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Internat. J. Math. & Math. Sci.

VOL. 18 NO. 2 (1995) 317-322 317

AN

APPLICATION OF

THE SAKAI’S

THEOREM

TO THE

CHARACTERIZATION OF

/’/*-ALGEBRAS

BORUTZALAR

(Received May II, 1993)

ABSTRACT.

Thewell-knownSakai’s tl,eorem,xvl,ichstates ll,atevery d.rivalionactingon avon

Neumann algebra

isinner,is,sed1o el,lain a newelegant pro,fof the

Saworotnov’s

characteriza-tion theorem forassociative

ll*-algel)ras

via two-sil,d ll*-algebras. This

proof completely

avoids structuretheory.

KEY WORDS AND PIIRASES. H’-algel,ra,

involulion, a,,tomorphism, derivation, centralizer,

yon

Neumann

algebra.

1991

AMS SUBJECT CLASSIFICATION CODES.

16K15, .16II20,46L10, ,16L57.

1.

PRELIMINARIES.

One

of the important

(and

very

d(’ep)

rem,lls in tl,o

tl,eory

ofvon

Neumann algebras

states that all derivalions ofvon

Neumann algel,ras

are inner. Tlis result wa estal)lisled

by

Sakai in

SAKAI’S

TIIEORE.I. L! 7"

bc a yon

Ncumann

algrbra and

D

derivation

of

TO.

Then there

existssome a

T

such that

D(.r)

ax- xa

for

allx

..

This result has many

applications

and we presenl an()ll,er one.

,

show how the Sakai’s theoremcan be usedto give a new

proof

of the characterizalionof

H*-alg(’l)ras

in theterms of two-sided

H’-algebras.

This characterizalion was estal,li,],,(1 ly

Saworotnov

in

[8]

who described the structure of two-sidedH*-alget)ras and lhen compareit wilh the strctre of

(one-sided)

H*-algebras

obtainedbyAmbrose in

[1].

(2)

in ordor toobtain an aI)stra,! ]ar,cl’i/alio1 f 11’ llill-’rl-Slill

ol’ralors.

Later various

generalizations appeared uc]a lwo-sil,,] il’-alg’la, col]’enleialg’lras and llillwrt

nod-ules.

We

referto

[8], [9], [10].

[i 1]

a,,,!

[12].

N,,,,a,o,iali,,, Zl’-,,lgebra, ,,’orealso studied.

recent paperson thiss,l)je(’l aro

[3].

[I]. [3]. [6]

an(!

[1:{]..,k,,,l,rosc

i,,lro, l,,,,’(! l],e following

DEFINITION.

A

collcx

al,e,’tra

A

xxl,il is aloa liill,,ll space will, lleinner prolcl

(,)

and an

algebra

invollion

A

A

i, call’l an ll’-,lq,I,’a if llefollmvig //’-conditions

are satisfied for all a’,ll.z

A.

A

no,l,’l for s’l alg,’l,ra is lle algel)ra

llS(Tff)

of all

llilberl-Schmid! operalors actingon a llill,,, pace7fflogollcr il I1’inerl)ro,i,’t

{T,

,q’)

lrace(TS*).

Ifwetakealook at tileH*-con(litios, wecan ask lal

Iappens

if tlealgel)ra

A

w’oldhave

twoinvol.tions: one leftan! one rigllt.

DEFINITION.

A

complex alg’l)ra

iscalled a two-sided II*-algbva if for every r G

A

leree\i,l elements

.r..r

G

A

call’d aright andaleft adjoin! of tleelemel .r

holds forall y,z

A.

A

model for a two-sided ll’-algelra wli,’l is not. an

ll’-alg,bra

can I)’ fo,nl in

[8].

llowever

Saworotnow

provedthai lwo-sil’l ll*-alg.lra arevery1e1o

H*-alg,lra.

Explicitlyhe

proved

TIIEOREII.

Let

(.4,

),l,,’)

be a t,,.o-i,te,t aoc,,,li,’e H*-algebr, ,,’ilh zero ann,hilator. There ezistsanotherinnerprod,,ct

(,),

o,, .,4,,,’h t/,,t

(.A.

(,),, r)

becomes ,n H’-alg6ra.

Beforewestartour newproofof tlisresllwe inlro,llce tl(, fi)llowingnotationfor the annihi-latorsof

A:

left

annihilator:

Lann(A)-

{.,’G

:.,’

(0)},

right

annihilator

Rann(A)=

{.r

A:Ar= (0)}.

annihilator:

Ann(M)= l.ann(A)

2.

RESULTS.

First weshall collectsomesinpleol,ervalionson lwo--i,l’,l

//’-algebras

in llefollowing:

LEIIMA. Let

(..4,

(,),l,,’)

be a t,,’o-.’i,h,1 asoei,,ti,’c !l’-al,jebra with zero ,nn,hilator. Then

(3)

SAKAI’S THEOREM TO THE CHARACTERIZATION OF

H*-ALGEBRAS

319

(;) L,,,(A)= R,,,,(A)=

(0).

