Derivations of Operator Algebras Author(S): Richard V

Annals of Mathematics

Derivations of Operator Algebras Author(s): Richard V. Kadison Reviewed work(s): Source: Annals of Mathematics, Second Series, Vol. 83, No. 2 (Mar., 1966), pp. 280-293 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970433 . Accessed: 11/10/2012 00:05

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http://www.jstor.org Derivationsof operatoralgebras By RichardV. Kadison*

1. Introduction This paper is concernedwith resultsdescribing the natureof derivations of operatoralgebras - especially,derivations of von Neumannalgebras. Neg- lectingconvergence questions, which can be dealt witheffectively in thiscase, the exponentialof a derivationwill be an automorphism.The adjoint-preserv- ing automorphismsa of a C*-algebra f actingon a Hilbertspece SC cannot,in general,be implementedby a unitaryoperator U on SC (as a(A) = U*AU, for each A in A1). It followsfrom [9; Cor. 2.3.1] that a extendsto the weak clo- sure %- of f (forSC separable)if and onlyif a preservesthe nullideal of the representationof 9finvolved. The workof Murrayand von Neumann[17; Th. XI, 18; Th. X] indicatesthat automorphisms of von Neumannalgebras with no part of typeIII tendto be spatial (i.e., implementedby a unitarytransforma- tion). Griffin'sresults [7, 8] extendthis to the type III situation. Kaplansky [12] has noted that automorphismsof type I von Neumann algebras which leave the centerelementwise fixed are inner. It is well knownthat automorphismsof factorsnot of typeI will not usually be inner. In fact, N. Suzuki [25] showsthat each countablegroup is isomorphicto a groupof outerautomor- phismsof the hyperfiniteII, factor. By analogywith his type I automorphismresults, Kaplansky [13; Th. 9] establishesthat each derivationof a type I von Neumannalgebra is inner. He proceedsfrom the resultof I. M. Singerthat each derivationof a commutative C*-algebrais 0. Singerand Wermer[24] provedanalogous results for commutative Banach algebras. An extensionof Singer's result(cf. Theorem2) estab- lishes that derivationsof a C*-algebra annihilatethe centerwhich accounts forthe fact that Kaplansky'sresult (which plays a key role in our work) does not requirethe normalizationon the centerpresent in the automorphismcase. Kaplanskywas led to conjecturethat each derivationof a C*-algebra is continuous. This was provedby S. Sakai [21]. Using these results,P. Miles [26] showsthat each derivationof a C*-algebrais inducedby an operatorin theweak closurein somefaithful representation of the algebra. That representsthe state of our knowledgeabout derivationsof von Neumannalgebras not of typeI (cf. [2; p. 257]). The relationof derivationsto * Research conducted with the support of NSF GP 1604 and in part with the support Qf ONR contract NR 043-325and NSF GP 4059, DERIVATIONS OF OPERATOR ALGEBRAS 281 automorphismsand our relativelycomplete information about automorphisms of operatoralgebras makes numerous "informed" guesses available. By analogy withthe case of automorphisms,we say that a derivations of a C*-algebraa acting on SC is spatial when thereis a boundedoperator B on SC such that 6(A) = BA - AB (=ad B(A)), foreach A in W. If B can be chosen in A, we say that s is inner. The guesseswould be that thereare non-spatialderivations of C*-algebrasand non-innerderivations of vonNeumann algebras. Our results establishthe negationof the firstguess and indicaterather strongly that the negationof the second holds. In particularwe show (cf. Theorem7) that each derivationof a hyperfinitevon Neumannalgebra is inner (cf. this with N. Suzuki's resultsquoted above). It shouldalso be notedthat certainfactors of typeIII fall withinthe scope of this assertion[20; ? 7, 19; p. 95].

2. Preliminary results We say that a state p of a C*-algebraa is definite[11; p. 398] on the self- adjointoperator A in f when p(A2)= p(A)2. In this case, p is multiplicativeon the C*-subalgebraof f generatedby A. The followinglemma is a combination of Singer's argumentthat derivationsof commutativeC*-algebras are 0 and results[10; Lemma]on the multiplicativeproperties of definitestates.

