.
N°274|)LEURS— ALGEBRESD’OPERATEURSET

IP_SPACESASSOCIATED

WITH

Colloques Internationaux C.N.R.S.
PHYSIQUE MATHEMATIQUEAPPLICATIONSEN

ARBITRARYVONANNEUMANN

ALGEBRA

by|oo|

Abstract:To

any

which

Denmark.

M,

LYM)=M,

Ll@y=M,,thea measure

tothe

HilbertspaceofM

issemifinete,the

LF (M) -spaces

andto

theKunzea unintroduced
LP (M)-spaces

byform.

||Dixmier,are||con-beindependent(upchoiceofeo.

|ooo0)

LPS,uw )defonctions

sur un espace(M) | &uM,La)

My,

que

LY

introduits
sontfixeunitaire)les12M)
SiLP (1)sontisométriquement

isomorphes

auxpar

Dixmier,deM. Laprés)construction

du choixde?

—175

bornés

non—

Les

semi.

|

auxassociééqui-(a une|oo:espacesaffiliésmodulaire:.

isomor-est

espacesindépendantes'avérepw ).

(SL,estM

1953-1958.

en

1'automorphisme

avec

M

Kunze

d'opérateurs

espacesrM,e?)@ suretSegal

demesuréet que-es-desespaces

standard.

comme2généralisentreprésentation

construits

croisés

poidsqui||de lade Hilbertespaces,|

unitarytoassocionsnous

M

NeumannVonde1¢p§|M.

on¢weight,

lesvalence1)

algébre1'espace

produits|

is iso-isometricThe|(Q,u).

standardaretoutemontrons

LP (m)

itsLP)

classiques>inA

BanachLP,

Banachtooutturnsof thedefinieM

a fixedwithassociatedgroup

paces

a

classical
space

BanachRESUME~ NousLPM,1953-1958.

Banachas certain spaces of unbounded operators affiliatedM with the modularofR(M,c®)crossed productconstruction

Thephespaces

inassociate

and that L®(M)

morphic|)

wegeneralizeon-|algebrafunctionsequivalence))

Neumannofautomorphism|von1<p<~,

structedwith the|Universitet,LP),

isomorphic|

OdenseLP(o,u)Segal~

UFFE HAAGERUP

spacesWhen||spaces

We show that©

|||

Introduction

Thisnote|

contains

an-outline

oCof a forthcoming

paper.

ER|

EE

In [4], [11] and [8] J. Dizmier, I. Segal and R. Kunze have comstructed the IP-spaces IP(M,7) associated with a semifinitevon Neumann algebra M, which generalize the classical Banach

spacesLP(N,u). The IP-spaces we construct in this note willconsistof operators affiliated not withbigger|

algebra,namelycrossedproductM,

witha modular

automorphismgroup.

Too?

=

wheretheeSyofisdualoo?

~~

hes

itself but with aM,=hT

1P (M)

of|M

satisfying
isdefined

affilisted with

8L%h=nR(M, of )a traceaction.

as the set of T-measurable operators
satisfying

MtheM,||oopee|

Sooo

equipped with a suitable norm. Since the triple(H,,T , 6") isindependent (up to unitary equivalence) of the choice of @, theIP-gpaces are independent of @.

We haveL°(M)=M

and|LY(M) 2 My.

12M)

IREoo

is a Hilbert

space,»

end the representation ofL(M)on
12(M)
defined by left
multiplication is standard. If M is semifinite the IP-apeces
constructed in this way are isometric- andIP (M7)for any n.f.s. trace 75 onM.

|=

176 —

orderisomorphic|

ooooto|||§ 1Construction of the IP-spacesLetM

be a vonNeumannnaturalinjectionin the crossed

is a fixed weight
anda oneIetTefx)
B

on|

on

where= [estexists a trace
the positiveby hit = A(t)Ttimesonthedualnormalend we let=eT,2)

If

ing | @|T|eh

is theCored)he

1.3 Theoremof

MyisIE@ ~» he,ytowithwhereMj

we puthisdetermined

CCh|

isA
seR

T-measurable

iff|

toof

set

|

: |

hy,

of

» whichthen.

.

@isbounded|Er

Ee|extension

operators© ed h.

||SE—177 —theM,

CLa unigue

Ohoo:

with|

avo.

,

§ withequal

decomposition==@oT7DE

affiliated= qn),

ah|

@=

¢=T (hy).