(ii)

Alcft

and rigl, adjo,,,l

of

,,,,!j ,I,,,,(ot a,’, ,,,,,q,,,

are both

algcbwc

in’olotions.

(iii)

The producl

of

A

is

(iv)

A

is dns in

A.

(v)

Both involutionsx .’ a,d.r .r arc

Proof.

(i) Suppose

tl,at aE

l,an,,(..4).

Takean)" .r.VE

.A.

Tl,en wehave

(.,.

,)

<,..’)

<,,:,

.,.’)

o

wherez is some left aljoilt ofx and !/r some rigl! alj()i,,l of y

(note

l,at wedon’t need the uniq,cncss of tile left and rigl,t adjoin,! i tl,ismome.!

).

TI,is i,npliesa

Rann(.A)

and therefore

Lann(.A)

C

Rann(.A).

In

asimilar way we

ge

Rann(A)

C

I.ann(A)

and ll,e,’eforeleftand right

annihilator of

.A

arcboth

cqal

[otle anililalorof

.A

wli(’l iszero

by

assumption.

(ii)

Let

at, and at, be two left adjoi,Is of lle clcmen! a. Take any x,y

.A.

Tlcnwe have

Using

(i)

wegeta t’ at a

I.

In

a si,ilarwaywe can provell,attherigl,t adjointisunique and

holds foralla

.A

and allcomplex

.

Alowehave for all a,

b,z,y

c:_...4

(iii)

The operators

L(x)

ax and

R(.r)

xa boll, ],ave adjoints

(as

operators actingon aHilbert space

4)

and arcthereforeconlinms.

By

standard

arguments

wecan now prove the continuity of the

product.

(iv)

Takeanya6

.A

+/-.

For

all x.y6

Awc

have

and using

(i)

we

get

a 0.

(4)

(vi)

First wetake y ab. ’il(,wl we liar(,

By (iv),

(v)

an!

tileconiiwlity 1"

!.

itl’r

l,r,llcl

il.al,,wei,lentityh,l,l., forall x,y E

.A.

Thefi)llowingproposi!ion coutailan apl,licaliot f ll. Sakai’s theorelnan! will becrucial theproofofTleorem.

IROIOSITION.

Let

A

be a. abort an,l I)

A

A

a continuous d;’ir,lion. ’l’hn thc’e ezist continuoslinearoperalor.

To

and ,% such Ih,I Ihc

fidio,’n

9holds:

(#

D

T0-

S0,

(iO

To(.y)=

o(.):

fo

,,

.,.,

u

(iii)

5;’o(.ry)=

ZSo(9)

for

all .r.y

_

.A.

(iv)

a’To(9)= So(.r)9

for

all .r, y

A.

If

D

is self-adjoint

(a.

an Ol,r,’,,to," acti,,9 on a llillu,’!

.l,acc),

To

and

.%

can al.o bc taken

self-adjoin t.

REMARK. In

thetheory ofC*-alg.lraslhe pair

(To..%)

islstallycall’,l a doublecentralizer.

PROOF.

Define

L(A)

and

R(.A)

1ol,etile

algelras

of colllinuols left alll rigllt, multipliersi.e.

L(A)=

T

B(A)" T(.ry)=T(.r)y

for all

z,U },

Then

L(A)

and

R(M)

arcyon

Ncumann

alg,,1)rasand

L(M)’

R(M)

hol,ls.

It

is easy to s that

L(M)

an

R(M)

are

alg(,1)ras

will identity.

From

Lemma(iii)

we get that

L(A)

and

R(A)

are closedi le slrong operalor

lop()logy.

Ve

m,l ee tlat

they

arealso sclfadjoint

subalgebras

of

B(M).

Takeany cotinosl’fl mulliplicr

T

a(1 compute

(T’(.,-u),

:)=

(.u,

())= (.,

Tf:).v

)

(.,.,

T(:u"))

(V’(.,.),

:,")

(r(.,’)u,

).

Hence

T"

isaleft multiplier andso

L(A)

isa vanNew,mann

a]gelra.

In

asiilar waywecanprove that

R(A)

isalsoa van

Neumann

algel,m.

(5)

SAKAI’S THEOREM TO THE CHARACTERIZATION OF

H*-ALGEBRAS

321

and

Lemma(iv), T

and

,q

comn,,le, rl’l,i sl,ows tl,al

I?(A)

C

L(A)’.

To

prove;l,e converseobserve that operators

L

y xy are cottinos left m,llii,lier for all

A.

Take any

S

L()’.

Then

showsthat

S

isarigl,tm,ltiplier.

Now define

& (A)

(A)

i,y

(T)=

DT-

TD. %Vm,,st showll,at

(T)

is in fact left multiplier:

A(T)(xy)

DT(.r!I)-

TD(.ry)

D(T(.r)!I)- T(D(x)y

+

.rD(9))

(DT)(x)y

+

T(.,.)D(y)

-(TD)(.r)y

T(.,’)D(y)

A(T)(.r)y.

