2-local triple derivations on von Neumann algebras
Karimbergen Kudaybergenov, Timur Oikhberg, Antonio M. Peralta, Bernard Russo
aa r X i v : . [ m a t h . OA ] J u l KARIMBERGEN KUDAYBERGENOV, TIMUR OIKHBERG,ANTONIO M. PERALTA, AND BERNARD RUSSO
Abstract.
We prove that every (not necessarily linear nor continuous)2-local triple derivation on a von Neumann algebra M is a triple deriva-tion, equivalently, the set Der t ( M ), of all triple derivations on M, isalgebraically 2-reflexive in the set M ( M ) = M M of all mappings from M into M . Introduction
Let X and Y be Banach spaces. According to the terminology employedin the literature (see, for example, [4]), a subset D of the Banach space B ( X, Y ), of all bounded linear operators from X into Y , is called alge-braically reflexive in B ( X, Y ) when it satisfies the property:(1.1) T ∈ B ( X, Y ) with T ( x ) ∈ D ( x ) , ∀ x ∈ X ⇒ T ∈ D . Algebraic reflexivity of D in the space L ( X, Y ), of all linear mappings from X into Y , a stronger version of the above property not requiring continuityof T , is defined by:(1.2) T ∈ L ( X, Y ) with T ( x ) ∈ D ( x ) , ∀ x ∈ X ⇒ T ∈ D . In 1990, Kadison proved that (1.1) holds if D is the set Der( M, X ) ofall (associative) derivations on a von Neumann algebra M into a dual M -bimodule X [18]. Johnson extended Kadison’s result by establishing thatthe set D = Der( A, X ) , of all (associative) derivations from a C ∗ -algebra A into a Banach A -bimodule X satisfies (1.2) [17].Algebraic reflexivity of the set of local triple derivations on a C ∗ -algebraand on a JB ∗ -triple have been studied in [24, 9, 12] and [14]. More precisely,Mackey proves in [24] that the set D = Der t ( M ) , of all triple derivations ona JBW ∗ -triple M satisfies (1.1). The result has been supplemented in [12],where Burgos, Fern´andez-Polo and the third author of this note prove that Mathematics Subject Classification.
Primary 46L05; 46L40.
Key words and phrases. triple derivation; 2-local triple derivation.Second author partially supported by Simons Foundation travel grant 210060. Thirdauthor partially supported by the Spanish Ministry of Science and Innovation, D.G.I.project no. MTM2011-23843, Junta de Andaluc´ıa grant FQM375, and the Deanship ofScientific Research at King Saud University (Saudi Arabia) research group no. RGP-361.
KUDAYBERGENOV, OIKHBERG, PERALTA, AND RUSSO for each JB ∗ -triple E , the set D = Der t ( E ) of all triple derivations on E satisfies (1.2).Hereafter, algebraic reflexivity will refer to the stronger version (1.2) whichdoes not assume the continuity of T .In [6], Breˇsar and ˇSemrl proved that the set of all (algebra) automor-phisms of B ( H ) is algebraically reflexive whenever H is a separable, infinite-dimensional Hilbert space. Given a Banach space X . A linear mapping T : X → X satisfying the hypothesis at (1.2) for D = Aut( X ), the set ofautomorphisms on X , is called a local automorphism . Larson and Sourourshowed in [22] that for every infinite dimensional Banach space X , every sur-jective local automorphism T on the Banach algebra B ( X ) , of all boundedlinear operators on X , is an automorphism.Motivated by the results of ˇSemrl in [31], references witness a grow-ing interest in a subtle version of algebraic reflexivity called algebraic 2-reflexivity (cf. [1, 2, 10, 11, 21, 23, 25, 26] and [29]). A subset D of the set M ( X, Y ) = Y X , of all mappings from X into Y , is called