AN INVARIANT FOR SUBFACTORS IN THE VONNEUMANN ALGEBRA OF A FREE GROUP

C. R. Acad. Sci. Paris,t. 316,Série I, p. 983-988, 1993 983 Théorie des groupes/Group Theory

An

invariant

for

subfactors in the

_von

Neumann algebra

of

a free group

FlorinRÂDULESCU

Abstract_— Inthis Note_weareconsidering_anewinvariant forsubfactors inthevonNeumann algebra

if

(Ft)of_afree_group.Thisinvariantis obtainedby Computingthe Connes's%invariant

forthe enveloping_{von Neumann}algebra in theitérationofthe Jone'sbasicconstruction for the

given inclusion.In the_caseofthe subfactors considered in[22], [24]thisinvariantiseasilycomputed

asarelativex invariant,in theform considered in[14].

One considerstheinclusion

if

(F„)_£se₌

if

(F„)xieZt2, where6 is_aninjectivehomomorphism

from Zt intoOut(if(FJ) (i.e. a Z„-kernel) with minimalperiodk2 [in Aut(if(F„))].Then there

exists_{a canonicalcopy}8 oflk _{in x}(*&)which can be liftedto Aut(sE)[4].The décompositionof

thegeneratorofthedual action ofÈk_on

_si

x!jZfcas theproductofa centrally trivialautomorphism

and an almost inner automorphism,gives _an action of_{Zt2©Zt2 on the algebra} sexl_jZt. The

algebraicinvariants[9]forthis last actiongiveamoresubtleinvariant for9.

As_{an application}we showthat, contrary to thecaseoffinitegroupactions(or moregênerai

G-kernels)_{on the}hyperfineII,factor(settled in[2], [9], [18]),thereexists_{non outer}conjugate, injective

homomorphisms(i.e. two Z2-kernëls)fromZ2 into Out(if(F,.)),with non-trivial obstruction to

lifting_{to an action on} £?(Fk). Moreover the algebraic invariants[3] do not distinguish between thèse two Z2-kernels. Also,thereexists twonon-outerconjugate,outeractionsofZ2on S(Ft)®R

that_{areneither almost inner or}centrally trivial.

Un

_{invariant pour}

les sous-facteurs de

l'algèbre

de_von

Neumann

associée

à

Un groupe

libre

Résumé_— Nous introduisons_{un nouvel}invariant_pourles sous-facteursdel'algèbre_{de von}Neumann

if

(Ft) _{d'un groupe libreFk, en} calculant l'invariant% de Comtespourl'algèbre enveloppantede la

constructiondebasedeJonesassociéeàl'inclusiondonnée.Dansle _casdessous-facteurs considérés .dans [21], [24] cet invariant peut être facilement calculéen utilisant une version relativepour%,

considéréedans[14].

Onconsidère_{une inclusion}de typei?(F„)_£se=£?(F„)xl e^j.-2>où0estunhomomorphismeinjectif de ï.k dansOut(i?(F„)), depériode minimalek1 [dansAut(if(F,,))]. Alors il existe _unsous-groupe canoniqueZtsAut(«e/)dansx(s/)[4].Ladécomposition du générateurde l'action duale de_{Zt sur}

s4xlZ,., _{comme le} produit d'un automorphisme centralement trivial et d'un automorphisme dans

Int(.s?),donne_uneaction deZ^SZ^surs4xlZt. Onobtient ainsiuninvariantplussubtilpour6,_en

considérant les invariants algébrique[9]del'actiondeZt2©Zt2.

Comme application nous montrons que, contrairement au cas du facteur hyperfini de type

II_j (analysé dans [2], [3], [9], [18]), il existe deux homomorphismesinjectifs de Z2 dans

Out(if(Fl))₌Aut(^7(Ft))/Int(ïf(Ft)),qui ontles mêmes invariants algébrique[3],mais qui_nesont

pasextérieurementconjuguées.

Versionfrançaise abrégée_—Le but de cette Note est d'introduire un nouvel invariant

pour les sous-facteurs des facteurs de type

II

_x associés à _{un groupe libre. L'invariant que}

nous proposons est obtenu en calculant l'invariant x de Connes[4] _{pour l'algèbre de von} Neumann enveloppante de la construction de base _{de Jones, pour l'inclusion donnée.}

L'invariant% a étéintroduit par A. Connes([4], [5]) pour résoudre un problème très difficile

posé _par Sakai[27] _{concernant l'existence d'un facteur} de _{type II1 non anti-isomorphe à} lui-même.

