Jones proved a striking result (seeand laid the foundation of index theory for a very important class of operator
algebras (II, -factors). Since then remarkable progress in the theory of operator
algebras has been made. Furthermore, based on Jones’ theory new invariants
betweenfor knots and links were discovered [33], [19], and the relationshipoperator algebras and various other fields (such as Virasoro algebras, solvable
writlattice models, and so on) has been clarified. (For this, see the exposition
beyondisfieldsothertoconnection
ten by T. Kobayashi.) This unexpectedjust onthe scope of the present exposition; hence, we will focus our attention
subsequentwith
operator algebras and explain the Jones index theory togetherasdevelopment. Although we tried to collect literature related to index theory
the mamuch as possible, we may have overlooked some literature. However,referencesthe(see
textbooks
terial that has already been covered in standard[67]-[73]) has been omitted.
1. VON NEUMANN

ALGEBRAS

The theory of operator algebras was initiated by Murray and von Neumann289-304.
This article originally appeared in Japanese in Stigaku 41(4)(1989),
46135.
46L10,
Primary
Classification.
1991 Mathematics Subject©1991 American Mathematical Society
0898-9583/91 $1.00 + $.25 per page

;

177

178
in the 1930s.

HIDEKI KOSAKIIn the introduction of their first article, as motivation for open-

ing the new field, quantum mechanics, representation theory, operator theory,
and abstract ring theory are mentioned. An operator algebra is a *-algebra of(bounded) linear operators on a Hilbert space (over C). Here, * refers to theadjoint of an operator, which is a correct generalization of an adjoint matrix.
Consequently, our theory deals with infinite-dimensional and noncommutative
objects. The algebra structure being available, results are often formulated inalgebraic language. However, due to infinite dimensionality, approximation arguments (i.e., analysis) are unavoidable, and we have to deal with several topologies. When an operator algebra contains the identity operator 1 and is closed
relative to the strong operator topology (can be replaced by other topologies),
it is called a von Neumann algebra. Here, a net {x,} of operators converges tox in the strong operator topology if and only if

lix¢&—x¢ — 0
for an arbitrary vector £. On the other hand, an operator algebra that is closed
under the topology determined by the operator norm[lx] = sup{[lx<]l; [IS] < 1}
is called a C"-algebra.

(Index theory for C*-algebras has been developed by

Watatani [62].) We remark that a Hilbert space-free characterization of these
algebras is also possible. In fact, in the late 1940s C*-algebras were introducedby Gelfand and Naimark in such an abstract form.
A von Neumann algebra can be expressed as the direct integral of factors(central decomposition)—this is analogous to decomposing unitary representations into irreducible representations. Here, a von Neumann algebra is calleda factor if its center (as an algebra) reduces to C1.
M,(C) and B(H)obviously factors.

The full matrix algebra(the set of all bounded operators on a Hilbert space H) are
When a von Neumann algebra M (C B(H)) is given, its

commutant M’ is defined by
M' ={xeBH);

xy=yx,

Vy e M}.

(Note that M’ makes sense only when the Hilbert space on which M acts is
specified.) Remark that M C M" is trivial. An important fact is that an operator algebra M is a von Neumann algebra if and only if M = M”. This
characterization (the double commutation theorem) obtained by von Neumann
in 1929 was a starting point of our theory. This result guarantees that thepolar decomposition(x = u|x|,|x| = (x* x)" ?) and the spectral decompo-

sition (|x| = Jo 4. de,)—fundamental tools for dealing with operators—canbe performed inside the von Neumann algebra in question. In particular, avon Neumann algebra contains an abundance of projections. (This is false
for a C*-algebra.) By analyzing the structure of the projections (under a certain equivalence relation), Murray and von Neumann classified factors into the179

INDEX THEORY FOR OPERATOR ALGEBRAS
following types:

type I,

n=1,2,...,0,

type II, ,type I,
type III.
A factor of type I, is isomorphic to M,(C) (B(H) if n=), and a factorof type II can be written as the tensor product of a II -factor and B(H).
A 1I,-factor has “continuous dimensions”, which was a surprise to specialists.
=
For example, consider M,(C) equipped with the normalized trace tr([a;])27 Ya, + a,,) . The (normalized) dimensions (i.e., trace values) of the projections (i.e., subspaces) are obviously 0, 5, and 1. From the tensor product

M,(C)® M,(C) equipped with tr@tr we get 0, 1,4,

3, 1. The more tensor

components we add, the more trace values of projections we get.

