Proceedings of the International Congress of Mathematicians
Berkeley, California, USA, 1986
Subfactors of Type Hi Factors
and Related Topics
V. F. R. JONES
1. Introduction. In this article we use the definitions of von Neumann algebra theory appearing in Haagerup's paper [H] in these proceedings.
So let M be a type Hi factor with (unique) normalized trace tr: M —• C
(tr(l) = 1, tr(a&) = tr(6a)). Whenever M acts as a von Neumann algebra on a
Hilbert space #, there is a uniquely defined number (ÌìUìM(^) £ [0, co] satisfying
d i m M ( © ~ i % ) = £ ~ i d i m M ( ^ ) , d i m M ( ^ ) = dim M (K') iff M and W are
isomorphic M-modules, and dim^(L 2 (M, tr)) = 1 where L2 (M, tr) is the Hilbert
space obtained from M by completion with respect to the inner product (a, b) =
tr(6*a). This number dimjtf(W) is the coupling constant of Murray and von
Neumann [MvN] and was originally defined as tTM{P[Mfç])/toM'{P[Mt]) where
f ^ 0 is an arbitrary vector in )1 and P[Mç\ denotes the orthogonal projection
onto the closure of the subspace M£ Ç #. (The other symbols have their obvious
" meanings, M' being the commutant of M.) This definition presupposes that M'
is also a Hi factor. If this is not so, one puts dim.M{M) = oo.
It is important to note that, for any Hi factor M, {dmiM(#) | M a Hilbert
space over M} = [0, oo]. This is a formulation of the "continuous dimensionality"
that so fascinated Murray and von Neumann. It is also important in Connes's
noncommutative integration theory [CI] where real-valued Betti numbers are
associated to foliated compact manifolds with invariant transverse measure.
The elementary example of this paragraph will serve as motivation for the
definition that follows. If T is a discrete group, all of whose (nonidentity) conjugacy classes are infinite (an i.c.c. group), and To is an i.c.c. subgroup of T, then
the von Neumann algebras UT and [/To on Z 2 (r), generated by left translations
by the appropriate group elements, are both Hi factors, and the coset decomposition of T over TQ shows immediately that dim[/r 0 (/ 2 (r)) = [T: To]. Note also
that Z2(r) is the same as L2(UT) so we have
dmW£2(t/r)) = [r : r 0 ].
(i)
© 1987 International Congress of Mathematicians 1986
939
940
V. F. R. JONES
Note that the left-hand side of equation (1) only involves Hi factors. So we
are led to make the following definition.
DEFINITION 2. If N Q M are Hi factors (with the same identity), define
[M: N], the index of N in M , by
[M:
N]=dimN(L2(Mitr)).
With this definition one may interpret the following result of M. Goldman [G]
as being an analogue of the fact that a subgroup of index 2 of a group is normal.
THEOREM 3 [G]. IfNQM
are as in Definition 2 and [M: N] = 2, then
there is auE M with uNu* = JV, u2 = 1, and M = N © Nu.
Both the examples from To < T and Goldman's theorem give the impression
that the index [M : N] is a discrete object, but its definition suggests an arbitrary
real number between 1 and oo. The next result shows that neither impression is
correct.
THEOREM 4 [Jl]. (a) / / [M: N] < 4, there is an n e Z, n > 3, with
[M: N] =4cos 2 7r/n.
(b) If r = 4 cos2 7r/n, n as above, or r G R, r > 4, there is a pair N Ç M of
III factors with [M: N]= r. One may suppose N and M hyperfinite.
The appearance of these numbers 4 cos2 7r/n was not at all expected a priori.
Note that 1 = 4cos 2 ?r/3, 2 = 4COS 2 TT/4, and 3 = 4COS 2 TT/6. The first "new"
index value is 4 cos2 7r/5 ~ 2.6180339, the square of the golden ratio.
We would like to add that, as first pointed out by Connes, the continuous
variation of [M : N] may be illusory since the examples that realize the numbers
between 4 and oo are "reducible" in the sense that Nf CiM contains elements
other than scalars. If one imposes the irreducibility condition i V ' n M = C, then
the smallest known value of [M : N] (greater than 4) is 3 + \/3. The current
feeling is that there should be a gap between 4 and the next irreducible index
value.
2. Proof of Theorem 4(a). We shall outline a proof of Theorem 4 which
makes a connection with Coxeter-Dynkin diagrams. The proof relies on an analysis of inclusions of finite-dimensional von Neumann algebras.
