C. R. Acad. Sci. Paris, t. 309, Série I, p. 923-927, 1989

923

-Analyse fonctionnelle/ Functional Analysis

Index theory and Galois theory for infinite index inclusionsof factors
Richard H. HERMAN and Adrian OcNEANU
Abstract
— We

study

inclusionsof vonNeumannalgebrasin the casewhenthe index

isinfinite.
Some of the main notions of the index theory initiated by Jones such as the standard
projections and Pimsner-Popa bases can be extended to this context.
The corresponding structural
results, such as Takesaki duality, hold.

We develop in the infinite index context results from the Galois theory for algebra inclusions of
the second author and obtain an intrinsic characterization of twisted crossed products by a discretegroup in terms of the position of the initial algebra in the crossed product algebra.

)

Théorie de Galois et de I'indice de Jones pour les inclusions de facteurs
Résumé — On étudie des inclusions d’algébres de von Neumann dans le cas ou Pindice est infin.
Dans ce contexte on peut généraliser les constructions les plus importantes de la théorie de Pindice deJones, telles que les projections standard et les bases de Pimsner-Popa. Des résultats structuraux
correspondants, comme la dualité de Takesaki, restent vrais dans ce contexte.

On étend au cas d’indice infini des résultats de la théorie de Galois pour les inclusions & algébres du.

second auteur, et on obtient une caractérisation intrinséque du produit croisé par un groupe discret entermes de la position de Ialgébre initiale dans I'algébre produit croisé.

Version francaise abrégée — On commence par étude des inclusions des algébres de vonNeumann d’indice infini. On travaille avec des facteurs semi-finis .# 2 4" 4 predual séparable
ayant des traces tr et T telles qu'il existe un poids opératoriel T avec tr=t°T. Dans ces
conditions il existe une base de .# comme module sur A". Si .# est un facteur fini et si
Pindice est fini on retrouve la base de Pimsner et Popa.
THEOREME.

—II existe une collection au plus dénombrable d'éléments {\}de Mtelle queT (x A¥) A.
chaque x € M vérifie x=)
On utilise un’ procédé de type Gram Schmidt(x, y) > T(xy*). Pour v dans .# bien choisi, toutr=T (rA*) A, avec A=T (vv*)~ 1/2 2,par rapport i la forme bilinéaire
élément r de [Av] s’écrit comme

INCLUSIONS DISCRETES ET COMPACTES. — Une inclusion de facteurs semi-finis MoS MH,
est appelée discréte si la trace de .#; se restreint (i un scalaire prés) en la trace deMo.
L’inclusion
MoS Myest appelée
compactesi, pour[DextensioncanoniqueMoS MS

M,,inclusion

A, S M,est discréte. Uneinclusion i la fois discrete etcompacte est d’indice fini.
THEOREME. — Soit JM le produit croisé de My par un groupe localement compact abélien
G. Linclusion MoS My est discréte (compacte) si et seulement si G est discret (compact).
On caractérise de maniére directe les inclusions compactes comme suit.
THEOREME. — L'inclusion de facteurs semi-finis MoS My est compacte si et seulement sipour une (et alors pour toute) base { Ni} de My sur My la somme Y Aud; est finde.
L’argument principal est que .#,, définie comme J sy Mo Ju, sur espace de Hilbert
standard de .#, est égale au produit croisé de .#, avec elle-méme au-dessus de .# o- Avec
ce résultat on montre que dans la tour (.#,) la propriété d’étre discréte (compacte) est depériode 2 par rapport i x.

Note présentée par Alain CONNES.
0249-6291/89/03090923§ 2.00 ©

Académie des Sciences:

924-C. R. Acad. Sci. Paris, t. 309, Série I, p. 923-927, 1989

APPLICATION AUX PRODUITS CROISES. — On
du produit tensoriel par un groupe discret.

obtient la caractérisation intrinséquesuivante

THEOREME. — Soient Mo S My une inclusion de facteurs et MoS My S My S My son
extension canomique. L’algébre M | est un produit croisé de Mo par une action extérieure perturbée
par un cocycle d'un groupe discret si et seulement si Mo
My est scalaire, MoM,estabélienne ex MyM M5
est un facteur.
::In the first part of this Note we study inclusions of von Neumann

the index is infinite.

We extend to this framework

algebras for whichsome of the main results of thetheory initiated by Jones [4] and developed in [10] and [2].
Previous generalizations ofthe theory by Kosaki [5] and Phan Loi [6] study the case of general algebras with finiteindex.