LEMMA1. If a is a derivationof the C*-algebra f and p is definiteon A in A, then p(6(A)) = 0. PROOF. Note that U(I) =(12) = 26(I), so that U(I) = 0. Thus 6(A) (A -p(A)I); and we may assume p(A) = 0. In this case, 0 = p(A+) = p(A-)y whereA = A+- A- A+ and A- are the "positive"and "negative"parts of A; for A+A A+2, so that 0 = p(A-1)p(A)= p(A+A) = p(A+2) - p(A+)2. Since 6(A) = 6(A+) - 6(A-), it will sufficeto show that p(6(A+)) = p(6(A-)) = 0. We may assume A > 0 and p(A) - 0. Let B = A"12. Then p(B) = 0. Hence p(6(A)) = p[L(B)B] + p[B6(B)] = p[V(B)]p(B) + p(B)p[L(B)] = 0, from [1; Lemma]. The substance of the foregoinglemma is that each derivationof a C*- algebra maps each self-adjointoperator in the algebra onto an operatorthat has 0 diagonalrelative to a diagonalizationwhich diagonalizes A. THEOREM2. Each derivationof a C*-algebraannihilates its center. PROOF. Let s be a derivationof the C*-algebra f withcenter C. Let p be a pure state of A, and C an elementof C. The representationof f associated withp is irreducible[23] and thereforemaps C intoscalars. Togetherwith the Schwarzinequality, this yieldsthat p is multiplicativeon C. From the preceding lemma,p(6(C)) 0. Since the pure states of f separateA, 3(C) 0. 282 RICHARD V. KADISON

LEMMA 3. If a is a derivationof the C*-algebra f acting on theHilbert space SX, then 3 has a unique ultra weaklycontinuous extension which is a derivationof sat-.

PROOF. We show that for each x,y in SC,() ? is stronglycontinuous at O on 5, the positiveoperators in the unitball 31 of A. Now

A > ([A65(A)+ 6(A)A]x,y) (=(6(A2)x, y)) is stronglycontinuous at 0 on Si, the set of self-adjointoperators in the unit ball of a, since I ([A6(A) + 65(A)A]x, y) I _ 11U 11(Hl Ax I I y II + IIx lIiiAy I1), where 11U 11< cO by Sakai's theorem[21]. Moreover,A A112is stronglycon-

= I - tinuousat 0 on positiveoperators, since IIA112x 112 (Ax, x) I < 11Ax 1.11 x 11- Thus A - Al/2- (6(A)x, y) is stronglycontinuous at 0 on 1+. We note next that 3 is weaklycontinuous on 31 to f in the weak operator topology.Since Ax = Atx - A-x withA+x and A-x orthogonal,IjA+xjI < IjAxjI and II A-x II _II Ax 1; so that A > A+ and A A- are stronglycontinuous map- pingson the self-adjointoperators in f at 0. Thus

A > (j3(A+)x, y) - (6(A-)x, y) = (6(A)x, y) is stronglycontinuous at 0 on 5,*. By linearitythis mappingis stronglycon- tinuousat 0 on 25,*and fromthis, everywhere on 31*.Hence the inverseimage of a closedconvex subset of the complexnumbers under A - (Q(A)x,y) has an intersectionwith 53* which is stronglyclosed relativeto i,. This intersection being convex,each weak limitpoint is a stronglimit point [3, 15], so that it is weaklyclosed relative to 51*. Since the closedconvex subsets of the complex numbersform a subbase forthe closedsubsets, A - (6(A)x,y) is weaklycontinuous on 5,*. Now A (A+m A*)/2and A (A-A*)/2i are weakly continuous mappingsof 51 into S1*;so that

A - (s(A A*)x,y) + i(0(A A*)xI ) = (6(A)x, y) is weakly continuous on 51. Thus s is weakly continuous on 51. The linearityof 3 now yieldsits uniformcontinuity relative to the weak- operatoruniform structure on 51. From the Kaplanskydensity theorem [14], 5--is the unit ball in %-, and is compactin the weak-operatortopology. Thus 3 has a (unique) weak-operatorcontinuous extension to 3i, and this extension has an obviousextension j from3; to T1-. It is easilychecked that this extension is well-definedand linear. For x in SC,

(A, B)-3 ([3(AB) - j(A)B - Aj(B)]x, x) is stronglycontinuous on * x 3i*,by strongcontinuity of operatormultiplica- DERIVATIONS OF OPERATOR ALGEBRAS 283 tionon boundedsets, weak continuityof s on 5y and boundednessof 6 (hence a). Since this mappingis 0 on e$*x e$*,a stronglydense subset of 51*x 5-*;it is 0 on 5,>*x , for each x, so that 6ais a derivation on f-.