A finite-dimensional von Neumann algebra is semisimple, so is a direct sum
of full matrix algebras over C. We shall represent such an algebra by a finite set
-of vertices^correspondingtorthe^simple directsurnmandsptogetherwith^integers—
giving the size of the matrix algebras. For instance, C©Ms(C)©M2(C) would be
1 3
2
represented by • - •. With this convention a pair A Ç B of such algebras can be
represented by a graph (Bratteli diagram) where the number of edges connecting
a vertex of the smaller algebra to a vertex of the larger one has the obvious
"multiplicity" meaning. For instance, the diagonal inclusion of Af2(C) ®Ms(C)
SUBFAGTORS OF Hi FACTORS
941
in M ö ( C ) would be represented by the diagram
5
and the inclusion
of M2(C) in MQ(C)
would be represented by the diagram:
i
In general the matrix A^ is defined as the matrix whose rows are indexed by the
vertices of A, whose columns are indexed by those of B, and whose entries are
the multiplicities.
The other ingredient of the proof of Theorem 4 is the iteration of a certain
basic construction which is made as follows: given N Ç M finite von Neumann
algebras with the same identity, and a faithful normal trace tr on M , one lets N
and M act on L2 (M, tr) as before and one considers the von Neumann algebra
JN'J where J is the extension to L 2 (M, tr) of the * operation. Since M = J M'J
[D], one has N Ç M Ç JN'J. If there are several algebras present we will use
JM to denote the J on L 2 (M,tr).
In the case that TV and M are finite factors, one has
PROPOSITION 5.
(i) JN'J
is a Hi factor <& [M: N] < oo.
(ii) If (i) is satisfied, then (a) [JN'J: M] = [M: N]\ (b) N' DM is finitedimensional] (c) [M: N]> dim(iV' DM).
In the case that M is finite-dimensional, the following result holds.
PROPOSITION 6. Affl'J = (A%y (independent of the trace). (Here we have
identified the center of N with that of N' and so JN'J, which allows us to make
the correspondence between rows of kj^ J and columns of Ajyf and vice versa.)
To illustrate Proposition 6, suppose N Ç M were given by
942
V. F. R. JONES
Then N Ç M Ç JN'J would have the diagram
3
\
2'
4
/
33
\
1
1
Given a subfactor N Ç M of finite index, the next step in the proof is to
iterate the above construction to obtain a tower Mi of Hi factors with MQ = N,
Mi = M, and Mi+i = JMìM-^JMH
where JM{ is the involution on L 2 (Mi,tr).
By finite dimensionality one obtains a tower dMi = N'nMi of finite-dimensional
von Neumann algebras with corresponding matrix A^ = AdMi+1.
The proof of
Theorem 4(a) will follow easily from the following result.
PROPOSITION 7. (i) There is an isomorphism of JdMi(dMi-i)'JdM{ onto a
two-sided ideal of <9M^+i which gives a containment of (Ai)1 as a submatrix of
Ai+i-
(ii) dimöM; = trace((nl=i A*)ttlU A,)*).
(iii) IfAk
= (Afc-i)*,- then Ap = (A p _i)* for allp>k
and [M: N] = \\Ak\\2.
It follows from Proposition 7(i) that ||Ai|| is a nondecreasing function of i,
and then from 7(ii) that dimM^ grows asymptotically at least as fast as ||A^||2A:
for any i. But [Mk: N] = [M: N]k, so by 5(c) ||A;||2 < [M: N] for all i. If
[M : N] < 4, the A^ are then a nondecreasing sequence of 0-1 matrices of norm
< 2. By [Bo, G H J ] the possible values of these norms are precisely the set
{2 cos7r/n | n = 3 , 4 , . . . } . So by strict monotonicity of the norm there must be
a k for which A& = (Afc_i)*. By 7(iii) we are through.
In fact, one obtains more information from this proof than just the values
4 cos 2 ir/n. It follows from [GHJ] and the connectedness of the Bratteli diagram
that if k is such that A& = (A^-i)*, then Ak must be the adjacency matrix for
a bipartite structure on one of the Coxeter-Dynkin diagrams An, n > 3, Dn,
n > 4, E6, E7, or Eg, and then [M: N] = \\Ak\\2 = 4cos 2 7r/r where r is the
Coxeter number of the diagram. For instance, one might have the inclusion
(dM)2 Q (dMs) given by the Bratteli diagram
which corresponds to the Coxeter-Dynkin diagram EQ, [M: N] = 4cos 2 7r/12.
The question of which diagrams arise from subfactors is interesting. Ocneanu
has added to dM a "comultiplication" coming from the J^'s which completely
943
SUBFACTORS OF III FACTORS
axiomatizes their structure even in index > 4. He claims that D& is impossible,
whereas An and D4 are relatively easy to construct. Bion-Nadal has shown that
the construction of the next section realizes E$.