algebraically 2-reflexive when the following property holds: for each mapping T in M ( X, Y )such that for each a, b ∈ X, there exists S = S a,b ∈ D (depending on a and b ), with T ( a ) = S a,b ( a ) and T ( b ) = S a,b ( b ), then T lies in D . A mapping T : X → Y satisfying that for each a, b ∈ X, there exists S = S a,b ∈ D (de-pending on a and b ), with T ( a ) = S a,b ( a ) and T ( b ) = S a,b ( b ) will be calleda 2-local D -mapping. If we assume that every mapping s ∈ D is r - homoge-neous (that is, S ( ta ) = t r S ( a ) for every t ∈ R or C ) with 0 < r , then every2-local D -mapping T : X → Y is r -homogeneous. Indeed, for each a ∈ X , t ∈ C take S a,ta ∈ D satisfying T ( ta ) = S a,ta ( ta ) = t r S a,ta ( a ) = t r T ( a ).ˇSemrl establishes in [31] that for every infinite-dimensional separableHilbert space H , the sets Aut( B ( H )) and Der( B ( H )), of all (algebra) au-tomorphisms and associative derivations on B ( H ), respectively, are alge-braically 2-reflexive in M ( B ( H )) = M ( B ( H ) , B ( H )) . Ayupov and the firstauthor of this note proved in [1] that the same statement remains true forgeneral Hilbert spaces (see [20] for the finite dimensional case). Actually,the set Hom( A ), of all homomorphisms on a general C ∗ -algebra A, is al-gebraically 2-reflexive in the Banach algebra B ( A ), of all bounded linearoperators on A , and the set ∗ -Hom( A ), of all ∗ -homomorphisms on A, isalgebraically 2-reflexive in the space L ( A ), of all linear operators on A (cf.[27]).In recent contributions, Burgos, Fern´andez-Polo and the third authorof this note prove that the set Hom( M ) (respectively, Hom t ( M )), of allhomomorphisms (respectively, triple homomorphisms) on a von Neumannalgebra (respectively, on a JBW ∗ -triple) M , is an algebraically 2-reflexivesubset of M ( M ) (cf. [10], [11], respectively), while Ayupov and the firstauthor of this note establish that set Der( M ) of all derivations on M isalgebraically 2-reflexive in M ( M ) (see [2]). -LOCAL TRIPLE DERIVATIONS 3 In this paper, we consider the set Der t ( A ) of all triple derivations on aC ∗ -algebra A . We recall that every C ∗ -algebra A can be equipped with aternary product of the form { a, b, c } = 12 ( ab ∗ c + cb ∗ a ) . When A is equipped with this product it becomes a JB ∗ -triple in the senseof [19]. A linear mapping δ : A → A is said to be a triple derivation whenit satisfies the (triple) Leibnitz rule: δ { a, b, c } = { δ ( a ) , b, c } + { a, δ ( b ) , c } + { a, b, δ ( c ) } . It is known that every triple derivation is automatically continuous (cf. [3]).We refer to [3, 15] and [28] for the basic references on triple derivations.According to the standard notation, 2-local Der t ( A )-mappings from A into A are called .The goal of this note is to explore the algebraic 2-reflexivity of Der t ( A )in M ( A ). Our main result proves that every (not necessarily linear norcontinuous) 2-local triple derivation on an arbitrary von Neumann algebra M is a triple derivation (hence linear and continuous) (see Theorem 2.14),equivalently, Der t ( M ) is algebraically 2-reflexive in M ( M ).2. We start by recalling some generalities on triple derivations. Let A be aC ∗ -algebra. For each b ∈ A, we shall denote by M b the Jordan multiplicationmapping by the element b, that is M b ( x ) = b ◦ x = ( bx + xb ) . Followingstandard notation, given elements a, b in A , we denote by L ( a, b ) the