3. The main results J. Schwartz[22] has introduceda propertyof vonNeumann algebras which he uses to establishthe existenceof a thirdisomorphism class of factorsof type IIL. We recall that Fo-(A), foran arbitrarybounded operator A on SC and fR a von Neumannalgebra, is the weak closureof co,(A), the finiteconvex com- binationsof operatorsUAU* with U a unitaryoperator in ?k. We say that A is mobile(relative to AZ)when co, (A) has non-nullintersection with Wi'; and we say thatRJ is mixing when A is mobilefor each boundedA. It is notedin [22] that each hyperfinitevon Neumannalgebra is mixing. THEOREM4. Each derivation a of a C*-algebra f acting on the Hilbert space SC is spatial. We may chooseB commutingwith an assigned maximal abelian subalgebraof A' or with an assigned mixing von Neumann subalgebra of W' so that a6 adB I W. PROOF. Let a be a maximalabelian subalgebraof W', and let c' be the lat- tice of projectionsin (T. From Sakai's theorem,a is boundedon f; and from Lemma 3, a has an extensioni to 2-. For E1, ..., En in gPand A1, ... , A. in a-, define 61(AlEl + *-- + ARE,) to be 8(A1)El + *-- + 5(An)En. If

A1El + *- + AnEn = ? then there exist centraloperators Cj,k, k,Ic 1, 2, ***, n in W-, such that Ekl CJkEk = Ej and jX1 AjCjk 0, from[9; Lem. 3.1.1]. Since o annihi- lates the centerof A- (Theorem2), 0 = J(Aj)Cjk; so that E, a(Ai)Ej 0,. again from[9; Lem. 3.1.1]. Thus a, as definedis single-valued.The linearity of a, is clear. We notethat the set %, of operatorson which6, is definedis an algebra and that a, is a derivation. In fact, since the operatorsAE generate W,0linearly, it sufficesto check the productrelation on AE BF AB EF, a routinecomputation. Observenext that 86is boundedon %0. In fact, each A1E, + *-- + A.En in %, can be expressed in the form B1Fl + **- + BkFk with B1, ..., Bk in a- and F1, ... Fk mutuallyorthogonal projections in CP; for if El,---,E3 are orthogonal,we replaceEj+1 by

Ej+l(El + *-- + Ej) + Ej+1 - Ej+?(El + *-- + Ej) Now A1E, + Ai+,E1,E1 -A1(El - Ej+1E) + (A1 + Aji41)EJ11E1.In this way we replace AE, + . . + Aj+,Ej+, by a sum in whichall the projectionsare 284 RICHARD V. KADISON orthogonal. We then deal with Aj+2Ej+2,*, ** EA successively, in the same way. WithE2, * , En mutuallyorthogonal, x = En Ejx and I x II = 1,

(A2E2+ A2E2 + + AnE)x112 II AjEjx112 since{AjEjx} ={EjAjx} are mutuallyorthogonal vectors. Now

112 I 112 j 112 11 AjEjx _ AjEj Ejx _ max {I AiEi 112:j = 1, ,n} since >l Ejx 112= 11xll2=1. Thus 11AlEl+ **+ AE11 ? max{llAjEj11}. On the otherhand, max {I1AjEj II} A2E2+ * + AnEnIII by orthogonality of {Ej}. Thus 1l631(A1El + *- + A.En) II max{11a(Aj)Ej II}. With Q, the centralcarrier of E in af',

a(Aj)Ej =I (A)Q | (AjQj) _ 115 1111 AjQj11 = 11 11 11AiEj 11 I since the mappingAEj AQj is a *-isomorphismof V-Ej onto f-Qj (see [9; Lem. 3.1.3], forexample), *-isomorphisms between C*-algebras are isometries [6; Cor. 6], i annihilatesthe centerof f- (cf. Theorem2), and j is boundedon V-(Sakai's theorem).Hence 1161(A1El + . . . + AE.) lII llIII max{ll AjEj II} = |I 0 1111 A1E2 + - - - + AnEnI1; and 61is bounded. It followsthat a, has a bounded extension(which is a derivation)from %,I to the uniformclosure of a,0(since by linearity,it is uniformlycontinuous on aJI)and, fromLemma 3, it has an extensionj, to ?o-. Now af- is a von Neumannalgebra containinga- and a (since it contains CP,the projectionlattice of (1). Thus a' lies in a' and commuteswith (a. Since (a is maximalabelian in a', af = a; and a- is of typeI. FromKaplansky's theorem,o, is inner. Say J, = adB I af-. Then a = adB I af, and B e d' since BE-EB =a(E) =a(I)E= 0, forE in P. Supposethat $Ris a mixingvon Neumannsubalgebra of a'. In particular, this is the case if R is hyperfinite[22; Lem. 2]. With U' a unitaryoperator in A', we note that ad U'BU'* I 2- = adB I a-; for U'BU'*A - AU'BU'* = U'(BA - AB)U'* = BA - AB with A in A-, since BA - AB is in a-. It follows that convex combinations a, U,'B Uf* + - - - + an UnB U'* induce the same derivationas B on a-, and hencethat operatorsin the weak closureof such convexcombinations induce the same derivationas adB on 2-. Since ?R is mixingsome such closurepoint (of convexcombinations with unitaryoperators in A) lies in Sk', so that a is adC I 2 forsome C in ok'. LEMMA 5. If ad B induces a derivationof the C*-algebraa, thenit induces a derivationof a'. The derivationad B of 2- is inner if and only if it induces an inner derivationof a'. DERIVATIONS OF OPERATOR ALGEBRAS 285

