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 avonNeumann algebra
isinner,is,sed1o el,lain a newelegant pro,fof theSaworotnov’s
characteriza-tion theorem forassociativell*-algel)ras
via two-sil,d ll*-algebras. Thisproof 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
veryd(’ep)
rem,lls in tl,otl,eory
ofvonNeumann algebras
states that all derivalions ofvonNeumann algel,ras
are inner. Tlis result wa estal)lisledby
Sakai inSAKAI’S
TIIEORE.I. L! 7"
bc a yonNcumann
algrbra andD
derivationof
TO.
Then thereexistssome a
T
such thatD(.r)
ax- xafor
allx..
This result has many
applications
and we presenl an()ll,er one.,
show how the Sakai’s theoremcan be usedto give a newproof
of the characterizalionofH*-alg(’l)ras
in theterms of two-sidedH’-algebras.
This characterizalion was estal,li,],,(1 lySaworotnov
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].
in ordor toobtain an aI)stra,! ]ar,cl’i/alio1 f 11’ llill-’rl-Slill
ol’ralors.
Later variousgeneralizations 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 followingDEFINITION.
A
collcx
al,e,’tra
A
xxl,il is aloa liill,,ll space will, lleinner prolcl(,)
and an
algebra
invollionA
A
i, call’l an ll’-,lq,I,’a if llefollmvig //’-conditionsare satisfied for all a’,ll.z
A.
A
no,l,’l for s’l alg,’l,ra is lle algel)rallS(Tff)
of allllilberl-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)raA
w’oldhavetwoinvol.tions: one leftan! one rigllt.
DEFINITION.
A
complex alg’l)raiscalled a two-sided II*-algbva if for every r G
A
leree\i,l elements.r..r
GA
call’d aright andaleft adjoin! of tleelemel .rholds forall y,z
A.
A
model for a two-sided ll’-algelra wli,’l is not. anll’-alg,bra
can I)’ fo,nl in[8].
lloweverSaworotnow
provedthai lwo-sil’l ll*-alg.lra arevery1e1oH*-alg,lra.
Explicitlyheproved
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)},
rightannihilator
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. ThenSAKAI’S THEOREM TO THE CHARACTERIZATION OF
H*-ALGEBRAS
319(;) L,,,(A)= R,,,,(A)=
(0).
(ii)
Alcft
and rigl, adjo,,,lof
,,,,!j ,I,,,,(ot a,’, ,,,,,q,,,are both
algcbwc
in’olotions.(iii)
The produclof
A
is(iv)
A
is dns inA.
(v)
Both involutionsx .’ a,d.r .r arcProof.
(i) Suppose
tl,at aEl,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,npliesaRann(.A)
and thereforeLann(.A)
CRann(.A).
In
asimilar way wege
Rann(A)
CI.ann(A)
and ll,e,’eforeleftand rightannihilator of
.A
arcbothcqal
[otle anililalorof.A
wli(’l iszeroby
assumption.(ii)
Let
at, and at, be two left adjoi,Is of lle clcmen! a. Take any x,y.A.
Tlcnwe haveUsing
(i)
wegeta t’ at aI.
In
a si,ilarwaywe can provell,attherigl,t adjointisunique andholds foralla
.A
and allcomplex.
Alowehave for all a,b,z,y
c:_...4(iii)
The operatorsL(x)
ax andR(.r)
xa boll, ],ave adjoints(as
operators actingon aHilbert space4)
and arcthereforeconlinms.By
standardarguments
wecan now prove the continuity of theproduct.
(iv)
Takeanya6.A
+/-.For
all x.y6Awc
haveand using
(i)
weget
a 0.(vi)
First wetake y ab. ’il(,wl we liar(,By (iv),
(v)
an!
tileconiiwlity 1"!.
itl’rl,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 Ihcfidio,’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, yA.
If
D
is self-adjoint(a.
an Ol,r,’,,to," acti,,9 on a llillu,’!.l,acc),
To
and.%
can al.o bc takenself-adjoin t.
REMARK. In
thetheory ofC*-alg.lraslhe pair(To..%)
islstallycall’,l a doublecentralizer.PROOF.
DefineL(A)
andR(.A)
1ol,etilealgelras
of colllinuols left alll rigllt, multipliersi.e.L(A)=
T
B(A)" T(.ry)=T(.r)y
for allz,U },
Then
L(A)
andR(M)
arcyonNcumann
alg,,1)rasandL(M)’
R(M)
hol,ls.It
is easy to s thatL(M)
anR(M)
arealg(,1)ras
will identity.From
Lemma(iii)
we get thatL(A)
andR(A)
are closedi le slrong operalorlop()logy.
Ve
m,l ee tlatthey
arealso sclfadjointsubalgebras
ofB(M).
Takeany cotinosl’fl mulliplicrT
a(1 compute(T’(.,-u),
:)=
(.u,
())= (.,
Tf:).v
)
(.,.,
T(:u"))
(V’(.,.),
:,")
(r(.,’)u,
).
Hence
T"
isaleft multiplier andsoL(A)
isa vanNew,manna]gelra.
In
asiilar waywecanprove thatR(A)
isalsoa vanNeumann
algel,m.
SAKAI’S THEOREM TO THE CHARACTERIZATION OF
H*-ALGEBRAS
321and
Lemma(iv), T
and,q
comn,,le, rl’l,i sl,ows tl,alI?(A)
CL(A)’.
To
prove;l,e converseobserve that operatorsL
y xy are cottinos left m,llii,lier for allA.
Take anyS
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,mannalg(’l)ra
is inner. T])ere existsa l(’fl multiplierTo
such thatA(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 seethatxTo(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,tmultil)liers
areclosed for adjoints(the
secondparagral)h
oftlisproof),
il f(llows latq
D=D’=(To+T)-(.o+S)
isaself-adjoint decompositionwith properlies
(i)-(iv).
Nowwearereadyfor the final step.
PROOF OF
TIlETIIEORE.I.
Define a linear operalor (l) on .,4 ,.vilh(I)(x)
(x)
I.
By
Lemma(v)
(I) is continuous andby
Lemma(iii)
(I) is an a,tomorphism ofA.
UsingLemma(vi)
weobtain
<(.r),.)
<.’,
.>
(.,:,
,-"’>
-II.,.ll _>
0which shows that (I) isselfadjoit and l],at ils
spectrum
iscontained inR
+.
By
[2,
Theorem 15, page91]
(I)exp(D)
forsomecontinuo,,s(obviously self-adjoint)
derivalionD
of.A.
Accordingtothe Proposition, wemaydecompose
D
To
5’o.
SinceD
is selfadjoint,wecanpositive 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,vM
(compare
Proposition(iv)
).
Now
define(z,U)
(t(.r),9}.
It
ro,,ai,, 1o vo,’ifv1},,1(M,
(,),,’)
is an//’-algebra.
Thisisdone 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
ourproof
the associalivityis used once in the proof of the Proposiiion namelywlen weproved thatL(.A)
is a comm,tantofR(.A).
This is true if and onlyifA
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),
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