A one-parameter group of automorphisms of L(F∞)⊗B(H) scaling the trace

Analysefonctionnelle/.Fwwcfr'0Ha/Analysis

A

_one

_parameter

_group

of

automorphisms

of if

(F00)(x)B(H)

scaling

_{the trace}

FlorinRÂDULESCU

Abstract — Weprovein thispaperthe existenceofaone realparameter, continuous group a,of

automorphismsof_{the von Neumann}algebra S£(Foe)®B(H), scaling the trace by(. Hère

if

(Foe)

isthe_vonNeumannalgebra associatedtoafree_{nonabeliangroupwith infmitely many}generators,

while B(H) is the algebra ofail boundedlinear _{operators on a} separable, infinité dimensional

Hilbert_{space H,}andxisthecanonical trace.

In particular it follows that there exists _a type III, factor _{having a core isomorphic} to J5f(Foe)®B(H), obtainedby takingthe crossproduct of.S?(FM)®B(H) by thegiven_group of

automorphisms.

Un groupe

à un paramètre d'automorphismes

de

module

R*

_{sur l'algèbre du}

_groupe

libre

_{à une}

infinité

de

générateurs

Résumé— Nous montrons l'existence d'un groupe a,,àunparamètre teR%,d'automorphismesde l'algèbreSC(FJ®B(H),tel_{que x}(O,(X))₌tx(x), t>0,

xeif

(FJ®B(H).IciSC

(FJ

_{est l'algèbre}

de vonNeumann associéà_{ungroupelibre non}commutatifavec une infinité degénérateurs, tandisque

B(H) _{est l'algèbre des opérateurs bornéssur un espace hilberlien separable} H et % est la trace

canonique.

En particulier, il s'ensuitqu'ilexisteunfacteurM,detypeIIIj telquel'algèbredes élémentslaissés

fixés_parle_{groupe modulaire (associé}à_{unetrace généralisée sur M) soit isomorphe}àS£(Foe)®B(H)

[i._{e.M est}leproduitcroisé deS£(Foe)®B(H)_parle_groupeà_unparamètred'automorphismes].

Versionfrançaiseabrégée_{—Dans cette Note nous utilisons une description de l'algèbre}

de von Neumann .S?(F00)®B(H)à l'aide des _{travaux de} Voiculescu [8] _sur les algèbres de

von Neumann associées à des groupes libres.

On montrera que ^f(Foe)®B(H) _{est engendrée par une algèbre abélienne} diffuse A

₌

L°°(R) (avecla _{trace donnée par} la _{mesure de Lesbegue) et par} des parties finies

pXp,

où _{p est une projection}finie dans A, _{et X un élément symétrique non-borné. En outre,} la

trace sur toute sous-algèbre engendrée parpL°°(U)et

pXp

(où _{l'unité p est une projection} finiedans A) _{est uniquement déterminée par}les _{valeurs qu'elle prend sur />L°°(R) et X} (les

deuxalgèbres _{sont indépendantes par rapport}àla _{trace, ausens des probabilités non}

commu-tatives de [7]).

De façon intuitive, si _{Y est} le générateur de A, _{on peut dire que} .5?(Foe)®B(H) est

l'algèbredes quantités mesurablesassociéeà deux observablesY, X, la _{deuxième ne}

permet-tant de mesurer que des _{quantités qui sont} finies _{par rapport} à Y (i.e. de la forme

pXp,

où _{p est un idempotent de mesure} finie, dans l'algèbre de Y).

De plus, pour toute projection finie_{p dans l'algèbre de} Y, les observables

pYp

et

pXp

sont indépendantes pour la _{mesure(au sens où} la _{mesure est uniquement déterminée par ses}

valeurs _sur les algèbres _{engendrées par}

pXp

et pYp).

