Fields Institute Communications
Volume 13, 1997

Bimodules,

Higher Relative Commutants and the Fusion
Algebra Associated to a Subfactor
Dietmar

Bisch

Department of Mathematics
University of California,

Berkeley

Berkeley, CA USA 94720

Abstract. We prove in this paper that the tensor product of reduced
bimodules associated to a subfactor can be recovered as a product ofcertain projections in the higher relative commutants associated to thesubfactor. After giving an elementary introduction to bimodules of II;
factors and their relative tensor product, we prove various formulas relating the representations of the Jones tower coming from different k-stepbasic constructions and show that the natural shift on the higher relative commutants, defined by two consecutive modular conjugations of
the tower, can be computed in terms of orthonormal bases and the Jones
projections e;. We give a detailed account of how the principal graphsof a subfactor can be recovered by calculating dimensions of intertwiner

spaces of certain (reduced) bimodules and show that each vertex of theprincipal graphs represents a unique reduced bimodule. Then we define the (full) fusion algebra associated to a subfactor and prove that
this fusion algebra can be calculated by computing products of certain
projections in the higher relative commutants of the subfactor. Explicit
formulas for these products are given. Finally we discuss reduced subfactors and give a procedure to compute the fusion algebra. of a subfactor
in those situations, when the principal graphs are simple. We show the
relation to reduced subfactors and discuss in detail the example of a
subfactor with principal graph Fs to illustrate the general algorithm.

’

Introduction
In this paper we prove various facts about bimodules associated to a subfactor, someof which are known to experts, but whose proofs are not readily available in theliterature. We refer to (Connes [1994], Sauvageot [1983]), (see also Ocneanu [1988],
Ocneanu [1991(a)], Popa [1986]) for some of the original papers on bimodules,
and to (Anantharaman-Delaroche [1993], Denizeau and Havet [1993(a)] [1993(b)],Ocneanu [1991(b)], Sunder [1992], Yamagami [1993]), for additional material (see1991 Mathematics Subject Classification. Primary: 46L10.
Supported by a Heisenberg fellowship and NSF-grant DMS-9307234©

131997 American

Mathematical Society

14

DietmarBischalso Bisch [1994(b)], Bisch and Haagerup [1996], Evans and Kawahigashi [1996],
Haagerup [1994], Goodman and Wenzl [1990],Izumi [1991], Jones and Sunder [1996],
Longo [1989][1990],
Popa [1994],Sunder and Vijayarajan [1993], Wassermann [1995],
and Wenz! [1988]). The above list of citations is by no means intended to be complete.
Here is a detailed description of the sections below. We begin in Section 1 with
an elementary introduction to bimodules associated to a pair of II; factors, define
Connes’ relative tensor product of two such bimodules and prove that this bimodule
tensor product is associative (Proposition 1.12). We discuss bimodule intertwiners
and state the Frobenius reciprocity theorem (Theorem 1.18). Finally, we show thatthe direct sum of bimodules is compatible with the bimodule tensor product. Mostof the material in this section can be found in (Connes [1994], Sauvageot [1983],
Ocneanu [1988] and [1991(a)], Popa [1986], see also Sunder [1992]).
Section 2 contains material that is needed in Sections 3 and 4 to establish thebimodule interpretation of the principal graphs associated to a subfactor and to
identify the tensor product of reduced bimodules as a product of certain projectionsin the higher relative commutants of the subfactor. We discuss the representationsof the tower of II; factors associated to a subfactor N C M coming from the k-stepbasic constructions, i.e., the basic construction for the inclusion N C Mj (resp.
M C My) and prove various formulas relating them (Lemma 2.4, Propositions 2.2,
2.5). Next, we discuss the natural shift on the higher relative commutants. We
show that the “spatial” definition using the modular conjugations is the same as
the “abstract” one using the Jones projections e; and orthonormal bases (Theorems
2.6 and 2.11). We do this for an inclusion of II; factors of the form A C B Cc By,
which we apply then to A = N, B = M,, (resp. A = M, B = M,) in Section
4. We give explicit formulas of the J - J-map and the shift in terms of the e;’s
and orthonormal bases, which are useful for the computation of tensor productsof reduced bimodules associated to a subfactor. We also discuss briefly Ocneanu’s
Fourier transform and give a simple application to illustrate the usefulness of thismap.
In Section 3 we show that the higher relative commutants associated to a
subfactor can be viewed as spaces of N-N resp. N-M resp. M-N resp. MM bimodule intertwiners. After recalling the definition of the principal graphsof a subfactor in some detail, we prove that each even (resp. odd) vertex of theprincipal graphs represents in a unique way a reduced N-N (M-M) (resp. N-M(M-N)) bimodule, i.e., a bimodule of the form pL?(M,,), where p is a projection in
N'NMapi1 (resp. M'NMay, NN May, M'N Ma, 1). This is done by writing down
an explicit isomorphism between these bimodules. We identify the contragredient(or conjugate) of a reduced bimodule as another reduced bimodule by calculating
the projection in the higher relative commutants to which this conjugate reduced

bimodule is associated (Proposition 3.11). We give then the definition of the (full)
fusion algebra associated to a subfactor, including all possible bimodule products,
i.e. the products of N-N (resp. M-M) bimodules with themselves (called the evenpart of the full fusion algebra, or simply the fusion algebra) and the products ofN-M (resp. M-N) with M-N (resp. N-M) bimodules, called the odd part (where
we form the relative bimodule tensor product over N resp. M of course). We list
briefly the properties of the structure constants (which are dimensions of spaces ofbimodule intertwiners) appearing in this definition.

Bimodules

andHigherRelative

Commutants

15

We prove in Section 4 that the edges (including multiplicities) of the principal
graphs can be recovered by computing dimensions of certain intertwiner spaces of(reduced) bimodules, thus, together with the results of Section 3 (the identificationof the vertices of the principal graphs as reduced bimodules), establishing the bimodule picture of the prinicipal graphs as “principal” fusion rule matrices, due to
Ocneanu. This uses some of the results of Section 2, in particular various formulas
involving the representations of the tower of II; factors associated to a subfactor
coming from different k-step basic constructions. We have to work spatially all
the time, since reduced bimodules are obtained by an action of a projection in acertain higher relative commutant on a Hilbert space L?(M}) (Propositions 4.1 and
4.3). In Theorem 4.6 we prove that the bimodule tensor product of two reduced
bimodules {over N or M), associated to projections p and q say, can be calculated
as a product of projections in the higher relative commutants, involving the modular conjugations and the shift from NN My,41 to My, ; N My, 3 for example.
Roughly, we obtain the tensor product of these two reduced bimodules by fixing
p and shifting ¢ far enough in a higher relative commutant so that the shifted ¢commutes with p and consider then the reduced bimodule associated to this new
projection. This reduced bimodule turns out to be the bimodule tensor productof the two reduced bimodules associated to p and q. We do this for all cases in
Theorem 4.6. Combining this theorem with the explicit formulas that we proved in
Section 2 for the shift and the J - J-map(in terms of e;’s and orthonormalbases),

we obtain an explicit procedure, that allows us to calculate the fusion algebra of
a subfactor whenever the higher relative commutants are well understood. Appli-

cations of this to the subfactors in (Bisch and Jones [1995]) for instance will bepresented elsewhere.
Section 5 contains a discussion of the basic construction of reduced subfactors
and their relation to reduced bimodules. We give a simple method to compute the(full) fusion algebra associated to a subfactor by solving matrix equations and thencalculating products of the resulting matrices. When the principal graphs of thesubfactor are not too complicated (for instance if they contain at most triple points),
the fusion algebra can be computed completely in this way. For instance, all the

calculations in (Bisch [1994(b)]) were performed using the method presented in thissection. We show how the principal graphs of the reduced subfactors associated to
an inclusion of II; factors can be determined from the fusion algebra and remarkthat their fusion algebra can be read off the fusion algebra of the original inclusion.
Let us point out that this is rather obvious, when we use the endomorphism picture

for bimodules (see for instance Longo [1989] and [1990], Izumi [1991]) (properly
infinite case). However, to keep the paper self-contained, we stay in the II; setting.
To illustrate the method presented in this section, we discuss in detail the example
of a subfactor with principal graph Eg, calculate the full fusion algebra associatedto such a subfactor, and give a full discussion of the associated reduced subfactors.
1
B°?

Preliminaries on bimodules

Let A and B be II; factors.

We denote by B°P the opposite algebra of B, i.e.

=

the

B

as Banachspacesandmultiplication

is definedbyb; - by=baby,

bi, bo € B. B°P is of course a II; factor. Recall that an A-B bimodule H is bydefinition a pair of commuting normal (unital) representations of A and B°P on theHilbert space
H. We usually denote the left action of A and the right action of B

Dietmar

16

Bisch

eH.
(which is by definition the left action of B°?) by a-£-b, where a € Abe B,
The notion of (unitary) equivalence of bimodules is recalled in the next definition.
Definition 1.1 Let A and B be II, factors and let H and K be two A-B
bimodules. We say that H and K are (unitarily) equivalent if there is a unitaryuw: H— K such that u(a-€-b) =a-u(€)-b, foralla € A, be B, { € H. We write
AHg = sKpg. Furthermore we denote by

Homu.p(H,K) = {T € B(H,K)|T(a-£:b) = a-T(£)-, for alla € A, b€ B, { € H}
the space of A-B

intertwiners from H to K.

If H = K, we write Homyg(H) forH).
Homa.p(H,
Observe that Homa. g(H) = A’ N (B°?)’ Nn B(H) is a von Neumann algebra.
Recall that an A-B bimodule H is called irreducible, if A’ N (BP?) nN B(H) = C.
Suppose that (B°P)’ N B(H) is again a II; factor (i.e. the coupling constant of B°”
on H is finite), then one defines the index of the A-B bimodule H to be the Jones

index of the subfactor A C (B°PY N B(H). If [(B°P)’ : A] < oo, then Homy.5(H)is a finite dimensional C*-algebra (Jones [1983]). We will sometimes say that the

two A-B bimodules H and K are isomorphic (as A-B bimodules), which meansthat there is a bijective A-B intertwiner 7 from H to K. The unitary in the polar
decomposition of T implements a unitary equivalence as defined above. We will
mostly be concerned with equivalence classes of A-B bimodules.
Let A C B be an inclusion of II; factors with finite Jones index and denote bytr the trace on B. As usual we let L2(B) be the completion of B in the norm | - [|2induced by the trace, in other words, L?(B) is the GNS Hilbert space with respect
to tr. Then 1 = 1p € L%(B) is the cyclic and separating vector, and we write b

for b(1), i.e. b € B viewed as a vector in L?(B) is denoted by b. As usual we let
J : L*(B) — L?*(B) be the conjugate linear isometry obtained by extending themap b — b* to all of L2(B) by continuity. L?(B) is a left B-module (hence a left
A-module),

where B acts by left multiplication.

L?(B)is also a right B-module(and hence a right A-module) with the action £ -b = Jb*J(£), b€ B, § € L*(B).
L?(B) becomes in this way a B-B (resp. A-B, B-A, A-A) bimodule.
Lemma 1.2 Let A, B be II; factors and let m; : A — B(H;), ¥; : B® —B(H,), © = 1, 2, be (nonzero) normal representations of A and BP such thatmi(A) C (BP), i = 1, 2. We define two A-B bimodules aH1p and sHap by

a-&-b=m(a)i(b)E, forallEi) If aH1p

=a

Hap

as A-B€ Hi, a € A, be B®, i=1, 2.
bimodules,

then there is a surjective *-1somorphism

8 : 1 (BP) — 4o(B°P) such that 8(m1(A)) = m2(A).

Furthermore, the Murray-

von Neumann coupling constants of ¥1(B°P) on Hy and of 12(B°P) on Hy coincide.
ii) Assume that the Murray-von Neumann coupling constants satisfy dim, (gery Hi
Suppose that there is a surjective x-isomorphism 0 := dimy, (gory Ho < 00.

1 (BP) — 13(B°P)’ such that 6(my(A)) = m2(A). Then there are automorphisms
Y € AutA, ¢ € AutB and a unitary uv : Hy — Ha, such that u(m (a)ya(b)€) =m2(w(a))2(d(0))(w(€)), for all € € Hy, a € A and b € B®. Thus 4H p is equivalent to aH p, with A and B actions twisted by automorphisms of A resp. B.

Proof If 4H, 5 = sHap, then by definition of unitary equivalence of bimod-

ules, there exists a unitary u : Hy — Hj such that u(m(a)y1(b)¢) = m2(a)¥2(b)u(§),
foralla€ A, be B°P, ¢ € Hy. Thus § = Adu does the job.

andBimodules

HigherRelative

17

Commutants

Suppose that ii) holds. The condition on the coupling constants implies that
6 is spatial, i.e. @ = Adu, u : Hy — H» a unitary. Hence Adu is a surjectives-isomorphism from 1; (B°P) onto 95 (B°P) and therefore ui; (b)u* = 1h2(¢(b)), forall b € B°P and some automorphism ¢ € Aut B°® = Aut B.

Similarly, there is a[J

¥ € Aut A such that um (a)u* = m2(¥(a)), a € A. The rest is clear.

Note that the lemma shows that the Murray-von Neumann coupling constants
and the subfactor associated to a bimodule determine the bimodule only up to

If we consider the B-B bimodule L*(B) with action z-¢ -y =automorphisms.

zJO(y)*J(€), =, y € B, £ € L*(B), 6 € Aut B an outer automorphism, we see thatthe automorphisms are indeed necessary.
We will define in Section 3 the fusion algebra associated to a subfactor anddescribe in Section 4 its multiplication law in terms of projections in the higher relative commutants. To understand this multiplication, let us start with recalling thedefinition of the relative tensor product (Connes [1994], see also Sauvageot [1983],
Popa [1986], Ocneanu [1991(a)]).

Definition 1.3 Let A be a II; factor with trace tr and let H be a left A-module.
A vector £ € H is called a (left A-) bounded vector in H if there is a constantc= c(€) such that ||a€|| < cllall2, for all a € A, where ||all2 = tr(a*a)z.

We denoteby HO the set of left A-bounded vectors in H. Similarly for right A-modules
left A°P-modules) and right A-bounded vectors.

(i.e.

Remark 1.4 It is easy to see that AH® C H® and A’H® C H®. From this one
deduces immediately that H® is dense in H.
Proposition 1.5 Let A C B be an inclusion of II, factors with [B: A] < 00.
Consider the B-B (resp. A-B, B-A, A-A) bimodule L*(B) as defined above. Thenthe left (right) A-bounded vectors and the left (right) B-bounded vectors coincideand are given by B.
Proof Let us prove that the left A-boundedB.

vectors in L?(B)are given by

If b € B, then |lab|22p, = tr(abb*a*) < [bl|*[all3, for all a € A.

Thus

B c L%(B)°, the set of left A-bounded vectors. Conversely, let £ € L?(B)°. Then
there is a constant c¢(€) such that |laéllz < c(€)l]lallz, for all a € A. Thus Re :A — L2(B), R¢(a) = a€, extends by continuity to a bounded linear map L(A) —L?(B).

Let A C B C°® B; be the basic construction and consider the composition

Ree : L2(B) — L*(B). Note that a1R¢(d) = mag = Reai(d), a, a1 € A, sothat Ree € A’ N B(L?(B)). But B; = JA'J and hence there is an z € By with
Ree = Jz*J € B(L*(B)). By (Pimsner and Popa [1986]) there is a unique y* € Bwith z*e

=

y*e.

Thus

Ree

=

Jz*Je

=

Jz"eJ

=

Jy*Je,

which

implies that

Ree(l) = RD) =¢= Jy*e(1) = 4, so that indeed & € B as desired.— The same
proof (with e = 1) shows that the set of left B-bounded vectors in L?(B) is also
given by B. From this it follows immediately that the set of right B-bounded (resp.

right A-bounded) vectors in L*(B) is again B.
Remark

1.6

0

The following slightly more general statement (which can be de-duced from 1.5) holds: If H is an A-B bimodule with finite indez, then the left
A-bounded vectors in H and the right B-bounded
instance Sunder [1992], II. Proposition 4).

vectors in H

coincide(see for18

DietmarDefinition 1.7 Let A be a II; factor and H

a left A-module,

Bischlet £, n € HC.

We let {(£,m)4 € A be the operator in A defined by(z€,m)g = tr(z(&,n)a),

for all z € A.

Note that (£,71)a exists and is unique by the Radon-Nikodym

theorem(notethat |(z*z€,m)| < c(€)c(n)||z||2). The uniqueness of the Radon-Nikodym derivative
implies immediately the following lemma.

Lemma 1.8 Let A be a II, factor and H a left A-module.
a € A.

Let ¢&, n € HY,

Then we havei) (-,-)a is C-linear in the first variable.

ii) (€,ma=n&%iii)(at, naiv)(€,€)a

=
>

all,

ma.

0.

Observe that (£,an) a = (£,n) aa*, which is an easy consequence of ii) and iii). Furthermore, it is easy to show that if n;,...,n, € H?, then (nism) a) 1<i <n €EA®
M,,(C) is a positive operator (considerA® M,,(C) on H" and identify ((n;,7;) A), ;as the Radon-Nikodym derivative ((n1,...,m)%(M1, ---,7n)") agm, (cy, Which is
positive by 1.8 iv), applied to the left A ® M,(C)-module H").
Similarly one can define a right Radon-Nikodym
the next definition.
Definition 1.9

Let A

be a I, factor and H

A°P_module), let £, n € H°.

derivative, which is done ina right A-module(nz,&) nm = tr(z(€,n)%),a leftforall ze A

Let us collect the properties of the right Radon-Nikodym
Lemma 1.8 for the left Radon-Nikodym derivative.

Lemma 1.10 Let A be a II, factor and H a right A-module.
a € A.

(i.e.,We let (£,m)% € A be the operator in A defined byderivativeas inLet ¢, n € H?,

Then we havei) (+,

is C-linear in the second variable.

if) (€,ma = (na)
iii)

(la,

ma

=

a*{€,m%-iv) (§,6)3 = 0.
Observe that (£,ma)% = (£,1)%a, which follows immediately from ii) and iii). As
above, we have that if n1,...,7, € H°, then (6:10) 1 <i jn €AQ M,(C)is a
positive operator (use again an amplification trick).
Co
If A C Bis an inclusion of II; factors, then L?(B) is a natural A-B bimodulewith the action a -¢-bseen above.

=

aJb*J(£),

preserving conditional expectation.
and(b1,b2)5a € A,b € B,¢ € L?(B),as we have

We have that (b1,b3)4 = E4(b1b}), where E4 : B — A is the trace=

E4(b1bs),bi,bySimilarly, L?(B)is a natural B-A

bimodule€ B.