operatoron A defined by L ( a, b )( x ) = { a, b, x } = ( ab ∗ x + xb ∗ a ). It is known that hemapping δ ( a, b ) : A → A, given by δ ( a, b )( x ) = L ( a, b )( x ) − L ( b, a )( x ) , is a triple derivation on A (cf. [3, 15]), called an inner triple derivation.Let δ : A → A be a triple derivation on a unital C ∗ -algebra. By [15,Lemmas 1 and 2], δ ( ) ∗ = − δ ( ) , and M δ ( ) = δ ( δ ( ) , ) is an innertriple derivation on A and the difference D = δ − δ ( δ ( ) , ) is a Jordan ∗ -derivation on A, more concretely, D ( x ◦ y ) = D ( x ) ◦ y + x ◦ D ( y ) , and D ( x ∗ ) = D ( x ) ∗ , for every x, y ∈ A. By [3, Corollary 2.2], δ (and hence D ) is a continuousoperator. A widely known result, due to B.E. Johnson, states that everybounded Jordan derivation from a C ∗ -algebra A to a Banach A -bimoduleis an associative derivation (cf. [16]). Therefore, D is an associative ∗ -derivation in the usual sense. When A = M is a von Neumann algebra,we can guarantee that D is an inner derivation, that is there exists e a ∈ A satisfying D ( x ) = [ e a, x ] = e ax − x e a, for every x ∈ A (cf. [30, Theorem4.1.6]). Further, from the condition D ( x ∗ ) = D ( x ) ∗ , for every x ∈ A, we KUDAYBERGENOV, OIKHBERG, PERALTA, AND RUSSO deduce that ( e a ∗ + e a ) x = x ( e a ∗ + e a ) . Thus, taking a = 12 ( e a − e a ∗ ) , it followsthat [ a, x ] = [ e a, x ] , for every x ∈ M. We have therefore shown that for everytriple derivation δ on a von Neumann algebra M, there exist skew-hermitianelements a, b ∈ M satisfying δ ( x ) = [ a, x ] + b ◦ x, for every x ∈ M. Our first lemma is a direct consequence of the above arguments (see [15,Lemmas 1 and 2]).
Lemma 2.1.
Let T : A → A be a ( not necessarily linear nor continuous ) ∗ -algebra. Then ( a ) T ( ) ∗ = − T ( );( b ) M T ( ) = δ (cid:0) T ( ) , (cid:1) is an inner triple derivation on A ;( c ) b T = T − δ (cid:0) T ( ) , (cid:1) is a 2-local triple derivation on A with b T ( ) = 0 . (cid:3) In what follows, we denote by A sa the hermitian elements of the C ∗ -algebra A . Lemma 2.2.
Let T : A → A be a ( not necessarily linear nor continuous ) ∗ -algebra satisfying T ( ) = 0 . Then T ( x ) = T ( x ) ∗ for all x ∈ A sa . Proof.
Let x ∈ A sa . By assumptions, T ( x ) ∗ = { , T ( x ) , } = { , δ x, ( x ) , } = δ x, { , x, } − { δ x, ( ) , x, } = δ x, ( x ∗ ) − { T ( ) , x, } = δ x, ( x ) = T ( x ) . The proof is complete. (cid:3)
Lemma 2.3.
Let T : M → M be a ( not necessarily linear nor continuous ) T ( ) = 0 . Then for every x, y ∈ M sa there exists a skew-hermitian element a x,y ∈ M such that T ( x ) = [ a x,y , x ] , and, T ( y ) = [ a x,y , y ] . Proof.
For every x, y ∈ M sa we can find skew-hermitian elements a x,y , b x,y ∈ M such that T ( x ) = [ a x,y , x ] + b x,y ◦ x, and, T ( y ) = [ a x,y , y ] + b x,y ◦ y. Taking into account that T ( x ) = T ( x ) ∗ (see Lemma 2.2) we obtain[ a x,y , x ] + b x,y ◦ x = T ( x ) = T ( x ) ∗ = [ a x,y , x ] ∗ + ( b x,y ◦ x ) ∗ = [ x, a ∗ x,y ] + x ◦ b ∗ x,y = [ x, − a x,y ] − x ◦ b x,y = [ a x,y , x ] − b x,y ◦ x, i.e. b x,y ◦ x = 0 , and similarly b x,y ◦ y = 0 . Therefore T ( x ) = [ a x,y , x ] ,T ( y ) = [ a x,y , y ] , and the proof is complete. (cid:3) We state now an observation, which plays an useful role in our study. -LOCAL TRIPLE DERIVATIONS 5
Lemma 2.4.
Let a and b be skew-hermitian elements in a C ∗ -algebra A .Suppose x ∈ A is self-adjoint with ax − xa + bx + xb = 0 . Then ax = xa, and bx = − xb. Proof.