PROOF. AssumingadB inducesa derivationof A, we observe,for each A in Wand A' in V',

(BA' - A'B)A - A(BA' - A'B) = BA'A - A'BA - ABA' + AA'B -(BA - AB)A' - A'(BA- AB) = 0 Y since BA - AB lies in K. Thus BA' - A'B lies in W. If adB induces an inner derivation of a-, say adB = adC on a-, with C in a-; then B - C commuteswith %- and, therefore,lies in A'. But since C lies in %-, ad(B - C) = adB on ?'. Thus adB induces an inner derivation of A'.

REMARK6. Since the questionof whetheror not a derivationis inner is clearlyan algebraic one; i.e., independentof the representationchosen, since all derivationsof concretelyrepresented C*-algebras are spatial,and sinceeach semi-finitevon Neumannalgebra has a faithfulrepresentation in whichthe commutantis finite;it wouldsuffice to demonstratethat each derivationof a finitevon Neumannalgebra is innerin orderto establishthat each derivation of a semi-finitevon Neumannalgebra is inner,by virtue of the preceding lemma.

THEOREM7. Each derivationof a mixing von Neumann algebra is inner. In,particular, each derivationof a hyperfinitevon, Neumann algebra is inner. PROOF. If 9Jacting on XCis a mixingvon Neumann algebra and a is a derivationof 9Z,then from Theorem 4, a = ad B J9Z. FromLemma 5, ad B induces a derivationof a9k'which is innerif and only if a is inner. However,from Theorem4, adB I aik'is adC I ', whereC may be chosen commutingwith an assignedmixing von Neumannsubalgebra of fi"(=9i). In particularC can be chosencommuting with 9Z. Thus ad B I 9X'is inneras is a.

4. Derivations by special operators From Theorem4, each derivationof a von Neumannalgebra is the re- strictionto it of adB for somebounded B. Undercertain assumptions on B, we can take the finalstep and establishthat thisderivation is inner. We begin by notingthat each of a large class of operators(a stronglydense *-algebra,in the case of a factor)is mobileunder a von Neumannalgebra.

LEMMA 8. Each operatorA1AJ + - AA' with A, *-* , Anin 9Uand A***, A' in RI'is mobileunder 9Z. 286 RICHARD V. KADISON

PROOF. Accordingto Dixmier'sapproximation theorem [2; Th. 1, p. 272], we can findunitary operators U1, * **, Un in R and non-negative real numbers ca, *... an with Ln=,aj -1 such that E n. oajUjA1Uj*is close to a central operatorof 9Zin norm. We consider

n=l j Uj(AA' + * + AnA I) UP - = 2>n= ajUjAUj*A'Uj*AU + a UjAn and locate V1, , V, unitaryoperators in 9kand non-negativereal numbers > /39 .,fi with sum 1 such that E flkVk( a6jUiA2Up*)Vk is close to a central operator of 9Z in norm. We note that Ek=> f3Vk(jn i a U3AUp)Vp is as close to the centraloperator near Li 3,U3A1Up as = orU3A1Uy is. We now consider M fkVk(En(1 aj U3A1US Af + * + ES> raiU3A, UjpA')Vk and continueas before. It followsthat some element of coa (AA' + * + AnA') is close in norm to C1A' + + CnA' with C1, ... , C, in the center of fR,this last operatorlying in fR'. From the proofof Theorem4 (the last paragraph),we see that if ad B maps 9Jinto fk and B is mobileunder 9Z', then adB I 9Zis inner. Combining this remarkwith the precedinglemma, we have:

THEOREM9. If ad B maps the von Neumann algebra 9Z into JRand B

AlAJr *** + AnA, withAj in U and A'i in R',j = 1, ***,n; then adB I 9Z * is inner. Note that, if fRis a factor,operators having the formdescribed for B lie stronglydense in the algebra of all bounded operators. The theoremwhich followswill be subsumedin the theoremfollowing it. However,Theorem 11 has an analyticproof and is effectedby passingto groupsof automorphisms, whilethe theoremwhich follows can be givena proofin terms of derivations and morealgebraically. We feel that the proofand statementare of sufficient interestto give separately.