Les résultats de [7], _{permettent de retrouver} le _{comportement de X,} Y _{par rapport}à la

trace, en considérant X comme une matrice aléatoire, quand Y _{est contenu dans l'algèbre} des matrices diagonales. Si l'on considèrela limite de la représentation matricielle,la _trace

est maintenant calculée comme la composée de l'espérance conditionnelleavec la trace (non normalisée) sur les matrices.

Note présentée par Alain CONNES.

Tout automorphisme de A _{donne un automorphisme de}

if

(F00)®B(H) qui laisseX fixe.

En particulier on prouve le résultat suivant:

THÉORÈME. _— II existe _{un groupe} à _un paramètre (oc,)IeK*+ d'automorphismes de

5£ (Foe)®B(H), tel _{que les a,} multiplient_{par t la tracecanonique x sur}

if

(Foe)®B (H) [i._e.

T(a,00)

₌

fT(x),

*ei?(FJ®B(H),

t>0].

La classification [1] de A. Connes des _{facteurs de type}

III

_{montre que l'existence d'un}

automorphisme de

if

(Foe)®B(H), multipliant la _{trace par t est équivalente} à l'existence d'un facteur M de typeIII,, _{tel que l'algèbre} des éléments fixés _par le _{groupe modulaire} (associé à _{une trace} généralisée _{sur M) soit isomorphe au facteur} £C(Fco)®B(FL)

[/e(0,

1)].

Comme nous avons par ailleurs trouvé un groupe (a,),eM»+ d'automorphismes de

if

(Foe)®B_{(H) nousavons} montré (à l'aide du travail de M. Takesaki[6]) qu'il existe aussi un facteur M de type

IIIj,

tel que l'algèbre des éléments fixés par le _{groupe modulaire} (associé à _{une trace généralisée sur M) soit isomorphe au facteur}

if

(Foe)®B(H).

COROLLAIRE. _— 77existe _unfacteur M de _type III1 tel_que l'algèbredes éléments invariants

par le _{groupe modulaire} (associée à _une trace généraliséesur M) soit isomorphe au facteur »2'(Foo)®B(H). Ici M est leproduit croisé de <£(F00)®B(H) _parle _{groupe d'automorphismes}

(ai)t

6 K +•

Inthis _{paperwe find}

_a

_{new description} for_{the von}Neumannalgebra «S?(Foe)®B(H)

which is _{a conséquence}

of

Voiculescu'srandom matrix picture for _{the von} Neumann algebras associated_{tofree groups. We will prove}

that

if

(Foe)®B(H) is generated by _a diffuse abelian algebra

A=

{Y}"

₌

L°°(IR) and the finite parts

pXp,

of

_{an unbounded}

symétrie élément X [which is

_not

affiliated to

if

(F00)®B(H)]. Hère

_p

_{runs over the} finite projections in A (offinite Lebesgue measure).

From an heuristical point

of

_{view, we may say}

that

we are given two symmetric observables X, Y and

that

if

(Foe)®B(H) is the algebra

of

measurable observables, where only finite

"parts"

pXp

(with respecttoY; _{i. e.}

_p

is _{afinite measure} projectionin thealgebra_{generated by Y) may be measured}out

of

X. In additiontheposition

of

the observables is so

that

_{for any}finite trace projection

_p

_eA,

_p

Yp

and

_p

Xp

behave freely withrespectto the measure (in the sensé

of

the noncommutativeprobability introduced by D. Voiculescu in[8]).

The matrixpicture

of

D. Voiculescuforfreeproductsshows

that

_{wemay assume,}

that

asymptotically,Xis _{a matrix}

of

_{very large size with} independent random entries, while Yconsists

of

constant diagonalmatrices. The_{measure is} then computedby composing the expectation value with the (nonnormalized)matrixtrace.

The mainresults

that

_weobtain_{in this wayare}the following:

THEOREM. _— There exists _{a one parameter continuous group} (a,)(eK*+

of

automor-phisms

of

i?(Foe)®B(H)

such that ex, scales the trace x on

if

(F00)®B(H) by t [i._e.

x(a,(x))

₌

tx(x),

xeif

(FJ®B(H),

t>0].