-(Note that the set of 7 -measurable operators
algebra with respect to strong sum and strong

|

there

|

C=e"

has|

ooo¢,+ Hence

LoM

semifinite}

FEPE Msatisfying

|

oog

on

onto the setof T-measurable
M,by|¢ oT =T(h*)

M , i.e,spectrel(et[9) p.m).

with||

Mj (cf [6],[7]1). The weight¢

operators

omap

onnormal,( A dedThe\|.

~ In particular©

MFRE

be the Radon-Nikodym derivative ofselfadjoint

3.3

xe

seRweight

oo| satisfies=xeM,|oosemifinite)..2 Theorem
setM

such that for

operator affiliated| TeOg4

hypositive’@oby(cf [12], proof of lemma 8.2). The trace T satis-

respect to the trace

1) Theits

Me —p M, , given

weight1.1 Definitionanyo®)

EA

M_ , such that

selfadjoint|ForA(t)is the dual action onis 27

R(H,M_ is generatedasRTMj

N=weight, T:

valuedT(x)-where 0 = 0%dTwithCo

the operator|

We will identify | M
productgroup of unitaries= ABNEY
bealgebra.

M. By construction

parameter|

Co

tohCs
a linearmap

affiliated

SoLo

oo||| Co|;

:

on
Mj
is an
preduct, of [1i]),

||

non

any

|

oo|

|
infinite on

isT

traceThea)||(mM).

on

1.6 Remarks~

is an isometry with respectLm)

onto

MN,ofWhil; = tr(lh|)

normto the|

oo:

B|

oo|

tr(h,) =0(1).

byIL:(M)

ontroo¢ -» hg

The maph6h =e 5h , seR.

11) We define a linear functional
1.5 Proposition

rsoperato
Aable
T : -measur-the set ofaffiliated with M, , for which||||
denote1.4 Definition
IY(M)1) We : letvenishingoperator

It (Mm),inb) If M is a factor of type III,, M, is a IL ~ factor. In thistr
case any normal trace on M, is proportional to T. Henceis not in general the restriction of a trace on M,.
1.7|

Definition

LP(M)
LPM)

We putCand|a) If p + q then IP(M)NI(M) = {0}.
¢) I'(M)anyonlyooL°°(M)|;
1.9LetPropositionpelle] andaffiliatedThenonthenconsists<eopIfb)

M_withvanishingof bounded= {nen|letoo
be a closed,&, and let

a = ula]following conditions are equivalent:(2) wel (H)—defined|E
operator|
densely

be its polar decomposition.

| I|
|
oooo|

bythat|B|||

lalg=lan
1l,%|)=ACo
p=Ce 5||||later,oo|=x.

lal? ertm).

and1.10 Definition
On IP(M) we define“Forunbounded.

HenceSoa(1) a € LP(M)-.
R}2=h,se| | ogh

is-|

oo|

operator

1P-operators.

= exp(-2)h }
|
|=h }

8heh

M_|
M_|oIEoo

1.8 Remarks

aff. withaff. with:

oo|

EE.

T -measurable
v -measurable= {n,= {n,11| lly isisa normalsoa norm|oooo(ef.

prop.

for— 178 —1.5).

Itwillbe proved

1l<p<goo.,

|

:||

1.11

lemma|

Lot h,k € T*(M),.

1Proposition~(1)oobeSo||(M):tr(un®*) = tr(h)(2)

Por any=xeLf(M):

1.14Theorem|

Letp,q €[1, 2], 2 + x = 1, and“then|Nall, iol,Ntr(x*x) = tr(xx*)|ll abl, <B

1.15 Remarks

|

Boo|

SEER

let a € IF(M)(HoldersaThe proofof
line theorem|LI(W),
owel

any unitary

andh e Lt (m)anyForooo||

1.13 Corollary|tr(ab) = tr(va).

andab, ba € L}(M)thenis analytic inanda eI’)

, 3+%=1, and let

Let p,q el,a|{xeC|Re x€10,11} .

the open strip
1,12BEThe function x-H'k * € L'(M)||andbe 1(M),inequality)

oohs|

|

|

|

oTheorem 1.14
is based on lemma 1.11 and the three
for analytic functions (compare with [9]p. 113).

|

Dixmiers proof of Holders inequality in [4] can not be applied,because in aur case the spectral projectionsof an IP-operator
is not in IP.
|oo|
:
1.16

Proposition

|

EE

(1)

Tet

’ 5 + g = 1, For

anypq€[1,0e]a(2)a,beIP(M)

Hence | I,Hasoll

to thePorany(3) L%(M)||

topologyofpel1,o0da € IPM).

is a Hilbert

oo:

|Lo
oothe norm topologyonIP(M)is equalconvergence(cfin measurecomlete179 —EE||in thespace with immer product|bly <1}

|

EE

ooE

is+ loll,|

inequality)

IP(1M)

Sola,|(Minkowskis

is a norm.