THEOREM10. If ad B induces a derivationof the von Neumann algebra IR with B a projection,then ad B I R is inner. PROOF. We notefirst that if AC = 0, withA and C in Rk,then ABC e A; for AB - BA e fRso that ABC - BAC = ABC C A. From Lemma 5, adB induces a derivationof 9J';so that A'BC' C 9' with A'C' = 0 and A', C' in J'. Thus 0 = A'ABCC' for such A, A', C, C'. In particular,with E and E' projections in R and W', respectively; DERIVATIONS OF OPERATOR ALGEBRAS 287

0 = EE'B(I - E')(I - E) = EE'B2(I - E')(I- E) = EE'B[E' + (I - E')]B(I - E')(I - E) = EE'BE'B(I - E')(I- E) + EE'B(I - E')B(I - E')(I - E) = E'EB(I - E)E'B(I -E') + E'B(I - E')EB(I - E)(I - E') = 2EB(I- E)E'B(I- E') .

It followsthat the centralcarriers of EB(I - E) and E'B(I - E') are or- thogonalfor all projectionsE and E' in URand Rk',respectively. Let Q be the unionof the centralcarrier of EB(I - E) for projectionsE in fR,and let P be theunion of the centralcarriers of E'B(I- E') forprojections E' in k'. Then QP=O, fromthe foregoing;and O-=PEB(I-E)==EPB(I-E), foreach projectionE in R. Thus PB leaves I - E invariant,for each such E in fR;and PB lies in fR'. Similarly(I- P)B lies in fR.Now B PB+ (I-P)B; so that ad B I URis ad (I - P)B I gR,and ad B inducesan innerderivation of R. Concerningthe theoremwhich follows,note that,if adB maps the C*- algebraK intoitself, then -(BA* - A*B)* (=B*A - AB*) lies in A, foreach A in W; so that adB* maps K intoW. Thus each of the self-adjointand skew- adjoint parts of B induce derivationsof W. If each of these derivationsis inner,adB I K is inner. The question of whetheror not all derivationsof a von Neumannalgebra are inneris reducedthen to the question of whether or not spatial derivationsby self-adjointoperators are. Additionof a scalar multipleof I to this operatordoes not affectthe derivationit produces. By judicious choice of this scalar, we may arrange that our operatoris positive and singular. Our next resultstates in essence that, if our positivesingular operatorannihilates a vector,the derivationto whichit gives riseis inner.