Since 0 = ax − xa + bx + xb . Passing to the adjoint, we obtain ax − xa − ( bx + xb ) = 0. Conclude the proof by adding and subtractingthese two equalities. The proof is complete. (cid:3) Let M be a von Neumann algebra. If x ∈ M sa , we denote by s ( x ) thesupport projection of x – that is, the projection onto (ker( x )) ⊥ = ran ( x ).We say that x has full support if s ( x ) = 1 (equivalently, ker( x ) = { } ). Lemma 2.5.
Let M be a von Neumann algebra. Suppose u ∈ M + hasfull support, c ∈ M is self-adjoint, and σ ( c u ) ∩ (0 , ∞ ) = ∅ . Then c = 0 . Consequently, if u and c are as above, and uc + cu = 0 ( or c u = − cuc ≤ ,then c = 0 . Proof.
For the fist statement of the lemma, suppose σ ( c u ) ∩ (0 , ∞ ) = ∅ .Note that ( −∞ , ⊇ σ ( c u ) ∪ { } = σ ( c · cu ) ⊇ σ ( cuc ) . However, cuc is positive, hence σ ( cuc ) ⊂ [0 , k cuc k ], with max λ ∈ σ ( cuc ) = k cuc k . Thus, cu / u / c = cuc = 0 , which means that cu / = u / c = 0 andhence s ( c ) ⊂ − ( u / ) = 1 − s ( u ) = 0 , which leads to c = 0 . To prove the second part, we have c u = − cuc ≤ , hence in particular, σ ( c u ) ⊂ ( −∞ , . The proof is complete. (cid:3)
In [2, Lemma 2.2], Ayupov and the first author of this note prove thatfor every (not necessarily linear nor continuous) 2-local derivation on a vonNeumann algebra ∆ : M → M , and every self-adjoint element z ∈ M , thereexists a ∈ M satisfying ∆( x ) = [ a, x ] , for every x ∈ W ∗ ( z ), where W ∗ ( z ) = { z } ′′ denotes the abelian von Neumannsubalgebra of M generated by the element z , and the unit element and { z } ′′ denotes the bicommutant of the set { z } . We prove next a ternary versionof this result. Lemma 2.6.
Let T : M → M be a ( not necessarily linear nor continuous ) 2 -local triple derivation on a von Neumann algebra. Let z ∈ M be a self-adjointelement and let W ∗ ( z ) = { z } ′′ be the abelian von Neumann subalgebra of M generated by the element z and the unit element. Then there exist skew-hermitian elements a z , b z ∈ M , depending on z , such that T ( x ) = [ a z , x ] + b z ◦ x = a z x − xa z + 12 ( b z x + xb z ) for all x ∈ W ∗ ( z ) . In particular, T is linear on W ∗ ( z ) . KUDAYBERGENOV, OIKHBERG, PERALTA, AND RUSSO
Proof.
We can assume that z = 0. Note that the abelian von Neumannsubalgebras generated by and z and by and + z k z k coincide. So,replacing z with + z k z k we can assume that z is an invertible positiveelement.By definition, there exist skew-hermitian elements a z , b z ∈ M (dependingon z ) such that T ( z ) = [ a z , z ] + b z ◦ z. Define a mapping T : M → M given by T ( x ) = T ( x ) − ([ a z , z ] + b z ◦ z ) ,x ∈ M. Clearly, T is a 2-local triple derivation on M . We shall show that T ≡ W ∗ ( z ). Let x ∈ W ∗ ( z ) be an arbitrary element. By assumptions,there exist skew-hermitian elements c z,x , d z,x ∈ M such that T ( z ) = [ c z,x , z ] + d z,x ◦ z, and, T ( x ) = [ c z,x , x ] + d z,x ◦ x. Since 0 = T ( z ) = [ c z,x , z ] + d z,x ◦ z, we get [ c z,x , z ] + d z,x ◦ z = 0 . Taking into account that z is a hermitian element and Lemma 2.4 we get c z,x z = zc z,x and d z,x z = − zd z,x . Since z has a full support, and d z,x z = − d z,x zd z,x , Lemma 2.5 impliesthat d z,x = 0 . Further c z,x ∈ { z } ′ = { z } ′′′ = W ∗ ( z ) ′ , i.e. c z,x commutes with any element in W ∗ ( z ) . Therefore T ( x ) = [ c z,x , x ] + d z,x ◦ x = 0for all x ∈ W ∗ ( z ) . The proof is complete. (cid:3)
Complete additivity of 2-local derivations and 2-local triplederivations on von Neumann algebras.