On rappelle que si _{M est un facteur de type II1,} alors l'invariantx(M) est le _groupe

(commutatif,voir [4]) %(M)

₌

(Ct(M)

D

Int (M))/Int

(M)gOut(M)

où_{Aut (M) est}le _groupe

d'automorphismes de M, Int(M) est le _groupe des _{automorphismes intérieurs et} Ct(M) _est défini _par

Ct(M)

₌

{

_ae

_{Aut (M),} || <x(xn)_—xn\\2_{—>0, pour toute suite centralisantex„}}.

Note présentée par Alain CONNES..

L'INVARIANT. _— Soit

AgB

_{une inclusion} d'algèbres _{de von} Neumann finies, soit

E\

_:B—>A une espérance conditionnelle de B dansAparrapport à une trace fidèle, normalesur

B (invariante _{parE) et} soit seoe l'algèbre enveloppantede l'inclusion Aç=B i.e. le terme à

l'infinidans l'itération de la constructionde base deJones[7]_pour l'inclusiondonnée). Dans _ces

conditions,

X00£0ut(j0,

est un invariant pour l'inclusion

A£B.

(Comme dans[4], la force de l'invariant vient du fait que l'on considère%(séoe),non seulement commeun groupe abstrait, mais comme un sous-groupe de

Out^^,)).

Dans les exemples de sous-facteurs des _{algèbres de von Neumann} des _{groupes libres}

considérés dans[24], _{on peut facilement montrer que cet invariant coïncideavec} l'invariant relatif x[14], _{pour une inclusion} des facteurs hyperfinis associés à l'inclusion donnée.

THÉORÈME. _— Soit N

g

M _{une inclusion} d'algèbres finies, à prédual séparable et telle qu'il

existe une base orthonormale deM _sur N, de _{type Pimsner et} Popa[20], _{pour une espérance} conditionnelle de _{M sur N. Soit} SE(Fk) lefacteur de _type

II

l associé à un groupe libre Fk,

k?±2.

Si sé

₌

(SE(Ft)*NM _est leproduit libre, amalgaméréduit_{des algèbres de von} Neumann introduit dans[22], alors

XM=X(N,M).

Ici

x(N,

_{M) est l'invariant relatif de Connes pour l'inclusion}Ni=M, considéré dans[14].

Comme applicationdes _{résultats précédents, nous}allons construire deux homomorphismes injectifs de Z2 dans Out(SE(Fk)), _{non extérieurement conjugués, tels que tous} les invariants algébriques (voir [9], [3], [18]) coïncident.

THÉORÈME. _{— Pour tout}

£eZ,

k^.2, il existe deux automorphismes_<x(eAut(SE(Fk)), tels

que les images de at dans Out(SE(Fk)) sont non conjugués et d'ordre deux. De _{plus, tous}

les invariants algébriques[3] _pour les deux homomorphismes injectifs de Z2 dans Out(SE(Fk))

coïncident.

COROLLARY. _— 77 existe des sous-facteurs de SE(Fk), de profondeur finie _{([7], [17]), non}

conjugués et tels que tous les invariants des deux inclusions _provenant des _{commutants relatifs}

dans la construction de base ([7], [18], _{[21]) sont} identiques.

THE_{INVARIANT. —} Given A<iB_{an inclusion}

of

type II1factors,

of

finite index, _we let

Boe be the enveloping algebra

for

_{the tower}

of

algebras in the (iterated) Jones's basic construction

[l]for

AgB.

Then

X(BJSOut(BJ

is_aconjugacyinvariant

for

A£B.

Hère,_{for a}type

II

_xfactor_{M, x (M) dénotes}the Connes'sinvariant forM[4].The_{x (M)}

invariant_wasintroducedby Connes [4]in connection_{to a breakthrough construction}

that

showed the existence

of

_atypeII-L factor

not

antiisomorphic_{to itself.}

Recall

that

%(M)

=

(Ct(M)

D Int

(M))/Int(M). Hère

Ct(M)

is the set

of

ail

auto-morphisms

_a

of

M

that

hâvethe property

that

[|a(x„)

—x„\\2->0,wheneverxn,

neZ

isa

central_{séquence in M} (z._{e. a boundedséquence in M with} ||[x„,y]\\2_{-> 0forany}

veM).