We are now ready to recall the definition of the relative tensor product of two
bimodules. Let A, B, C be II; factors, let H be an A-B bimodule and K a B-C

Bimodules

andHigherRelative

Commutants

19bimodule and denote by H® (resp. K°) the right (resp. left) B-bounded vectorsin H (resp. K). Consider the algebraic tensor product H° ® K° and define forYi®m, 2&5®n; € Ho

K°

Seem)
i
J

gon)

=>

(&mn})

5, Eu,(1.1)

4dwhere (n;,7}) 5 is the Radon-Nikodym derivative of z € B — (zn;, n}) with respect

to trp as above (definition 1.7) and (-,-),, is the inner product on H.

Similarly,

let (£:,€;)% be the Radon-Nikodym derivative of z € B — (57,&:) with respect totrp. Then we have for ¢;, £; € HO, ni, 7f € K°

Elm,Ea= tre ((mi, 15) BE), €)%)= tra ((&}, &) 5mnj) B)= ((§,&)
Bm mj) K
and thus we have on H° ® K°

Oem,
Y gon) =>i

J((& 5mm).

(1.2)

J

We leave it to the reader to check that (-,-) as defined in (1.1) or (1.2) is a(possibly non-degenerate) inner product on H® ® K°.

Note that (1.1) is actually

defined on H ® K° and (1.2) on H° © K.
We let Ni.

y be the null space of this inner product and define H ®g K to be

the completion of H° ® K°/N.., in the norm induced by (-,-) on H® ® K°/N;. ,(Connes [1994], see also Sauvageot [1983]).
Definition 1.11 Let A, B and C be II; factors, thenH®p K as defined above
is called the bimodule tensor product (or the relative tensor product over B) of theA-B bimodule sHg and the B-C bimodule gKc. We denote the equivalence classof the the vector 3, & ®n; in HOO K°/N.yby >, & ®@p mi or [3 & @ mi).

It is easy to see that H®p K is an A-C bimodule: If welet ). &;®n; € HGK?C,

then we have an A-C action viaa- (3, & ®m) -c =), (a&;) ® (nic). To show that
this defines a left A-action and a right C-action on the relative tensor product,

one proves that |la-Y", & ® mill < [alll 35; & @ mill and similarly for the right Caction.

This inequality is shown by using an amplification trick and the remark

about positivity of ((m:,7;)4), ; after Lemma 1.8 (resp. Lemma 1.10). We leave thesimple details to the reader. Since this induces in a natural way an A-C bimodule
structure on the relative tensor product H ® g K, we denote this A-C bimodule by
AH ®p Kc.

20

DietmarConsider the quotient map ¢ : H*

Bisch

© K° — 4H ®p Kc, defined by ¢(5>_, & ®

ni) =»; & ®p mi. ThenIY 6 ®pmlkesn

<O

&®mY

&®m)

= (&mom)B:&i)y1]<> l&i(masms) Bla12%]< (maxi; (mim)1) D_II&ill zr lI; ari,jThus ¢ extends by continuity to a map, still denoted by ¢ : H © K 0» AH ®p
Kc. Similarly, using (1.2) instead of (1.1) in the above estimates, we see that ¢extends by continuity to a map ¢ : H 0 K —» 4H ®p Kc. In particular, we

get that ¢(H © K°) and ¢(H° © K) are dense in 4H ®p Kc and we could havedefined 4H ®5 Kc equally well by using (1.1), defined on H © K° and taking
the separated completion as above, or by using (1.2) on H° ® K and taking theseparated completion. As we have just shown, all three ways of defining 4H ® Kc
coincide (see also Popa [1986]). Associativity of the bimodule tensor product is now
immediate and we include a proof for the convenience of the reader.
cLp

Proposition 1.12 Let A, B, C and D be II, factors and let \Hp,be bimodules. Then

sKc

and

(aH ®p Kc) ®c Lp = AH ®5 (Kc ®c Lp)
as A-Dbimodules.

Proof

Tt follows from the paragraph preceding the proposition that the com-position of the quotient maps H° © (K°®L%) — HO (K®c L) —» H®s(K®cL)
is continuous with dense image.

Similarly, the composition of the quotient maps

(HOOK ®L® —» (Hep K) © L® —» (H ®p K) ®c L is continuous with dense
image. The result follows now from (H° © K®) © L° ® Ho (K° © L%). OO
Lemma

1.13

Let A, B, C be II, factors and let s4Hp and pKc

be A-B resp.

B-C bimodules. Then £b®gn = E@pbn forallé € HY, n€ K° (or€ € H, n € K°,or£ € H°, ne K), where H® (resp. K°) denotes the right (resp. left) B-bounded
vectors in H (resp. K).
Proof By definition of £®p7 (Definition 1.11 and the remarks afterwards) weneed to show that £&b®n — £ ® by € Ny. y, which follows immediately from Lemma

1.8 (resp. 1.10).
Lemma

1.14

O
Let A and B be II; factors and let 4Hp

be an A-B

Consider the A-A bimodule L2(A) and the B-B bimodule L*(B).

bimodule.

Then 4L*(A)®4

Hp = sHp and 4H ®p L?*(B)p = sHp (all equivalences as A-B bimodules).
Furthermore, if A C B is an inclusion of II; factors and we regard L*(B) as a B-A
bimodule, then 4Hg ®p L?(B)a = aHa (as A-A bimodules).

Proof We know that L2(A)° = A. Define a linear map T : A® H — H by
T(@®E) =at, a € A, £ € H Observe that (4;,a;)% = aja, for &;, a; € A (weBimodules

andHigherRelative

Commutants

21

regard L?(A) as a right A-module here). Thus

ITO ai @&)lh
i

=

(ab

a)m =>

i,j(a, 8:96)
ij= OY a ®&, > a ® &i)s

which shows that T' is well-defined, factors through N(. ., and hence induces asurjective (since A is unital) isometry, still denoted by T', from 4L?(A) ®4 Hp —aHpg, which is clearly an A-B intertwiner. This establishes the first equivalence
and the remaining ones are shown in the same way, using the left Radon-Nikodym
derivative.
[OO
Next we discuss briefly morphisms (intertwiners) between bimodules.
Proposition 1.15 Let A, B, C be II) factors and let sHp, sKgp, aH;p,Kio, 1 =1,2, be bimodules.
Let
R: aHp — aKp and S: sHig — aHyp be
A-B

intertwiners

and letT : gKyc—

pKa

be a B-C

intertwiner.

Theni) R maps left A-bounded (resp. right B-bounded) vectors of H to left A-bounded(resp. right B-bounded) vectors of K.
ii) If n,n are left B-bounded vectors in K;, then the Radon-Nikodym derivatives

satisfy (T(n), T(n'))g = (T*T(n),n")s = (n, T"T(7))5.
ili)

There is a unique A-C intertwiner

S®pT : aH1 ®

Kic —aH: ®p Ka¢

such that S@p T(E ®p Nn) = [SR T(E ®n)], for all £ € H® (right B-bounded
vectors) and nn € K° (left B-bounded vectors), where |] is as in Definition
1.11. Furthermore, if S and T are bimodules isomorphisms, then so is S ®p
T.
Proof

leR(E)x

The proof is straightforward.

If £ € H is a left A-bounded vector, then

< |RIlla€lla < c(€)l|allz, for all a € A (we used the notation of 1.3).

Thus R(£) is a left A-bounded vector in K.

The same argument applies for right

B-bounded vectors and hence i) is shown.
Let us show ii). We have seen in i) that T'(n), T(r’) are left B-bounded vectors
in K5. By definition of the Radon-Nikodym derivative we have (bT'(n),T(n’)) Ky =tra (5(T(n), T(7))5), for all b € B. But (¢T(n),T(n))., = (Tn). TG),

= (T*T(n),b*n’) = tra (b(T*T(n),7')8), b € B, which implies the first equality
by uniqueness of the Radon-Nikodym derivative. The second equality is shown inthe same way.
We proceed with the proof of iii). Let us denote by (-,-); the inner product on
H

6 K°,

i = 1,2 (see Definition 1.11).

Let}

be the usual tensor product of S and T. Theni) and using ii) we obtain,.&®mn;€ He

K,°, and let ST

S®T(},& ®@ mi) € H.' © K,° by

22Dietmar

(STD

Bisch

&em),SeT()& em):= (SENT),3(1) 5, SE) , = (SSE (TT (0), 13) 5).&5)
1,3

4,3

= (3 SSE) @T*Tm), Y_&@mh = (SST TD<IS* ST_& © mill Fro0k.0;

TID_&®m),)&®mhso that SQT(N¢..y,) C N¢.,,- Thus we get an induced map, still denoted by S®T,
from H:° ® Ki°/N..y, — Ho K3°/N,.y,, which is continuous by the above
computation and therefore extends to a map from 4H; ®p Kic — aH2 ®p Ka,
denoted byS®pT.
S®p T is clearly an A-C bimodule morphism and satisfies bydefinitionS®p T(Y_;& ®m) = [S®T

(YX; & ® m:)]. The uniqueness is clear by adensity argument. Finally, if § and T are bimodule isomorphisms, S$ ®p T is also,
which can be easily deduced from the above calculation.
O
Definition 1.16 Let A, B be II, factors and let H be an A-B bimodule withactions denoted by aéb, a € A, b€ B, { € H. We define a B-A bimodule pH 4 (alsodenoted by 4 Hp ) as H, the conjugate Hilbert space, with the B-A action defined byb-§-a=a*{b*, where { denotes the vector ¢, considered as an element in H. The
B-A bimodule gH 4 is called the conjugate (or contragredient or adjoint) of the AB bimodule nH. An A-A bimodule H is called selfcontragredient if sHa = aHa
as A-A bimodules.
Remark 1.17 ) Let H be a right A-module with right A action denoted by &a,
Ec H, ae A Then Hy is a left A-module with left A action a-£ = £a* (Definition
1.16).

Clearly, the right A-bounded vectors H 0 and the left A-bounded vectors "coincide. Thus tra(a(€, Ma) = (a £1) 5 = (n€a*)y = (18,6); = tra(a(€&, na),
for allé, ne H°, a € A. Hence (£,7)a = (€,1)%, for all €, n € HO.
i) Let A, B, C be II, factors, and let sHp, Kc be bimodules. Then 4H ®p Kc =cK®pH 4 as C-A bimodules. Let us briefly sketch a proof of this statement. Define
a linear map TK
0H— AH ®g Kc, by T(7®E) =£R®p 1, Ech, feR’.

Then we get for & € H°, i; € K°, i = 1,2,(&®8m,&®s)enka= (£®8M2,6 ®BM)
= ((61,€2) Bm), pox,== E2®Mm,& OM) = ((€1,&2)Bm2m)(n2, ((€1,6)B)'m) x = (61,&)B)

Mm, M2) == (7: (61,62), M2) = (Mm ®&,Mm® &2) 50 om
ThusT is well-defined and induces an isometry K ©H ING) z0 0 — aH ®p Kc,
which is clearly a C-A bimodule morphism (note that c- (7® &) - a = 7c* ® a*§,Bimodules

andHigherRelative

Commutants

23

which maps under T to a*£ @pnc* =c-£ Rp n-a, c€C,a€ A) and extends bycontinuity to an injective C-A bimodule morphism ¢K ®g
Hy — 4H ®p Kc. It
15 easy to see that this map is also surjective.
11) It is obvious that 4Hg = sHpg as A-B bimodules.
BA
Recall that if A C B is an inclusion of IT; factors, then 4L?(B)g=gL?(B)4,
where the bimodule equivalence is realized by the modular conjugation J : L? (B) —L?(B), since J, viewed as an operator L2(B) — L?(B), is a linear surjective isometry, satisfying J(b- £-a)= J(a*zb*) =bz*a = bJ(2)a=0b-J(Z) a, for all a € A,
be Band & €BC L*B) (note that the inner product on L2(B) is defined as(& 7) =(n, €) 125)&, n € L*(B)as usual).

Finally, let us state the Frobenius

reciprocity theorem.
Theorem

1.18i) Let AHp,

Let A, B and C be II; factors.

AKp

be A-Bnaturally isomorphic:

bimodules.

Then

the following vector spaces

Homa. p( AHp,

4Kp)

= Homp_4( BK a, Ha).

Let AHg, pKc and oLc be bimodules with finite index.
vector spaces are naturally isomorphic:

Then the following

if)

Homuy
c( aH ®p Kc,

ale)

= Hom
p( aHg,aL ®c Kp).

iii)

Homa.c( 4H ®p

Alc)= Homp.c(sH ®a Le).

Kg,

Kc,

Proof We will only give a hint of the proof.

are

Let T € Homag( 4aHg, 4aKB)

and consider T* : K' — H defined by (§,T*(n)), = (T(¢),n),, £ € H, n € K.
Then it is immediate that T*(a-n-b) = a-T*(n)-b, forall a € A, b € B and
ne

K.

Thus

T*

€

Homu.g(

Kg,

AHg).

Now

we

considerT*

as an operator

K — H, defined by T*(f7) = T*(n). T* is clearly linear and satisfies T*(b- 7 - a) =T*(a*nb*) = T*(a*nb*) = a*T*(n)b* = b-T*(n) -a = b-T*(7) - a, for all a € A,
be B,ne€ K. Thus T* € Homp s(aKg, aHg) and the map T — T* is thedesired natural isomorphism in i), which is an algebra isomorphism if H = K. This
proves i).
Using the properties of of the contragredient bimodule and i), it is immediatethat ii) and iii) are equivalent. Using again i) and Lemma 1.14, we see that it isenoughto show

that

Homy

g{ 4Hp,

AKg)

=

Homa

a(

aL*(A) 4, AK

®p

pHa)

which is left as an exercise (see for instance Ocneanu [1991(a)], Sunder [1992] fordetails).

O

Remark 1.19 Let sHp, sKp be A-B bimodules with finite index.
Then
Homu.g( aH, AK) and Homap( AKp, aHg) are naturally anti-isomorphic andhence isomorphic. From the first part of the proof of Theorem 1.18, we see that the
map T € Homag( aHp, aK) — T* € Homa.
p( aKp, AHB) is a conjugate linear, surjective isomorphism, which implies the claim.
Before we end this section, let us introduce some notations, which will be usedlater on.
Definition 1.20 Let A, B, C be II; factors and let a = sHp,
B= oKg, v=
Lc be A-B resp. B-C bimodules. The A-C bimodule sHr®p gLc will be denotedby ay. Furthermore, we let (a,3) = dimg Homa.g( AHp, KB).

24

Dietmar

Bisch

Observe that if o and 3 were endomorphisms, one would write ya = yo a,rather than ay. Frobenius reciprocity can now be rewritten in the following way.
Corollary

1.21

Let A, B, C be II; factors.

i) LetsHp

and B =a =AKp

be A-Bbimodules

with finite index.

Then(a, 8) = (B,a), {a,8) = (B, a).
Consider

1.18.

the

bimodules

« =

sHg,

3 =

Kc

and v =

aL¢

as in

Theorem

1.19.

iv)-vi)Thenii) (8,7) = (a, 70).

iii) (aB,7v) = (8,a).
From these identities it follows that,iv) (8,7) = (v,aB) = (7,af).
v) {v,aB) = (B,7a) = (Ba).
vi) (v,@f) = (@ #7) = (a,7B).
Proof i)-iii) is just a rewriting of Theorem
follows by applying the rules i)-iii). O

1.18 and Remark

Finally, let us show that the relative tensor product is compatible with directsums of bimodules.
Definition 1.22 Let A and B be II, factors and let sHg, AKp be A-B bimodule.
The Hilbert space H @ K becomes an A-B bimodule with the action

a- (Edn)-b=(a-&E-b)®d(a-n-b),€called the direct sum of the bimodules sHg

Hneand sKp

K,ae

A, be

B.

Itis

and denoted by AHp $4 Kp.

Recall that the inner product on H & K is given by (€®n,& @&n') = (§,6)u +(mn)

kK, &, 1€ H,

Proposition

bimodules.

ug n

1.23

be II; factors and let AHp,

AKp

and gL¢beThen(4aHp®

as A-C€K.

Let A, B, C

4aKp)®p Lc = (4H ®p Lc) ® (4K ®5 Lo)

bimodules.

Proof First observe that (H & K )° = H° ® K° (right B-bounded vectors).
Then, let us define a linear map T': (H & K)° OL — (HL) & (K°©L°) byT(Eem®¢)=(E®)dn®(), £€ H, ne KC ( € L°. We have,Bimodules

andHigher( (>=Relative

Commutants(& On) ® Gi), TO)i25(& 7) ® Gi) (Hoo LO) (KOO LY)iE®G) em), (eg) em eG)
i,j

=) (6

® 6,& ® {moore + (© Giymj ® GG) koro)

1,5
=

>

(8:6

Gi),

&)y

+

(m:{<s, $5) B, m5) kK)

1,3=> ((&®m)(6:8) B&G © Mm) pax

1,5=O
Note that Ni.

(Eom) ®G)(Eon) 8G) maker

00 0160020,= Me gogro © Ny gogro-

Thus Tis well-defined and induces a map between the quotient spaces, which is continuous andhence extends by continuity to a map T : (4H ® aKp) ®p Lc — (4H ®5Lc) ® (aK ®p Le). Tt is clearly injective by the above calculation and obviously
an A-C bimodule morphism. Surjectivity of T' can be shown in various ways. For

instance, one can define amap S: (H°® L%) @ (K° © L°) — (H° ® K°) ® L® by

S(Ci(@em)e(X,vi®u))=2(&o0) en +X,0®v)®u; & € Hm,pi € L° v; € KO, which is clearly the inverse of T. As above, one shows that S
extends to an injective A-C bimodule morphism from ( 4H ®5 Lc) ® (aK ®p Lc)
to ( AH
2® aK B) ®p Lc, and hence the surjectivity of T follows.

[OO

Shifts, Fourier transforms and the k-step basic construction

We discuss in this section the k-step basic construction, various natural representations of the higher relative commutants associated to it, and the naturalshift on the higher relative commutants. Some of the material in this section is
known to specialists and can be found partially in the literature (see, for instance(Ocneanu [1988], {1991(a)] and [1991(b)], Pimsner and Popa [1986] [1988], David,
Choda and Hiai [1991]).
Let us fix an inclusion of II; factors N C M with finite index and let
NcMco*"
be the associated

My Cc®*MC...

Jones tower with e; € Mj,

obtainedby iterating the basic con-struction My_; C My C+" Myy1 = { My, ex+1}” C B(L?(My)). The next result
is quoted from (Pimsner and Popa [1988]) and follows from the abstract characterization of the basic construction given there.
Proposition

2.1

Let

N C M

be an inclusion of II, factors with finite index.

Let
fx

=

[M

: NIEE+D/2(ep| exa. e1)(ext2€k41

gr

=

M

: NJEE=1/2(ep| exa. e2)(er+2€k+1.€2) cee (ear “es er) (ezk+1
ee e3) cee (e2k—-1.

“on €k+1)sex) (eax SN €k+1)-

Theni) Let N C My Cv Moy: be the basic construction, i.e., fe : L2(My) — L3(N)is the orthogonal projection and Maxi = {My, fr}” C B(L*(My)). Then
there is a surjective x-isomorphism ¢ : Mag+1 — Magysuch that ¢( fx) = fr

26

Dietmarand ¢(z) = x, for all x € My. We will say briefly thatis the basic construction with projection fx.
il) M C Mj C9My

N C My Cf

Bisch

Moy:

is the basic construction with projection gy.