Straightforwardverification sl)owstl,at A is indee(l a (l(’,ivalion.

Now

we ,,ea well-known result that every derivation of ayon Ne,mann

alg(’l)ra

is inner. T])ere existsa l(’fl multiplier

To

such that

A(T)

ToT-

TTo

DT-

TD.

Thus for all

T

(.A)

wehave

(To-

D)T

T(To-

D).

So

So

To-

D

(.A)’

(M).

X, m,,t seethat

xTo(y)

So(x)y

holds for allx,y 4:

T0u)-

6’0(),j

,,(u)

..%(u)

+

..%(u)- ,,(.,’)u

+

To(x):

%(’),j

xD(y)

+

So(xy)-

To(xy)

+

D(.r)y

xD(y)

D(xy)

+

D(.r)y

O.

Finallysupposethat

D

iss(.lf-adjoil.

From

tile facttl,a! left and rigl,t

multil)liers

areclosed for adjoints

(the

second

paragral)h

oftlis

proof),

il f(llows lat

q

D=D’=(To+T)-(.o+S)

isaself-adjoint decompositionwith properlies

(i)-(iv).

Nowwearereadyfor the final step.

PROOF OF

TIlE

TIIEORE.I.

Define a linear operalor (l) on .,4 ,.vilh

(I)(x)

(x)

I.

By

Lemma(v)

(I) is continuous and

by

Lemma(iii)

(I) is an a,tomorphism of

A.

Using

Lemma(vi)

weobtain

<(.r),.)

<.’,

.>

(.,:,

,-"’>

-II.,.ll _>

0

which shows that (I) isselfadjoit and l],at ils

spectrum

iscontained in

R

+.

By

[2,

Theorem 15, page

91]

(I)

exp(D)

forsomecontinuo,,s

(obviously self-adjoint)

derivalion

D

of

.A.

Accordingtothe Proposition, wemaydecompose

D

To

5’o.

Since

D

is selfadjoint,wecan

(6)

positive invertihle operalors and

P

I’I

"-I lohls. I’

is a i.ft multiplier,

I"

is a right multiplier and therefore they commute.

It

iseasy 1oseethat

.rl;(!/)

I’(

r)9

holds for all .r,v

M

(compare

Proposition(iv)

).

Now

define

(z,U)

(t(.r),9}.

It

ro,,ai,, 1o vo,’ifv1},,1

(M,

(,),,’)

is an

//’-algebra.

Thisis

done inthe following computation"

((.)v(u),-)=

(t’-’(.,.)(u).-)

(’t.v-’(x)u,>

Theproofisthus completed.

REMARK.

It

is not known whether le nonassociative

//’-algebras

can lw characterized in thesameway.

Saworotnow’s

original proof

isentirelyassocialive.

In

our

proof

the associalivityis used once in the proof of the Proposiiion namelywlen weproved that

L(.A)

is a comm,tantof

R(.A).

This is true if and onlyif

A

isassocialive.

REFERENCES

1.W.AMBROSE,Structurelheorems foraspecialcla.sof Banachalgebras, Trans.A,wr lalh.See. 57(1945),

364-386.

2. F.F. BONSALandJ. DUNCAN, Complete normed a]gebra., Erg-b. Math. Grengob. 80 (1973)

3. M. CABRERA,J.MARTINEZand A.IODIIGUEZ,Nona.,ociaiverealH*-algebras, Publ.Mat.32(1988), 267-274

4. M.CABRERA, J.MARTINEZandA. NODRIGUEZ,Mal’eev H’-algebras, Malh. Proe Camb. Phil. See. 103(1988),463-471

5.M. CABREIA,.].MARTINEZandA. IODRIGUEZ,Strueturabl’//’-algebras, J._. Algebra 1,t___.7(1992),19-62 6. J.A.CUENCA,A. GARCIA andC.MAI1TIN,Structure theoryforL’-algebras, Malh. Prec. Camb. Phil.

See. 107.(1990),361-365

7. S.SAKAI,Derivations ofW*-algehra.s, Ann. Mah. 83(1966),273-279

8. P.P.SAWOROTNOW,Ongeneralization ofnolilionofII"-a|gcbra, Prec.Amer.Mlh. See. 8(1957),56-62 9. P.P.SAWOROTNOW, Ontwo-sidedH*-algebras, Pac.J. Math 16(’1966),365-370

10. P.P.SAWOROTNOW, Ageneralized Ililbert space, Duke Math. 3,5(1968),191- 197

11. P.P.SAWOROTNOW, Representation ofacomp]ement’d algebron a locally compactspace, Illinois J.

Math.14(1970),674-677

12.J.F. SMITHandP.P. SAWOROTNOW,On.someclass ofscalar-product algebr, Pac.J.Math. 11(1961),

739-750

13. B. ZALAR,Continuity of derivationson Mal’eev H’-alg.br., Malh. Prec. Camb. Phil.See. 110 (1991),

(7)

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