The principal role in our study is not played by the algebras, which are kept

semifinite.
Instead we study for the first time what happens when the index becomesinfinite.
The main ingredients of the finite index theory, such as the Jones projections,no longer make sense in this context. We show however, using appropriate substitutes,
that the main relations and technical properties of the finite index constructions can begeneralized to the infinite index setting. Consider as a model the case of a sequence ofinclusions with finite index in which the index tends to infinity.

The trace of the Jones

projection tends to zero and the corresponding idempotent does not exist in the infinite
index case. A scalar multiple of the Jones projection, normalized as in statisticalmechanics [15] so that its trace is one, would yield at the limit an operator-valued
weight [3] satisfying an analogue of the Jones relations.
We then show that the basis ofPimsner and Popa [10] can be obtained by a Gram-Schmidt orthogonalization processwith respect to an operator valued inner product and is naturally associated to modules
rather than algebra inclusions. The spanning property of the Jones projections isreplacedbytheSauvageot [12].

densityofatensor

product

construction

asin

Connes [1]and

Using these tools, we prove that properties such as discreteness andcompactness of inclusions, which we introduce in this framework,

display a period twooccurrence in the tower of extensions which generalizes Takesaki duality [14].
The second part of this Note develops in the infinite index context results from theGalois theory for algebra inclusions of the second author ([7], [8]).

As in that context,

we obtain an intrinsic characterization of the twisted crossed product by a discrete group.in terms of the position of the initial algebra in the crossed product algebra. We use,
however, a different approach,in which information coming from the tower of relative

commutants is translated into the property that the normalizer of the initial algebra inthe larger algebra is spanning.

This yields then the crossed product structure following

Sutherland [12].

GENERAL THEORY.

— We take as our setting a pair of semifinite factors .# 2 A" withtraces tr and t related by an operator valued weight T such that tr=toT.

Under thecondition that the predual be separable we prove the existence of a basis for .# over A.
THEOREM

1. —There exists a collection of elements { A; } in JM, at most countable,

such that every x in M can be written as x=T(xA*)\;.
Given any such basis {AN},the matrix (T (AM; A}));; is a projection in the matrices over N°, uniquely determined up toprojection equivalence.
Moreover if the index is an integer or infinite, the basis can bechosen to satisfy T (A A})=5;LC. R. Acad. Sci. Paris, t. 309, Série I, p. 923-927, 1989

925

In the case where .# is a finite factor there exists a projection E, from .# onto Aandcorresponding

Hilbertspace

projectione,.

constructed by Pimsner and Popa.

In that case the basis is the one:TaeorREM 1’. — Each element x in M can be written as x=Y E(x A\}) A; with respect
to a basis { A; } as in the Theorem 1. Moreover the basis satisfies YA¥e, h;=1.
The proof of Theorem 1’ relies on the following result.
ProposITION 2. — If n is the left ideal associated to an operator valued weight between
factors M and A on a separable Hilbert space, then the identity can be written as I=) p,

where the p; are in n.
This proposition is applied to the pair of factors # and ( .#,e,-the fundamental construction of Jones [4].

> where the latter isIn the case of infinite index (4,e,> is asemifinite non-finite factor.
Theproof

orthogonal(x,y)

of Theorem

1 is basedbasisas= T(xp*).

in .#ona module

The argumenta Gram-Schmidt

over4

with

for a cyclic submodule

can be made into a formal argument.

procedure

respect

tofor choosing

thebilinear

anformuses the following idea, which

For v in 4 carefully chosen, any element r in

the left #-module [Av] can be written as r=T(rA*) A, where

A=T (vv*)"

2p.

Theproof follows from the equality

T (xo0* T (09%) V2) T (00%) 12 p =x T (00*) T (00*)~ Y2 T (00%) ~ V2 T (00%) "12 v= xv.
With additional arguments
one gets T (A, A¥)=9;;LTo show the relation between two bases { A, } and { A; }, we express one in terms of
the other as A,=) T(A;A¥) A; Then the matrix (T (A; A¥));; is a partial isometry W andWW*=(T (A;A}));; and W*

W=(T (A; A}));; are the support projections for the respectivebases.
DISCRETE AND COMPACT INCLUSIONS.

—We

study

twodifferent

types

of inclusions,modelled on the idea of crossed products by outer actions of compact and discretegroups. The test used is whether the trace on the algebra extends to a semifinite traceon the crossed product.
Given an inclusion. of semifinite factors #, < .#; we call the inclusion discrete if
the trace of .#, restricts (up to a scalar multiple) to that of .#,, or equivalently, if
there exists a conditional expectation of .#, onto .#, preserving the trace. Theinclusion #,< .#, will be called compact if upon forming the standard extension
MoS My S M,, the inclusion AH; S A, is discrete. An inclusion is both discrete andcompact if and only if it has finite index, which is shown using the lower bound for
conditional expectations ([11], Theorem 2.2).
Aa.
These definitions are justified by the following properties of crossed products, statedbelow for abelian groups.
TuaeEOREM 3. —abelian group” G.