We will have to work with explicit representations of the higher relative commutants N' N Mag; and M MN May, on the Hilbert space L2(My). If x € My, we
denote by # the vector z(1p,) € L?(My), where 14, is the cyclic and separating

vector in L?(My). Furthermore, let Ji : L*(My) — L?(Mj) be the modular conjugation, Jy (£) = x*, £ € My. Ji is then an antilinear isometry on L?(M) satisfying
J, 2 = id. Let us denote by 7, the (necessarily faithful) representation of My, Marq

obtained from the basic construction of N C M; C B(L?*(My)).
precise (see also (David, 1.5.6), (Jones and Sunder [1996])).
Proposition

2.2

The representation my of My, Magi

Let us be more

on L?(My)as defined

above satisfies mi (x)(2) = TZ, for all 2 € My, C L?*(My), x € My, and mi (2 fry) (2) =TEN* (yz)", for all fry € Magy, x, y, 2 € My, fr as in 2.1 and Ey* : My —N the trace preserving conditional expectation.
Thus Moky1 = mr(Moky1) =Jkme(N)Ji. More generally, we have mp(z)(§) = [M : N** Ey, (zy fi)”, for allTE

Mok 41, ES

M;.

If zx € My,y € My,then mr (2)(§)=(M: N*Ep,(zyge)”and hence Jpmi(M)Ji, = mi (Mag).
Proof The first part is precisely the definition of 7, written in detail. Since
Msk+1 = span{afrb|a,b € Mi}, it is enough to show the formula for mx (z)(§) for
elements x = afib, a, b € Mj (we omit the hats to keep the notation simple).

But En, (afkbyfi) = aEN* (by) Ear, (fe) = [M : N]=*+DaEMx (by). Thus indeedm(afkb)(y) = aEM (by) = [M : N*1 Ep, (afeby fr) as desired.

Finally, sinceMy, = span{agib|a,b € My}, it is enough to show the last formula for elements
T = agrb, a, b € My. Observe that fi = [M : N)*gi(e;...ex)(eaks1...€xs1), SO
thatme(ageb)(y)= [M : NPP**1 Ep, (agrbygi(er -- - ex)(e2k41=[M: N)2k+1g

EMs (by) Em,

x41)(gx (ex en er) {e2k+1[SPN €k+1))= [M : NI*"'aEp*(by) Er, (fi) = aBjg* (by)= [M : N]*En, (agbygr)Thus

my My,_is equal to the representation

comingfrom the basic construction

M C My, C My, € B(L?*(My)) and hence Jymp(M)'Jy, = mp (May) as claimed.

[J

Observe that if x € Mog41, y € My, there is a unique element w € Mj, such thatzyfi = wfi, namely w = [M : NIFH
Thus mk (z)(§) = wb.

EVE(oy fy) (Pimsner and Popa [1986]).

The fact shown above, that the representation coming from the basic construction of M C Mj, is equal to m|as,, (note that my is a priori in the representation
coming from the basic construction of N C Mj), will be used several times lateron. Note also that by choosing a tunnel M > N OD N; O ... and performing thebasic construction for N; C My C B(L?*(My)), we can actually represent each M,,
on L2(Mj). The next lemma contains some useful identities, which will be used inBimodules

andHigherRelative

Commutants

27

Proposition 2.5 and Lemma 3.6 to show how the above defined representations are
related.
Lemma
Then

2.1.

2.3

Let fibe the projection for the k-step basic construction as ini) frerti€k...e1fip1 = exqr€k...€1fkr1, k>0.
ii)

€k+2€kiif)€k+2€k+3 - - - €2k fk€ok -+3. €2k+1frezk+1

ceCk43€k42€k4+3€k+2=€1€2... ex frex

= €2€3...€, [reg...€2€1,... e363,

k >1.

k > 2.

Proof i): We have fi = [M : NIF=+D/2(¢p
1 egpi1)--- (ey... exy1), since
fr = fr. Before we start the proof, let us do a special case, which illustrates themechanism of the proof:
frezezerfs= [M : NP3[M : N|8(eseqes)(ezezeq)(ereaes)(ezese;)(eseseser)(esesezez)(eseseses)(ereseses)=[M: N|¢[M. N|(eseses)ez(eseq)er(esesezer)(eseseser)(eseseqes)(ereseseq)=[M: Nlbes(eses)ezer(ezezer)(eseseses)(eseseqes)(ereseses)= [M : N]Seseqer(esezener)(eseseser)(eseseqes)(eregeses)= ezezey f3.
The proof in general works the same way:
frers1

ex...e1 fet= [M : NPD

NJEFDCD2(0 1 ee)(e1 PEN er+1){€k+1

TN e1)(ext2

ce e1) ee(eak+3(es. esa)

ce ek+2)= [M : NJ*FDEFD/2[Ar . NJEE=1/2(gp 1 egy)e1(er42“es esezeq)cee (e2k+3(ea(es.. . exq2))

IN €k+2)= [M : N=+HDEED/2[pr NjE-DE=2)/2(g,Co€gkgy)(es(esq AP ek+3))ezer(eseser) {enya “en €2) see (e2k+3

-

eks2)

=[M: ND) (k+2)/2[ pp : N)E=2(k=3)/2(¢,
Ce .egpg1)(es(es ...erta))eseser(esezeser)(esesesen)(exra --. €3)
tet (e2k+s3 .e .€k4+2)
=

(M

: NCD

R42) /2[

pp: N|E=3)(k=0/2(¢,ee€2k+1)

cee(e5...exts5)esezezei(eseq ...e1)(eges ...ex)(er...e3)(ek+s5 ves €4) ce

=[M:

ND

(e2k+3(ej+2 “o €k+j+2)€j+1€5(€k+j+2

ce ek+2)

E+2)/2[ pp : NJE=Dk=i=1)/2(¢,

ces €i+1)te“oe e1(ej+2€541(eak+3= [M : NJ6+DE+D/
20,

1| Ce egry1)

ee e1)(ej+3a. ea)en €k+2)e(ensr-.-e1)(€pyz...€1)(e243 - -- €x42)= €k41---€1 frst.
Thus the proof of i) is complete. ii) and iii) are proved in a similar way, using
the commutation relations of the Jones projections e;. Let us just indicate theproofof it).

hi is [M€k+2

Set Jue
k—2

: Nm

C241

k€2%t1.-

(ex+1
—1

a

es)(ex+2

Cen es) ce (eak—2

IN er )(eak—1

x the Jones projection for M;
Chan =€1... exfrer...e1

C= [M : NI"

Mj,

cen €k+1),

C Moy,_1,

hiiie.

i.e.

then

O

Next we review the shift on the higher relative commutants. Let us change
notation for a while and consider an inclusion of II; factors A C B with finite

28

DietmarBischindex. Let A C B C°* Bj be the basic construction and denote by (my, L*(B))
the standard representation of A C B C B; on L*(B). Let Jy : L?*(B) — L*(B)be the modular conjugation and define vo(z) = mgt Jomo (2)* Jo, rz € AN By,ie, mo(v(x)) = Jomp(x)*Jy. The properties of the basic construction imply that
JoBJy

=

B’

and

JoA'

Jy

=

By

(Jones[1983]),

so

that

Jo (A

n Bi) Jo=

(note that all algebras are represented on L?(B), so that we omit 7.

A

n By

If we treat

A" N By as an abstract (finite dimensional) algebra and we want to emphasizethat it is represented on L?*(B), we write mo(A’ N B;)). Thus 5, defines a map
Yo: A'NBy — A'NB; (viewed abstractly as finite dimensional algebras). It is easy to
see that yo is a surjective, unital, linear *-anti-isomorphism from A'NB; — A'NB;
such that v§(z) = z, for all z € A’ NB).
Note that one often writes brieflyoz) = Joz* Jo, x € A’ N By, when the representation is understood.
The next lemma contains a number of identities which will be useful in thenext section.

Lemma 2.4 Let A C B be an inclusion of II, factors with [B : A] < co. Let
A C B C* By be the basic construction and denote by (mg, L?(B)) the standard
representation of AC B C By on L%(B). Let yy: ANB,
Yo(z) = m5 Jomo (z)* Jo, xz € A'N By as above. Then,— A'NB;be defined byi) mo(z)(a1-€a2) = a1-mo(x)(€)-az, for allay, az € A, £ € L*(B), x € A'NB,
where ay - £ - ay = mo(a1)Jomo{az)*Jo(€). Thus mo(z) is an A-A intertwinerof the A-A bimodule L*(B).
ii) Let x € By, y € B, then mo(z)(§) € B. If we view the element mo(z)(§) asan element in B, denoted by mo(x)(y), then mo(z)(y)er = zye; (in By).
iti)

We have vo(z)yer = yzey for allziv)

We have mo(yo(p))(y*)
y € B.

=€ A'NB,y € B.

(mo(P)(®))", for all projections p € A’ N By

Proof We set m7 =m, e =e;, vy =i): Let 2 € ANB,a1, a2 € A

and
J = Jj.
and b € B.

Thenn(z)(a;andA- bay)=

m{x)m(a1)Jm(az)*J(b) = m(a1)Jn(az)*n(x) (b), since z € A’ N By.
ii): By Proposition 2.2 we have that 7(z)(§) = [B : A]Eg(xye;,)", for all y € B,z € By. Thus n(z)(y)e, = [B : A]Eg(zye1)e;Proposition 2.2.
iii): Let= zye; by the comment afterz € A'N B and y € B.

We verify the equality in the representation 7.

B,|pHence 7(y(z))n(y)m(e)(8) = Jr ()* JWEA®)) = J(z*Ea (8) y*) = yaBa(b),
b €

where

we

used

that

is left multiplication.

Onthe otherhandm(yze)(b) = yzE4(b), b € B, so that indeed Y(z)ye = yxe as claimed.

iv): Letp € A’ N B; be a projection and let y € B.

T(x (p)@) = Ir (@)IG") = n(+(0)) 3").

Then

(7(p)(®))"=

0
The following proposition shows how the representations my, mx, (see Proposition 2.2) are related, when restricted to the higher relative commutants associated
to
NC M.

Bimodules

andHigher

Proposition

2.5

Relative

Commutants

Let wy(resp.

wpy1)

29

be the representationsof My,

Magi(resp. Myy1, Magis) on L2(My) (resp. L?(Mgy1)) as in 2.2. Let x € N'N Magy1,y € My and 2 € M. Then mi(x)(y)ext1€k ---€12 = Tpr1(z)(yexsi1€k -.-€12) as
vectors in L?(Myy1).
Proof First observe that mi (z)(y) € My and mxi1(z) (vers -- .e12) € My
(proposition 2.2), so that the above equality indeed makes sense in My; (to keep
the notation simple we omit “hats” as usual). Furthermore, as in 2.4 i), we see that

Te1
(2) (w2) = Thr1(@)er1Tr41(2)Terr (w) = Jip1Tet1(2) Torre (2) (w) =Tr+1(x)(w)z, for all x € N' NM Magi1, w € Mii, 2 € M, since mi41]ps is left multiplication and Jg+1mk41(M)Jerr = Ter1 (Mago) (by the last part of Proposition
2.2). Thus we have mi41(z)(vert1€k -..€12) = Trr1(x)(yerr1€k...€1)2, and it istherefore enough to show the identity in the proposition for z = 1. Next, recall that
Msy.+1 = span{afib|a,b € Mi}, so that it will be sufficient to show the identity
for x = afb, a, b € My. By Proposition 2.2, we have,me(afib)(y)ertier...e1=[M: NIFH EL(afebyfr)ensiex - en

= [M : NFL aEN* (by) By™
=aEN*(by)er+1ex(f)ensiex . -. e1

cL. €1.

The right-hand side is calculated using Lemma 2.3 as follows,
rrr {afib) (vek+iex...61)=[M:

NIF+2 E22(a fibyeg

rex co.e1fres1)=[M: NJk+2 ppreesa (afrbyfreriren- e1furr)= [M : NJ**2aEN* (by) Epy™

srk. €1 fri)(ex

= aEN* (by)ex tix ...€e1,
which completes the proof.

O

Next we give a short proof of a well-known formula for vg (see for instance(Ocneanu [1991(a)], David)).
Theorem 2.6 Let A C B be an inclusion of II) factors with finite index, let
A C B C® Bj be the basic construction and let {m;},c; C B be a finite basis of B
over A, such that b=, m;Es(m}b), for allbe B. Let x € A'N By, thenv(x) =[B: AY Ep(eym;z)eim;.

(2.1)

sel
Proof As above, let m = my denote the representation of By on L?(B) comingfrom the basic construction, let e = e¢; and J = Jy. Then we have for all b € B,

Dietmar

30

ze

Bisch

A’ NB,[B: A] 3 m(Es(emz))m(e)n(mf)(b) =[B: A] _m(Es(emiz))(Ea(m{b))=[B: AY(Es(emizEa(m;b)))" =[B: Al(Ep(ed_miEa(mib)z))"= [B: AlEp(ebx) = [B : A)(Ep(e(m(z*)(b))*))" = (r(z*)(b"))*= Jn(z)*J(b) = ©(7(2))(b),where we used that (ebz)* = z*b*e = m(z*)(b*)e by Lemma 2.4 ii). O
Recall that {m;}iz1,..n+1 C B (n <[B: A] <n+1) is called an orthonormal
basis of B over A if Eg(mim;) = 0, i # j, Ea(mim;) = 1,1 <i <n, andEa(m} mp1) is a projection in A (Pimsner and Popa [1986]). Any such basis
satisfies 3, mierm} = 1, b = >, miEa(m;b), for all b € B. Note that in theover A.
above theorem we do not require that {m;}:cr is an orthonormal basis of B
not
Any finite basis will do. Furthermore, observe that the formula for ~yo(z) does
depend on the choice of the basis, since the left-hand side in (2.1) is independentof such a choice.
Proposition 2.7 Let A C B be an inclusion of II factors with finite index andlet {m;}icr be a finite orthonormal basis of B over A. Let tras be the normalized

trace on A’ = A’ nN B(L*(B)) and consider the map¢(z) = [B: AI7HY mami,z € A.

Then ¢ is the unique tras -preserving conditional expectation from A'—B

(C B(L*(B)).

Proof We first show that ¢(A’) C B'. Let A C B C° Bi be the basic construction, J : L2(B) — L?(B) the modular conjugation, and recall that [J,e] = 0,
B, = JA'J, By = span{aeb|a,b € B}. Let aeb € By, a, b € B, then ¢(Jaebd) =[B : A", miJaJeJbJm; = [B : A|7' 3; JaJmiemiJbJ = [B : A" JabJ €B’. Thus indeed ¢(4A’) C B'. We clearly have ¢(b') = ¥', for all b’ € B’ (since= bg(a)c,S.mimt = (B : A)), ¢(z*) = ¢(z)*, ¢((4)+) C (B')+ and ¢(bac) tra(¢(z))b,c € B', a € A".

Furthermore,if

x = JaebJ€ A’, a, b € B, then

= [B : A tra(JabJ) = [B : Al rp, (J(JabJ)*J)= trp, (b*ea*) = tra (JaebJ) = tra (z). O=[B : Al" trp, (b*a*)

Since the tr 4/-preserving conditional expectation is unique, the above formula
does not depend on the choice of the orthonormal basis.
Corollary 2.8 Let A C B be II; factors with finite index and let yo : A'NB; —A’ NB, be defined as above.

Then trp, (vo(z)) = tra(z), for all z € AN By.

Bimodules

andHigherRelative

Commutants

31

Proof By Theorem 2.6 we have,
trp, (vo(z))= [B : A)

trp, (Ep({eym;z)eymy)= >tr, (ermizm])= [B: Altrp, (e1¢(x)) = [B : Altra (Jo(z)*e1J)= [B: Altra (Jo(z)* Jer) = [B: Altra (EE (Jp(z)*J)ey)= [B: Altrg(Jo(z)*J)tra (er) = trp(Jo(z)*J),
since J¢(z)*J € A’NB and hence EE (J¢(z)*J) = trg(Jo(z)*J). Thus trg, (vo(x))= trp, (Jo(x)*J)) = tra (¢(z)) = tra (z) by the previous proposition. [0Next we will discuss the shift from A’ NB; — Bj N Bs. Let J; : L*(B;) —L%(B;) be the modular conjugation and consider the basic construction A C B; C
m1 (Bs) = Jy A’ Ji, where we denote, as before, by 7; the representation of By, Bs on
L?(B;). Then, as above, we get a linear *-antiisomorphism +; : A’ Bs — A’ N Bs
such

that

m1 (v(x)=Jim{x)* J,

x € A

n Bs.

Since

Jim

(A

Nn BJ:=

m1 (BjNnBs), we have that vy, (AN By) = B] N Bs. Hence the composition vv, : A’ NB; —Bj N Bj is a surjective *-isomorphism, which is called the shift from A’ N B; to
Bi N By (here it is just the 2-shift, but later we will take for instance A = N,
B = Mj and we will obtain a shift from N' nN Magy) to My, ; N Myry3) (seeOcneanu [1988], Ocneanu [1991(a)]).

Note that one sometimes writes Jy JozJoJ1

instead of v;-yo(z), when all the representations are understood.

We will deduce an

explicit

projections

formulafor the shift in termsof a basis andthe Jones

ej, esand ez.
Lemma 2.9 Let A C B be an inclusion of II; factors with finite index and let
A C B Ct Bj be the basic construction. Theni) ES (ze1yb1) = Ea(zEg(eiybh)), for all z, y € B, by € By (Es = E§,

Ep = EB").

ii) Ea(zEg(e1yb1)) = Ea(Ep(bize1)y), for allz, y€ B, by € A'N By.
Proof The proof of i) is trivial and ii) follows from the uniqueness of the trace
preserving conditional expectation E% Bi — A. O
Next we prove some identities which will be needed later on.
Proposition 2.10 Let A C B be an inclusion of II) factors with finite index
and let A C B C® Bj be the basic construction. Let {m;}ic; be a finite orthonormal

basis of B over A. Then {[B: Alzmieim;}i jer is a (not necessarily orthonormal)
basis of By, over A satisfyingbi=[B:A]> mierm;Ea(m}Ep(eimiby))
J
=

[B

: A

“mieym;1,JER((mieym;)*by)=[B: A]> Ea(Ep(bymier)m;)mieim;,

for all by € By.

J

Futhermore, if x € A’ By, then x = [B : A] dij Es(m;Eg(eim;z))mjerm;.

