A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS

Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs Shanx Unversty Tayuan 030006 P. R. Chna e-mal: xaofeqsxu@alyun.com Abstract Let M be a von Neumann algebra wthout central summands of type I 1. Suppose that ϕ : M M s an addtve map and s a scalar wth ±1. Then ϕ satsfes ϕ ( AB BA) = ϕ( A) B Bϕ( A) + Aϕ( B) ϕ( B)A for any A, B M wth AB + BA = 0, f and only f ϕ ( I ) s n the center of M, ϕ( I ) = 2 ϕ( I ) + ϕ( I ) and ϕ ( AB) = ϕ( A) B + Aϕ( B) ABϕ( I ) for all A, B M. Partcularly, f ϕ s lnear, then ϕ s a dervaton. 1. Introducton Let R be an assocatve rng (or an algebra over a feld F ). Recall that an addtve map δ on R s called an addtve dervaton, f δ ( AB) = δ( A) B + Aδ( B) for all A, B R; s called an addtve Jordan dervaton, f δ ( AB + BA) = δ( A) B + Aδ( B) + δ( B) A + Bδ( A) for all A, B R 2010 Mathematcs Subect Classfcaton: 47B47, 46L10. Keywords and phrases: von Neumann algebras, dervatons, Jordan products, -Le dervatons. Ths work s partally supported by Natonal Natural Scence Foundaton of Chna (11101250) and Youth Foundaton of Shanx Provnce (2012021004). Receved December 18, 2013 2014 Scentfc Advances Publshers

2 JIA JI et al. 2 (equvalently, δ( A ) = δ( A) A + Aδ( A) for all A R, f the characterstc of R s not 2); s called a Le dervaton, f δ( [ A, B] ) = [ δ( A), B] + [ A, δ( B) ] for all A, B R, where [ A, B] = AB BA s the Le product of A and B. The structures of dervatons, Jordan dervatons, and Le dervatons had been studed ntensvely for many researchers (for example, see [1, 4, 6, 11] and the references theren). For a scalar F and for A, B R, we say that A commutes wth B up to a factor, f AB = BA. The noton of commutatvty up to a factor for pars of operators s an mportant concept and has been studed n the context of operator algebras and quantum groups ([3, 8]). Motvated by ths, a bnary operaton [ A, B] = AB BA, called -Le product of A and B, was ntroduced n [13]. An addtve map L : R R s called an addtve -Le dervaton, f L ([ A, B] ) L( A), B [ ] + = [ A, L( B) ] for all A, B R. Ths concepton unfes the above three notons. It s clear that a -Le dervaton s a dervaton, Le dervaton and Jordan dervaton f = 0, 1, 1, respectvely. The structure of -Le dervatons on varous operator algebras was also dscussed by several authors (for example, see [9, 13, 17, 18]). Recently, the queston of under what condtons that an addtve map becomes a dervaton attracted much attenton of many researchers. For example, Q and Hou [14] characterzed the lnear maps L between J -subspace lattce algebras that satsfes L ([ A, B] ) = [ L( A), B] + [ A, L( B) ] for any A, B wth [ A, B] = 0. Let X be a Banach space wth dm X 3 and B ( X ) be the algebra of all bounded lnear operators actng on X. Lu and J n [10] showed that, f δ : B( X ) B( X ) s a lnear map satsfyng δ ([ A, B] ) = [ δ( A), B] + [ A, δ( B) ] for any A, B B( X ) wth AB = 0 (resp., AB = P, where P s a fxed nontrval dempotent), then δ = τ + ν, where τ s a dervaton of B ( X ) and ν : B( X ) CI s a