THEOREM11. If adH maps the von Neumann algebra R into itself, H is positiveand Hx, = 0 for a vectorx0 such that [xJ] has central carrier I in R', thenad HI R is inner. PROOF. Note that HAxo = (HA - AH)x$; so that HAx, is in Rx0,when A lies in fR. Thus HE' = E'H, where E' is the projection(in R') withrange [Rx0]. Since E' has centralcarrier I, A - AE' is a *-isomorphismof 9z onto RE' (cf. [9, Lem. 3.1.3]). Thisisomorphism carries adHi l 9onto ad HE' i9E'; so that the latteris innerif and onlyif theformer is. Now HE' > 0, HE'xo =0 and [kRE'x0]= E'(X5). We may assume x0is cyclicfor iR. Witht and s real,define Ui, as exp (itH) exp (-sH); so that z-)Uz is an entireoperator-valued function of the complexvariable z(= t+ is). SinceHxo= 0, Uzx,= x. foreach z. Note that IIUI II = IIexp (-sH)I 1; so that II Uz I _ 1 if a ! 0, sinceH ? 0. Note also that A m UtALULis an automorphismof 9Z(and 288 RICHARD V. KADISON of 9k')since adH maps 9k(and RJt')into itself. If A and A' are self-adjointoper- atorsin R and R', respectively,then (A'U0Ax,,,x,) = (A'UAUAUx,,,x,) = (U0AU.A'x,, xj) = (AU0,A'x,,Ux,,) = (AU0tA'x.,xo) = (xoA'UtAxo) = (A'UtAxoxO). Thus theentire function f definedby f(z) (A'UAx,, x0)is real-valuedfor real z. From I UfI 1 forz in theupper half plane, we see that f(z) I IAxIoll1 A'x 1I for such z. From the Schwarz reflectionprinciple, f(z) = f(z); so that f is boundedin the entireplane. Liouville'stheorem now yieldsthat f is constant; so that (UtAx,,A'x.) (Axo,A'xo) = (Ax., UtA' Utx.), forall real t and each self-adjointA in Rt. Since [Rix,]=K7, (UtA'Ut-A')xo=O. However,U0tA'Ut - A' lies in fR'. Withx0 separating for XR', we concludethat UtA'Ut = A' for all real t and each self-adjointA' in fR'. Thus Ut lies in kR, forall real t. But iH is the normlimit of (Ut - I)/t as t >0. Thus H lies in 9R,and a is inner. REMARK12. The argumentabove works equally well to show that a strongly-continuous,one-parameter unitary group with an invariantvector, and with positivespectrum which induces automorphisms of a von Neumann algebra forwhich the invariantvector is cyclic,consists of unitaryoperators in the von Neumannalgebra. This is the case whereH above is possiblyun- bounded. It is the one-dimensionalanalogue of the result provedin [1; see Props. 1 and M]that the representationof thetranslation subgroup of the Poin- care group associated with a local quantum fieldtheory which has a cyclic vacuumstate has its image in the weak closureof the algebraof local observables of that theory.Our resultcould be adaptedto give anotherproof of this fact. We are gratefulto H. Araki forthe privilegeof seeinga pre-publication copyof [1]. 5. Related results If 9Z is a finitevon Neumannalgebra and Tr is its center-valuedtrace, then,with A, B, and C in fRand AC equal to CA we have Tr (C(BA - AB)) = 0. Thus if the derivationa of 9Zis to be innerand C commuteswith A, we should have Tr (Ca(A)) 0. As furtherevidence that derivationsof von Neumann algebrasare inner,we prove: THEOREM13. If a is a derivationof the finitevon Neumann algebra R and A and C in R commuteand A is self-adjoint,then Tr (C@(A)) = 0. In DERIVATIONS OF OPERATOR ALGEBRAS 289 particular, Tr (a(A)) = 0, for each A in R. PROOF. From Theorem4, a = adB I R, forsome bounded operator B. Re- call that if T and S are in R and TS = 0, then TBS = T(BS - SB) lies in R. In particular,with E, F orthogonalprojections in R, EBF lies in R. Let (dbe a (self-adjoint)maximal abelian subalgebra of R containingA. In [19; Ch. II], von Neumannintroduces the conceptof a "diagonal part" of an operatorin 9i relativeto (S. In [11,Lem. 1] it is shownthat a diagonalprocess 9D (not unique in general) exists which maps each boundedoperator B onto an operator@)(B) in (' whichis a weaklimit point of operators BIElI .En withE1,... *En projectionsin (a,where BIE EBE + (I - E)B(I - E). Thus B - D(B)is a weak .. limitpoint of operatorsB - BlEl ..En. But BJlEl1lEn J FjBFj with {Fj} a familyof mutuallyorthogonal projections in G having sum I (sinceE1, - - -, En .. commute). Hence B -BlEl JE,lEn FjBFk. With B givingrise to a derivation,each termof this sum lies in k,as notedin the firstpart of the proof. Thus B - 9D(B),a weak limitpoint of operatorsin 9k,lies in Ai. Since @D(B) lies in (', adB and ad(B - 9D(B))are the same on ai. Thus Tr (C3(A)) = Tr (C ad B(A)) = Tr (C[ad (B - 9D(B))](A)) = 0 by the remarkspreceding this theorem. REMARK14. In [5] the existenceof maximalhyperfinite subfactors of a factorof typeII, is established. One could extendthe argumentto show that theirrelative commutant is commutative.It may well consistof scalars in all cases, though this is not known. At any rate, examplesof hyperfinitesub- factorswhose relativecommutant consists of scalars are known (makinguse of the groupmeasure space examplesof [16; pp. 192-209]and the fact that, if the groupis commutative,the resultingfactor is hyperfinite[18; Lem. 5.2.3, 4; Cor. 4.1]). Let 9l1' be of typeII, and OR' a hyperfinitesubfactor with relative commutantscalars. If a is a derivationof OR,we can choose B so that adB IOR = a and B is in OR). Now the relativecommutant of 9TOlin Oil' is the relativecommutant of 'DR in 9t0. If a subfactorof a factorhas an abelian subalgebrawhich is maximalabelian in the larger factor(many examples of this exist) it will have relativecommutant the scalars. The converseof this may well hold; viz., if a subfactorof a factor has relative commutantthe scalars,then some (maximal) abelian subalgebra of it is maximalabelian in the larger factor. If this does hold,say a in 'DRis maximalabelian in 01%,then a diagonalpart 9D(B) of B relativeto (a lies in (a,and hencein OR. With B giving rise to a derivationof OR,B - 9D(B)lies in OR(from the proofof the preceding theorem)as does @(B); so that B lies in OR and a is inner.