Int

(M) is the closure

of Int

_{(M) in the pointwise convergence topology on M,} while

Aninvariant for subfactors in the algebraof_{a free group} 985

We willshow

that

for the subfactorsin free _groupalgebras, considered in [24] (or for

those considered in [22]), this invariant _{may be} computed and

it

coincides with the relativeform

of

_{the invariant x} [14]_{for an}associâted

pair

of

hyperfinite factors.

Using the full strength

of

the invariant_{(which cornes as} in [4] by consideringx(Boo)

as a subgroup inOut(Boe)

=

Aut(B00)/Int(Boe),

rather than

_{as an abstract abelian group}

(see also [8], _{[12], [27])) we} obtain _{as an application}

that

the classification theory for

finite _{group actions (and for} G-kernels)

_on

type

II

_x factors _{such as} SE(Fk) (the vonNeumann algebra

of

afreegroupFk) isdifférentfrom the correspondingclassification

theory

_on

the hyperfiniteIIX factor([2],[3], [9], [18]).

THEOREM. _— There exists _{two injective}homomorphismsfrom Z2 zzzZoOut(SE(Fk)),

for

any

keZ,

k^.2,

non-Hftable to an action of~L2 on SE(Fk) [i.e. two Z2-kernelson SE(Fk) with non-trivial obstruction]_{that are not outer conjugate} (in Out) and

for

which

ail

the

algebraic invariants([2], [9]) coïncide.

Explicitly this _means

that

there exists two (non-inner) automorphisms_a;,

z'=l,

2 _on SE(Fk), having

_non

conjugate images

(of

order two) in Out(SE(Fk)). Moreover; both automorphisms are

not

liftable to actions

of

Z2 _{on SE(Fk)(i.e}

af

₌

Adgi, &i(gi)=_—g;,v

gi=g?

~g/~, withg?- _projections

_of

_{trace 1/2in} SE(Fk) (see[3], [9])).

By [19], _{the actions on} SE(Foe), obtained by multiplying the generators by complex

numbers

of

modulus1 _are distinguished (modulo outer conjugacy) by the topology

induced_onZfrom

Aut

SE

(FJ.

This is easily seen

_not

_{to work in thecase}

of

finitecyclic

groups.

For

example ail Z2 _{actions on} SE(F2)

that

_{are obtained} by multiplying the generators with

±1

_or by switching _{the generators, are conjugate} via éléments in

AutJ5f (F2), whichbelong to theimage

of

the

0(2)

_{action on}SE(F2),consideredin [29].

COROLLARY._— There exists _twonon-conjugate,

finite

depth subfactors in SE(Fk) having

the_{same higher relative commutant invariants}([17],[7], [21]).

It

isplausible

that

_{an invariant}

of

the type considered in this Note could beused to

settle Kadison's problem in [13] (see also Sakai's

book

[26], Problem

4.4.44)

_on

the isomorphismclass

of

the algebrasassociâted

to

free_groups.

We_{are indebted to D.} Bisch,V. F.R. Jones,S. Popa and D.Voiculescuforuseful_{commentson this Note.}

For

the

proof of

the nextlemma_seethe

proof

of

lemma

4.3.3

in[26].

LEMMA 1. _— Let

NçM

fe

_{a pair}

of

séparable,finite _{von Neumann algebras so} that

there exists an orthonormalPimsner-Popa[20] basis

for

the inclusion

N£M,

with _respect to somefaithful, normalconditional expectation

from

M ontoN. Assume that N, M hâve centerswithtrivial intersection.

Then the vonNeumann algebrareduced

free

product

sé

₌

(SE(Ffc)(x)N)*NM {which by

[22] is _a type M^

factor)

has the _{property that ||Egf(y)—}

v||2^14

_max\\[gi,y]\\2, with

;₌i,2

respect to the inducednormal

faithful

conditional expectation E^ from

se

into N. Hère gh

i=l,

2 are two

of

the

k

generatorsin Fk, while

||x||

2

₌

r(x*x),

xesé

and_{T is the trace}

on

se.

