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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 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. 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 automor- phismsof 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 commuta- tive 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 con- tinuous. 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. Fromthe preced- ing 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 weaklyconti- nuous 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 exten- sion 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 subalge- bra of W' so that a6 adB I W.
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