By Connes's classification

of

type

III

factors [1] it is well known

that

_{any such}

automorphism_{a, on} if(F0O)®B(H) scaling trace by

te(0,

1), gives _{a type}III, factor with a core isomorphicto

if

(F00)®B(H).

The fact

that

_{we hâve a continuousone parameter group}

of

automorphismsshows (by the work

of

M. Takesaki[6])

that

_{we may now in} addition find _a type

IIIj

factor having_{acore isomorphic}to

if

(Foe)®B(H).

COROLLARY. _— There exists

_a

typeïïï1

factor

M having

_a

core [1] isomorphic to

if

(Foe)®B(H). Equivalently M is thecrossedproduct

o/if

(Foe)®B(H)with the action

of

U_{induced by the oneparametergroup}

of

automorphisms.

In particular_{we reprove in this way}

that

the _{fundamental group}

of

if

(Foe) is R+/0 [equivalently

that

_{for any selfadjoint nonnull} idempotent _{e in} i?(Fa>)>

e&ÇF^e

^£C(Fm)]._{This covers}partlythe main resuit in [9].

The results suggests

that

_apossible connection could be realized betweenthe noncom-mutativeprobability theoryrecently introduced by D. Voiculescuand thework([3], [4], [5], [2])_{on algebraicquantum}fieldtheory.

The (verynatural)question about thepossible existence

of

_atype III1 factorhaving

_a

core isomorphic to

if

(Foe)®B(H) was first raised to me by M. Takesaki. We are indebtedtoD. Voiculescuforseveral usefulremarks.

We will first construct

a

semifinite algebra

se

_{which we show} to be isomorphic to

if

(Foe)®B(H) and then show

that se

has _{a one parameter group}

of

automorphisms scalingtraceby t, t

e

U.

The idea istoconsider firstthesubalgebra

J/0

of

thefreeproductL00(R)*C[X](X is

hère an undetermined variable, assumed to be selfadjoint) generated by L00(U) and the

finite parts

pXp

of

X, where

_p

_{runs over the set}

of

_{ail finite Lebesgue measure}

projections in Loe(U).

If

_{one requires}that

_{p Xp}

is semicircular (withthecorresponding normalization conditions coming from Proposition2.3 in [7]) andfree with the algebra

pLx(U),

_{one gets in this way a} well _{defined trace x on} sé0. We réfèreto [7] for the définitions regardingnoncommutativeprobability.

We let

se

be _{the von} Neumannalgebra coming from G.N.S. construction forsé0 and

x. Clearly anyautomorphism

a

of

L00(U),scalingtraceby / inducesan automorphism

oc0ofstf0 (mapping Xinto t~1,2X). Moreover, by proposition2.3 in [7],<x0also scales

trace byt.

It

remainsto show

that

se

isisomorphicto

if

(Foe)®B(H). Thisis done exactly by the same technics as the one used in the

proof

ofTheorem 3.2 in[7].

Werecall first

_a

fewdéfinitions from [7], [8]. Let (A, cp) be a C*-algebra withunit1

equiped with

_a

trace cp. A family

of

subalgebras 1_e

A,£

A

(ieï)

iscalled_afree family

of

subalgebras

if

q>(a1a2.

. .a„)

=

0 whenever0,-eA,-.with

ij^ij+1,

l^j-^n—l,

(p(aj)

=

0,

7=1,

2.

..n. A family

of

subsets(n;),-eI is called free

if

the family

of

the subalgebras

generated by theCîtand 1 is free. Moreoverafamily(/j)i6lis free

if

the family

{f,

1}"

is free. A free family (fi)i is called semicircularif

f

_areselfadjointand

if

thedistributions

of

theyj-,

ieï,

_{are given by}thesemicirclelaw

In particular it follows

that

_{the von} Neumannalgebra generated by _an infinité free semicircular family C/î),eS is isomorphic to if(FcardS). To observe that, _{one has} to consider unitaries F;

that

_{generate the same abelian (diffuse) von} Neumann_{algebra as}

f,

for each

ieS.