32Dietmar

Bisch

Proof It is easy to see that {[B : AT me Vier is an orthonormal basis of
By over B. Thus, if by € By, we get by = [B : A]>_, me; Eg(eym}b;).
But
Egleym?b) = 2 m; E4(m}Eg(eim;by)), so that b, = [B : A] di; mieim;
Es(mjE B(eymb;)), which shows the first equality. The second one follows from
Lemma 2.9 i).
Since by = [B : A]>.. Ep(bim;e1)eym! and Eg(byme;) =>; E4(Eg(bimier)m;)ms, the third formula follows.
The expression for z €A’'N By follows from this and Lemma 2.9 ii).

[J

The next theorem gives an expression of the “spatially defined”

shift (usingJ's) (Ocneanu [1988] and [1991(a)]) in terms of an orthonormal basis and the Jones
projections e; (Pimsner and Popa [1986]). The second formula below is well-known(Ocneanu [1991(a)], Pimsner and Popa [1986], see also David).
Theorem 2.11 Let A C B be an inclusion of II) factors with finite index andlet y17vo be the shift from A’ N By to BN Bs as defined above. Then
MY0(z)er = [B: Al%ejeazezeze,
for all z € A'N By.
have

Thus, if {m;}ic1
my(z)is a finite orthonormal basis of B over A we=[B: AJ? > miererzezererms,
iel

for allz € A’ N B;.
Proof

Let

{m;}ic;beanorthonormal

basis

of B

over

A

with

m;

=

1, so

that Ea(m]) =0, if ¢ # 1, and E4(m}) = 1 (this can be assumed without loss ofgenerality since A is a factor). Then {[B : AF mierm;}i je 1 is a basis of B; overA satisfying the hypothesis of Theorem 2.6 (with B; in place of B) by Proposition
2.10. Set ry; = [B: Alzmeym;, and let A C B; cf Bj be the basic construction
with projection f = [B : Aleseieze; (Proposition 2.1). If z € A’ N By, we have by
Theorem 2.6 that,ner= (Br: AY

Es, (frum)
frie
i,j

= [B: AP(Q_ Es, (Nriyn(@)fri)ei,j=[B: A] OO “miesmjvo(z) fmierms)eriJ=[B: ARS “mierm; Ep(ermiz)eimjeer eseamiermyes
ijk
=

[B

: APD

"erm;

Ep(eymyz)ermyesermeseze;jk= [B: Al%erex([B: A]"Ea(m;Ep(eymyz))mjeim])eseze;
Jk

=

[B

: A)%ejeszezeqe,

where the last equality follows from Proposition 2.10. The formula for -y;7g is now

immediate since }_, mem?

=1.

OO

Corollary 2.12 The shift v1 : A’ N By — B{ MN Bj is trace preserving, i.e.,
tra, (myo(z)) = trp, (z), for allz € A’ N By.

Bimodules

33

Commutants

Relative

Higherand

Proof We calculate trp, (m170(z)) = [B : AP*Y; trp, (miei eazezeseym;)=: AY, trp, (miei Ep(x)ezerm})[Bmi)trp, (z) = trp, (z), forallz € ANB.

=

= trp, (3, mie

Yoitre, (m: ES" (x)erm})O

Although not needed in this paper, we include for the sake of completeness a
few words about the 2-shift and the Fourier transforms, both of which are particular operators in the affine Hecke algebra associated to a subfactor (Jones [1994],

Jones [1996]). For the next theorem, see (Pimsner and Popa [1988]).

Theorem 2.13 Let A C B be an inclusion of II; factors with [B : A] < oc.
AC B C® By C2 By C ... be the basic construction and define maps Ty, :Let=

by T(z)MN Bop+t3

A—

Al MN Ban€92€1,

ey cn 42T€2n43€2042: Ante[B

for all x € A’ N Bony. Let {m;}ier be a finite orthonormal basis of B over A andset Sp (x) = 3, miTn(z)m}, x € AN Bany1. Then we have,(Tn(z)) = [B : Al"Yrp,,

Bani1, Tu(1) = er. Furthermore trp,
alze

AN

for allz

2 0, T,. (zy) = To(z)Tn(y),Ta(z*z)= (Tn(x))*,i) To(z*)€ A'N(x), for

Bony.

Thus Sy, : A'N Bangs —ii) Sp(x) € By N Banys for all xz € A’ N Bangi.
B| N Banys is a unital x-isomorphism, which is onto and trace preserving,iii)=(Sn(T))Sp(x)S Bj N Bopt2

BN

Bany2

ze

1 (x),

tre,tT Bynss

A'n

Bony.

: A'N

Sn ay:ThusnN By,.

x € A’

for allBg,—which is onto and trace preserving.

is a unital x-isomorphism,iv)

The definition of Sy does not depend on the choice of the orthonormal basis.

v)

Sn+1|A'NBonis-

Sn+1lanBan

Sn,

Sp|A'Ban=(Sn)nen

Thus

.

—

trace-preserving surjective x-isomorphism S : U,A'NB,

definesa

UnBiNB,

such that S(A’ 0 By) = By N Byya.
So: ANB;vi)—

Bj N Bs

satisfies So = 71%.

To show ii), let = €Proof The easy verification of i) is left to the reader.
AN Banyt, bi, ba € B. Then,[B

bie1ba Sn (x) =
=

(B

"by

: APH
: APY

“mE

Ba (bomi)eres

es€1Mm

E27 42T€2043

a(mbyEa(bami))eres

co

X24

ese

Ea (m;b1Ea (bam; ))m;

3. exerm;

i,j
=

[B

: APY

“mene

“E204

“mies

ce

2Z€243

i,J
=

(B

: AP)

€2n42T€2n

43. - ezeym;bierbs

J= Sp (z)bre1bs,
S,,since YE A(bom;)m; = ba. Hence S,(z) € BiNBanys as claimed. The fact thati).
is a unital *-homomorphism, which preserves the trace, follows immediately from
Thus

S, is injective.

Since dim A’ 0 Bypy1and the proof of ii) is complete.
ii).

But

€2n4+2T€2n4+3€2n+2—[B= dim Bf N Bap y3, Sp is also surjective

If z € A’ N Bay, then S,(z): Al" lzeana,so thatindeed€ B] N Bangs

Sn (x) 1S Bj

byNn Banyo.

The remaining statement in iii) follows as in ii). The fact that S, does not dependDietmar

34

Bisch

on the choice of the orthonormal basis can be deduced easily from the fact thattwo orthonormal bases of B over A differ by a unitary in an amplification of A

(Pimsner and Popa [1986]). The first two equaltities in v) are straightforward andthe rest of v) is a consequence of ii) and iii).
2.11. O

Finally, vi) is obvious by Theorem

Definition 2.14 The map Sn : A’ 0 Bypy1 — Bj N Bangs (resp. Solan,
A’ N By, — BN Banta) is called the 2-shift on A’ N Bony (resp. on A’ 0 Bay).
Remark 2.15 One can of course define in a similar way a 2-shift from B'NB,
to By, N Bnt2. The “orthonormal basis approach” to the 2-shift can be found in(Pimsner and Popa [1986]) and the “spatial approach” using J's in (Ocneanu [1988],
Ocneanu [1991(a)], see also Popa [1990], Choda and Hiai [1991]). It is shown in(David) that both points of view coincide. Note that the 2-shift actually defines
AC B anda trace preserving *-isomorphism between the standard invariants of
Bi C B; (resp.
BC By and Bo C Bs).

Let us discuss briefly another set of special operators contained in the affine
Hecke algebra associated to a subfactor, the so-called Fourier transforms(Ocneanu [1988] and [1991(b)], see also Bisch [1994(a)]).
Definition 2.16
let

AC BC

By

C®

Let A C B be an inclusion of II; factors with finite index and
By

C...

be the basic construction.

¢, ¢1 and ¢r : B'N Brgy — A'N By

[B

=

é(z)

: A" Ep, (exs1ek...e2e17)
k+4+2

We

definelinear mapsby é-(z) = [B: AF Ep, (zeres . .. exert),

and d(x)

: A"

= [B

k42

...Exexy1). We call gr, Py and ¢y, the left, resp.
k.. €x€1T€1€2(exi1€
2-sided Fourier transform on B' MN B41.

Ep,

right, resp.

Let us remark that other normalizations of the above defined maps can also befound in the literature and that these maps are the classical Fourier transforms forfinite groups in the case where the subfactor is obtained as a crossed product bya finite group (see Ocneanu [1991(b)]). Various combinations of shifts and Fouriertransforms will give interesting linear maps on the higher relative commutants. For
instance, the maps 7x can be written as such a composition. To see the usefulnessof the maps defined in 2.16, let us prove the following (well-known) proposition.
AC BC
Proposition 2.17 Let
By, = dim B' N Biya, for all k.

By C By C ...

Then dimA’ N

be as above.

Proof Let ¢, : B' N Byy1 — A’ N By be the Fourier transform as defined in
2.16. If ¢.(x) = 0, then exioze; ...exr1ex+2 = 0 and hence exyoTe; = 0. Usingan orthonormal basis of B over A, we get exo = 0 and hence z = 0. Thus
dim A’ Nn Bx > dim B’ N Biy1. Using the Fourier transform on A’ MN By, we seethat dimA} N Bix; > dimA’ N By > dimB’ N Bg, where B D AD A, is thedownward

basic construction.

and B’ N Biy1

3

M,

But the 2-shift is an isomorphism between A] NBy_1(Theorem 2.13), so that all dimensions coincide.

[J

Principal graphs, reduced bimodules and the fusion algebra
associated to a subfactor

Let N C M be an inclusion of II; factors with finite index and let N C M C*!
C®2 M, C ... be the associated Jones tower of II; factors. We have seenBimodules

andHigherRelative

Commutants

35

above that the Hilbert spaces L%(M,,), n > 0 (My = M), are natural N-N, MM, M-N resp. N-M bimodules via the action a -%-b = azb, a, bin N or M,& € M, C L?*(M,) (extended to all of L?(M,) by continuity). Observe that if
Jn + L3(M,) — L?(M,) denotes the modular conjugation, we could have written
this action as a - & - b = wm, (a) Jom, (b)* Jo (£), where m,, denotes the representations
introduced in Proposition 2.2, since 7, | 5s is left multiplication. In this section, p will

denote the N-M bimodule x L?(M) with the above action a-£-b = aJob* Jo(€), a €N, be M and ¢ € L2(M). As we have seen after Remark 1.17, the contragredientbimodule p is then just L2(M) as M-N Hilbert bimodule with the actions as above(exchanging N and M of course). Throughout this section we will work with NN and N-M bimodules. It is a trivial exercise to rewrite all the statements for
M-M and M-N bimodules. We will use the notation for bimodule multiplication
introduced in 1.20.
The following two results are well-known. We include the proofs for the convenience of the reader.
Proposition 3.1 Let N C M be an inclusion of II, factors with finite index
and associated N-N resp. N-M bimodules L?(M,,) as defined above. Thena) NL*(M,) ®u mL*(M)y = yL?*(M,)n as N-N bimodules.
b)~L?(M,)ON

NLE(M) py >Thus, if we let p=

NLA (Mp1) um as N-M

ny L*(M)pbimodules.

be as above, we havei) (pp)™ = NL? (M,,_1)n as N-N bimodules.
ii) (pp)™ = ML? (Mp)nas M-Mbimodules.

iii) (pp)"p= NL?*(M,)m as N-M

bimodules.

iv) (pp)"p= ML*(M,)nbimodules.

as M-N

Proof As in the proof of Lemma 1.14 we see that yL?>(M,) ®p L*(M)y =~L?(M,)nas N-N bimodules.

Let us prove b). By Proposition 1.5 we know thatthe right N-bounded vectors in L?(M,) (resp. left N-bounded vectors in L?(M))
are given by M,,(resp.

M).

As usual, we will abuse notation and write just =instead of # for the vector & = x(1ar,) € M, C L*(M,).
Define a map T : M,, ® M — L?*(M,;,) by

T(>

=

® Yi)

=

M

: NF

Note that {(a,b)ny = En(ab*), a, b € M.

(x1 ®y1, 22 Qy2)>

ziensien...€2€1%Y;.

We compute,= (z1 (Y1,92)N: 22) 2a,= (21 En(1193),72) = tra, (2321
En (1195)),

Dietmar36

for all z; € My,y;€ M,i=1,

2. On the other hand we have,+ €192) 13 (a)(z1€n+1 ...€1Y1,T26n41(M : NJ!

Bisch= [M : Nt tr(z3z1€nt1-= [M : N|"* tr (aba1519361 - + en+1)

En (Y195)ent1 - -- €2€1€2-

en+1)(1133)En z1
1 = tr(z3En (1195 )ens1)= [M : Nltr(e52= (z1 ® 1,72

@ Y2)-

denoted by T,
This implies that T is well-defined and induces an isometry, again

N-M
from xy L2(M,)®n nL*(M)p to ~L2(Mj, +1) a which is clearly a morphism of
bimodules.

Observe that Mp1= Me,

the linear span of elementsof the form aens1b,

a, be(or even M-M)

which implies that M41
Mpenii€n...exM.

From= Mpept1€nMn_1
this we deduce(by which we mean

1M,

My),

M;,

= M,_1e.Mp_1,

and by induction we obtain Mp1

that

our map

T is surjective

and

=

hence

b) is shown.

(and of course
The proof of i)-iv) is now immediate by induction using a), b)
O
associativity of the bimodule tensor product, Proposition 1.12)
to an inclusion
Proposition 3.2 The higher relative commutants associatedintertwiners. Moreof II; factors N C M with finite index are spaces of bimodule
precisely, one has,

N'O Manyyn(n L*(Mz)n),2 Hom

N' nM,=~ Homa(nL? (Mp) um),

M' nN Ms,

= Hompg-n(ma LA(Mp) um),

MNMyw

= Homan(mm LA(Mp)n)-7, the represenProof Let us prove the first identity. Denote as in Section 2 byntation of N C M,, on L?(M,,). Then by (Pimsner and Popa [1988], see Propositio
Thus,2.1), Tn(N) C mp(Mp) C Tn(Man41) C B(L2(M,,)) is the basic construction.
p n)ny (NL2(Mn)Hom=ma(N)'N Tn (NPYNM B(L*(My))= tn (NY N (Jomo (N) JR) 0 B(L* (My)
= 1 (N' 0 Mapy1) = N' 0 Many,

where J, denotes as usual the modular conjugation on L?*(M,).

For the second

the proof is identical.
identity we use the last statement of Proposition 2.2, and thenO
The remaining two identities are shown in the same way.
can be viewed
We will show next that the principal graphs of a subfactor
to the vertices of theseas “principal fusion rules” for certain bimodules associatedSunder [1996]).
graphs (see also Ocneanu [1988], [1991(a)] and [1991(b)], Jones andT' will describe theIf we denote by (I',T’) the principal graphs of N C M, thenBimodules

andHigherRelative

Commutants

37

principal part of the Bratteli diagram of C =and Mthatf
C= MnNnMc
M NM
call first the definition of the principal graphs

NNN Cc NNNM Cc N' NM, C
Cc MNM,C....
Let us re(Goodman et al. [1989], see also

(Popa [1990], 2.1, 2.2, 3.5), (Wenzl [1988], 1.1, 1.2)).

We will describe the con-struction of I. Since N' N Ms,1, n > —1, (resp. N' N Maz, n > 0) are finite dimensional C*-algebras, they are multimatrix algebras and we suppose that
their simple summands are indexed by a set K,, n > —1 (resp. L,, n > 0) (set
M_; = N, My = M). Let 23,13 be the central support of eany3 in N' 0 Mopys(observe that 29,43 = V{uezntsu*|u € U(N' N Mani2)} (Popa [1990], 2.1),
then

(N'

Nn Mos, 13)2on+3=span(N’

Mn Mop0)eant3(N'Nn Mo,y0)

def

Yonts

and

N' N Mapa & (N'N Mopy2)zonts C (NN Mayys)zanys is (isomorphic) to the(algebraic) basic construction of N' N Mapy1 C N'N Mayo.
In other words,
N' N Mas, 3 has a direct summand (which is the 2-sided ideal Y,43 generated
by

esny3

in NN

Ma, .3),

thatis isomorphic

to the basic

constructionof the fi-

nite dimensional algebras N’ N Ms,11 C N' N May,2. Furthermore, note that
ean+3(N' MN Manys)eants = (NN Mani1)eznys. Due to the properties of the basic

construction we get a natural isomorphism ¢ : Z(N' N Mp,41) — Z(Yon43) (thecenters), which sends a projection ¢ € Z(N' N Man41) to the unique projection
G € Z(Yan 3) satisfying gesnt3 = deants (§ = JqJ, where J is the modular conjugation on N’' N Mz, 2). Thus, if gx is the identity of the k-th simple summand in
N'N Manga, k € Ky, then § = ¢(qx) is the identity of a simple summandin Y2n3
(and hence N'NMzp 43), indexed by some k € Kyi. Themapk € K;, = k € Kn
identifies K,, with a subset of K, 1; and we will henceforth regard K,, as a subset
of K,,1 using this identification. Recall that K,1\K,, is usually refered to as the“new stuff” (Goodman et al. [1989]). If pr € N' N My, is a minimal projection< gk, then preany3(N' N Manis)preonts = Pr(N' 0 Mani 1)preants = Cpreznys,i.e., preanya

prGean+s

is a minimal

projection in N' N May, 13.

Since preoni3= (Prean+3)q, where § = ¢(qx), we have that pgeanis= prGreansa=< g,_so that

Prean+3 is contained in the simple summand of N "MN May,3 indexed by k. Thus,
the identification k¥ € K,, — k € K,y1 as described above, can be obtained in
the following equivalent way: Let py be a minimal projection in the k-th simple
summand of N' \ Ma, 11, k € Kp, let k € K, 1 be the index of the simple summand in N' gd M My, +3, which contains the minimal projection pgezny3, then the map

ke K, — ke K,y, is precisely the above described identification.
The same analysis can be carried out for
N' N Ms, C N' N Myp,41, and we get
an identification of L,, as a subsetof L,;.

Namely,if p; is a minimal

projectionin the I-th simple summand of NN Ma,, then pie2,12 is a minimal projection ina simple summand of N' N May, +g, indexed by some | € L,+1. The map! € L, —le L,4,

identifies as before L,, with a subset of L,;.

we setK =|J,-_; Kn and L = J,;5q Ln. Note that K_; = {k_} is a singleton,

Using these identifications,since NNN = C. Thus, if we denote by Gr, = (Gki)kek,., leL,,, the inclusion
matrix of the unital inclusion N’' N Ma,.1 C NN Ms,12, where Gg; denotes themultiplicity with which the k-th simple summand of N' N Ma,4 sits in the [-thsimple summand of N’ N My, 12, then the transpose matrix GY, is the inclusion
matrix for
NN Ma,yo — Yan. 3 (this follows again from the properties of the finite

dimensional basic construction (Jones [1983])). In particular, G%, is a “submatrix”
of the inclusion matrix Hny1 = (Hik)icL,ye, k€Knyrs Or N'NMopyo C N'N Map ys,
where Hj; denotes as before the multiplicity with which the I-th simple summand of

BischDietmar38

NN Map4g sits in the k-th simple summand of N "MN Man 43. More precisely, we have
Hy = Gri, l € Lyy1, k € Kp, C Kpy1. Thus, there is a unique “Kx L” matrix G =(Gri)kek, 1eL, such that (Gri)kek,, iL, 11 1S the inclusion matrix for N'NMo,+1 C

N' 0 Many2 and ((Gri)keKoir, tern)’ is the one for N' N Manio C N'N Mans

(see Popa [1990]). We construct now a bipartite graph I" in the following way. The
vertices of I are defined to be the set KUL and there are Gi, k € K, | € L, edges
between a vertex k € K and a vertex | € L. T is then called the principal graph of
N C M. Recall that T has a distinguished vertex *, which is the vertex denoted byk_, above. Equivalently, we could have constructed I' in the following way: The
vertices of I are {k_1 }ULoUKo\K_1UL1\LoUK1\Kp"-- = KUL as before. Let x
denote the vertex k_; and connect * to each [ € Lg according to the multiplicity Gu,| € Ly. Then connect each | € Lo to the vertices k € Ko\K_; according to theirmultiplicity Gri, k € Ko\K_1, | € Lg, then each k € Ko\K_, tol € Li\Ly etc.