A CHARACTERIZATION OF ADDITIVE DERIVATIONS 3 lnear map vanshng at commutators [ B] A, wth AB = 0 (resp., AB = P ). Let A be a untal prme algebra over a feld F and F. In [16], Q et al. consdered the addtve maps L on A satsfyng the condton L ([ A, B] ) = [ L( A), B] A, L( B ) for all A, B A wth [, ] = 0. B + [ ] A For other related results, we refer to [2, 5, 7, 10, 14, 15] and the references theren. The purpose of the present paper s to gve a new characterzaton of addtve dervatons on von Neumann algebras wthout central summands of type I 1. Let M be a von Neumann algebra wthout central summands of type I 1 and s a scalar wth ±1. Denote by Z ( M) the center of M. We prove that an addtve map ϕ : M M satsfes ϕ ( AB BA) = ϕ( A) B Bϕ( A) + Aϕ( B) ϕ( B)A for any A, B M wth AB + BA = 0, f and only f ϕ( I ) Z ( M ), ϕ( I ) = ( 2 + 1) ϕ( I ) and ϕ ( AB) = ϕ( A) B + Aϕ( B) ABϕ( I ) for all A, B M (Man Theorem). Furthermore, t s shown that ϕ s a dervaton, f ϕ s lnear (Corollary). 2. Man Result and ts Proof In ths secton, we wll gve our man result and ts proof. Man Theorem. Let M be a von Neumann algebra wthout central summands of type I 1. Suppose that ϕ : M M s an addtve map and [ ] s a scalar wth ±1. Then ϕ satsfes ϕ([ A, B] ) = ϕ( A), B + [ A, ϕ( B) ] for any A, B M wth AB + BA = 0, f and only f ϕ( I ) Z ( M ), ϕ( I ) = ( 2 + 1) ϕ( I ) and ϕ ( AB) = ϕ( A) B + Aϕ( B) ABϕ( I ) for all A, B M, where Z ( M) denotes the center of M.

4 JIA JI et al. Partcularly, f ϕ s lnear, then we have the followng Corollary: Corollary. Let M be a von Neumann algebra wthout central summands of type I 1. Suppose that ϕ : M M s a lnear map and [ ] + s a scalar wth ±1. Then ϕ satsfes ϕ([ A, B] ) = ϕ( A), B [ A, ϕ( B) ] for any A, B M wth AB + BA = 0, f and only f ϕ s a dervaton. Proof. The f part s clear. For the only f part, by Man Theorem, we have ϕ( I ) = 2 ϕ( I ) + ϕ( I ) and ϕ ( AB) = ϕ( A) B + Aϕ( B) ABϕ( I ) for all A, B M. To complete the proof, we only need to check ϕ( I ) = 0. Indeed, f ϕ s lnear, then ϕ ( I ) = ϕ( I ) = 2ϕ( I ) + ϕ( I ), and so ϕ( I ) = 0 as 1. Before provng the Man Theorem, we need some notatons. Let M be any von Neumann algebra and A M. Recall that the central carrer of A, denoted by A, s the ntersecton of all central proectons P such that PA = 0. If A s self-adont, then the core of A, denoted by A, s sup{ S Z ( M) : S = S, S A}. Partcularly, f A = P s a proecton, t s clear that P s the largest central proecton P. A proecton P s called core-free f P = 0. It s easy to see that P = 0, f and only f I P = I. For more propertes of core-free proectons, see [12]. We frst gve two useful lemmas, whch are needed to prove Man Theorem. Lemma 1 ([12]). Let M be a von Neumann algebra wthout central summands of type I 1. Then each nonzero central proecton C M s the carrer of a core-free proecton n M. Partcularly, there exsts a nonzero core-free proecton P M wth P = I.