In particular

ail

centralséquencesin

se

_are asymptotically containedinN.

Thislemma together with lemma

1.4

in[22] andlemma 3in[11]gives:

THEOREM2. _— Let

NçM

be _a

pair

of

séparable,

finite

_{vonNeumann algebras so} that, with _{respect some}

faithful

normalconditional expectationfrom M intoN, there exists ah orthonormal Pimsner Popa[20] basis

for

the given inclusion. Assume that N, M hâve

centerswith trivial intersection.LetsE

₌

(SE

(Ft)®N)*NM.

Then

X((^(F/£)®N)*NM)^x(N,M).

Moreover in this identification, correspondingéléments in %(sé) and

x(N,

M) hâve tlie

same obstructionsto liftingfrom thequotient.

Hère%(N, M) is therelative _{x invariant}[14] for theinclusion

NçM,

where N, M

of

-séparable finite_{von Neumann}algebras:

X(N, M)

=

(Ct (N, M)

H Int(N,

M))/Int(N, M),

where Ct(N, M) is the set

of

ail automorphisms

of

M, leaving

N

invariant and acting

asymptotically trivial _{on centralséquences}forM

that

_{are contained}in

N

and

Int

(N, M)

is the setofail inner automorphisms

of

M

that

_{are implemented by}unitariesin N. ; _\;

COROLLARY3. _— Let

^

₌

(A^B^C; A^C^D)

be _{a commuting square} [21]

of

finite

dimensional algebras, _{which is irreducible (i. e. the} centers

of

A, B and respectivelyC, D

hâve trivial intersection)andX-Markov ([7], [6])i.e. there existsa X^Markovtrace, in the sensé

of

Jones [7]

for

C^A,

which restrictsto aX-Markov trace

for

D£B).

By[24]

(y(Fl)®B)*BA2(^,(Ft))®D)*DC

is_afiniteindexinclusionoftype

ll1factors

[that _are isomorphic SE(FN)

for

_some

N>1].

Let seoe _{be the envoloping von} Neumann algebra

for

the tower

of

algebras in the iterated Jones's basic construction

of

the given inclusion.Notethat

se

x is isomorphic to (SE(Ft)®Boe)*B Aoe. HèreBM,Aoe areobtained

byiterating thebasic construction

for

the inclusionsDi=B andrespectively

CgA.

Then

LEMMA 4 ([17], [14], [15], [1], [23]).

-

Let 6 be _a Z4 action _on

_a

copy R_x

of

the

hyperfiniteJl1

factor

_so that 9 induces _a Z2-kernel_on R_x with obstruction —1 [3]. Let R0

_{= R_1XeZ4.}

Then

%(R_UR0)^Z2®Z2.

Moreover,

for

exactly _one

of

the two copies

of

Z2 in %(R_1, R0), there exists _no obstructionto liftingfrom the quotient.

We_mayassume(see [10])

that

The fc-thstep inthe iteratedbasicconstruction forthisconstructionis:

where (Uk)keZ is _a family

of

unitaries with four

point

spectrumso

that

each spectral projection for Uk has trace 1/4 and

_g=g+—

g-'is

_a selfadjoint unitary whose spectral

projections _g± hâve trace 1/2. We hâve the following relations

Tj£=l,

keZ,

U,£gU*=

-g~ifk=0,

1, UkgUj?

₌

_g,

if/ceZ/{0,

1} and

UtUt+1Ujf

₌

y=ÏUt+1

(the

trace onthis constructionis specifiedbyrequiring

that

eachnon-trivial monomialin the U;t's and

_g

_{hâve zéro trace).} With this notations, _{a commuting square for} the inclusion

R

_ !