:::

Suppose that JM, is the crossed product of My by a locally compactThe inclusion Myo < #, is discrete if and only if the group G isdiscrete.
The inclusion M , = M is compact if and only if the group G is compact.
To prove that the group must .be compact if the inclusion is, we use the existence ofthe conditional expectation from .#, onto .#, to obtain a finite Haar measure.
It isinteresting to verify that if u, are the unitaries in the crossed product implementing thegiven action, then the corresponding spectral projections serve as a basis, albeit not theobvious one when the group isfinite.

:

926
C. R. Acad. Sci. Paris, t. 309, Série I, p. 923-927, 1989
ET...
- We now provide a global condition on the basis for .#,inclusion is compact.

over .#, such that theTHEOREM 4. — Given an inclusion of semifinite factors My < M,, the inclusion iscompact if for a given (and hence for any) basis { A; } for JM, over My the sum Y
AFA, isfinite.

::

The first argument in the proof is the fact that the extension .#,, defined as
Jus #4) 4, on the standard Hilbert space of .#, is equal to a tensor product of .#,with itself over .#,,.
Suppose that .#, S #, is a pair of semifinite factors with traces t, and t,. Let n,be the dense ideal in .#, for the operator valued weight T (in the sense of [3]), relating
the traces of .#; and .#,. On the Hilbert space of .#, we consider the operators y ® xwhere y, x belong to nr, nf respectively.

Their operation is defined by y ® x (2)=y T (xz)for z in #,.
These operators extend to bounded operators on the whole Hilbert
space. Denote the algebra generated by these operators by 4; ® ,,#4, extending thecorresponding notion in [2].

Then we have

THEOREM 5. — Mo=] MI=M | ® 4, MH, Where I=], is the conjugate linear isometry such that J MH J= MH.
To get a trace on the semifinite factor 4; ® ,,.#, we use the basis for .#, overMy
In general the element Y A¥ A, is a scalar, which may be infinite.
THEOREM 6. — The map y — Y to (T (Ay (AF) defines a faithful, semifinite normal traceon MQ 4, My.
YA¥A,; is finite.

This trace restricts to a multiple of the trace on M, if and only if
Moreover the map Y= (0;();, where ®,;()=T (A; y (A})), provides

an isomorphism between M | ® 4,
My and Mat (AM).
Remark. — With { A; } a basis for .#, over #4, it is possible to write down a basisfor M;® 4,M#,

over M,.

In the finite index case this is ([7], [11]), up to a scalar,{ e); }, where e is the projection corresponding to the conditional expectation of .# Lonto .#,. In the infinite index case we have no such ¢, but in the tensor product picture{ p.®; } is a basis, with projections p, chosen as in Proposition 2, and the operator
valued weight T from 4 ®
,,.#, to .# is given by T(y ® x)=yx.
With these comments we do the same construction for the pair 4,4.
THEOREM

7. — Following the same construction of the map as in Theorem 6, we obtainan isomorphismof Mywith Mat(.#,).

Moreoverthe restriction of this mapto JM,coincides with the map of Theorem 6.

This shows that in the tower of extensions discreteness and compactness have period
two. The inclusion #, < .#, is discrete if and only if .#, < #, is compact, and theinclusion .#, = #4 compact if and only if #,; < #,is discrete.
APPLICATIONS TO CROSSED PRODUCTS. — We give a characterization of twisted crossed
products by discrete groups using the tools developed above. This extends the characterization obtained by one of us [7] for the case of finite index.

SoTHEOREM 8." — Let My S My be a discrete inclusion.
Then M, is a cocycle twisted
crossed product of My by an outer action of a discrete group if and only if MyM, isscalar, MyM, is abelian and MyM M4 is a factor.
The group is made to appear as unitaries normalizing .#, by using an extension of a
proposition of Pimsner and Popa [10] relating such unitaries to projections inMo

MyOf course

the necessityof theseconditions

is well known [14].

More-C. R. Acad. Sci. Paris, t. 309, Série I, p. 923-927, 1989927

-generally we conjecture the following characterisation if we drop the condition that thein .#, be abelian.

relative commutant of #CONJECTURE.

— Let MoS My

crossed productof #,byanNote remise le 21 septembre

outeraction

Then M,of a discreteKac

is a cocycle twistedalgebra

if andonly

if

M5 is a factor.

is scalar and My

Mo (YMbe a discrete inclusion.

1989, acceptée le 25 septembre 1989.

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