A CHARACTERIZATION OF ADDITIVE DERIVATIONS 5 Lemma 2 ([12]). Let M be a von Neumann algebra. For proectons P, Q M, f P = Q 0 and P + Q = I, then T M commutes wth PXQ and QXP for all X M mples T Z( M). Proof of Man Theorem. By a smple calculaton, t s easy to check that the f part holds. For the only f part, assume that ± 1 and ϕ : M M s an addtve map satsfyng ϕ([ A, B] ) = [ ϕ( A), B] + A, ϕ( B) for any [ ] A, B M wth AB + BA = 0. We wll prove the only f part by several clams. By Lemma 1, we can fnd a core-free proecton P M wth P = I. In the sequel fx such a proecton P. By the defntons of core and central carrer, we have P = I P = 0 and I P = I. For the convenence, wrte P1 = P, P2 = I P and M = P MP,, { 1, 2}. Then M = M + M + M +. 11 12 21 M22 Clam 1. P ϕ( I ) P = P ϕ( I ) P 0 and P ϕ( P ) P = P ϕ( P ) P 0. 1 2 2 1 = Snce P + P P = P P + P P 0, 1 2 2 1 2 1 1 2 = 2 1 2 1 2 1 = P we have [ ϕ( P ) P ] + [ P, ϕ( P )] 0 and [ ϕ( ) P ] + [ P, ϕ( P )] 0, and P that s, 2, 1 2 1 = 1, 2 1 2 = ϕ( P ) P P ϕ( P ) + P ϕ( P ) ϕ( P ) P 0, (1) 1 2 2 1 1 2 2 1 = ϕ( P ) P P ϕ( P ) + P ϕ( P ) ϕ( P ) P 0. (2) 2 1 1 2 2 1 1 2 = Multplyng by P 1 and P 2 from left and the rght n Equaton (1), respectvely, and multplyng by P 2 and P 1 from left and the rght n Equaton (2), respectvely, one gets P ϕ( P ) P + P ϕ( P ) P 0 and P ϕ( P ) P + P ϕ( P ) P 0, that s, 2 2 1 2 1 1 = 1 1 2 1 2 2 =

6 JIA JI et al. P ϕ( I ) P = and P ϕ( I ) P 0. 1 2 0 2 1 = Multplyng by P 2 from both sdes n Equaton (1) and multplyng by P 1 from both sdes n Equaton (2), one obtans ( ) P ϕ( P ) P 0 and ( ) P ϕ( P ) P 0. Note that 1. So 1 2 1 2 = P ϕ( P ) P = and P ϕ( P ) P 0. 1 2 1 0 2 1 2 = 1 1 2 1 = Now defne a map δ : M M by δ( A ) = ϕ( A) + TA AT for all A M, where T = P ϕ( P ) P P ϕ( P ). Clearly, δ s also addtve and 1 1 2 2 1 P1 satsfes AB + BA = δ([ A, B] ) = [ δ( A), B] + A, δ( B) It follows from Clam 1 that [ ]. 0 P δ( I ) P = P δ( I ) P = P δ( P ) P = P δ( P ) P 0. 1 2 2 1 2 1 2 1 2 1 = In addton, we have δ ( P 1 ) = ϕ( P 1 ) + TP 1 P 1 T = P 1 ϕ( P 1 ) P 1 = P δ ( P ) P P ( TP P T ) P = P δ ( P ) P, (3) 1 1 1 1 1 1 1 1 1 1 M11 and δ ( P 2 ) = ϕ( P 2 ) + TP 2 P 2 T = P 2 ϕ( P 2 ) P 2 = P δ( P ) P P ( TP P T ) P = P δ( P ) P. (4) 2 2 2 2 2 2 2 2 2 2 M22 In the followng, we wll prove that δ s a generalzed dervaton by Clams 2-5. Clam 2. δ ( M ), = 1, 2. M We only gve the proof of δ ( M ), and the proof of another ncluson s smlar. 11 M 11