ing the infinite tensor product ®;-,

M,(C)Complet-relative to ®:2, tr, we obtain ansII,-factor. It is not so hard to imagine that the trace values of the projectio(unique)afill up the entire closed interval [0, 1]. A II, -factor always admitsasnormalized (tr(1) = 1) trace with this property, and this can be considered
(with
C
—
M
tr:
l
functionalinearA.
an alternative definition of a II,-factor
suitable continuity requirement) on a factor M is called a trace if(i) tr(x"x)>0,

x € M(positivity),(ii) tr(xy) =tr(yx), x,y EM (tracial property).
the
From a highly noncommutative countable group (ICC group such as S_,
can
one
s)
generator
two
finite permutations on N, or the free group F, withconstruct a II,-factor (the von Neumann algebra generated by the left regular
representation, i.e., a certain completion of the group ring). Also (by making
from ause of the group-measure-space construction) one obtains a II -factor
ation.
(probability) measure-preserving ergodic transform
if
On the other hand, a factor of type III does not possess a trace (evenfrom
one allows +oo as a trace value like the usual trace on B(H)). Startingon
completi
certainang
performi
by
III
type
oo; M,(C) one obtains a factor offromnaturallyarisingfactorsofdifferent from the previous one. Most examples
fromquantum statistical mechanics are known to be of type IIL. A factor arising
4 nonsingular (but not admitting an invariant measure) ergodic transformationis also of type III. However,

due to the fact that traces do not exist, type III

Tomitafactors had been very difficult to handle before the appearance of thefactorsIII
typeon
Takesaki theory (1967). It was their theory that made analysis
intopossible. Based on this theory Connes further classified type III factors
type I,

[orem

(0<i<

1),

type III, .
Also several structure theorems were obtained by Araki, Connes, and Takesaki.
of(The final step was marked by Takesaki [59].) It can be said that the analysis
N

180

HIDEKI KOSAKI

type III factors reduces to the study of II -algebras and that of automorphismgroups on these algebras. Partly due to this fact, the study of automorphismgroups has been (and is still being) actively carried out by several authors.
When a factor is generated by an increasing sequence of finite-dimensional
subalgebras, it is called AFD (approximately finite dimensional). Almost all
factors appearing in applications are AFD. Very recently, the classification ofAFD

factors has been completed:typel,
type II,
type II,
type III;type III,type III,

M (C), B(H),unique R
unique R® B(H)equivalent to classificationof ergodic flows (flows of weights)
unique (for each 1); the Powers factor
unique(Murray, von Neumann)(Connes)(Connes, Krieger)(Connes)(Haagerup)

The uniqueness of an AFD II, -factor (this factor is denoted by R) is especially
important. It was further proved by Connes that R is the smallest infinitedimensional factor. Recall the II,-factor constructed from the infinite tensor
product of M,(C) and the one arising from S_. Obviously, both of thempossess a structure of finite-dimensional nests so that they are isomorphic to R.
On the other hand, the II, -factor arising from F, is not AFD. From a given
type III factor a certain nonsingular ergodic flow (the flow of weights [17]) can
be constructed in a functorial way. In the AFD case the flow of weights serves
as the complete invariant for the isomorphism class. A factor is of type III,
(resp. III) if and only if the associated flow of weights is trivial (resp. periodic
with period —logA). All other ergodic flows correspond to type III,-factors.
The study of non-AFD factors is equally important; however, definite results (such as the above classification) have not been obtained. Since AFD factors
have been classified, we would like to attack the classification problem for subfactors (in a given AFD factor). Jones’ index theory serves as an important tool
for such a problem.
As indicated above, the theory of von Neumann algebras is closely related toergodic theory on nonsingular transformations. From the viewpoint of operator
algebras the notion of orbit-equivalence (which is weaker than the usual notionof equivalence in ergodic theory) plays an important role. Such ergodic theory
has been developed by many authors (Dye, Krieger, Connes, and many others).
2. JONES’ INDEX THEORY
Here, Jones’ theory on index for II -factors will be explained. Let M 2 N
be a pair of II,-factors, and Tr = Tr,, be the (unique) normalized trace on
M. When M is acting on a Hilbert space H, one can regard H as an Mmodule. Hence, the quantity “dim, H” can be introduced by using Tr (theory
on coupling constants). Jones defined the index [M : N] based on coupling
constants, but in this exposition we will explain the index without mentioning
dim, H so explicitly.

181

INDEX THEORY FOR OPERATOR ALGEBRAS

To motivate the reader, let us assume that AM, (C) is acting on C? in the usual
way. The commutant is C1, which is too small compared with M,(C). By
letting M,(C) act on a larger Hilbert space, we will obtain a larger commutant
having the “same size” as the original algebra M,(C). Let Tr be the usual(unnormalized) trace on M,(C). Then M,(C) is a 4-dimensional Hilbert space
under the inner product(x,y) € My(C) x M,(C) — Tr(y"x) € C.
Forgetting the original action of M,(C) on C?, we let M,(C) act on the above
Hilbert space M,(C) by left multiplication. If we denote a vectorTa] € M,(C)
2
X22

1
[

by (xy, » Xy15 X12 X,,) € ct , the action (i.e., left multiplication) of[Gnayin the algebra
4 x 4 matrixaz
ap

on the Hilbert space

M,(C)

ay

ay

ay

9x

0

M,(C)

0

ain

4p

ay

4p

(- [Gn
dy

= C*

2
Gp

is expressed as the® 1) X

A matrix commuting with the above matrices must be of the form
0
b,
0
b,
id)
(-1e [
by0b;
0
by, by
O
by
0
by
by
0
by,
0
(which is a matrix representing right multiplication). Note that M = M,(C)and M’' have the same size now. Let us make this statement more precise.