UNIVERSITY OF PENNSYLVANIA 290 RICHARD V. KADISON

REFERENCES 1. H. ARAKI, On the algebra of all local observables,to appear. 2. J. DIXMIER,Les algebres d'operateursdans lespace hilbertien,Gauthier-Villars, Paris, 1957. 3. , Les fonctionelleslineaires sur l'ensembledes operateurs bornes d'une espace d'Hilbert,Ann. of Math., 51 (1950), 387-408. 4. H. DYE, On groupsof measurepreserving transformations: II, Amer. J. Math., 85 (1963), 551-576. 5. B. FUGLEDE and R. KADISON,On a conjectureof Murray and vonNeumann, Proc. Nat. Acad. Sci. U.S.A., 37 (1951), 420-425. 6. I. GELFAND and M. NEUMARK, On the imbeddingof normedrings into thering of operators in Hilbert space, Rec. Math. (Mat. Sbornik), N.S. 12 (1943), 197-213. 7. E. GRIFFIN, Some contributionsto the theoryof rings of operators,Trans. Amer. Math. Soc., 75 (1953), 471-504. 8. , Some contributionsto the theoryof rings of operators:II, Trans. Amer. Math. Soc., 79 (1955), 389-400. 9. R. KADISON, Unitary invariants for representationsof operatoralgebras, Ann. of Math., 66 (1957), 304-379. 10. , The tracein finiteoperator algebras, Proc. Amer. Math. Soc., 12 (1961),973-977. 11. R. KADISON and I. SINGER, Extensionsof pure states,Amer. J. Math., 81 (1959),383-400. 12. I. KAPLANSKY, Algebrasof type I, Ann. of Math., 56 (1952),460-472. 13. , Modulesover operatoralgebras, Amer. J. Math., 75 (1953), 839-859. 14. , A theoremon rings of operators,Pacific J. Math., 1 (1951), 227-232. 15. E. MICHAEL, Transformationsfrom a linear space withweak topology,Proc. Amer. Math. Soc., 3 (1952), 671-676. 16. F. MURRAY and J. von NEUMANN,On rings of operators,Ann. of Math., 37 (1936), 116- 229. 17. , On rings of operators:II, Trans. Amer. Math. Soc., 41 (1937), 208-248. 18. , On rings of operators:IV, Ann. of Math., 44 (1943), 716-808. 19. J. von NEUMANN,On rings of operators:III, Ann. of Math., 41 (1940), 94-161. 20. , On infinitedirect products,Comp. Math., 6 (1938), 1-77. 21. S. SAKAI,On a conjectureof Kaplansky, T6hoku Math. J., 12 (1960), 31-33. 22. J. SCHWARTZ,Two finite, non-hyperfinite,non-isomorphic factors, Comm. Pure Appl. Math., 16 (1963), 19-26. 23. I. SEGAL, Irreducible representationsof operatoralgebras, Bull. Amer. Math. Soc., 53 (1947), 73-88. 24. I. SINGER and J. WERMER, Derivationson commutativenormed algebras, Math. Ann., 129 (1955), 260-264. 25. N. SUZUKI, A linear representationof a countablyinfinite group, Proc. Japan Acad., 34 (1958), 575-579. 26. P. MILES, Derivationson B* algebras, Pacific J. Math., 14 (1964), 1359-1366. (Received March 3, 1965)

Added June 14, 1965. Since the precedingresults were obtained,several furtherfacts related to derivationshave been established. J. Ringroseand the author[28] provedthat each derivationof the von Neumanngroup algebra of a countablediscrete group is inner. S. Sakai [30] then completedthe argu- mentsof the presentpaper to obtain: THEOREM15. Each derivationof a von Neumann algebra is inner. DERIVATIONS OF OPERATOR ALGEBRAS 291

J. Ringroseand the author[29] gave a simplifiedproof of thisresult, using a deviceof Sakai's, in a paper which derives the corollarythat each norm- continuousrepresentation of a connectedLie groupby *-automorphismsof a von Neumannalgebra is a representationby inner automorphisms.Reducing to this result, H. Borchers[27] showed that each strongly-continuous,one- parameterunitary group with spectrumbounded below, which induces automorphismsof a von Neumannalgebra, induces inner automorphisms.With this same reduction,G. Dell' Antonio(private communication) showed that a weakly-continuous,one-parameter group of *-automorphisms of a vonNeumann algebra is inducedby a strongly-continuous,one-parameter unitary group in the algebra if it satisfiesa certaincondition akin to the semi-boundednessof the spectrum. (These last results state, roughly,that the energyand mo- mentumof a quantumfield are observablewithout the assumption of a vacuum state.) Since provingTheorem 4, we have feltthat the stepto Theorem15 should be a straightforwardmatter of showingthat co,,(B) containsan operatorin fit,when B inducesa derivationof R. The presentaddendum is promptedby findingsuch a proof. Thoughwe give theproof for an arbitraryvon Neumann algebrarather than for factors alone, the essential ideas are foundin the latter case. In sketch, Zorn's Lemma yields a minimal,non-null, convex, weak- operatorcompact subset X of co,,(B) invariantunder unitaryoperators in a'. By minimality,the elementsof X have the same norm. But ifB1 and B2 are distinctelements of SiC,cog, (B1 - B2) containssome aI, a =A 0 (this last is somewhatover-simplified). Thus B3-B4 BU= with b > 0 and B3,B4 (positive) operatorsin SiC;so that II B3 I > I B4 Hence XKiconsists of a single operatorwhich, by invariance,lies in R. PROOFOF THEOREM15. From Theorem 4, our derivation has the form ad B IR; and fromthe discussionpreceding Theorem 11, we mayassume that B ? 0. FromTheorem 2, B is in C', whereC is the centerof ?R. Thus if {Q.} is an orthogonalfamily of projectionsin C with sum I such that ad B I9ZQa, adA, IRQa, whereA,, is in 9RQ, and supa {IIAaI}