(“the new stuff is connected only to the old new stuff” (Goodman et al. [1989])).

Observe that this procedure results in the same bipartite graph with distinguishedvertex x as the previous one, since for example each k € Ky, viewed as an elementof Kn41,

is connected

to | € L,withthe same

multiplicity

Gx;as | €

Lyis tothat same k, viewed now as an element of K,, (this is just a rewording of what was
described above as “G?, is a submatrix of Hy11"). The details of all this can befound in (Goodman et al. [1989] or Popa [1990]).
The principal graph I" is constructed in the same way, using sets K/, n>0,
indexing the simple summands of M' N My, and L!,, n > 0, indexing the simple
summands of M’ N Ma, 41. We obtain a “K’ x L"” matrix G' = (Gyyp)vek', ver
as above. Observe that I" has again a distinguished vertex *' = ky, if we denote

Kj = {kj} (i-e., kj is the index of the simple summand C in MnNnM=C).
Suppose now that we are given the principal graphs (I',)I") of N C M with

distinguished vertices x resp. *'. Then we obtain the matrices G, G' via!
7) where Ar (resp. Ar)5) x. resp. Arp = (eyAr = (orare theIf we denote by I'even (resp.

Thyen)

adjacency matrices of the graphs I" resp.

I".

and Taq (resp. I 4,) the vertices with even (odd) distance from * (resp. *'), thenit is obvious that K = even (resp. K' =T(,.,) and L = Todd (resp. L' =TL4q)In particular, K_; = {*} and K,, n > 0, is the set of even vertices with distance< 2(n+1)

from *, whereas Ly, is the set of odd vertices with distance < 2n+1

from%. Similarly for K/ and L!, (caution, the index of Kj, is shifted by one comparedto Kp).

We proceed with showing that each index k € K (resp. | € L) can be viewed asa uniquely determined irreducible N-N (resp. N-M ) bimodule and we show in thenext section that the numbers Gy; are dimensions of spaces of certain N-N resp.
N-M bimodule intertwiners.
Let p € N' 0 Myny1 (resp. ¢ € N'N My) be a projection and let 7, be therepresentation of My, Ma,41 on L?(M,,) coming from the n-step basic construction
for N C M,, (Proposition 2.2). Recall that 7n|n,, is equal to the representation of
Mj,

on L?(M,)

comingfromthe n-stepbasic construction

for M

C M,,

(Propo-sition 2.2, last statement). The Hilbert space m,(p)L*(M,) (resp. mn (q) L2 (My)is then an N-N (resp. N-M) bimodule in a natural way. Namely, since 7n(D)(resp. mn(q)) is an N-N (resp. N-M) intertwiner of the N-N (resp. N-M) bimodule L2(M,,), we have that a m,(p)(£) - b = Tn (p)(azb) = am, (p)(z)b, for allBimodules

andHigherRelativea, be Nand z € M, (resp.
a € N, be M).

Commutants39

a-mn(q)(Z)b= Tn (q)(azb) = amn(q)(x)b, for all

Since m,|p is left multiplication, this action can be rewritten as

a mp (p)(€) - b= mn(a)umn (b)* Jnr (p)(£), a,b€ N, € € L?(M,) and similarly for

mn (q).

Definition 3.3 Letp € N'NMany1,

resp. ¢ € N'NMay,, n > 0, be projections.

We denote by m,(p)L?(M,), resp. m,(q)L?(M,) the N-N resp. N-M

bimodules

with above defined actions and we will call them the reduced bimodules associatedtoNC M. Similarly for M' N\ My, and M’' N My, 11.
Note that one sometimes simply writes pL?(M,,) when the representation isunderstood. Also, observe that the space of left resp. right N (resp. M) bounded
vectors of a reduced bimodule clearly contains 7, (p)(M,) resp. m,(q)(M,) as adense subspace, a remark that will be used later on, when we define bimodule
morphisms on various tensor products of reduced bimodules.
Lemma 3.4 Let p € N'N Mani (resp. p € N' N Ma, ) be a projection. ThenTn(p)L2(M,) is an irreducible
N-N (resp. N-M) bimodule if and only if p is
minimal in N' 0 Manyq (resp. N' 0 May,).

Proof The space of N-N intertwiners of mo, (p) L*(Mp,) is isomorphic to (NV p)'N
PpMa,41p,
ment.

which is equal to Cp, iff p is minimal.

Similarly for the second state-

[J

Lemma

Ma,).

3.5

Let p; and ps be equivalent projections in

Then mo (p1)L3 (My) = ma(p2)L*(M,)

as N-NN' May;(resp. N'N

bimodules (resp.

as N-Mbimodules). Furthermore, if p € NN Ma,41 (resp. p € N' N May) is a projection
which is an orthogonal sum p = Yr pi of projections p; € N' Nn Many; (resp.

pi € NN My), 1 <i <k, then mo(p)L3(M,) = @r_, mn (pi) L2(M,,) as N-N(resp. N-M) bimodules.
Proof If v €N'N Mant (resp. N'N My,) is the partial isometry between p;
and po, then m,(v) gives the desired bimodule equivalence. The rest is obvious.
[J
Next we will show that each k € K (resp. | € L) labels precisely one irreducible

N-N (resp. N-M) bimodule contained in @&,(p5)"™ (resp. @n(pp)"p). This will beaccomplished by choosing a minimal projection in the k-th (resp. [-th) simple
summand of a higher relative commutant and considering the associated reduced
bimodule. The previous lemma shows that this reduced bimodule will not dependon the particular choice of the minimal projection as long as they are in the same
simple summand. What we need to show however is that this reduced bimodule does
not depend on the level of the higher relative commutant, i.e. minimal projectionsindexed by k (resp. l) in any higher relative commutant N’ N My, (resp. N'N
Ms,.), n > 0, give equivalent reduced N-N (resp. N-M) bimodule. It is of course
enough to establish this equivalence for reduced bimodules in two consecutive higher
relative commutants N' N May, 1 C NN Maps (resp. N'N My, C N' NM Mayo).
Lemma

3.6

a) Let p € N'N Ma,1

be a projection, n > 0.

i) (n=0). m(pes)(y) = pyer, for ally € My.
ii) mni1(peents)(y)
Mpa,

n >

= [M

: Nr Ep"(pyes .enfn)en...€1,1.

b) Let g € N' N Ma,

n > 0, be a projection (Mg = M).

for ally€Dietmar

40

Bisch

i) (n=0). m(ge2)(y) = ¢Em(y), for ally € M,.
ii) (n=1).
iii)

m(ges)(y) = qyez, for ally € Ms.

Tn+1(g€2n+2)(Y)
Moy,

n >

=

(M

: NPL

Eyer

for

(ayes ---enfn)en--€2,

ally

€

2.

Proof The proof uses the explicit expression for the representations mx given

in Proposition 2.2. We prove first a). If y € Mi, then mi(pes)(y) = [M :N2E, (pesyfi) = [M : N>pyEn, (esezeresea) = pyer and a), i) is shown. Let(pean tay frt1) byy € Muy, n> 1. Then mni1(pe2n+3)(y) = [M : NI" 2 Ey"

We compute

2.2. Observe that fre: = [M : N|""le,ia...eani2fn€oni3-- enya.

the conditional expectation Ep(peansay fn+1). Let z € Mp4,

then=[M: NI" ltr(zpyesntsensz - €2nt2fn€onts. cent)tr(zpyesn+afnsr)= [M : NI" tr(zpyent2 - - - €2n+1fne2nt1 e€nt2€2n43)= [M : NI" %tr(zpye; ...en fren =[M:

NI"

er)

tr(zEpt (pyey ...enfn)en --. e1),

where we used Lemma 2.3 ii) in the third equality.

Thus mn41(peant3)(y) = [M :ETovine (pyer ...enfn)en...€1 as claimed.
We prove b) next. If y € Mj, then m(ge2)(y) = [M : NEw, (geayfr) =[M : NPqEwm(y)En, (e2e1esez) = qEn(y) and b), 1) is shown. If y € My, thenmo(gea)(y) = [M : NPE, (qeay fo) = [M : NPqyEn,(eaf2) = [M : N°qyEns,(eq(esezer)(esezer)(eseses)) = qyez and the proof of b), ii) is complete.

n>2andy€ M,,,.

(Proposition 2.2).
Mata,

Then mpy1(geant2)(y) = [M : NI"? Epi

Finally, let(geznt ay fut)

We compute the conditional expectation as in a), ii). Let z €thentr(rgesni2yfnr1)=[M: N|"Pltr(zqyesn santa. - - €2nt2 fronts. -€nt2)= [M : NI" %tr(zqyeni2 - - -€2nfnezn -€n+2€2n+2)=[M:

NI"—3tr(zqyes c..enfnén... e2)=[M:

NI*Str(z Eyre (ques... enfn)en -.. ez),

where we used Lemma 2.3 iii). Thus m,41(ge2nt2)(y) = [M : NPL Eye...€nfn)én ...e2 as claimed.

Theorem 3.7 Let p € N' N Mani (resp. ¢ € N' N May)
Tn, Tnt1 be as in Proposition 2.2 and let Mon 1 C+? Map
part of the tower associated to N C M. Thenas N-N

resp. N-M

2

(M,)

NN
=

TQ)
LA (My)

2

n(p)L

(ques

O

N-M

Tn+1(Pe2n43)L

2

(Mra)

Tapa(geznt2) LAM)

bimodules, n > 0 (My = M).

be a projection, let
C28 Moa,3 be a

Bimodules

andHigherRelative

Commutants

41

Proof We will use Lemma 3.6 to define explicit bimodule isomorphisms. These
maps will be related to the Fourier transforms defined in Definition 2.16. Let us
begin with showing the equivalence of the N-N bimodules.
Suppose first thatn > 1.

Define a linear map T : Tni1(peoni3)L?(My1)antl
T (mp s1(peanta)(@). = [M : NI*F
Epp n

—

later extended by continuity to all of mm, 1(pean+3)L?(My41).
tation,we will omit

hats as usual.

wa(p)L3(M,)

Recall thatby

T will) be(pyer ...enfn)", y € Moy.

To simplify the no-M, 1, = M,e,t1€,...e1M.

Thus,

ify =ae,i1...e1b, a € My, b € M, we get Ep, (paepyi€n...e1bey...enfn) =Ey, (paEn(b)enyr...e2e1€2...€n41fn) = [M : N] "En, (paEn(b) fr) = [M:N=Or(p)(aEn(b)).

Hence

T(mn+1(Pe2n+3)(Mny1))-

Tn (p)L*(M,)

is asubspace, which is clearly dense. Note that e,i1f, = faént1 = fn (see Proposition 2.1). Recall that if w € M,41, then there is a unique w’ € M, withweny1 = Wept1, namely w = [M : N|Epy, (wenq1).
Thus we,= [M :N]Eum, (went1)ent1.

This will be used in the computations below.

Let now z,y € My,1. Then, using Lemma 3.6, we calculate the inner product,(Tn+1(pean+3)(E),
Tni1 (Penta) (D))= [M : N*"r(ey ...en

(Em,

(pyer

enfn)) En,= [M : NP" tr(en(En,,, (yer... enfr)ent1)(pre; co enfn)en...€1)(En,(per... enfr)ens))

Moni
Mop41
3n+3 tr(en (Ex.
=— [M:N](Bp?
(pyer ...enfr)ent1)ens1)*(Be (54 per enfoonsr) ens)=[M: NP3tr(enent1 Ey(pyes .. cenfn)

Eni"(prey .o-enfn)ent1)

1

Monta
= [M :. Nj 3n+1 tr(Ey""(pyer . . . en fn) + EpMan (pe_enfn))-

Thus T extends to an isometry 7,11 (pegn+3)[2 (Mnt1) — 7, (p) L?(M,,), which
is onto as seen above. T' is an N-N bimodule morphism because,T(a- Tni1(Peant3)(d) -b)= T(ns1(peznts)(ayd))= [M : N72ntl Ej, (paybe; ...en fn) A= [M : N)*2* (aE, (pyes . . . en fn)D)"=a

T(mn+1(pean+3)(9)) - b,

for all a, b € N. This shows the equivalence of the

N-N bimodules in the theorem

for n > 1. If n = 0, we define a linear map T' : m (pes)L2(M1) — mo(p)L2(M)by T(mi(pes)(®)) = [M : N|2Euy(pye1)”, y € M;. Since M; = Me; M, we
have

for

y =

aeib,

a, b €M,

that

Ey(paeiber)=

Ep(paEn(bler)

=

[M

:

N]7'mo(p)(aEn(b)) (fo = €1). Hence Em(pyer)” € mo(p)L*(M), for all y € M,
and T(r (pes)(Mi)) is clearly dense in mo(p)L*(M). If x, y € M;, we computeBischDietmar42(using Lemma 3.6),= tr(ery™prer) = tr((pye1)* (pze1))(m1 (pes) (2), m1 (pes)(@)=[M: N2tr((Em(pyer er)

Em(pzen)er)=[M: Nltr(Enm(pyer)* Em(pzer))bimodule morAs before, T extends to a surjective isometry, which is an N-N
N-N bimodules in thephism and establishes therefore the equivalence of the abovecase n = 0.
N-M bimodWe proceed with the proof of the equivalence of the reduced
3.6, we have twoules as in the statement of the theorem. According to Lemma
Let us start with thespecial cases n = 0, n = 1 and the general case n > 2.

L? (My)) = mn)D2 (Mnt1)
general case. We define a linear map T': Trni1(qeans2byT(mns1(ge2n42)(@))

=

[(M

: N)%

Epp

(que: . cenfn)™

y €be later extended to all of Tny1(gezn+2)L?(Mny1) by continuity.
M,eni1...eaM;, we have for y ==(qaent1 ...exbes...enfn)
Ep

T

will

Since Mp1

=

Mpta.

aepy1...€2b, a € Mp1, b € My, that=ens1fn)Eu, (qaEy (bent: -- - e3e2€3.

we get that
[M : N]™"*'Ey, (qaEn(b) fn) = [M : N]=?"m,(q)(aEn(b)). Thus
clearly dense.
T (nt1(geans2)(Mns1)) C mn(q)L*(M,,) is a subspace, which is
Let z, y € M1,

and compute, using Lemma 3.6, the inner product,(Trns1(gezn+2)(E), Tni1(ge2n+2)(9))=[M:N])*~2tr(e; .. cen (EM= [M : NJ**tr(en (EM... (qye2

enfn)ent1)(gre coienfn)en..- ez)

En,

En,

(qzes... enfn)ent1)(Exe? (ques... enfn)ent)ent1)”

r (Ep= [M : NP***2t(enEy(ques .. cenfn))(Ere? (qzez... enfrn)en+t1)€nt1)=[M: NP"2tr(enent1 Eye (ques - cenfn)= [M : NP tr(EM2 (qyer...enfn)* Eni

Epi(qzes co enfn)ent1)(aze2

en fn))-

mn (q)L*(M,), which
Thus T extends to an isometry Tnp1(qeznt2) L*(Mny1) —since
is surjective as seen above. T' is an N-M bimodule morphism,
T(a : Tns1(ge2ns2)()’ b)= T (nt1(gezn+2)(ayd))=[M: N|% Ep, (gean+20ybes

en fu)"= [M : N|% (aE, (geant2ves - - enfn)b)"= a T(mnt1(ge2n+2)(D)) +b,

Bimodules

andHigherRelativefor all a € N and b € M.

Commutants

43

This establishes the desired N-M

bimodule equivalences

for n > 2. If n = 0, define a linear map T : m1(ge2) L2(My) — mo(q)L(M) byT(mi(ge2)(9)) = ¢Em(y) = mo(q)(Em(y)), y € Mi. By Lemma 3.6 b), i), T extends
to a surjective isometry m1 (ge2) L%(M:1) — mo(q) L?(M), which is an N-M morphism
by a similar computation as above. This completes the proof in the case n = 0.

Finally, the case n = 1. Define a linear map T : m2(geq) L2(M3) — m1(q)L?(My)by T(m2(qes)(d)) = [M : N13 E32 (qye2)", y € M,. Note that Ey (qyf) =[M :N]1E}f* (qyes), for all y € M,. Hence Ey (qyez) € m1(q)L%(M,), for all y € My,
by a similar argument as above, using Ms = Mies M;. Thus T (m2 (ges) (My) C
71(q)L?(M) is a dense subspace. Let z, y € M; and compute the inner product(m2(qed)(2), m2(qea)(§))= tr(eayqrez)

=

[M

:

NPtr((Ewm, (que2)es)

En, (qzez)ez) = [M : Nltr(En, (qye2)* En, (qze2)) (Lemma 3.6). Hence T extends to an isometry m(gez)L*(M:1) — mo(q)L%(M), which clearly onto and an
N-M bimodule morphism since T'(a - m2(geq) - b) = T(m2(gea)(ayb)) = [M :N]% En, (gesaybes) = [M : N]zaEw, (geaye2)b = a-T (m2(ges)(§)) -b, for alla € N,b € M.

Thus the proof of the theorem is complete.

Remark

3.8

The

above proof simplifies

if

[OJ
p =

1 or ¢ =1.

Let us con-sider p = 1. Since En, (fn) = [M : N] "eps1, we get for y € Myyq that
Tnt1(e2n+3)(y) = [M : NJ*"ye1...enEum,,,(fa)en...e1 = yer (Lemma 3.6). It
is then easy to see that the map T : mny1(eany3)L? (Mp1) — L3(M,), definedin the proof of the previous

theorem,

N|*F

My yi. Similarly in the case q = 1.