The tracial property Tr(x*x) = Tr(xx")

implies that the map

x — x" isaofunitary involution of the Hilbert space M,(C). The matrix representation
this unitary involution J is
0
0
0
J
©
J,
0
0
I=
0}°
0
J,
0
J,
0
0
0
JMJ =M "thatwhere J, denotes complex conjugation. It is easy to see that
"
M
onto
M
from
is, x — Jx*J gives us an anti-isomorphism
Let us return to II,-factors M 2 N. As above, M is a pre-Hilbert space
relative to the inner product(x,y) e MxM —tr(y*x) € C.

182

HIDEKI KOSAKI

Let us denote the Hilbert space completion by L*(M } (= L*(M ; ir), the standard Hilbert space of M . As before M acts on L*(M ) by left multiplication,and from now on we understand that A (hence the subfactor N as well) is
acting on L*(M) . Again x € M — x* € M (after the extension) gives us
a unitary involution J on L*(M ) satisfying
JMJ = M'. We obviously get

N' 2 M’'. Generally, the commutant of a II,-factor is a factor of type either
I, or II, . The commutant N' (C B(L*(M)) is of type II when N is verysmall in AM . Therefore, in this case the index[AM : NJ

is defined to be+oo.

Let us now assume that N' is a factor of type II, (with the unique normalized
trace tr,.). Let e, be the orthogonal projection from LYM ) (= M by thedefinition) onto the closed subspace N. This subspace being invariant under
the action of N, the projection e,, belongs to N' and try: (ey) € (0, 1] makes
sense.
Definition (Jones index [M : N]). When

N’ is of type IL, , we set[M: N= (try (ey).
Let us take N = C1 in the previous example M = M,(C) (although thisis not of type II,;). Then N = M,(C), and the closed subspace N and the
projection e,, are
Xx
0f.
x

1

0
0

01
0 0

1

0

01

1{0

respectively. Thus, the (normalized) trace value of ey is (4 +0+0+1)=1"
and the index is 4 as expected.
The Jones index [M : N] satisfies all the expected properties such as (1)M2 ND L implies [M : L]=[M : N][N : L], (2) when a finite group G
acts on N as (outer) automorphisms, the crossed product (semidirect product)
M = N x G satisfies [M: N] = #G.
An obvious (and probably the most important) question is what the possible
values of [M : N] are. When M = M (C), then [M : NJ has to be the squareof an integer (since M and N have the same unit). On the other hand, thecharacteristic of a II -factor being of continuous dimensions, an arbitrary value
seems to be possible. Jones proved the following surprising result:
Theorem [32].

The Jones index [M : N]belongs to{4c0s” 2 :n=3,4, um,+00].
nFurthermore, any value in the above set can be realized as the index value of a
subfactor in the AFD 11 -factor R.
The last half of the theorem being quite technical, in what follows we will

briefly explain the first half. It is not so hard to prove that N' is generated (asINDEX THEORY FOR OPERATOR ALGEBRAS

183

a von Neumann algebra) by M and ey. Since N is invariant under J (NJ = ey. Consequently, J N'J is generated by
is a =-subalgebra), we see Je,

JM'J=M and Je,J =e, . Weset M, = JN'J, the basic extension of M 2

N . Hence, starting from M 2D N, we have constructed (basic construction) the
new inclusion M; 2 M of II -factors (if [M : N] < o), and this procedure
preserves the index value ([M; : M] = [M : N]). The extension M, wasobtained as a factor acting on LI? (M). However, by letting M, act on its own
standard Hilbert space L*(M 1)» one can construct the next basic extension AM,
of MDM:
NCMCM=(M,ey)SM,=(M,ey, ey).

Tterating this procedure, we obtain the following canonical tower of II, factors:
NCMCM,CM,CMC.

The index value between any neighboring factors is always [M : N]. Note thateach basic construction produces the projection and that we get the sequence{eitico1,.. of projections (ey =¢,, €;; =€,, and M, is generated by M,_,and e,_, with My = M ). These projections satisfy the following beautiful
relations:ee; =ee,li—jlz2,ee,6; =[M: NJ]~1e,trew) =[M : NT! tr(w),where w is a word
one, e,...,€_-

Similar relations appear in various fields of mathematics (Hecke algebras, braid
groups, etc.). This is the reason why the Jones theory became a bridge between
operator algebras and various other fields.
The index [M : N] being finite means that N is quite large in M. Thus,
the relative commutant M NN’ turns out to be finite-dimensional. Furthermore, N is irreducible in M, ie, MN N' =Cl aslongas [M:N]<4. Avery interesting question is: What are the possible index values of irreduciblesubfactors (when [M : N] > 4)? These values form a closed set, but at thisBased on representations ofstage we do not know how large this set is.!
a sequence of index
constructed[64]
the Hecke algebra of type 4, , H. Wenzl
values tending to n® (n =3,4,...). As far as values close to 4 are concerned, Jones obtained 3 + 31/2 (2 4.73205...) . Very recently, Haagerup andSchou [25] obtained infinitely many values in the open interval (4,5) (such as2715 + VT7) > 4.56155... and 27'(5 ++/13) 22 4.30278...) . Also Ocneanu

obtained a value approximately equal to 4.02642... (by using the graph Ej).
In these examples a larger factor is always R. A factor-subfactor pair arises asa certain limit of increasing pairs of finite-dimensional algebras, and the notion! For the footnotes' to 5 see the Appendix which has been added in the translation.