Proof of Theorem 4(b). Let us first dispose of the case r > 4. The hyperfinite Hi factor R has fundamental group = R [D], so choose a projection p G R
with tr(p)" 1 + tr(l - p)""1 = r and an isomorphism 0: pRp —• (1 — p)R(l — p).
Let M = R and N = {x + 0(x) \ x e pMp). One checks [M : N] = r. Notice
though that this proof relies on the fundamental group. Pimsner and Popa [PP]
have shown that for Hi factors with Connes' property T (and hence countable
fundamental group—see [C2]), the set of index values for subfactors is countable!
We now suppose r = 4 cos2 n/n, n = 4,5,6,
Let A C B be an inclusion of
finite-dimensional von Neumann algebras whose Bratteli diagram is a CoxeterDynkin diagram with Coxeter number n (to obtain 4 cos2 ir/n). Let q — e2ni/n.
FIGURE
13
There is a unique trace tr on B which admits an extension to
tr(xgs) = ztr(x)
îOTXEB,
JBA'JB
satisfying
(8)
where QB '• B —• B is q on A and —1 on the orthogonal complement of A, and
z=(q + l)-\
Iterating this process as before, one obtains a tower Bi of C*-algebras together
with elements gi = CB{ which define an endomorphism $ : \ji Bi —• |J^ Bi by
§(x) = lim/c_>oo(0i02 • • • 9k)%(gi92 • • • 9k)*- Then $ preserves the trace on |J; Bi,
so applying the GNS construction one obtains a Hi factor M from |Ji B% a n d
$ : M —• M. One may show that [M: $(M)] = 4cos 2 w/n.
Hecke algebras and braids. In the proof of Theorem 4(b) we used a sequence of elements gi in the tower construction. It is easy to see that they satisfy
the relations
9Ì
= {Q-
l)9i + q,
9i9i+i9i = g%+i9%gi+ii
9i9j = 9j9i
for \i - j \ > 2.
(9)
(10)
(H)
944
V. F. R. JONES
If q is a prime power, relations (9), (10), and (11) are known [Bo] to present
the commutant of G = GL n (F g ) acting on the complex-valued functions on G/B,
B being the subgroup of upper triangular matrices. This is called the Hecke
algebra H(q,n-\-1) of type An\ the name also applies to the algebra presented
by (9), (10), (11) for any value of q. Thus the Hecke algebra is represented
(not faithfully) in the tower. When q = 1, relations (9), (10), and (11) present
the group algebra of the symmetric group, and one may deduce much of the
structure of the Hecke algebra from that of the symmetric group.
Relation (8) suggests that there might be traces on H(q, n + 1) defined by
tr(l) = 1 and
tr(a;0n) = ztr(x)
for x G H(q, n)
(12)
for arbitrary values of z. This was proved by Ocneanu (see [HKW, J 2 , W]) who
also determined the values of (q, z) for which the Hecke algebra admits a von
Neumann algebra structure for which tr(a*a) > 0. Wenzl calculated the indices
of the corresponding subfactors defined using limfc->oo g\gi'-'Qk as in the proof
of Theorem 4(b). See [W].
Artin showed that relations (10) and (11) present the braid group Bn on n
strings where the n — 1 g^s correspond to the n - 1 o^'s as in Figure 13.
<* € B3
(/ )
^
FIGURE
a
14
Thus for q ^ 0 there is a representation 7r of Bn in H(q, n) defined by nfa) =
gi. At present there is no geometric interpretation of this representation though
it does contain (as a direct summand) the Burau representation which can be
lïeHïïœdn^m^^^
cover of the disc minus n points. It is not known whether or not 7r is faithful for
n > 3. The special values q = e±2W™ correspond to the values for which ir may
be unitarized (though one must take a quotient of the Hecke algebra—see [W]).
The definition of $ in the proof of Theorem 4(b) was suggested by the wellknown braid group relation (G\-- • 0n)o~i(ai • • • a n ) _ 1 = 0^+1 for i < n.
SUBFACTORS OF Hi FACTORS
945
Braids and links. A braid a G Bn may be closed to give the oriented link
à as in Figure 14.
Any tame oriented link in Ss may be obtained in this way (Alexander) and the
equivalence relation on braids defined by isotopy of their closures was described
algebraically by Markov. (For a general reference see [Bi].) It is generated by
types I and II Markov moves which are the following:
type I : a G Bn & ßaß"1
G Bn,
type II : OL G Bn <* aa*1 G B n + i .
One may consider the function of q and z on the disjoint union of the braid
groups defined by a —• tr(7r(û;)) where tr is defined by relation (12). This
function is invariant under type I Markov moves, and because of the similarity
of (12) and type II Markov moves, it may be renormalized to give a link invariant.