Ey, (yer ...ent1)”, y €simplifies

to T(7ns1(e2n+3)(D))=[M

Corollary 3.9 Let N C M be an inclusion of II, factors with finite index,
and letT' be the principal graph of C = NNN C NNnM c NnM; C ....
Then there is a bijection between Teyen (resp. Toga) and the set of equivalenceclasses of irreducible N-N (resp. N-M) sub-bimodules of @,,~_, nL*(M,)n (resp.
@D,.>0 NL (Mp) ), where M_y = N, My = M. More precisely, if a € Teven (resp.
B € Toda), choose a minimal projection p, (resp. qg) in the a-th (resp. [(3-th)
simple summand of N'N Map+1 (resp. N'N Ma, ), where n is such that a (resp. 3)
occurs as the index of a simple summand of N'N May(resp. N'NMs,,). Then the(equivalence class of the) irreducible N-N (resp. N-M) bimodule 7, (ps )L?*(M,)(resp. Tn(qs)L%(M,)) does not depend on the choice of n and the choice of theminimal projection in the a-th (resp. B-th) simple summand of N' N Many:(resp.

N'N Ma,,) (where mm, is as in Proposition 2.2).
Proof The fact that 7, (ps) L?(M,,) does not depend on which minimal projection one chooses in the a-th simple summand of N' N My, ; was shown in Lemma
3.5. Furthermore, this bimodule is irreducible by Lemma 3.4. If we choose a minimal projection p, in the a-th simple summand of N' N Ms, ; and a minimal
projection p, in the a-th simple summand of
NN Mas, ,1, some n < m, then p,
is equivalent to py€2,13€2n+5-.- €2nr2k+1 (for an appropriate k), which is a minimal projection in the a-th simple summand of
N' N Ms,1. Applying Theorem

3.7 k times, we obtain m, (pa) L*(M,) = Tm (Po)L? (My) as N-N bimodules. Notethat by Propositions 3.1 and 3.2 we get all the irreducible N-N sub-bimodules of
D,.>_1 ~L?(M,)y in this way. The statement about the N-M bimodules is shownin the same manner.
[J:Dietmar

44

Bisch

for theAll the statements in 3.4-3.8 and their proofs can of course be rewritten
M-M and M-N bimodules.
higher relative commutants M "My, and M'"Mzni1,
representation
Note that we work then with the same representations mn, since theto Ma,
restricted
M,,,
C
N
of
ion
construct
basic
the
from
comingL?(M,,)
onof My,
M, (Propositionis equal to the one coming from the basic construction of M C
for the sake of
2.2). Let us state the result analogous to the previous corollarycompleteness.
finite index,
Corollary 3.10 Let N C M be an inclusion of II, factors with
C MnNM C....
and let I" be the principal graph of C= M'NM C M'0NM,;of equivalence
Then there is a bijection between T'. (resp. Tiga) and the set
L2(Mpy)wm (resp.
classes of irreducible M-M (resp. M-N) sub-bimodules of D0 um®,50 MLA (My)N), where Mg = M. More precisely, if @ € To, (resp.

8 €Thqq),

simple summandchoose a minimal projection po (resp. qg) in the a-th (resp. B-th)B) occurs as(resp.
a
that
such
is
n
where
Many),
0M'(resp.
My,N
M'
ofThen thethe index of a simple summand of M' N My, (resp. M' 1 Many).
L? (Mp)(equivalence class of the) irreducible M-M (resp. M-N) bimodule Tn (Pa)of the(resp. m.(qe)L*(My)) does not depend on the choice of n and the choiceminimal projection in the a-th (resp.
M' 0 Many).

of M' N Ma,

B-th) simple summand

By Corollaries 3.9 and 3.10 it makes sense to talk about the N-N

N-M, M-N) bimodule & € Leven (resp.
is the terminology we will use below.
bimodule.

(resp.

(resp. M-M,

8 € I ven> ¥ € Todd, 8 € Thyq), which

We discuss next the conjugate of a reduceds andProposition 3.11 Let p € N' 0 Mapi1, ¢ € N' 0 Ma, be projection(2) In, T €let vn : N'N Many1 — N' 0 Mania be defined by mn (n(x) = Jn
N' 0 Mani (see 2.4). Then,
NNnp) L2(M,)
_ __

TL?

Thus

Dla.

(My)

= Tn(m(p))L?(M,)
M-N=

ma(1a(e))L*(M,)

conjugation ~ defines an involution on Coven

as N-N bimodules and,as M-N bimodules.
andand a bijection between Todd

ted by the
Proof We will show that the bimodule equivalences are implemen

for allmodular conjugation J,, : L?(M,) — L*(M,). Recall that 7a (p)(2) € M,
re,x € My. Then Jo (ma (p)(2)) = Jn7n(p)Jn(2") = Tn (Yn (p))(&"). Furthermo( Tn (7a ()(@)),

In (72) (D))) 1. (on (py £2000)) ) = trar, (7a (1) (0) (TaTn ((0))(@)— trag, (Tn (3 (P)(")(P)(2))")= (ma (@)(@), Ta (P)(E)) 1 (pyr2(Mn) = (tn (0)(@), 7 (0) 9) sorry
where the third equality follows from Lemma

for all z, y € M,,

2.4 iv).

Thus

LetJ,, is a linear, surjective isometry from Ta) L2(My) — Tn(va(p))L? (My).
=
(aT)
(p)
(mn
Jn
=
br)
()
a, be N, then J, (b-m,(p)(®) - a) = Ju(a*- (0)((ra(p)(@*2b*))")"
*

«\)*

A

= ((@* ma (P)(@)67)7)"
*

*\*

A

= (b(mn(p)(2)) a)"
*

A

=b-Jn

(ma (p)(2))a
NAY

Bimodules

andHigherRelative

Commutants

45

for all z € M,,. Thus J, implements an N-N bimodule equivalence and the first
identity is shown.
To prove the second one, observe that m,(vn(q)) = Jumn(q)Jn € Jumn(N'N
Mop)Jn = mo(M' 0 Many), since Jomo (M)' Jn = mp(May) by Proposition 2.2.
Thus 7, (y,(q)) is an intertwiner of the M-N bimodule L*(M,,), so that the second
equivalence indeed makes sense as an equivalence of M-N bimodules. As in thefirst part, one now shows that J, implements this desired equivalence.
The first bimodule equivalence shows that, given the N-N bimodule a € I'gyen,
the conjugate N-N bimodule is again a reduced bimodule and hence labelled bysome @& € even. This map is well-defined by 3.9 and clearly an involution. Similarly,
since v,(q) € M’'N Map 1, the conjugate of an N-M bimodule 8 € T'oqq is an M-Nbimodule indexed by a 3 € TI ;,. Again, this map is well-defined by 3.9 and 3.10
and clearly a bijection.
O
It goes without saying that we also have an involution on I',,,. The conjugation(or contragredient map) v € Toqa — 7 € I'Lyq gives an identification of T'oqq andI" 44 and induces a permutation on the even levels as we have shown above. Observethat the fact that the conjugate of a reduced N-N resp. N-M bimodule is indexedby the vertices of the principal graphs follows of course already from Proposition 3.1
and the Remark 1.17. Let us point out that Proposition 3.11 in conjunction with
Theorem 2.6 (using A = N, B = M,) gives an explicit formula for the conjugate
bimodule by calculating a projection in a higher relative commutant.
Let us now define the (full) fusion algebra associated to a subfactor. Observethat this can be done without knowing what exactly the result of the relative tensor
product of two reduced bimodules is. Namely, if @, 8 € Teven (resp. Tyeq), then aand (is an N-N (resp. M-M) sub-bimodule of 5 L?(N)n, (pp) (resp. mL*(M)n,(Pp)¥) for some k > 1. Hence af = a ®n 3 (resp. a ®u 3) is a sub-bimodule of(pp)?* (resp. (pp)**) and can therefore be decomposed as (N-N case)

af

=

>

YET evenN, asswhere the integers Nos denote the multiplicity of the irreducible N-N bimodule~ in the N-N bimodule ap, i.e., N,s = dim Homy_n(7, @8). Similarly for M-M

bimodules with Teven replaced by I'l...
Unfortunately, we cannot multiply two N-M

bimodules«, 8 € ['caq-

However,the products @ ® y 3 and a ® ps 3 can be formed and will, with the same reasoning

as above, be sub-bimodules of (5p)* resp.

(pp)*, for some k. Hence they can be

decomposed into an integer linear combination of the irreducible bimodules in I...
resp.

Teven-

Rather than giving a definition of an abstract fusion algebra and identifying
the one coming from subfactors as such, let us stay with subfactors and define what
we mean by the fusion algebra associated to a subfactor.
Definition 3.12 Let N C M be an inclusion of II, factors with finite index
and denote by (T',T") the principal graphs as above.

Dietmar

46

Bisch

€
i) (N-N part) Let ZT even be the formal Z-linear combinations of the set {alaLeven}. We define a multiplication on ZYeven by

aB=ad@nB=», Nim
YET even

where N]; = dim Homy_n (v.28), a, 8, 7 € Leven, and extend it to all of

ZT even by linearity, respecting the distributivity law.

ii) (M-M part) Let ZT',, be the formal Z-linear combinations of the set {aja €Ten}. We define a multiplication on ZT en bY

aB=a®upB=

)
YE

My,
even

of
where M], = dim Hompr—m (7,28), a, B, v € Then, and extend it to all
ZI. by linearity, respecting the distributivity law.
ZlevenRecall that the contragredient map induces an involution ~ : ZT even —and ~ 1 ZT,op — Zhen.

nem = (ZT evens ZT fens) the two

We denote by § =(or fusion
Z-algebras with involution defined as above and call § the fusion algebraring) associated to
Observethat

N C M.
we

have

two

selfcontragredientdistinguished(=x L2*(N)n) and *' (=n L?*(M)py) in Teven resp.

Tiven:

bimodules*

Clearly, Zl even andbimodule
ZI". are unital (the units are * resp. «'), associative algebras (sincesquaremultiplication is associative !) over Z with a natural trace defined using theusual wayroot of the index of the bimodule (i.e., of the associated subfactor) in the(see remark after Definition 1.1). We can replace Z by C to get complex algebras.

If we let L (resp. R) be the left (resp. right) regular representation of §, i.e.,Lo(8) = of (resp. Ra(B) = Ba), a, B € Zl'eyen (resp. ZI',,,), then the matrix

representation of L,(resp.

Ra), @ € Teyen, in the basis B = {77€ Teven} isand similarly for M-M
and Ro = (Na), ger...
given by Lo = (N26), ser...,bimodules and TI, (7 is the row index and 3 the column index of the matrix).
Observe that NJ, = NZ. Furthermore, since NJ; = NZ, we have Ls = (La)tand similarly Rs = (Ra)? (as matrices as above). In particular, the N-N bimodulea € Toven is selfcontragredient iff the matrix L,, is symmetric.

Similar statements

hold for T',,, and M-M bimodules.

andWe let ZT'oqq be the formal Z-linear combinations of the set {aa € Toda}
similarly for ZI",,. We define a multiplication ZT oqq X ZT 34 — Zl even by

af=a®upB=D

Pls

Elevenwhere P); = dimHomy_n (v.08), a € Toad, B € I

4a» 7 € Leven, extended bylinearity as before. Similarly we define a multiplication : ZT 4g X Load = Llgyen
byaf =a®nN f=>
Qs
YET Grenwhere Q);by linearity.

= dimHomyum (v,aB),
Furthermore

we have

ZT 44 — ZT 44, Zlodd X Llyena € Tygproducts

B € Toad, 7 € Deven, extended

ZlevenX Zl'oaa—

ZT oad,ZY en %— Zlodd and ZI"j4 X ZT even — ZI',44 definedBimodules

andHigherRelative

Commutants

47

in a similar way. Thus we can regard for instance a € ZI'oqq as a matrix L, :ZI"44 — ZT even (the matrix representation is with respect to the obvious bases) or
as a matrix Ry : ZI 44 — ZI, etc. This point of view allows us to perform fusion
rule calculations by multiplying matrices and solving matrix equations (Section 5,
see also Bisch and Haagerup [1996]).
Definition 3.13 Let N C M, T', TV be as above. We call Fran = Frun(N CM) = (ZT even, LT lyons ZT oad, ZL 44,7) with involution and various multiplications(“even x even”, “even X odd”, “odd x even”, ‘odd x odd” and “odd x odd” )defined as above the full fusion algebra associated to N C M. The
N-N and M-M
part of Fran (what we called the fusion algebra associated to N C M in Definition

8.12) is called the even part and the N-M(resp. M-N) parts are called the odd

part of run.
In the next lemma we collect some of the properties of the structure constantsof the fusion algebra associated to a subfactor.
Proposition 3.14 Let N C M be an inclusion of II, factors with finite index.
Let (N35) a,8,7€T even be the structure constants of the N-N part of the fusion algebra“3 NCM-Theni) (unit) NJ« = NJy = bay, for all a, vy € Teen.
ii)(associativity)> NN= YD Nas,
7,€€l evenv¥,€€ evenfor all a, B, 7 € Teven.
iii)(involution) NJ, = Nga for all a, B, ¥ € Teven-iv) (Frobenius reciprocity) N] 5 = No = NZ, for all a, B, ¥ € Leven.
Proof The properties of the structure constants follow immediately from theindicated properties of the algebra. iv) is a reformulation of Corollary 1.21 iv)-

vi).

O

It is clear that similar statements hold for the M-M part of the fusion algebra
and that various compatibility conditions between the
N-N, M-M, M-N and N-M
parts imply conditions on the other structure constants as well. We will leave it to

the reader to list them. Note that we will usually identify I', jy with ['oqq using thecontragredient map.

We end this section with a definition.

Definition 3.15
hd

Ng

=

(N

3 5),68€

The matrices

even(resp.

NY=(Nja)7,B€T

even )sac

Leven,o ME) = (M2) per, (resp. ME) = (Mga) per...) @ € Dien,lr

are called the fusion matrices associated to

N C M.

Note that the principal graphs I', I" of N C M do not determine the fusion alge-

bra completely in general (see, for instance (Bisch [1994(b)], 3 + v/3 example)) andadditional information is needed to calculate the fusion matrices (namely the “orthogonality information” contained in the commuting squares formed by the higher
relative commutants). Let us remark that the fusion matrices can be interpreted
as principal graphs of reduced subfactors - more on this in Section 5.

Dietmar

Bisch

48

ants
tensor products and higher relative commut
reduced bitwo
ofproduct
tensore
We show in this section how the bimodul
d as a product of projections inmodules associated to a subfactor can be recovere
will identify the multiplicities Gri
the higher relative commutants. Furthermore, wespaces of bimodule intertwiners.
ofons
dimensiasgraphsoccurring in the principal
Bimodule

4

Recall that the simple summandsof N 'N Map

are indexed by K,, and those

representation of May, Mani
N’ N My, by Ly. As before, we denote by 7, theof
N C M, and Gy, are the enof
tion
construc
basic
the
from
on L?(M,) coming
I' (see the discussion after
graphlprincipa
thetoedtries of the matrix G associat
Proposition 3.2).
be a projection. Then,
Proposition 4.1 Let q € N' 0 Ms, C N' 0 Map

rn (@ LA (My) ®pr LAM)

n= (mn(0) ®t id 12 (ary) (LP (Mn) Ont L*(M))= mag2 )(My)

5 0 minimal projection in the l-thas N-N bimodules. Thus, if g = a, | € Ln,
simple summand of N' 1 Man, then2NN
mn(@) LA2 (My) = ED Grmn(pr) I (Mn),k€EKn

as N-N

bimodules, and hence
dim Hompn-N (mn (pr) L* (My), Tn(q)where pike Ky.

in the k-th simple

is a minimal projection= Gi,

L?(M)n)

L* (My) Omsummandof N "Nn Mansi,(q) L? (My) is indeed an N-N
Proof Observe that since ¢ € NN Man, 7
es follows from Proposition 1.15
bimodule. The equivalence of the first two bimodul
e is immediate by Lemma 1.14.
iii). The equivalence of the first and third bimodul
Since the [-th simple summandion.
proposit
theofnt
statemetheNow let ¢; be as insummand of N "0 Mania
simple
k-th
the
in
Gi:
of N' N Ms, sits with multiplicity
equivalence is obvious by Lemma(by definition of the principal graph), the third(Mp) aswe mean Gy, copies of the bimodule on (pk) L2)
L2 (My)3.5 (by Gx mn (prirreducible N-N
are
Kn,
€
k
)
(Mn),
L*
(pk
Tn
es
bimodulusual). Since the reduced[1
the third bimodule equivalence.
bimodules, the last statement is immediate from(nL?
Homp.n—Mp)m)Corollary 4.2 The embedding from Homn-m(NLAH(
T € N' 1 Ms, coincides with the(M,)N), given by Ton (2) = mn (2) ®m dy L2(M)ns

l graph T.
inclusion N' 0 Man C N' NM Map 1, given by the principa
Proposition 4.3
Ton (p)

L2 (My)RN

be a projection, thenLet p € N' Nn Ma,
L*(M)

nm

=

Tn(P)

®N

id

L2 (Mm(L? (Mn)

Nn

L? (M))> 7541p)
L* (M11)is a minimal projection in the k-thas N-M bimodules. Thus, if p= pk, k € K,,,
simple summand of N' N Man 1, then2

Tong (Pr) LP (M1)

NM

=

PD
l€Ln+1

2

Gri Tna1(q) L* (Mn)

Bimodules

andHigheras N-Mbimodules, and hence

dim Hom pn.

Relative

Commutants

49(mn41(@)
L* (Mp1), 7 (pk)LP (Mn) @n L*(M)pr) = Ga,

where q; is a minimal projection in the l-th simple summand of

N' N Ma, 1.

Proof The equivalence of the first and the second bimodule is obvious by
Proposition 1.15 iii). We will show the equivalence between the first and the third
bimodule. Recall that 7, +1(p)(yent1...€12) = Tn (D)(Y)ent1 ...e1z, for ally € M,
and z € M by Proposition 2.5. Also, recall that 7,(M,) is dense in the space of rightN-bounded vectors of m,(p)L?(M,).

Define a linear map T : m,(p){(M,)

Tn41(P)L? (Mp1) by T(ma(p)(9) ® 2) = [M : NJ™HD 2my € M,,
have

2 € M

for y;[M : NJ"€(as usual we will omit

Mp,

© M —

(p) (yeni ren.

“hats” in the calculation below).

€12),

Then wez, €e M,i=1, 2:(Tng1(p)(y1€nt1 -- €121),
Tn 1 (P) (Y2€ns1 - -- €122)) 12 (ar)= [M : N]"*ltr(z5er ... entr (mn (P) (2) Tn (0) (Y1 ) Engi - - €121)= [M : N]""Ltr (mn (p) (12) Tn (D) (W1)Ens1 - . - €12173€1 - . . €nt1)= [M : NI" tr (mp) (y2))* mn (p) (41) En (2123)€nt1...€2€1€2...

€nt1)= [M = Ntr((m (0) (42) 7 (0) (31) En (2125 )ent1)= tr((mn () (42) * 7 (0) (1) En (2227)=(mn (0) (41) (21, 22) N, Tn (D)(¥2))Las)= (mn (p)(y1) ® 21, mn (P) (y2) ® 22),
which implies that T is well-defined, factors through N(.