It

follows

that

the family (F,)ieS is also free andhence

that

_{the von}

Neumannalgebra generated by them isisomorphicto

i?

(Fcards) (see[7]). BycardS _we

dénote the cardinality

of

S.

CONSTRUCTION OF THE ALGEBRA

JJ.

-

Let A

₌

Loe(IR) with the trace <ï> comming

from the _{canonical Lesbegue measure on} U and let

Lf(U)

be the subalgebra

L2(R)

_H

L°°(R). Let X be _{an arbitrary} variable and let sé0 be the subalgebra

of

the freealgebraicproduct

generatedby

Lf

(U) and

{pXp\p

_projectionin

Lf(U)}.

srf0 has

_a

canonical involutivestructure

if

_{we require}for X tobe selfadjoint.

We now definea traceT onsé0 which is uniquely determined subjectto thefollowing

conditions:

(1) _xrestrictedto

Lf

(R)

₌

O.

(2)

pXp

and

_p

L00(R) are free with respecttotherestriction

of

_x to

psé0p

(3) <!)(p)~ll2pXpis _{semicircular with respectto} thetrace(TQ?)-1)T

_onpsé^p.

In the conditions(2) and (3) above

_p

_{runs over the set}

of

ail finitetraceprojections in L00(R).

Theexistenceand the unicity

of

_x followsfrom the fact

_{that p Xp}

and

_p

L00(R) are

generating in

psi'0p

_an algebra stfp (with unit p) which is

_a

_copy

of

the algebraic free product

while on thislastalgebra (withunit p) conditions(2), (3)_{are defininga unique}normalized trace xp= (x(p))_1x [7]. Since s#0 is the union

of

the algebras sev, after ail finite projections

_p

in

Lf

(R), weobtain_{in this waya trace on}srf0.

The fact

that

the trace is well defined followsfrom proposition2.3 [7] _{which asserts}

that

wheneverD is _an abelian algebra, free to the algebra generated by _a semicircular élément Z, then for _any nonnull projection

qeD,

the algebras generated by qT> and

qZq

_are free, while %(q)~112qZq is _{semicircular with respect to} the normalized trace

(x(<7))_1_X(Xistne trace onthe algebra generated byD andZ).

To check

that

the traceis well _{defined we hâve to take two} nonnullprojections_{p, q} in

Lf

(R). _{We may assume}

that

q^P-

The tracexp= (T(/?)_1)T defined by the condi-tions (2), (3) _{on the} subalgebra sé°p

of

s/0

(with unit p) is _so

that

x(p)"ll2pXp

is semicircularandso

that

L00(U)p and

pXp

arefree.

But the propositionrecalled before (from[7]) shows

that

_{in q}stf°p_q,

(xp(q))-ll2q((x(p))-ll2pXp)q=(x(q))-ll2qXp

is semicircular and

that

(T(q))~1,2qXq and _{qL°° (R) are free with respect} to the trace

As for

_p

let srf\ be the subalgebra

of

s40 (with unit q) generated by Loe(R) q and

qXq.

Let_xq=(x(q)_1)xbe the tracedefined by conditions (2), (3) _{on ,s/° [Le. so}

that

(T(q))~1!2qXqissemicircularandfree with qL°°(R)]. Clearlysé°q

£

_qstfp_qand_{we only} hâve to check

that

_{xpq restricted} to

sé\

coincides with _xq [as this will imply

that

xp

restricted to

sé\

is

(0(^)/0

(p))xg]. Butthis follows from the fact

that

both _{traces are} making

qXq

semicircular and free with respect to qL™(R). Thus the restriction

of

0

(p)_xptoséq coincideswith4>(q)_xqandhence_xis welldefined.