A CHARACTERIZATION OF ADDITIVE DERIVATIONS 7 In fact, for any A 11 M 11, by the equaton A 11P2 + P 2 A11 = 0, we have [ δ( ) P ] + [ A, δ( P )] 0, A whch and Equaton (4) yeld 11, 2 11 2 = δ( A ) P P δ( A ) 0. (5) 11 2 2 11 = Multplyng by P 1 from the left n Equaton (5), one gets P 1 δ( A11 ) P2 = 0; multplyng by P 2 from both sdes n Equaton (5) and notng that 1, one has P δ ( A ) P 0. 2 11 2 = Smlarly, by the relaton P A + A P 0, one can check 2 11 11 2 = P δ( A ) P 0. Hence δ ( M ). The clam holds. 2 11 1 = 11 M 11 Clam 3. δ( M ), 1 2. M For any A M ( ), snce A A A A = 0, we have [ δ( ), A ] + [ A, δ( A )] = 0, A that s, + δ ( A ) A Aδ( A ) + Aδ( A ) δ( A ) A ( 1 )( δ( A ) A + A δ( A )) = 0. = As 1, the above equaton can be reduced to δ( A ) A + A δ( A ) = 0. (6) Snce ( P A )( A P ) + ( A P ) ( P + A ) = 0. By Equatons + (3)-(4) and Equaton (6), one gets 0 [ δ( P ) + δ( A ), A P ] + [ P + A, δ( A ) δ( P )] = = δ( P ) A δ( A ) P + P δ ( A ) + P δ ( A ) A δ ( P ) δ( A ) P. (7) Multplyng by P and P from both sdes n Equaton (7), respectvely, by Equatons (3) and (4), we obtan ( 1 ) P δ( A ) P = 0 and ( 1 ) P δ ( A ) = 0, whch mply P P δ( A ) P = 0 and P δ( A ) P = 0. (8)

8 JIA JI et al. Fnally, we stll need to prove P δ( A ) P = 0. To do ths, note that A ( P P ) + ( P P ) A = 0. So ( 1 + ) δ ( A ) = [ δ( A ), P P ] + [ A, δ ( P ) δ ( P )] = δ( A ) P δ( A ) P P δ( A ) + P δ( A ) A δ( P ) δ( P ) A. Multplyng by P and P from the left and the rght n the above equaton, respectvely, gets ( + ) P δ( A ) P = ( 1 + ) P δ( A ) P. Ths 1 mples P δ( A ) P = 0. By Equaton (8), one gets δ ( A ) M, completng the proof of the clam. Clam 4. δ( I ) Z( M). For any A M ( 1 2), by Equaton (7) and Clam 3, we have proved that δ( P ) A A δ( P ) = 0. Combnng the equaton and Equatons (3)-(4), one obtans δ ( I ) A = δ( P ) A = A δ( P ) = A δ( I ). It follows from Lemma 2 that δ( I ) Z( M). Clam 5. δ ( AB) = δ( A) B + Aδ( B) ABδ( I ) for all A, B M, that s, δ s a generalzed dervaton. We wll prove the clam by four steps. Step 1. The followng statements hold: (1) For any A M and B M, we have δ ( AB ) = δ( A ) B + A δ( B ) A B δ( I ), 1 2.

A CHARACTERIZATION OF ADDITIVE DERIVATIONS 9 (2) For any A M and B M, we have δ ( AB ) = δ( A ) B + A δ( B ) A B δ( I ), 1 2. Take any A M and B M. Snce ( A A B ) ( B P ) + ( B P )( A + A B ) = 0, by Clams 2-3, one gets + 0 [ δ( A ) + δ( AB ), B P ] + [ A + AB, δ( B ) δ( P )] = = = δ( A ) B δ( A B ) + A δ( B ) A B δ( P ) δ( A ) B δ( A B ) + A δ( B ) A B δ( I ). (1) holds. For any A M and B M, note that ( A B + B )( A P ) ( A P )( A B + B ) = 0. By a smlar argument to that of (1), t s + easy to check that (2) holds. Step 2. For any A M, we have δ ( A B ) = δ( A ) B + A δ( B ) A B δ( I ), = 1, 2., B For any hand, we have A, M and any S M, by Step 1, on the one B δ ( A B S ) = δ( A B ) S + A B δ( S ) A B S δ( I ). On the other hand, δ ( A B S ) = δ( A ) B S + A δ( B S ) A B S δ( I ) = δ( A ) B S + A δ( B ) S + A B δ( S ) 2A B S δ( I ). Comparng the above two equatons obtans ( δ( A B ) δ( A ) B A δ( B ) + A B δ( I )) S = 0,