., and induces an isometry

from 7, (p)L?(M,) ®n L*(M)ar t0 mpni1(p)L2(Mp41), which is clearly onto sinceM, 1 = span{aenii€e,...e1b]a € M,, b € M} (see proof of Proposition 3.1) isdense in L2(M,, 1). It is easy to see that Tis an N-M map, so that the equivalenceof N-M bimodules between the first and the third bimodule is established.
Now let py be as in the statement of the proposition. Since the k-th simple
summand of N' N My, 1 sits with multiplicity Gy; in the [-th simple summandofN'N May42, I € Ly, the third equivalence is again obvious by Lemma 3.5.

The last statement follows immediately from this equivalence and the fact that the
reduced bimodules 7,1 (q;)L?(M,41) are irreducible N-M bimodules.
[J

Corollary 4.4 The embedding from Homy_n(nL*(M,)n)—

Homn.p (nL?

(Mp1) uM), given by mp(z) — 7a(z) ON dy r2(ar)s, T € N' 0 Many, coincides
with the inclusion
N' N Map, C N' 0 Maya, given by the principal graph IT.
Remark 4.5 i) Propositions 4.1 and 4.3 characterize the entries in the matric G = (Gr) perc (K = Teven, L = Toaa) as dimensions of certain intertwiner spaces and we could of course define the principal graphs using these dimensions. Clearly, similar propositions hold for the principal graph T' and M' 1 Ma,
M' NN Mapi1, M' 0 Ms, 2 - the proofs are the same (use again Proposition 2.2,

Dietmar Bisch50

last statement).

Namely,we havethat the inclusion Homapm (mL (My) ym)—

given by mn(x) — n(x) ®a id, 12(aryy, and the inclusionn(Mm LE(M)N),Homa
rn Nn) — Homar
Homa(mu LEM)(mL? -ar(Mya) um), given by ma(z) — mo (z) Nn
idy L2(M)y» coincides with the inclusions M' N Ms, C M' 0 Ma,yq, resp. M'N
In the notation introduced in Definition 1.20, the last
Mypy1 C M' 0 Mapyo.
statements of Propositions 4.1 and 4.3 (also formulated for I) read
Gs

=

Gls

={v,8'p)(7, 8p)=(vp, 8),YE

Levens6 € Toda,= (vp, ¥),Y€ Tovensy

6" €Toaq-

CN NM, and M' 0 Mani Cii) The multiplicities of the embeddings M' NM,
N'N Mapn41 can be described in a similar way. More precisely, these inclusions coapr (mL? (My) um) — Homa (nL2(Mp) um), given
incide with the embeddings Homby mo(x)—idy ram),

OM T(x),n(mm L2(My)N)and the one from Homato

Hompn.n(NL2(Myp)N), given by mn(x) — idyr2(ar)y OM n(x) (i.e., multiplication
by nL2(M)p from the left). The multiplicities can then immediately be calculatedusing Frobenius reciprocity. We find that the multiplicity of the a-th simple summand of M' 0 My, (resp. M' N Ma,y1) in the B-th simple summand of N' Nn Ms,(resp. N' NM Many1), a € Tho, (resp. a € TV 44) B € Toga (resp. B € Teven) 18

given by (pa,B) = (a, Bp) = G, 5, since f € Tpyq (resp. (pa, 8) = (8,ap) = Gpa
since & € T'oaq) (see i) and Corollary 1.21), where p= NyL*(M)u

as usual. Thus

we have recovered all the multiplicities of the embeddings,
NM,U
MnNM,,

C

N' NM

C

C

U
M nM

C

N' NM MypyoU
M
0 MaioObserve that the role of theas dimensions of spaces of bimodule intertwiners.
conjugate ~ (or contragredient) map is displayed nicely in this bimodule picture of
the above embeddings.
We will now show how the bimodule tensor product of reduced bimodules canbe determined by computing certain products of projections associated to these
reduced bimodules. The idea of how to do this has been known to experts for sometime (see also Goodman and Wenz! [1990]), we were however unable to find any
references in the literature. We will obtain a procedure that allows us to calculate
the fusion algebra associated to certain subfactors quite easily (as for instance thesubfactors in (Bisch and Jones [1995])). Let vx : N' N Magy1 — NN Mag, be
Jk, = €
the surjective, linear *-antiisomorphism defined by mx (x(x) = Jxme(x)*N’ 0 Magy (apply Lemma 2.4 with A = N, B = My). Then shagi1 = Yor+17%
N'NMagt1 — Mj 1M Myyy3 is a trace preserving, surjective *-isomorphism (by theremark before Lemma 2.9 and Corollary 2.12 with A = N, B = My, By = Mag

and Bs = Myky3), which we call the ((2k + 2)-) shift on N' 0 Mag41.

Note that

A= N, B= M;. in Theorem 2.13,
shy 1 is a shift as in Definition 2.14: if we letthen shor is the 2-shift which we denoted there by Sp (2.13 vi)). Similarly, we geta trace preserving, surjective *-isomorphism shox = Yorvk : M "MMay — Mj NM
A= M, B = My), which(by the remark before Lemma 2.9 and Corollary 2.12 withis again a shift as in Definition 2.14. Note that Yor (Z) € Mj, MN My for all z €p Yor |MnMas
NM Mar).
r Jo = mar(Ma(MN Mag)
0 Moy, since mar (Yok (2) € Joxma
M'

is therefore a linear, surjective *-antiisomorphism M’ NM My, — Mj, N My.

Bimodules

andHigherRelative

Commutants

51

Theorem 4.6 a) Let p, ¢ € N' N Ma,y1 be projections and let
N'O Many1 — Mj, 1 0 Mypnys be the shift as defined above. Then,
2

2

NN

bimodules.

Furthermore p shoni1(q)Tn (p)L* (My) @n Tr(q)L* (Mp)
as N-N

trace oS.

(p shana (9) = trp,

7on41(p shant1(q))L*(Mani1)€ N' N My, 3 is a projection with

(PItrMaTarr

ys (9)-=

trMypbimodules.

a (pr2n(9))= tra, (Ptr,

as M-M2bimodules.

beTon (Yen(9)

L2 (May)

Furthermore, pyan(q) € N' N My,11 18 a projection with trace(y2n(9)).

c) Let p, g € M' N Ma, be projections and let
the shift as above. Then,
Tn (P)L?

0 Mynyqev
2
2=
Tn (P) LA (My) nm Tn (vn (q))L? (My)N-N

Tn (P) LH (Mp) @um Tn(Q)LAH (My)as N-N:2=b) Let p, ¢ € N' 0 Ma, be projections and let yon, : N' 0 May, — Mb
defined as above. Then,
2

shg,,

(Mn) @nr2
mn (q) LP (My)sha, : M' 0 My, — Mj

MM=

Furthermore, p sha, (q)

1 My,

be

2
Ton(p shan (q))L* (May)€ M' 0 My,

is a projection with trace

tra, (p shan(q)) = tras, (P)tras,, (9)d) Let p, ¢ € N' 1 My, be projections and shoni1 : N'N My, — Mj, 0 Mantothe shift as above, restricted to N' N Ma,,. Then,)
Tn(P)L? (Mn) Nn mn(Q)L* (Mn)

M-M
=
M-M
2

as M-Mbimodules.

Furthermore, v,(p)mn(1n(p))L*))(My) ®N mn(q)L*(M,)

mont1(Vn(p)shan11(q))

shan+1(q)be

L*(Many1)€ N' N My,10 is a projection

with trace ta,., (Yn (P) Shon+1(q)) = tTM,,4, (Vn (P)) 0M, (9).
Proof Let us start with the proof of a). Observe that p and sha, 1(q) arecommuting projections, so that p shop+1(g) € N' NM My, 3 is again a projection
with trace as stated in the theorem (by Corollary 2.12). We define a linear map
T : ma(p)(Mn) © mn(@)(Mn)—

T2n41(pshont1(q))L* (Many)by T (mn (p)() ®

Tn(q)(§)) = [M : N]®+D
2m, 1 (p shont1(9))(@ fay), 2, y € My, fn the Jones
projection for N C M, (Proposition 2.1 i)). The fact that T is well-defined will
followfromM, fo M,a computation

wef span{z fry |x,of theinnery € M,}

that

My, 1 =and that the Radon-Nikodym

products

below.

Recallderivatives

satisfy (mn (q)(91),Tn (@)(F2))~v = (G1, mn (a) (2)nv, Y1, y2 € M, (Proposition 1.15ii)). Furthermore, we haveTon+1(Pshon11(9)) (x fry)

= T2n4+1(V2n+1Vn(9))T2n+1(P) (2 fry)

= Jont172n41(1 (0) *) Jon+1 (PZ fry)=

Jon+1(((@)*y*= (Pz fry(@))",

faz*p)")

Dietmar

Bisch

52

Let zs, yi € Mn, i =1, 2. We compute

since Ton+1| Many, 18 left multiplication.

( Tn (P)(E1) ® Tn (q) (51), Tn (P) (£2) ®

(9) (92) == (00 (0) (31) (Ta (9) (§1)> Tn (0) (2) 2 Tn (PY (22) p21)

7 (P) (82)))ar,= (ra(21)B4" (31 (70) (02)))= tra, (23m (P)(@) EN" (41 (1 (9)(82))))that ma(p)where we used in the third and fourth equality
intertwiner. Next we calculate,is a right N-module( T{n(p) (81) © 7a) (3), T (RP)(22) & Tn(@) 2) gaat,: NH(mona (p shan

(M

: NJ

((pz1

fan

(0),

(@2fn¥2)")

= (M

: NI"

MM,

00 (43 fn 2pT1

fuy17m(0))=[2 (ayn 1)

1(9))(@1 fn¥1)s T2fa¥2)= (ML2( Manin)

Wn (@)93 Fn)
fot021= [M : NJ" Ltrag,, oo (@5= [M: NPM,= [M : NI"(z3mn(p) (21) fot (mq) (2))" fn)traps (z3mn (0) (21) EN" (v1 (7n(q)(92))*) fr)= tra, (23m (P)(@1) EN" (11 (mn(@)(32))%))= (map) (E1) ® Tn (a) (#1) Tn (Pp) (£2) ® Tn (9) (92),= Tn(p)(z1) fr and W(@Qys fn =where we used in the fifth equality that pzifnii) and iii) (applied to A =
2.4
Lemma
by
fn
y2))*
Tn(Y(@)(¥3)fn = (mn(q)(
T'
N,

B

=

M,,

is well-defined

B,

and

=

Msi,

induces

&1

=

fn).

an isometry,

Thus

the

above

still denoted

definedby T

linearmap: m,(p)L*(M,)®n® ma (9)(9))) = Tan 1(p Shans1(9))L? (Mzn+1). Since T(a- (mn (p)(®)
LA (My)7n(@= (apzfry1n(g))" = a:b) = T(ma(p)(@2) ® ma(Q)@h) = (Pazfuyb1n(@)”
we have that the induced map T'
T(mn(p)(2) ® 7n(q)()) + b, for all a, b € N,

on of the N-M bimodule strucis an injective N-N bimodule morphism (by definiti
after Definition 1.11), which isture on the relative tensor product, see the remark(Mant1)-shon+1(9))L%
onto, since mant1(P shont1(q)) (Many) is dense in Tony1(PThis completes the proof of a).
will be brief. To keep the noSince the arguments for b)-d) are similar, we(it should be clear from the proof oftation simple, we will omit all the “hats”

Recall that mn(q)L2 (My) =a) where they need to be used). Let us prove b).
which proves the first iden3.11),
tion(Proposi
esbimodulTn (Yn (q)) L? (My) as M-N(My) Om (Yn (@)) (Mn) —
(p)
7a
+
T
map
linearatity. To prove the second one, defineBimodules

andHigherTon {DY(9) L* (My)Relative

Commutants53

by(1a (9)(@) © 0 (10(@)(@)) = IM + NT" (p100(0)) (E00).

for all z, y € My, gn, the Jones projection forthat m,(7.(q))

M C M,, (Proposition 2.1ii)). Observe€ mn(M’' N Mani1) and mon (Y2n(q)) € Jonmon(N' N Man)JanTon (Mj, 0 Myny1).
above stated trace.

In particular, pyan(q)Furthermore,€ N' 0 Myni;

recall that Ms,=

is a projection with

= M,g,M,.

We

computefor z;,Yi € M,,, i =1, 2 (omitting “hats”){ Ta (P) (21) ® Tn

(2) (y1), Tn (P) (m2) ® mn (1m (9) (v2)

= (ma(p) (21 E27" (1 (mn (10.(0)) (12))*) ) , 22)= tr, (25m(p) (1) Epp (117 (9)(43)))
since 7, (Vn(q))(y2)* = mn(q)(y3) by Lemma 2.4 iv). We also used in the second
equality that 7, (p) is a right M-module map.

Since man |us,, is left multiplication,

we have T'(m,,(p)(£) ® mn (yn (9))(9)) = PZgnyy, ©, y € My. Thus
( T(r (p)(@1) ® Tn(¥(@))(11), T (Mn () (22) ® Tn (Va(@))(¥2)))
12 (at,= [M : N]"trar,,, (23PT190Y19Y59n)= [M : N|"trar,,, (23700 (0) (m1) Eng” (170(0) (43) 95)= tra, (2370 (0) (@1) Er” (3179) (43)= (mn (p)(€1) ® Tn (1n(@)) (41), Tn (P) (22) ® Tn (1m (@)) (52),where we used that pz1g, = mn (p)(21)gn, q¥39n = Tn(q)(¥3)gn, by Lemma 2.4 ii)(applied to A = M, B = M,). Note that we also use that =m, is the representation coming from the basic construction of M C M,, (Proposition 2.2). As before,
T is well-defined and induces an isometry m,(p)L?3 (My) ®n Tn (1a (q))L2(M,) —Ton (P¥2n(q))
L? (May), which is clearly an N-N bimodule morphism (note thatYn(q) € M' N Ma,+1) and surjective with the same argument as above.
Next

we provec).

It is clear that p sho,(g)is a projection in M' N My,

with trace stated above (proof as in a)). The proof of c) proceeds now as theproof of a), namely one shows that the linear map T : 7, (p) (My) © Tn (q)(My) —Tan(p shan (q))L*(M3n) defined by T (mn (p)(£)®7n(4) (5) = m2n(p shan (9))(Zgn¥) =(prgnyyn(g))", z, y € My, gn as in Proposition 2.1 ii), is well-defined and induces
the desired equivalence of M-M bimodules.

Finally, the proof of d).

Since 7,(p)L2(M,)

2 mn{vn(p))L*(M,)

bimodules, by Proposition 3.11, the first identity follows.

as M-NThe second one is now

shown precisely as in a) (with 7, (p) in place of p). Observe that the T" defined thereis actually an M-M bimodule morphism since v,(p) € M'N Msp: (thus 7, (vn (p))
is a left M-module map) and since ¢ € N’' N My, (thus m,(g) is a right M-module
map).
Since shont1(q) € Mj, 1 N Manto, it is clear that v,(p) shany1(q) €N'N My,42 is a projection with trace as stated above. This completes the proof of
the theorem.
[ODietmar

54

Bisch

Remark 4.7 i) Observe that a) of the previous theorem gives a formula for thebimodule tensor product of two reduced N-N bimodules, b) for the one of a reduced
N-M with a reduced M-N bimodule, c) for the one of two reduced M-M bimodules
and finally d) for the tensor product of a reduced M-N with a reduced N-M bimodule. Since we have explicit formulas of the antiisomorphisms vk and the shifts
shoni1 and sho, in terms of the e; 's and orthonormal bases (Theorem 2.6, The-

orem 2.11, Theorem 2.18), we can calculate the (full) fusion algebra associated toa subfactor by computing products of projections in the higher relative commutants(as given by the theorem) and decomposing the resulting bimodules into irreducibles
according to Lemma 3.5. This method proves to be very useful if the higher relativeincommutants of the subfactor are known in detail (for instance for the subfactors(Bisch and Jones [1995])).

We will give some applications elsewhere.

for#4) If we use Theorem 3.7 in conjunction with Theorem 4.6, we have explicit

maulas for the bimodule tensor product of any two reduced bimodules mn (p)L2 (My)and Tm (q)L?(M,,) (embed the projections in the same higher relative commutant
using 3.7 and compute the bimodule tensor product according to 4.6) and we could
abstractly define a product on the (equivalence classes of) projections in the higher
relative commutants in this way.
wi) Associativity of the bimodule tensor product of reduced bimodules can be proved

using 4.6 (see Remark ii). It amounts to showing that certain projections, obtainedby using the formulas in 4.6, are equivalent.

Fusion algebra calculations and reduced subfactors
N C M be an inclusion of II; factors and
5.1 Reduced subfactors Let(resp.
let M,, be the II; factors in the associated Jones’ tower. If p € N' NM,
reduced
a
pM,p)
C
Mp
(resp.
pMyp
C
Np
call
we
projection,aisM,)
p € M' 0
We will discuss in this section only reduced subfactors of the form
subfactor.
those ofNp C pM,p and leave it to the reader to reformulate everything forNote that (Np) NpMpp = p(N' N M,)p, so that thethe form Mp C pM,p.
We
reduced subfactor is irreducible iff the projection p is minimal in Nn M,.
will usually assume that p is minimal when we talk about reduced subfactors,
that
although this is not necessary for the statements below. Furthermore, observe5[pM,,p : Np| = tra, (p)try (p)[Mn : N] (which is called a local index of

N C M).

is to use
A good way to deal with fusion questions regarding reduced subfactors

the endomorphism picture (tensor N C M with B(H), H an infinite dimensional

andHilbert space, consider the resulting algebras in their standard representation
define an endomorphism using a common cyclic and separating vector), which makesTostatements regarding the fusion algebra of reduced subfactors rather obvious.
keep this article self-contained, we will however stay in the II; setting.
Let us start with the basic construction for reduced subfactors. The following

lemma is well-known (see, for instance Bisch [1994(a)] or Wenzl [1988]) - we include
a proof here for the convenience of the reader.
AC B C Bj be
Lemma 5.1 Let A C B be II; factors with finite index and let
B'nBy,
the basic construction. Let p € A'NB be a projection, let ¢ = yo(p) = JpJ €where J : L2(B) — L*(B) denotes the modular conjugation.

Then,

Apq C (pBp)q C paBipg
is (isomorphic to) the basic construction for (Ap C pBp) = (Apq C (pBp)q).

Bimodules

and

Proof

Let

HigherRelative

us denote

Commutantsby tra,

trp,

55

trp,

the uniquenormalized

traces on the

IT; factors A” = A’ Nn B(L*(B)), B, Bi.

Recall that trp, (Jz*J) = tra (x), forall z €

Furthermore,A’, by uniquenessof the trace.

we havethat[(pBp)q:

Apq) = [pBp : Ap] = tra (p)trg(p)[B : A]. Similarly, [pgBipq : (pBp)q] =trp, (Qtr (q)[pB1p : pBp] = trp, (q)trp(q)[B : A]. But trp, (q) = tra(p) andtrp: (q)=tryps(JqJ)=

trg(p),

so that bothindices coincide.