The algebra

se

is _now obtainedby takingthe G.N.S. constructionassociated to

s/0

and _x and by considering the weak closure

of

stf0 in this représentation. In _{this way} we obtain a semifinite faithfull tracex on

se

which extendsthe given trace. A system

of

generators for

se

is L°°(R) and

\pXp\p

projectionin L*(R)}. [Note

that

_now

Loe(R) is

_a

subalgebra

of se

(containing the unit

of

se) and

that

the restriction

of

x to

L00(R) is$.]

PROPOSITION. _— With the notations above _wehâve

J^

₌

if

(F00)®B (H).

Proof. _— Let

_eu

_e2,

. . . be

a

partition

of

the unity in LM(R) with projections

of

equal finite trace. Let _vt,

ieH

be the partialisometriesfrom the polar décomposition

of

eiXe1

₌

vibi,

ieN,

i^>2. The behaviour

of

Xwhencutted offby finiteprojectionsin L°°(R) [Le. properties (2), (3) above] and by the

proof of

Theorem 3.2 in [7], shows

that

vf_vi

₌

_e1;vtvf

₌

et and moreover

that

the family

(4) {e1Lco(U)e1}U{vfReXvi,

vfïmXvtliJeN,2Si^j}U{bi, a,|ieN,

z^2} is free. _{Hère at is a} semicircular generator for vfLao(U)vi. On the other hand, by construction

se

is generated as avonNeumannalgebra by L00(M)and

{pXp\p

projection in Lj?(R)}.

Hence it follows

that

the family (4) _{is indeed a System}

of

generators for _e1

se

_ex and thus

that

psép

is isomorphic to

if

(Foe) for _any projection

peLf(U).

Thus

se

is _a

factorisomorphicto

if

(F00)®B(H).

Using this new description

of

if

(Foe)®B(H)

(^sé)

_{we are going} to show

that

any isomorphism

of

L°°(R) induces an automorphism

of

if

(Foe)®B(H)^^/.

The automorphismis first defined algebraically_onsé0 and thenextended bycontinuity to se.

PROPOSITION. _— With the notations above let 0 be an automorphism scaling trace by

t>0

o/L°°(U). Dénote byS _{the automorphisminducedon L}00(R)*C[X],mapping L00(U)

onto L00(R) by 0andX into

t~m

X.

ThenS _{induces (by restriction) an automorphism}

of

sé0, also denoted by S, _{which scales}

the trace by t andthus an automorphismS

of se

scaling trace by t.

In addition the depencence8-»Sispoint-continuous.

Proof. _{— Clearly we only hâve} to prove

that

S scales trace by t.

It

is sufficientto

show

that

S _restricted

_to

_psé0p

_{scales the trace by t. We may also assume}

that

x(p)

₌

$(p)=l.

Let

q=Q(p).

Let

_eu

_e2,

. .

.,

enbe any nnonnullprojections in

p

L°°(R). Dénotebyxp

=

(x(p))

~J_x,

xq= (x(#))-1x the trace on

psép,

respectivelyon

qstfq,

given by the conditions(2),

(3). [Inparticular_x restrictedto

_p

se

_p

isx^while therestriction

of

x toq

se

qisx(q)xq.]

Moreover

for_any

iu

. .

.,

ime{1,

...,«}

since 0 scalestrace onL™(U)by t.

Thus

Because

of

property(3)

of

x,

t~1/2qXq

is semicircularandfreewith respectto Lco(U)q

in the algebra

qsé

_{q (with trace} xqand unitq).

The relation(5) and the fact

that

the trace is _{uniquely determined on a free} product

[8]

if

it

is _{determined on} his factors shows

that

we may continue the previous equality with

We hâve used hèrethefact

_{that p Xp}

is semicircularandfree with

_p

Loe(U)in se'p,with respectto the tracex _. ThusSscales the trace onsé0 andhenceforth on se.

Thelasttwopropositionsgivethe

proof

of

_{our main}resuit.

Noteremisele5mars1992,acceptéele 15avril1992.

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