1.17 Proposition(1) For any 1p € [1,02](2)|

all, = sup{ltr(ad)l | ve1d(m),
For|

97 ».

p-norm.

(ald)

ooi06).
oo= tr(b*a).

ILetpel[2,00l . For a,beIP(M)

Theprooflemmaofbaseda|oo

1.19 Theorem
Let .p e[l,mo]|
gq = 525. An operator‘a functional
isandbya(x)isomorphismofLY(mM)a€ L(M)= tr (ex).
ontoa=»,

Banachspacepiaell, =]CreIP(M) _.

For

aeM:
©

|

andx e L2(M)e(a)x= xa

Theorem(1)~

is a normal,A (resp. Qe )Menti-representation) of(2)Thevoneachother,andwhereJ

is the(resp.
representation
faithfulBELM).
on the Hilbert .space¢ (HM) areand|QU) = IAM
conjugate

ConooooA(M)algebras

ooaa.

,1.2)

Neumann

ee|

:

oo

we putEEooordering

is ‘the dual of the(mM),(sa =selfadjoint)..

= 1M)

eee

VY velP(y),tr(ab)z0i.e. the partial ordering ofofaThenand let aeli{M).

2 + F =1,

220 &>

|

|

oo|

|

1.20 Proposition
Letlinear|isometryofcommutants
|

|

|

SREP
x->x¥

in12M).

© (3) The quadruple (A(M), L2(¥), J, L2(M),) is a Stendard form|in thesenge

|

|

oo|

Same as [4] proof of theorem 7, =Proof:

defines

The map

dualtheIE

IP(M).

of|oooo|PM)on@,isometric

anput|

line

threetheand

1.1%

lemma

on

1.18istheorem.

||SoDEi?)(hap + Moll)- dl, <2€ Plage
Ip +eelle of?
I JasoP

CC|Co|p.30)(cf. [4] lemma 51.18 Lemmaof [57].

ooI— 180 —||§2 The semifinite caseIetM

be a semifinite

andletT

be a

U(R)

regularrepresentation

theERT.

NeumannL(R,H)

2(R)

oo=

Ce|

algebra associatedLetR.

(RR)=withdenoteF|

thelefttheFourier-~BEe [mist £(t) dt[|withoo|the ‘groupof(F£)(s)

Identifying= M®UR)

vonPlencherél operator|M.
|R(M, 0)isonoowhereSENATE

von Neumann algebra on a Hilbert space H,n.f.s. trace

have:

weHSL? (R)=| oo

AethenUR)

whereI™(R)

nacts

=
as

FL(R)F

multiplicationR(M,¢™)

For any borel function f(s)operators

nonaETheoren

Jetpell,ee[and the mapLP(M).
ontoon R

we letoperator.

Ifaa ©F m(exp(Z))FLele§3ApplicationstoIetMa vonooL%(R).

mn(f)

|

Henceon1°(R).
EE

aeIP(M,%.)

vonNeumann

NeumannSo

thena @ F'n(exp(£))F € IP (M)is an isometry|
Coalgebras

algebrawith||

withof | IP(M,T,)a periodic& periodic
qoweight.

NSF-weigt

be the action

GC => aut(M)definedits injection in theMted bybyk(t)LetSand a group of unitaries

denote

the|

her©

|

=_

is1

theovalued>
Neidentify

Mwithweight©" n,_y(x)|My —-»

M,pt
xeM

givenby||:action— 181 —|

xem.

=0

dual

We will

M, = R(M,x). Mo is generaoo
A(g), ge G, such that=a@xaE)

operator8(x)©" 7, neg,vhere= od «

crossed productal)|

R -»6. Let x+@=R/prz

Put

be the quotient map

a| Ps

and let«:°oo

end let T, be a period for @, , i.e. Op=1.
t -»t

@

denote the closed,g-»fg
|

2.1

beoo

= Me FL(R)F.

densely defined multiplication

|

|

ofGZ
on
M;. (cf. [71) .
8aZon|}.

||

Repeating

thearguments

frcm

thethepositivebyki

selfadjolnt= A(})

startofoperator

affiliated, teR . The trace

Te®

= AT

where:

Twe letgemifinite’of@=T(ke).

ooPropositicn

(1)

The

|

{io @ normal,operatorswhich

ek=2k~~

and|

©

ko

affiliated

with

it

tospacestheisnothard

IP (M)

toforofKg ,SU

decompositiona|iff@= Ak

prove

Sois bounded.

p <eo~ that
from

these

thespaces

general|1P-spaceswith My|.

|are isomorphicconstruction.