For brevity,we say that a set of operatorsstable under the mappings A AlE, A -> U*AU, with E a projectionand U a unitaryoperator in X' is 'stable'. The positiveoperators in the closedball of radius I B I with center o is a convex,weak-operator closed (in fact, compact)stable set containingB as is B + R'. Let X(B) be the intersectionof all such sets (one could show that UC(B) = coS,(B)). Zorn's lemmayields the existenceof a set DC minimal withrespect to inclusionamong the non-null,convex, compactstable subsets of XJ(B). If P is a projectionin C and B, is in X, {B2: IIB2PI I < 11B1P j, I B2 in IC} is such a subsetof K; so that 11B2P 11I 11B1P IJfor each B2 in XK. We provethat XJconsists of a single operator,which must lie in aR by stability;by showingthat the set JCOof differencesof pairs of operatorsin X containsonly 0. Note that X.i is a convex,compact, stable set of self-adjoint operatorsin X' (since B + X' containsXK). If Bo is a non-zerooperator in JC0, using -B,, if necessary,we may assume that B+ # 0, whereB+ and B- are the positive and negative parts of B,, respectively. If CB,+CBO-= 0 then cog, (B.) containssome C1 + B-, withCCB+ = C, C1 > 0 and C1 in C. (Apply the Dixmierprocess [2; Ch. III ? 5, 31; XXII p. 3.33 Lemma 15, 29; Lemma2] to B+ in 9?'C,- extendingthe unitaryoperators as I- CB+.) SinceXJC, is stable, C1 + Bo - B1 B2, withB1, B2 in TXi.For an appropriatecentral projection P and some positive a, B1P - B2P - C1P > aP. But then B1P I 11B2P + aP 11= 1JB2PIj + a > II B2P I I (recallthat operators in XKiare positive), contradictinga propertyof Xi. We mayassume that CB+CCB- # 0. In thiscase, withE the rangeprojection of B+, COER.E(BE) containssome non-zero C1E, whereC1 > 0 and C1is in CC+; and co(I-E)S(IE) (B,(I - E)) containssome C2(I- E) whereC2 < 0, C2is in CCB_ and C1C2# 0. Thus XK,contains C1E + C2(I - E) B1 - B2, with B, and B2 in tX. Now B I - B WE= ClE + CAI -E), andB", BIB are in DC. We may assume that B1 and B2 commutewith E. Since XJCis convexand containsB2 and (B2 + C1)E + (B2 + C2)(I - E), it contains(B2 + tCJ)E+ (B2+ tC2)(I - E) (= A,), foreach t in [0, 1]. For someprojection P in C, aP < C1P and C2P< bP, with a and -b positive numbers. Then IIAP I= max{I I(B2 +tC1)PE I, 11(B2 + tC2)P(I- E) 11};and, for small t, 11(B2 + tCJ)PEI I _ II B2PEJJ+ ta, tb + JIB2P(I- E)JI >I(B2? + tC2)P(I-E)JJ. If ]JB2PEJIl J'B2P(I- E)JI then for small t, J1APII > ]|B2PII. If JJB2P(I- E) lj > JB2PE II, then for very small t, IB2P(I - E) I > I (B2 + tC2)P(I - E) II jIj (B2 + tCJ)PE II; so that I B2P I >I AP I forsuch t. In anyevent, we haveoperators A, and B2 in XC with 11APII J 1# B2PJI- DERIVATIONS OF OPERATOR ALGEBRAS 293

SUPPLEMENTARYREFERENCES 27. H. BORCHERS, Energy and momentumas observablesin quantumfield theory, to appear. 28. R. KADISON and J. RINGROSE,Derivations of operatorgroup algebras,to appear. 29. , Derivationsand automorphismsof operatoralgebras, to appear. 30. S. SAKAI, Derivationsof W*-algebras,Ann. of Math., 83 (1966),287-293. 31. J. SCHWARTZ,Lectures on W*-algebras, NYU notes (mimeographed),1964.