Observethat

pq = gp and pges = pes (Lemma 2.4 iii)), where e4 : L%(B) — L?(A) is theorthogonal projection.
= trp, (p) "trp, (pg)Now define e = trp, (p)~'pgeapg.
trp, (eapq)= trp, (pg)~'[B: Al!

We get that tryq5, pele)= trp(p)tra (n)~'[BA]7'.
Furthermore, an easy calculation (using pges = pes) shows that e? =e € pgBipg. We will apply the abstract characterization of the basic construc-tion (Pimsner and Popa [1988]) to show that pgBipq is the basic construction for
Apq C (pBp)q.
check

Since we clearly have that e commutes with Apq, we only need to

EPILIP(e) = trp(p)Note that Eg'(peap)‘tra (p)'[B : Al lpg.

= [B : A"'p and tIpgBipe(PaZTPY) = trpp,,(pzp), for allx € B. Thus, if z € B, we havetTpeBipe(PGTPgEAPG)= trp, (zpeap)irp! (q)trs, (p)~!
= (B : A]

rp, (9)trp, (p)"tra, (zp)

= [B: A] "trp, (¢)"trpp, (zp)=[B:

A" trp, (9) 'ttpgB,
pe (PaTPY),

so that EPIZYP9(pge pq) = [B: A] trp, (q) "'pq as desired.

O

Recall that if p € A is a projection, the basic construction for pAp C pBp is
pAp C pBp C pBip with Jones projection pep.
Next we identify the iterated basic construction for reduced subfactors. Let
N C M be an inclusion of II; factors with finite index. By (Pimsner and Popa
[1988]) we have that N C M,, C Ma,41 C M3p40 C Manis C ... is the basic construction for N C M,,. Denote as in the discussion before Theorem 4.6 by Vr +1)n+r
the surjective, linear x-antiisomorphisms N'N Mri 1ynt2r+1 = NN Mari 1yns2rtt(note that N C M(;41yntr C Ma(rt1)nt2r41 is the basic construction), restricted to
Mon NM; 42)n+r+1, which yields a surjective *-antiisomorphism Moy Nn

Mai2ynrt1 =My,

ony O Migyngrst(Mrngr—1 © Mey1ynir C Mri2yntri

is the basic construction).
Corollary 5.2 Let N C M be an inclusion of II factor with finite index andlet p € N' NM, be a projection.
Consider the reduced subfactor Np C pM,p,set
A= Np and B = pM.p and let A C B C By C --- C By be the iterated
basic

construction.

Set 0

=D

1

= go¥n{g0), sees qr@ € (r—1Mrnir—1r-1) Ngr—1M(41yn4r@r—1 and(isomorphic to) Nqx C qe Mar C ... qeM(ky1)ntrd-= Gr—1Yrntr—1(gr-1).

AC BC

Then

By C --- C By is

Proof The proof is a straightforward induction, using Lemmacomment after the lemma.
[OQ

5.1.1 and the:Dietmar

56

Bisch

Observe that it follows that reduced subfactors of reduced subfactors are reduced subfactors of the original inclusion. In particular, the vertices of the principal graphs of reduced subfactors can be identified with subsets of the verticesof the principal graphs of N C M, and the fusion algebra of a reduced subfactor can be read off the fusion algebra for N C¢ M. Let p € N' N Many: (resp.
g € NN My,) be a projection and consider the reduced N-N (resp. N-M) bi-

module a = m,(p)L3(M,) (resp. B = mn(q)L*(My,)) (see Proposition 2.2). Then

Homy.n (a) & (Np) NpMzy11p and since ad = Tn (p)L2 (My) ®N Tn (p) L2 (My) =Tont1(P¥ons1(P))L?(Many1) by 3.11 and 4.6 a), we have Hompy.y(ad) = (NpYont1(P)) N PY2nt1(P) Man+3py2n+1(p). Similarly, Homp.m(8) = (Ng) NgMang
and

since

88

=

mn (gq) L? (My)Om

mn (q)L% (My)=Ton (@y2n (9)

0

=

4,

L* (Man)

4.6byb), we have Homn.n (88) = (Ngv2n(g))' N qv2r (Q) Man+1q72n(q). Thus, tensoring
repeatedly with a, & (resp. 8, 8) from the right we get (M_y =N, My=M).
Proposition 5.3 Consider the reduced subfactors Np C pMy, 1p, p € N'N
My, a projection, and Ng C qMang, q € N' N My, a projection and let thebimodules a and B be as above. Furthermore, define projections po = p, p1 =+s Pr
PoY2n+1(P0)s
Gr—1Y2rn+r—1(@r—1)-=

Pr-1Y2rn42r-1(Pr-1),

q1=

q0Y2n(Q0),-5

Gr

=

Then,

(i) Homn.n((a@)*) = (Npa—1)’ MN pok—1(Mak(2nt1)+2k-1)P2k—1-

D2(i) Homn.n((ad)ka) = (Np2x)’ 0 pak (M(2k+1)(2n+1)+2k)(iii) Homn-n((BA)F) = (Ngak—1)’ NM gok—1(Makn+2k—1)Q2k—1(iv) Homn.n((BB)F8) = (Nga) 0 gor (M24 1)n+2k) G2kAs in Propositions 4.1 and 4.3 one can then determine the principal graphs
for the reduced subfactors. Let us consider first a reduced subfactor of the form
Np C pMans1p, p € N' N Mani a minimal projection. By Corollary 5.1.2 (or5.1.3) we see that the even and odd vertices of the principal graphs are subsets ofTCeven (I, TV denote as before the principal graphs of N C M). The even (resp.

odd) vertices are obtained by decomposing (a@)* (resp.

(aa)*a), for all k € N,

into irreducible N-N bimodules. Let 7 be such an even vertex and é an odd one.
As in 4.1 and 4.3 one shows that the number of edges between these two vertices isobtained as (ya, 6). Note that all computations here involve only N-N bimodules.
N'N Ma,Next, let us consider a reduced subfactor of the form Ng C qMsnq, q €a minimal projection. Again, by Corollary 5.1.2 (or 5.1.3) we see that the verticesof the principal graphs of this reduced subfactor are (identified with) subsets ofToven and Toga (one graph) (resp. Tf. and I'[4,4 (the other graph)). The even(resp. odd) vertices are obtained by decomposing (BB)* (resp. (BB)*B), k € N,

into irreducible N-N (resp. N-M) bimodules. Let 7 be such an even vertex and 6
an odd one, then the number of edges is again given by (v3,6) (N-M bimodules).
We leave it to the reader to formulate the analogue of Proposition 5.1.3 and theabove discussion for the other principal graph of a reduced subfactor (replacing
by &, 8 by 3). Furthermore, as mentioned above, observe that the fusion algebra
can be read off the fusion algebra for N C M for both types of reduced subfactors.
5.2 Computation of the fusion algebra associated to a subfactor
The calculation of the fusion algebra for an inclusion of II; factors N C M cansometimes be carried out by purely (linear) algebraic methods (see, for instanceBimodules

andHigherRelative

Commutants

57

Bisch [1994(b)], Haagerup [1994], Izumi [1991], Ocneanu [1991(b)], Sunder [1992],
Sunder and Vijayarajan [1993], for other methods see Bisch and Haagerup [1996],

Goodman and Wenzl [1990], Kawahigashi [1996], Kosaki et al., Wassermann [1995],
Remark 4.7), since algebraic/combinatorial properties of the fusion matrices (forinstance the fact that they have only nonnegative integer entries) are sometimes
enough to calculate the bimodule tensor products. However, as soon as the principal graphs of the subfactor have multiple edges or quadruple points, these algebraic
manipulations may not be enough to determine the fusion algebra completely (see,for instance (Bisch [1994(b)], 3 + v/3 example) and additional information must be
used to determine the fusion algebra (see, for instance (Kawahigashi [1996]) for asolution for the 3 + v/3 example). We will give a method that allows us to calculate in many cases the fusion algebra associated to a subfactor by solving matrix
equations and we will illustrate this method with an example.
Let
N C€ M be an inclusion of II; factors with finite index and let p =~L?*(M)p be as before. Let (I, I") be the principal graphs of N C¢ M and denote by G and G’ the associated matrices (see the discussion after Proposition
3.2). As shown after Definition 3.12, we can regard R, (resp. Rj), right multiplication by p (resp. p), as a matrix ZI'oqqa — Zleyven (resp. Zleven — Zlogq). If
we let By = {a|a € T'even} and Bs = {3|8 € Tyqa} be the canonical bases of
ZT

yen

resp.

ZI'oqq, then the matrix representation of R,

to these bases is G (resp.
(ap,B)GY), i.e.

(resp.

Rj)

with respect(Ro) us = (a, 8p) = Gap, and (Bs) ga == Gap, @ € Teven, B € Toad (Remark 4.5).

Thus R50=(GGY)"(R(pp)» * ZL'even — ZT even) and Ripp)np = (GGY)"G (Rpp)np * Zlodd — Zl even).

Let N'N Mansi = Ger, Ap"
Dicer

n

APD

ABN

> po, (C).
a,

ATH) = Meni (C), and N' 1 My, =Denote the irreducible N-N (resp.

N-M) bi-modules indexed by K,, C T'even (resp. L, C I'oaq) (see the discussion after Proposition 3.2 for the notation) by ax, k € K,, (resp. 3;, | € Ly). Then Proposition 3.1
and Lemma 3.5 imply(a) (op)"=~ Pex, al?"b) (00)"0 = Pics. &*™B;

gy

as N-N bimodules, n > 0.

as N-M bimodules, n > 0.

The right regular representation yields the following matrix equations:(1) (aet)™** = 3eK.

a’R,,, n> 0(all matrices are “T'even X Leven”

matrices).

(2) (GGY"G => ler, a{*™ Ry, n>0matrices).

(all matrices are “Teven X Toda”

The matrices appearing in the equations (1.1) and (1.2) have nonnegative integer entries and the matrix R,, (corresponding to the distinguished vertex x) is
the identity matrix. Note also, that if N C M is irreducible, the irreducible N-M
bimodule indexed by ly € Lg = {lp} is precisely p = xL?*(M)ys, so that the correponding matrix Rg, is given by G. Furthermore, if the involution on Zl'¢ven istrivial(this happens

for instanceif the indices of the N-N

bimodulesa € Teven,i.e., the Jones indices of the corresponding reduced subfactors, are distinct, since
the indices of a and & coincide), then all matrices R,, are symmetric. Furthermore, the dimension vector @?**V) for the dimensions of the simple summand of

BischDietmar58

N' 0 May41 can be calculated as gn+l)— (GG)(x), n > 0, and the one forIn many situations, for instance= Gta?n+1).

N' 1 Mayo is obtained as d+?)when the principal graph has at most triple points, this information is enough tosolve the matrix equations (1) and (2). In particular, a solution of (1) gives theright regular representation of the fusion algebra ZT even. Note that one obtains
a similar set of matrix equations for the other principal graph I ’ with associatedmatrix G (replacing N' N My by M’N Mj, etc.). If we can solve both sets of matrix
equations, the full fusion algebra (Definition 3.13) associated to N C M can be
determined simply by multiplying matrices (i.e., we compute the fusion algebra inits right regular representation). The fusion calculations in (Bisch [1994(b)]) werecarried out using this procedure. Let us also point out that we can read off the principal graphs of the reduced subfactors from the matrices R,, resp. Rg. Namely,
the matrices associated to these principal graphs are precisely the matrices Ro,
resp. Rp, restricted to the vertices appearing in the decomposition of (ana),(aka) ax, n € N, and similarly for 8; (by Proposition 5.1.3 and the discussion
afterwards). Let us illustrate this algorithm with the example of a subfactor withprincipal graph Eg.

(See Figure 1)

B2

Ba

Bo

Figure

1

The

2},

ag

=

principal graph

Eg

1
We

let Ceven

=

{ao,

a1,

*, Codd={Bo,$1, B=}

and

G

=

:

00
1

)

.

0 01
Since the principal graph is finite, we will have to solve only finitely many matrix
equations. Note that a subfactor N C M with principal graph Fg has depth 4 (thedepth is the longest distance of a vertex in the principal graph to the distinguishedvertex x) and Jones index [M : N| = 4cos® f; (Goodman et al. [1989]). Recall that

there are two non-isomorphic hyperfinite subfactors with this principal graph and
N' NM arethat both principal graphs are given by Eg. The simple summands ofindexed by fo or Ly = {0}, those of N'NM; by So, Br, B2 or L1 = {0,1,2} and thoseof N' 0 Ma, by Bo, B1, Bz or L = L, = Li, n > 1. Similarly, the simple sumnmandof N'N N = C is indexed by ag or K_; = {0}, the simple summands of N' NM,

are indexed by ag, a1 or Ko = {0,1} and those of N'N Man, are indexed by ag,
ai, az or K = K, = K; ={0,1,2},n > 1. The dimension vectors @™ for N'NM,,

are 3) = GG») = (1,1,0), a = Gta® = (2,1,1), a® = Ga? = (2,4,1) anda@® = G'@® = (6,4,5). Since N C M has finite depth, the matrix equations (1)and (2) reduce to the following set of four equations:(i) GG* = Roy + Ra,(ii) (GGY)® = 2Raq + 4Ra, + Ray.

Bimodules

(iii)

and

HigherRelative

Commutants

59(GGYHG = 2Rp, + Rg, + Rg,.

(iv) (GG')’G = 6Ry, + 4Rgs, + 5Rz,.
100

Since Ra, = [1010

y , we get Ry, = f2)

and hence R,, = (Gat)? —

0

01
010
0 01
2R,, —4R,, = &1 )) Since N C M is irreducible we have Rg, = G as
1 00
remarked above. It is then easy to solve the remaining equations and we obtain
010
0 01

Rg =|{101),Rg, =

f

1

)

[1
f

1

1]. This yields the following fusion rules (we0)

write ag = 1):

BE

I

0]

US
1

For instance we have RZ = §

2

1

6

3

Va

= Roy + 2Ra, + Roy = Rag+20; 40m

1 21
The decomposition of R2 as a linear combination of R,,’s can be read off the firstrow of the matrix RZ. In the same way we calculate the remaining fusion rules.

BE
I

00000000]
EY:

Loe

|

8

EZ
|
|
BK
| Bo | 14a

[BB]
[BB]
[| on [orto|Note that the involution on the N-N part (and M-M part) of the fusion algebra
is trivial. Since I' = IY = Eg, the above tables actually determine the full fusion
algebra associated to N C M. Furthermore, as explained in Section 5.1, we canread off the principal graphs of the reduced subfactors easily. Before we do this, let
us briefly work out the local indices. Recall that if N C M is an extremal inclusionof II; factors with finite index, principal graph I" and associated standard matrix
G, and if we denote by § = (si) rex and f= (t) Jez, the normalized trace vectors
on the higher relative commutants

N' N Ma,4+1 resp.

N’ NM»,(normalized

suchthat s, = 1), then GG'§ = [M : N|§, G's = [M : N|t, Gt = §, G'Gt = [M : NJi.

Dietmar60

If we let € = (5,[M : N]zt), then Arf

Bisch=[M : N)z€ and £ is the normalized vector

the reduced
whose entries are the square roots of local indices, i.e., the indices ofscommutantrelative
highertheins
projection
minimaltosubfactors associated(see for instance Popa [1994]). Let us now come back to the Eg subfactor discussed+
above. If we let t = [M : N] = [|G||* = 4cos® &, then § = (1, ~ 1,12 — 4t2), © = (1,458, £=341) and =(,t—-1,t2-4t + 2, Vi, 5}, =,the entries of £ are in the order ao, a1, 2, Bo, Bi, Bz.

where

Note that t3 — 5t2 +5t — 1 = 0 (¢ is a zero of the characteristic polynomial of GG*).

Thus the local1)? =indices are the squares of the entries of £, which are given by ind = (1,(t —graphs
7.4641016 ...,1,4 cos? =, 2,4 cos? 75). Let us now determine the principalp; (resp.
of the reduced subfactors with these indices. Choose a minimal projection

N’NMs (resp. N'N M3) and considerg;) in the i-th (resp. j-th) simple summand of
the reduced subfactors Np;

C p;Msp;,i € K= {0,1,2}(resp.

Ng;

C g; Mag,j € L = {0,1,2}). Note that all other reduced subfactors (associated to minimal
projections) are isomorphic to one of these.

Their indices are given by the vector

ind above, where the first three entries are the indices of

1 =0,1,

Np; C p;Msp;,

2,

C poMapo
and the last three those of Ng; C ¢;M2q;, 5 =0, 1, 2. The index of Npy
is the identity
is one, so we are done (this also follows from the fact that R,,
ng thematrix). The even vertices of Np; C p1 Msp; are computed by decomposi
the odd ones
tensor powers (ra), k € N, into irreducible N-N bimodules andread from theby doing the same to (rar) ar, k € N (note that &; = a1). We
are therefore
first fusion table, that even and odd vertices of this reduced subfactor
identified with ag, cy

and az.

Thenumberof edges between«a; and«;are givengraph isby (asa, a) (Section 5.1), so that the matrix associated to the principal
principal graphprecisely R,, (both principal graphs actually coincide). Thus theis given by*

Figure2which are
The same analysis for Np, C pa Mzp2 shows that the even vertices,by ao,
calculated by decomposing (apa) = a?F = ap into irreducibles, are given

number of edges
and similarly, we get that the odd ones are given by just az. The

confirming our index
are {oo2, as) = 1, so that we have again an index 1 subfactor,

by the bimodule
calculation above. The reduced subfactor Ngo C goM qo is given
just get anotherBo, which is equal to p = ~L?(M) pr, as remarked above, so that we
The even
copy of N C M. Let us consider the reduced subfactor Ng; C qi M2q1.

(usingvertices are calculated by decomposing (53151) * into irreducibles, which yields

Bimodules

andHigherRelative

Commutants

61

the second and third table) the labels ag and ay. Similarly, decomposing (5 By)" Je
yields 3; and the edges are calculated as (apf, 51) = 1, {(@151, 81) = 1. Thus theprincipal graph is given by

ao

51

a1

Figure 3

which confirms again our index calculation (we have an index 2 subfactor). Finally,
we consider Ngz C ¢2Mgq,. Decomposing tensor powers of Jz and 3; into irreducibles results in the even vertices ag, a; and ap and the odd vertices fp, 3; andBz.

Calculating the edges, we get (of, 80) = 0, (82,51)= 0, {of2, 52) = 1,

(a1Ba, 82) = 1, {1fB2,51) = 1, (a1fa, Bo) = 1, {@2f2, fo) = 1, (202,41)= 0,{a2f8s, 82) = 0, from which it follows that the principal graph is given by Figure 1(thus the other principal graph is given by the same figure). The fusion algebras of
the reduced subfactors can be read off the above tables.
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