184

HIDEKI KOSAKIof commuting squares is important. This notion was originally introduced by
S. Popa to obtain his deep results on II,-factors, and is playing a fundamental role in index theory as well [22], [48], [54], [64]. In the construction ofthese examples, considering various graphs is crucial and the index appears as
the square of the maximal eigenvalue (the Perron-Frobenius theorem) of theincidence matrix of a given graph. The importance of graphs had already been
pointed out in Jones’ first article [32] and especially in [35].
The Pimsner-Popa article [48] has been playing an important role in thesubsequent development of index theory. Computation of the relative commutant is important, but unfortunately quite difficult. Therefore, as a substitute(see [54] for explanation) Pimsner and Popa investigated the relative entropy
H(M/N). (This notion was introduced by Connes and Stermer in the process ofobtaining their noncommutative Koimogoroff-Sinai type theorem.) Generally,
the inequality H(M/N) < log[M : N] holds. They obtained several necessary and sufficient conditions (which have been proved to be extremely useful)for the equality, clarified the relation between H(M/N) and M Nn N', andcomputed H(M/N) for several examples. Here, the so-called Pimsner-Popabasis is fundamental so that we will briefly explain this notion. The projection

ey: L*(M)=M — N sends M into N so that Ey = ey], : M — N canbe considered. This is called a normal conditional expectation and satisfies thefollowing properties (together with suitable continuity):
Ey(x)=x,x € N (projection property),{ E\ (y,xy,) = VEX);y; € N, x € M (bimodule property).

Such a map was originally introduced in [60] as a noncommutative version ofa conditional expectation in probability theory. We easily observe that ¢, = e,satisfies eyxey = Ey (x)e;, x € M. When [M : N] < 00, it was proved in [48]that there exists a family {4;},_,, , in M (rn <[M:N]+1) satisfyingn> Aegh; = 1.
i=1This family is called a Pimsner-Popa basis. Since x € Mnsatisfiesn

ex = egxl => exhiegd; => Ey(xA))egh;,
i=1

i=1

by dividing both sides by e, (as can be justified) we getnx=» XA,

Xx; =Ey(xA)€N.

i=1

In other words, {4;},_, , _, indeed forms a basis (when M is regarded as an
N-module). Based on this expression, Pimsner and Popa obtained the following
estimate (the Pimsner-Popa inequality):

Ey(x"x) > [M:N"'x"x,xeM.

INDEX THEORY FOR OPERATOR ALGEBRAS

185This means that the difference is positive as an operator. The estimate is very
powerful. In fact, it turns out to provide a characterization of the value [M :N]. The above-mentioned inequality on the relative entropy also follows from
this.
When [M : N] < co, the infinite-dimensional spaces M and N have a finitedimensional difference. Hence M and N are expected to share many properties
in common.

In fact, several results have been obtained in this direction, andalmost always the technique in [48] turns out to be a key.
3. CLASSIFICATION OF SUBFACTORS IN R (GALOIS THEORY FOR R)
Since AFD factors have been classified, we would like to classify (finite-index)
subfactors in R, for example. This problem has been actively investigated by
A. Ocneanu and S. Popa. Here we will explain Ocneanu’s approach (see [47]for details). A finite-index subfactor N of R is (as a factor) isomorphic to R
(by Connes’ theorem) so that classification of subfactors up to conjugacy is amain issue (i.e., Galois theory for R). This means that for an automorphismo of R the two pairs M 2 N and M 2D a(N) are identified. It is routine tocheck [M : N]=[M : a(N)]: hence, the Jones index is indeed an invariant forthis classification. For example, when [R : N] = 2, Goldman’s theorem (Jones
work was strongly motivated by this theorem) asserts that R is the crossed
product of N (= R) by a Z,-(outer) action. Hence, Connes’ classification ofZ,-action on R shows that an index 2 subfactor of R is unique up to conjugacy.
Generally, the classification of subfactors and that of automorphisms are closely
related. 2
Based on Jones’ index theory and deep analysis due to Pimsner and Popa,
Ocneanu introduced an invariant (for subfactor classification), called a quantized group (paragroup or coupling system). His invariant can be considered
as a “quantization” of a finite group. There are several equivalent ways of describing his invariant. Here, for simplicity we deal with the case [M : N] < 4and explain Ocneanu’s paragroup. A paragroup (&,,., tie0,1,.) consists of a graph &,,. , (called a principal graph) and a family {y,}, of antiautomorphisms on certain finite-dimensional algebras. Let us begin by explainM 2 N gives us the canonical tower of II,ing ®,,.. As remarked in §2,factors:
N=M_CM=MCM
CMM, CMC.
of
Setting 4, = M; nN N', we obtain the increasing sequence {4 },__; o,algebras)
matrixoffinite-dimensional von Neumann algebras (i.e., direct sums
that contains an enormous amount of combinatorial data. Since M NN '=Cl
(when [M : N] < 4), we have 4_, = A; = C1. However, 4;’s (k > 1) are
getting larger and larger. In fact, 4, contains at least ¢;,¢e;,...,¢._,- An
increasing sequence of finite-dimensional algebras is described by the associated186