We will instead use proposition 3.2 to prove the following
slight strengtliening of a result due to Connes and Takesaki(ef. [3 , Chap.II, corollary 4.10] ).
oo— 182 —||= 7 ( (eg =aT-measurableobtained|Cae

We could now continue as in§1and constructnew"consisting of the 7 -measurable operators affiliatedwhich satisfies:
EE+CoHoweverMyav) 20)

ek||

is equalto the set ofAax oi)
isendwith respectto-'k|a=2%ain particular¢|

g =@e3,ooame:be ‘the spectrala >0:

ooput(A=ex-ZEH.

= J mae ¢EBonsemifinite |gelfadjointkydetermined|REoothen for any|¢positive(2) Tet;M,|

derivetive

3.2

|

k is

oo

be the Radon-NikodymieeeweightM,

whereQve

T 3|

withHM,

set

that

|

normalk,getsatisfies:

3.1 Definitionanywe\ =exp(- £0.

.

‘For

§1

@o8 =T(k+)

7 such thathas a (unique) n.f.s. trace

||

|

3.3

Theorem|

LetM

be(1)

For

any

a g*-finite
twonormal,a unitary(2)|Foranyué&M
two,umethodof [3]
|lt

isA.

Connesnormalthat

toprove,

andE.

NL

©

R Von

M3!

Wyweights

¥||

oo

that

Theoremuis

R $yona§ -finite

sense,

that

there

sothat lly

M,CrReferences
[1]

|

||

©
[2]

.

A.

ConnesAnn.

Sc.

denotsS\

existsthereexists

|

M ,w

oo

Connesfactors
A.

& E.

np

typen

¥ul

|

valid

|

|
,

any

two

IIL, -factor

arealmostsequenceof=ofornunitaries

Ss -H0Q.

|||

desfacteursSupérieure,4

deser.

typet.

III,
6,

fasc.

2|
Stdrmer , Homogeneity& M,|

for A=1.
thata|

Normaleof type III.

Connesoo|

proved,existsclassification

1'Ecoleuon

there.

(1973) 133-252.
A.

[3], Unehe,|x $ A NZ
NSY in the above inequalities).

3.3

\p,

‘¢

Stérmer [ 2 JhaverecentlytheinYP.

|

states

5eINJoi

P u'sSu

ApS

Aygu

AE

statesequivalentinwl

|

III A

Ay

n.f.s.

Ve gives

easy

type

thatM , such

3.4 Remarkoffaithfulsuchunboundeda unitary€Thefactor

J. funct.

Takesakioftheanalysis, Theflow

ofstatespaceof||28 (1978) 187-196.
weights

onafactorof type III. Tohoku Math. J. 29 (1977) 473-475.
[+]

|J.

|
[5]

Dixmier

Bull.
U.

Math.

Haagerup

Math.

|, Formes

Soc.

, TheScand.

[6]

U.

Haagerup|To

appearin37

linéairesFrancet.

standard(1975)

sur81

anneauformofValuedJ.

analysis.

vonWeights— 183 —d'opérateurs.

9-39.

271-283.

, Operator
funct.

un(1953)|||

Neumann|algebras.

|

|

|

in

von

NeumannN

algebras.

|
oo

|

[7]

|U.

|
[8]

R On, Ly

Fouriergroups.

Trans.

Amer.

[9]|E.

|-thealgebras| R. Kunze~

Nelson , Notes

dual

weights

II.

Math.

for

transformMath.

crossedScand.

Soc.

on locally89

products

43,119-140(1958)

on non-commutativeof(1978).

compact}

unimodular

519-540.

integration.

aE

Journal‘of funct. anal. 15 (1974) 103-116.

[19]

G.K.

|
|

Haagerupvon Neumann

Pedersenvon Neumann[11]I.

|Ann.

[12]

-|

Segal& M,

Takesaki

Algebras.

, A non,

TheActa Math.

commutativeM.

Takesakiofvon, Duality

Neumannforalgebrasof

Nikodym

Theorem|for(1973)

53-87.

|

ofabstract

integration.

andthestructureMath.

131,extension|

crossedproducts

type

Acta

ITI.

(1973)|

Haagerup|ooMatematisk

Institut,

Niels

Allé,

Bohrs

Radon

130

of Math. 57 (1953) 401-457.

249-308.

Uffe|

Odense5000

Universiteit

Odense,Denmark.

— 184 —|:)

oo||