It is convenient to change variables by putting A = (1 — q + z)jqz. One then has
the result that
Xa{q X) =
'
(Jx(l-q)T * (V/X)6tr(7r(a))
depends only on a where the image of a G Bn in Z under abelianization is e.
Relation (9) translates into the fact that if L+, L_, and Lo are three links
with projections differing in only one crossing where they are as in Figure 15,
then
-^xL+-^qxL_
=
(^-^)xLo.
'+
->
graph
link
F I G U R E 17
(16)
946
V. F. R. JONES
The existence of such a link invariant was proved in [F+], Ocneanu using the
approach outlined above. Note that if L is the unlink with c components,
/
An — i
i
c-1
\VX(l-q)J
The invariant XL has been found to be quite powerful. It contains the Alexander polynomial via the specialization AL(£) = Xi,(t, 1/t), but is also very sensitive to mirror image asymmetry. There is no known nontrivial knot with the
same X as the unknot. It is easy to see that X can be made into a two variable Laurent polynomial PL via the popular substitution m = i(y/q - 1/^/g),
; = i/y/Xy/q. The invariant is multiplicative under connected sums and is unaltered if all orientations of a link are reversed. But it is very sensitive to reversal
of the orientation of a single component.
There is another specialization of XL which is proving to be of particular
interest. It is VL(£) = Xi,(t,t) and has the remarkable property that it only
changes by a power of t if the orientation of any component of a link is changed.
V L ( 0 comes from precisely the Hecke algebra quotient given by the proof of
Theorem 4(b), and was actually noticed before the discovery of XL (see [J3]).
Kauffman has given an explicit formula for VL(£) (a "states model") which
can be calculated from an arbitrary link projection. In the case of an alternating
link, this exhibits VL as a specialization of the Tutte polynomial of the graph
associated to the checkerboard shading of the link projection (Figure 17).
These ideas enabled Kauffman, Murasugi, and Thistlethwaite to develop powerful methods for handling alternating projections of a link—in particular, solving some century-old problems of Tait.
In another line of development, Figure 15 was extended to include the "Loo"
caseQC- This is not orientable and Brandt, Lickorish, Millett, and Ho defined a
polynomial QL(X) of unoriented links by QL++QL= Z ( Q L 0 + Q L O O ) - Kauffman
improved on this by first defining an invariant R of regular isotropy (the move
Ä.<=>— is not allowed) by the same formula as for Q and R(SL) = aR(-),
R(9>.) = a~1R(-). If one then gives the link diagram an orientation, one defines
w L
FL(CL,X) = a~ ( ÌR(a,x)
which is a link invariant (where w(L) is the sum of
the signs of the crossings).
REFERENCES
[Bi] J. Birman, Braids, links and mapping class groups, Ann. of Math. Studies, No. 82,
==Prineeton=Univ^Pressy=Princetony=N=r=jT|=1974T=
[Bo] N. Bourbaki, Groupes et algèbres de Lie. IV, V, VI, Hermann, Paris, 1968.
[Cl] A. Connes, Sur la théorie non-commutative
de l'intégration, Lecture Notes in
Math., vol. 725, Springer-Verlag, Berlin-New York, 1979.
[C2]
, A Hi factor with countable fundamental group, J. Operator Theory 4 (1980),
151-153.
[D] J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien, Gauthier-Villars, Paris,
1957.
SUBFACTORS OF Hi FACTORS
947
[F-f] P. Freyd, D. Yetter, J. Hoste, W. Lickorish, K. Millett, and A. Ocneanu, A new
polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 183-312.
[G] M. Goldman, On subfactors of factors of type Hi, Michigan Math. J. 7 (1960), 167172.
[GHJ] F. Goodman, P. de la Harpe, and V. Jones, Coxeter graphs and towers of algebras
(to appear).
[H] U. Haagerup, Classification of hyperfinite von Neumann algebras, Proc. Internat.
Congr. Math. (Berkeley, Calif., 1986).
[HKW] P. de la Harpe, M. Kervaire, and C. Weber, On the Jones polynomial, Enseign.
Math, (to appear).
[Jl] V. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25.
[J2]
, Hecke Algebra Representations of braid groups and link polynomials, Univ.
Calif., Berkeley, Preprint.
[J3]
, A polynomial invariant for knots via von Neumann algebras, Bull. Amer.
Math. Soc. (N.S.) 12 (1985), 103-111.
[MuN] F. Murray and J. von Neumann, On rings of operators, Ann. of Math. 37 (1936),
116-229.
[PP] M. Pimsner and S. Popa, Sur les sous-facteurs d'indice fini d'un facteur de type
III ayant la propriété T, C. R. Acad. Sci. Paris 303 (1986), 359-361.
[W] H. Wenzl, Representations of Hecke algebras and subfactors, Thesis, Univ. of Pennsylvania, 1985.
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