HIDEKI KOSAKI

Bratteli diagram. For example, let us consider the three cases (i) [M : N] =4 cos? £, (il) M = N x Z, (crossed product), and (iii) M = Rg ODN= Rg,(Rg, = the fixed point subalgebra of R under an S;-outer action). The corre- sponding Bratteli diagrams are
k=-1

(1)

(ii)

(iii)

0

1
2
3
4
5

Here, each vertex - represents a matrix algebra M,(C) (the size n to be determined shortly), and the number (could be more than one) of edges between
vertices indicates how many copies of the smaller matrix algebra are sitting inside the larger one. The sizes of matrix algebras are determined as in Pascal’s
triangle (with multiplicities of edges counted). Thus the sizes of the involved
matrix algebras in the three cases are
k=-1

(0) 1

(i)

\

0
1

A
11

2

21

\/\

3

2

4
5

5

5

n/

(iii) 1

\

A
11

3

2

3

3

8

\

aN
I-11

4

AY

3
\WAY

1

AWA

AN

2

AN

AWA

9

9

AWA

33
\V

3 1
\N/NY

5

99

5

:

1

4

9

4

Hence, for example the A4,’s in the three cases are1) MC) eC,(if) M;,(C),(iii) M,(C)eC,
while the A,’s are(i) My(C) ® M;(C),(ii) M;(C) ® M;(C) ® M,(C),(iii) M,(C)e M,(C)aC.
Furthermore, in each of the three casesA,

is sitting in 4;

as the following

187

INDEX THEORY FOR OPERATOR ALGEBRAS
subalgebra:

[4]
(1)

0

0

(ii)

0

0
«a

(NEES
[4]

;

{

Ae My(C)

os

[4]

[4]
i.
(iii)

[sieve]

A

|

l

0
A

|
0

0

cn]

y
«a

[a]

then its “mirror
Notice that if a certain edge appears between A,_; and 4,
In the firstimage” also appears at the next stage (ie., between 4, and 4, ,).
so thatimages”“mirrorthebesides
few steps new edges might also be added
diagram
each
steps
few
each diagram gets wider and wider. However, after a
is defined tostops getting wider (finite depth). In the above three cases depth, depthbe 3, 2, and 4 respectively. Depth might be generally infinite; however
diagramBratteli
the
of
part
the
Once
is always finite as long as [M : N] <4.
taking(byed
recoverely
denoted by thick edges is given, then the rest is complet
is
&,,.,
by
successive mirror images). This part of the diagram, denoted
the Dynkincalled a principal graph. The principal graphs in (i), (ii), (iii) areparagroupaof“order”
the=(Indexdiagrams 4,, D,, and A; respectively.
principalaofmatrixce— the square of the maximal eigenvalue of the incidenn), theorder
Nx G2 N (G being a finite group ofgraph.) When M =resulting principal graph is easily shown to ben—1 vertices

Hence,

4,isan n-dimensional abelian algebra (Ce CoC---®C).

This

shows up as
happens precisely because [”°(G) (with pointwise multiplication)bijectively
ondcorresp
1
A =Mn N', and the n vertices at the level k =a “quanisgraph
to the group elements. This fact explains that a principal
group. Without
tization” of the underlying set (i.e, #G-point set) of a finite
second ingrediTheinterest.
limitedofthe multiplication law this set alone iswas definedM,
that
ent {y,}, corresponds to the multiplication law. Recall(M = M,)), where J; is the unitary involution
as JyN "J, (acting on L
Therefore, the map y, : x — Jox* Jy is an anti-automorphismof LAM).
to g! , whereof 4, = Mn N'. In the above “group case” y, sends g

188{g} 2€G

HIDEKI KOSAKI
is the natural basis in /*(G).

eration” in a given paragroup.

OriginallyHencey, determines the “inverse op-

Af, +1

Was defined as J My,

J(on

L? (M,.)). But it was proved in [49] that M,,_, can be identified with thebasic extension of the (k + 1)-step inclusion M, DO N. In other words, whenM,,_ is represented on L? (M,) with the unitary involution J, , JNJ, can be
identified with M,,,. Consequently, as above we get the anti-automorphism
Ve i X — J x", of Ay, = My N N'. In the “group case” y, gives us information on the multiplication (and y,, 7, ... give no further information).
Let us be more precise. The composition y, oy, is shown to send 4, = [®(G)i

into 4, NA], and 4,N A; is isomorphic to 4, ® 4, 2 [(G) ® [*°(G) (since
depth= 2). The transpose of the linear map

1G) ® 1G) — I'(G)
w
g®h

—

w
gh

({g} 2€G also denotes the dual basis in / Le)words, y, oy, corresponds to comultiplication¥
is precisely

7, 0 7,.