£

R0 isgivenby

Since

U0eB

nomializesA and AdUo|c

₌

-A-dg|c,_while

_geD,

_{we get:}

LEMMA 5. _— With A, B, C, _{D as} before, (SE(Ffc)®B)*BA_{is isomorphic to the cross}

product

of

(SE(Fk)®D)*DCby the Z4 action _{on (SE(Fk)®T>)*DC induced by} AdU0. [Note that this action is in

fact

_{an injective Z2-kernel}on (SE (F/c)(x)D):ilcDCwith obstruction-l

Aninvariant for subfactorsin the algebra_{of afree group} 987

Moreover, by Corollary3, the enveloping algebra

for

the inclusion before is

sé

₌

(SE(Ft)®R_1)*R_1R0 andhence,by theprecedinglemma, %(sé)^,Z2®Z2is_gênera- '

tedby

AdU?^

\^

andby

AdU0U_1

\^.Thereexists_{no obstruction}tolifting to

Aut

(sf)

for

the

first

_copy

of

Z2, while

for

thesecond_copy

of

Z2, theobstructionis _—1.

LEMMA 6. _— With the notations before, let ÉS

₌

sé

MZ2, where the Z2 action on

se

is induced by:

Ad\3\i\sieri(sé)

(this action is _{an invariant}

for

se). Let s be the dual action on the cross product. Then SE isomorphic to

(^?(Ft)®Q_1)*Ql

Q0, where

Q_

_{= {R_ls}

UI-L}", Q0

₌

{R0, \)2-1}". Moreover _we hâve thefollowing décomposition

for

_s:

where

.5'1=.?AdU0U_1eInt(J'),

52=AdUgU*

1eCt(SS). In addition sf

_{= Adh,}

with h

₌

X5\gesé andsi(li)=—h.

The décompositionabove prevents SE from being isomorphic

to

Jones's example ([8],

inanalogy with the construction in[4]). We recall

that

in Jones'sexample one considers

N=(i?

(FN)®R0)x|TZ2 for

a

suitable action_y. Then

_{x(N) =}

Z2©Z2,where, forexactly one

of

the two copies

of

Z2 inx (N)<iOut(N), there exists no obstruction for lifting to

an action on N. By[4] (seealso[8]) _{this copy} (denotedbyy) corresponds to the dual Z2 action for

N=(if

(FN)®R0)X!YZ2.ByTakesaki dualitywe get:

PROPOSITION7. _— With the notations before, the dual action y on the cross

pro-duct

NxjçZ2,

has décomposition

of

the

form:

_y

_{= txt2AdW,}

where

z1eCt(N

XIçZ2),

Z2eInt(Nx]çZ2). Moreover there exists selfadjoint unitarieseI==e'+—e'_, with

e\

projec-tions

of

trace 1/2, so that tf

₌

Ade1,

tt(el)=

-^el, tt(e,-)

=

_ep ii=j.

COROLLARY8.

-

With thenotationsabove,

Nsrf

₌

(5'(Fl.)®R_1)*E_,Ro-Proof.

-

Assume the contrary. ThenSE

₌

sé

Y-Z2, N=(JSf(FN)(x)R0) !*YZ2 would be

isomorphic(sinceineach

of

thetwocasestheZ2 actionis canonical). Moreover

it

would

follow

that

the corresponding dualactions hâve thesanie image in Out, and thus by[3],

wemay assume

that

they areequal.

Since in N=(„S?(FN)®R0)KrZ2, the décomposition

of

_{an automorphism as} the

pro-duct

of

_a centrally trivial _{automorphism and an almost inner automorphism,}is unique

(moduloinner automorphism), itwould follow

that

_{the décompositions}

_{= sïs2}

(regarded

asah action

of

Z4©Z4)isouter conjugateto thesimilaractiondescribed by

tu

t2 in the

preceding lemma. Butthisisimpossible becausethetwoactionshâve différent

characteris-tic invariants (see [9]). [Since t1(e2)

₌

_e2, •(t^y2

₌

Adel, while, with the notations in

Proposition5,

st(h)=

—h, sf

₌

Aàh.]

PROPOSITION9. _— Let

N

be the

factor

constructedin [8]. Thenthere exists subalgebras

A<=B z'zz

N

so that Ais isomorphic to (SE(Fk), B is the crossproduct AX)^Z4, whereQ is

a Z2-kernelon A with non-trivial obstruction (to lifting). Moreover

N

is the enveloping

algebra

for

the inclusionA<iB and Ais isomorphic to SE(Fk).

Noteremisele 10février1993,acceptéele 12_mars1993.

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DepartmentofMathematics, UniversityofCalifornia_atBerkeley, Berkeley, Ca 94720,USA and InstituteofMathematicsofthe Romanian Academy.