10 for all S JIA JI et al. M as δ( I ) Z( M), that s, ( δ( A B ) δ( A ) B A δ( B ) + A B δ( I )) SP = 0, holds for all S M. Note that P = I. It follows from the defnton of the central carrer that span{ TP ( x) : T M, x H } s dense n H. So δ ( A B ) = δ( A ) B + A δ( B ) A B δ( I ). Step 3. For any A δ( A ) B + A δ( B ) A B δ( I ), 1 2. M and B M, we have δ( A ) = B For any A M and B M, snce [( A B + A + B P ) ( P + A + B + B A )] 0, 12 21 12 21 2, 1 12 21 21 12 1 = by Clams 2-4, one can get 0 = [ δ( A 12 B21 ) + δ( A12 ) + δ( B21 ) δ( P2 ), P1 + A12 + B21 + B21A12 ] [ A 12 B21 + A12 + B21 P2, δ( P1 ) + δ( A12 ) + δ( B21 ) + δ( B21A )] + 12 = ( 1) δ( A B ) + ( 1 ) δ( A ) B + ( ) A δ( B ) 12 21 12 21 1 12 21 + + ( 1) A12B21 ) δ( I ) + ( 1) δ( B21A12 ) + ( 1 ) δ( B21 ) A12 ( ) B δ( A ) + ( 1) B A δ( ). 1 21 12 21 12 I As 1, the above equaton mples and δ ( A B ) = δ( A ) B + A δ( B ) A B δ( ), 12 21 12 21 12 21 12 21 I δ ( B A ) = δ( B ) A + B δ( A ) B A δ( ). 21 12 21 12 21 12 21 12 I Step 4. For any A, B M, we have δ ( AB) = δ( A) B + Aδ( B) ABδ( I ). Take any A = A + A + A + A B = B + B + B + B. 11 12 21 22, 11 12 21 22 M Combnng Steps 1-3 and Clams 2-3, t s easly checked that δ ( AB) = δ( A) B + Aδ( B) ABδ( I ). So δ s a generalzed dervaton.

A CHARACTERIZATION OF ADDITIVE DERIVATIONS 11 Clam 6. δ ( I ) = 2δ( I ) + δ( I ). Therefore, the theorem holds. For any A, as A ( I P ) + ( I 2P ) A 0, we have 12 M 12 12 2 1 1 12 = [ δ( A ) I 2P ] + [ A, δ( I ) 2δ( P )] = δ([ A, I 2P ] ). 12, 1 12 1 12 1 It follows from Clam 3 and Equatons (3)-(4) that δ ( A ) = δ( A ) + ( + ) δ( I ). (9) 12 12 1 A12 On the other hand, by Clam 5, we have δ ( A ) = δ( I ) A + δ( A ) A δ( ). (10) 12 12 12 12 I Comparng Equatons (9) and (10), one acheves ( δ( I ) ( 1 + 2) δ( I )) A12 = 0, that s, ( δ( I ) ( 1 + 2) δ( I )) M P2 = 0. Note that P 2 = I. It follows from the defnton of the central carrer that span{ TP2 ( x) : T M, x H } s dense n H. It follows that δ ( I ) = ( 2 + 1) δ( I ). Clam 7. ϕ ( AB) = ϕ( A) B + Aϕ( B) ABϕ( I ) holds for all A, B M and ϕ ( I ) = ( 2 + 1) ϕ( I ). Note that ϕ ( A ) = δ( A) TA + AT for each A M. By Clams 5-6, t s easly check that the clam holds. The proof of the theorem s fnshed. References [1] M. Brešar, Jordan dervatons on semprme rngs, Proc. Amer. Math. Soc. 104 (1988), 1003-1006. [2] M. Brešar, Characterzng homomorphsms, dervatons and multplers n rngs wth dempotents, Proc. R. Soc. Ednb., Sect. A 137 (2007), 9-21. [3] J. A. Brooke, P. Busch and B. Pearson, Commutatvty up to a factor of bounded operators n complex Hlbert spaces, R. Soc. Lond. Proc. Ser. A Math Phys. Eng. Sc. A 458 (2002), 109-118.

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