In otherpovk)=Y goh=) gong'k.
gh=kgWhen depth is two (but 4, is not necessarily abelian), besides the natural
algebra structure (coming from AM, N N') 4 = Mn N’ is equipped withcomultiplication y,oy,: 4, — ANA; = 4,®4,. Consequently, 4, is a finitedimensional Hopf algebra. When depth > 3, we have higher-order products
7, as well so that a paragroup is more general than a Hopf algebra.
A paragroup satisfies certain properties, and it is possible to axiomatize thenotion of a paragroup by them. Conversely, starting from an abstract paragroup, one can construct a pair of (AFD) II, -factors (string algebras). Ocneanu
showed that a paragroup serves as the complete invariant in the classification of
subfactors (with indices < 4) of R and classified paragroups (whose orders areless than 4).°

The notion of downward basic construction (basic extensions

into a given subfactor) introduced by Jones [32] plays an important role. The
classification result is as follows:
Theorem [47]. For each of the Dynkin diagrams A, (n> 2), D,(n>2),
there exists a unique subfactor of R. For each of the Dynkin diagrams Eg and
Eg, there are exactly two subfactors of R.
A, (n>22)
D,, (n>2)
E,

——
o
——
...
_—

Eg

—t—

Here, the index is 4 cos Z—<<

with the Coxeter number

m=n+1
m=2n+2
m=12
m=30
m.

INDEX THEORY FOR OPERATOR ALGEBRAS

189

way of
For each of the principal graphs 4, and D, there is a unique
doing
of
ways
two
are
there
Eg
E;,
introducing y,’s whereas for each of
Ey.
E,,
of
each
from
so. This is the reason why two subfactors are available
4. INDEX THEORY FOR TYPE III FACTORSon
The index for II,-factors is defined by using coupling constants based
theoryspatialConnes’traces. Although general factors do not admit traces,
the index[11] gives us a substitute for coupling constants. Based on this theory
can be debetween arbitrary factors (the type III case is of primary interest)
related
some
and
index
of
notionafined. In this section we will explain such
results.
H.
Assume that a von Neumann algebra M is acting on a Hilbert space
not beWe do not require this action to be standard so that M and M' mayon Malfunction
linearpositive
normalof the same size. Let w be a faithfulsimilar
a
be
y'
Let
0).
=
(w(x*x) > 0 and the equality holds only when x
positivefunctional on M’. The spatial theory gives us a certain (nonsingular)e) on
selfadjoint operator (denoted by dy/ dy’ and called the spatial derivativ(see [11]H . This is a right substitute for a coupling constant in the general setup
M DO Npair
ubfactorfactor-sa
for the definition and detail). Let us assume that
arise
not
may
E
and a normal conditional expectation E : M — N (see §2, but

defined.
from a trace as in §2) are given. For this E the index IndexE will be

y =pokFixing a faithful normal positive linear functional p on N,weset
y , onebyinducedMonproduct(functional on M). Starting from the inneras inoncompleti
obtains the standard Hilbert space L? M) (= LX(M ; ¥)) byation. (This§2. Also let M (hence N as well) act on L*(M ) as left multiplic
representation of M

is known as the GNS representation.)

Note, however, thatlack of tracialx € M — x* € M does not give us a unitary involution as in §2 for
can talk aboutproperty. (This map is an antilinear closable operator so that one
the Tomitaof
point
startingaisthis
the polar decomposition of the closure—= J, is theJ
where
Takesaki theory. Let J A) 2 be the polar decomposition,
absolute value
phase part, a unitary involution, and A, is the square of thental theorem
part, a nonsingular positive selfadjoint operator. The fundame
(te
=M
MA"
Ay
of the Tomita-Takesaki theory states JMJ = M " andgroup {6)} cr
R). Hence, Ad A, gives us the one-parameter automorphism
N and theof M, the modular automorphism group.) So far the subfactor
a functional on
expectation E have not played any role. Recall that y' isto the spatial
thanksM'. Although N is not acting standardly on L*(M ),theory (and [23]) there exists a (unique) operator-valued weight F : NM
satisfyingd(poE)/dy' =dp/d(y oF).

expectation,
An operator-valued weight is a map similar to a normal conditional
as its value.
but the projection property is not required and “+00” is allowedin Haagerup’s
The general theory on operator-valued weights can be found

190

HIDEKI KOSAKI

original article [23], but the reader might as well think that in the present context
an operator-valued weight is just a normal conditional expectation multiplied bya positive scalar (could be +00). There is no canonical choice of ¢, y' (likeunique traces on II,-factors), but it can been shown that the above F (andL*(M ), J) is uniquely determined by E. We will denote this F by E'.
(This notation is justified after some sample computations.) Our IndexE is
defined by using El.
ifm 2 N happen to be II,-factors, the Jones index [M : N] is exactly Index E,, with the normal conditional expectation E,
constructed from the unique trace (see §2).

Besides the obvious property E™'(1) € (M') 4 (or +00), E ~1(1) satisfies

uE '(u* = E” (ulu*) = E7'(1)for an arbitrary unitary # in M’ (the bimodule property). Consequently,
E™! (1) belongs to the center of M', that is, E 1) is a scalar. We set

IndexE=E (1)(€]l,]).

Many important properties of the Jones index can also be proved for IndexE .
Details can be found in [38]. When M NN = Cl, a normal conditional
expectation (if it exists) is unique. However, there are generally many normal
conditional expectations, and the value Index E does depend on the choice of
E. When Index E < +oo (this property does not depend upon the choice ofE), there is a unique normal conditional expectation Ej: M — N such thatIndex E, = Min{Index E; E}. Furthermore this unique E, is characterized
by the property

E}! = (Index Ey)E,

on Mn N(see [31], [45]). One of the important properties of E
atraceonMNN'.

is that E, behaves like

(However, in the II,-factor case, this E,

is not necessarily

E,, arising from the unique trace. This phenomenon is related to the behaviorof the relative entropy H(M/N).)In the rest of the paper we will assume that M and N are factors of type
III. An obvious question is how well types (III), III,, III,) of A and N are
conserved under the assumption Index F < +oco. In [42] it was shown
M type III, & N type III, ,M type III; & N type III,

M, N type III, , I, ,0< A, u<1,respectively,= logd/logu €Q.
As mentioned in §1, types can be seen from flows of weights. Let (X,,, FM ),(Xy» EY )} be the flows of weights of AM, N, respectively. When Index EF <+00, each of the flows restricts the other in the following sense [27]. One canconstruct a flow (X, F,) that is a common finite-to-one extension of (X,,, EM)INDEX THEORY FOR OPERATOR ALGEBRAS
and

191

(Xp, EY ) in the sense that(i)there exists an

m-to-1projection map(m < Index E)

7, : X —X,,

such that EM om, =m, 0F,,
© (ii) there exists an n-to-1 (n < Index E) projection map 7 : X — Xp
such that EY ony ="nyokF,.
Under a certain condition (for example, the flows are periodic) we further get
mn< Index E. The common

extension(X, F,)

need not be ergodic, but wehave only finitely many ergodic components. Since a (partial) converse can beproved [28], the above restriction (on the flows under the assumption IndexE <+00) is optimal.
We now explain that an inclusion M 2 N of type III factors splits into an“essentially type II” inclusion and a “purely type III” inclusion. It is naturalto imagine that the (nonergodic) flow (X, F,) appearing above is the flow ofweights of a certain von Neumann algebra (which is not necessarily a factor).
Indeed we have the following result [40]:
Let M DO N be factors of type III, but not of type III,. Assume Index E< oo and E gives us the minimum index value. Then there exist two von
Neumann algebras 20, 9B such that
MD>ADB DN and A, B have the same center (they are factors if,(i)for example,

M

NN’ = Cl),(ii) 2%, B have the same flow of weights, which is exactly (X, F),(iii) E: M — N splits into the three normal conditional expectations

MEaSs

EN,

(iv)

G:? — B is an “essentially type II” inclusion.
The last statement (iv) requires some clarification. According to the structure
theorem for type III factors, the crossed product M=M x; R relative to themodular automorphism group {o, = a} is a von Neumann algebra of type
II admitting the special action {6,} cr » called the dual action. The original
factor M can be recovered as the crossed product M x o R [59]. (It is also
possible to express M as the crossed product by a Z-action if M is not oftype III, .) The above (iv) means that 242 B admit the simultaneous crossed
product (R or Z) representation: 2A = 2A x gR2 B= B xo R. Then (ii)means that % and B have the same center (2 L®(X)) so that we have thesimultaneous central decomposition3)
3)

a= | dw) do2%= | Bw) do.
bePeConsequently, {A(w) 2 B(w)} wey 18a field of inclusions of II _-factors. Since
we can pull out a type I_-factor from a type II__-factor, we actually obtain afield of inclusions of II, -factors.

Thus, for eachw € X

we can compute theJones index between the (fiber) II,-factors. This index value is constant on each
ergodic component of (X, F,).

192

HIDEKI KOSAKI
Although we have to look at the behavior of the dual action §, carefully, theabove consideration shows that analysis of 2 DB

reduces to that of inclusionsof II -factors (at least in principle.) * On the other hand,
M D2 2% and BDO N
are “purely type III” inclusions in the sense that the effect of inclusions appears
at the level of the involved flows of weights. These two inclusions are “dual” toeach other so that in what follows we will just consider % DO N . For simplicity
let us assume M = 2% = 8B DO N. (Cutting by a suitable projection, one canreduce B DO N into this situation.) In this case the projection map x, : X —X,, is one-to-one so that we simply obtain the factor map(Kyps FY

(Xy, F,)between the two flows. In other words, X,, is (as a measure space) isomorphic
to Xyx{1,2,...,n}, and FM can be expressed as the skew-product flow

Fl, i)=(F"(@), 9, ().
By direct calculation we observe M N N' = C1 and IndexE = n. For AFD
type III, (0 <A <1) factors, the classification of M (=2 =) 2 N up to
conjugacy is equivalent to that of the above factor map between the two flows[40]. Consequently, classification subfactors (with finite indices) of an AFD

type III,(0 <A <1)+factor reduce (at least in principle) to:classification of subfactors of R (together with the effectof the dual action)
classification of factor maps between ergodic flows.

Let us return to the general case (AFD not required, but
The following three conditions are equivalent:

M =A

=82

N).

(i) depth of M 2 N (in the sense explained in §3) is two,(ii) N is the fixed point subalgebra M of an outer action on M by a finite
group

G

of order=n,(iii) the number (generally < n) of ergodic components of the flow (w,i, Jj)— (FM), 0, (i), 0, (J) (on Xyx{1,2,...,n}?) is exactly n.
One can similarly find a necessary and sufficient condition (in terms of thefactor map) for M to be the crossed product N xT" relative to an outer action
on N by a finite group I" of order n. When this happens, I" is automaticallyabelian. Furthermore, the groups G and I" can be determined by looking at
the ergodic decomposition of the flow in (iii).
Let us explain the difference between index theory for type III, factors andthat for type III, (0 <A < 1) factors. Assume M =2A=B2 N as before.
Depth may or may not be two in the III, case. When depth is two, any finitegroup G (of order n) can occur. On the other hand, in the II, (0 <A <1)case depth is always two and the group G in (iii) is automatically the cyclic
group Z,. Consequently, we always get M = N x Z,, or equivalently, N =My . This fact was directly showed in Loi’s thesis [43]. He further proved theINDEX THEORY FOR OPERATOR ALGEBRAS

193

following uniqueness result: The AFD type III, (0 <A <1) factor M admits
a unique (up to conjugacy) subfactor N such that N is of type IIL andIndexE = n. In our language this corresponds to the fact that a periodic flow
(period = —(logA)/n) admits a unique (up to isomorphism as a factor map)n-to-1 extension.
Not only the classification of isomorphism classes of factors (see §1) but
also the classification of subfactors are related to certain classification problems
in ergodic theory. Several attempts of developing index theory in the context
of ergodic theory (more precisely in terms of ergodic equivalence relations)have already been made [18], [29], [58]. When an inclusion of AFD type III,(0 < A < 1) factors does not contain an “essentially type II” inclusion, it can be
described by certain ergodic equivalence relations.

Construction of various in-clusions based on subtle examples in ergodic theory and computation of various
invariants in terms of ergodic theoretical data seem to deserve investigation.
For a pair M 2 N of von Neumann algebras without finite traces, R. Longoconstructed the canonical endomorphism y : M — N (unique up to an innerperturbation) and obtained some remarkable results (see [44], for example). He
has also developed index theory for type III factors based on this technique [45].
Although the index in [38] and that in [45] are equivalent, each approach seems
to have its own advantage. Our approach explained so far is based on analysis offlows of weights so that results on type III, factors are quite difficult to obtain,
On the other hand, quantum field theory provides us with natural inclusions oftype III, factors, for which Longo’s approach seems to be fitting. 5
APPENDIXThe original exposition in Japanese was written more than a year ago. Index
theory for operator algebras initiated by V. Jones is a very active area, andsince then a lot of substantial contributions have been added to the subject
matter. In particular, some remarkable results were announced in the Satellite
Conference of ICM-90 on “Current Topics in Operator Algebras” (which washeld at Nara, Japan, Aug.

16-19).

The reader can find some of the latest resultsin the Proceedings of that conference to be published in the near future. To
update this exposition, we here add several endnotes.
1. Quite decisive results were obtained by S. Popa for this problem. If factors
are not required to be AFD, an arbitrary value (> 4) is possible [93]. For the
AFD

II,-factor, this problem is related to a certain combinatorial problem.

Inparticular, there is a gap for index values between 4 and 4.026... , the value
obtained by Ocneanu.
2. This has been completely justified in [91]. A subfactor approach seems tobe very powerful for analysis of actions (even for analysis of minimal actionsof compact groups as was shown by Popa and Wasserman).
3. To be more precise, the result was announced, for example, in [47] without detailed proofs. A complete proof (with a slightly different approach—commuting squares and the Pimsner-Popa inequality) was published by Popa194

HIDEKI KOSAKI[89]. Popa’s approach also works for certain (= strongly amenable) infinitedepth cases. In particular, the complete classification of index 4 subfactors of
R was announced in [91].
4. Some classification results were obtained in [87], [85] when 2 and 9B are
AFD

and

of typeur, , O0<i<l.

5. Jones and Wasserman have constructed some inclusions of type III, factors based on representations of loop groups.
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Translated by H. KOSAKIDEPARTMENT OF MATHEMATICS,FUKUOKA, 810, JAPANCOLLEGEOF GENERALEDUCATION,

KyusHu

UNIVERSITY,