Let P ( M ) denote the lattice of projections in a von Neumann algebra M. Let X be a Banach space. A mapping µ : P ( M ) → X is said to be finitelyadditive when µ n X i =1 p i ! = n X i =1 µ ( p i ) , for every family p , . . . , p n of mutually orthogonal projections in M. A map-ping µ : P ( M ) → X is said to be bounded when the set {k µ ( p ) k : p ∈ P ( M ) } is bounded.The celebrated Bunce-Wright-Mackey-Gleason theorem ([7, 8]) states thatif M has no summand of type I , then every bounded finitely additive map-ping µ : P ( M ) → X extends to a bounded linear operator from M to X . -LOCAL TRIPLE DERIVATIONS 7 According to the terminology employed in [32] and [13], a completelyadditive mapping µ : P ( M ) → C is called a charge . The Dorofeev–Sherstnevtheorem ([32, Theorem 29.5] or [13, Theorem 2]) states that any charge ona von Neumann algebra with no summands of type I n is bounded.We shall use the Dorofeev-Shertsnev theorem in Corollary 2.8 in orderto be able to apply the Bunce-Wright-Mackey-Gleason theorem in Proposi-tion 2.9. To this end, we need Proposition 2.7, which is implicitly appliedin [2, proof of Lemma 2.3] for 2-local associative derivations. A proof isincluded here for completeness reasons.First, we recall some facts about the strong ∗ -topology. For each normalpositive functional ϕ in the predual of a von Neumann algebra M , themapping x
7→ k x k ϕ = (cid:18) ϕ ( xx ∗ + x ∗ x (cid:19) ( x ∈ M )defines a prehilbertian seminorm on M . The strong ∗ topology of M is thelocally convex topology on M defined by all the seminorms k . k ϕ , where ϕ runs in the set of all positive functionals in M ∗ (cf. [30, Definition 1.8.7]).It is known that the strong ∗ topology of M is compatible with the duality( M, M ∗ ), that is a functional ψ : M → C is strong ∗ continuous if and onlyif it is weak ∗ continuous (see [30, Corollary 1.8.10]). We also recall thatthe product of every von Neumann algebra is jointly strong ∗ continuous onbounded sets (see [30, Proposition 1.8.12]).Suppose X = W is another von Neumann algebra, and let τ denote thenorm-, the weak ∗ - or the strong ∗ -topology of W. The mapping µ is said tobe τ -completely additive (respectively, countably or sequentially τ -additive )when(2.1) µ X i ∈ I e i ! = τ - X i ∈ I µ ( e i )for every family (respectively, sequence) { e i } i ∈ I of mutually orthogonal pro-jections in M. It is known that every family ( p i ) i ∈ I of mutually orthogonal projections ina von Neumann algebra M is summable with respect to the weak ∗ -topologyof M and p = weak ∗ - X i ∈ I p i is a projection in M (cf. [30, Definition 1.13.4]).Further, for each normal positive functional φ in M ∗ and every finite set F ⊂ I, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p − X i ∈ F p i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φ = φ p − X i ∈ F p i ! , which implies that the family ( p i ) i ∈ I is summable with respect to the strong ∗ -topology of M with the same limit, that is, p = strong ∗ - X i ∈ I p i . KUDAYBERGENOV, OIKHBERG, PERALTA, AND RUSSO
Proposition 2.7.
Let T : M → M be a ( not necessarily linear nor continu-ous ) ( a ) The restriction T | P ( M ) is sequentially strong ∗ -additive, and consequentlysequentially weak ∗ -additive; ( b ) T | P ( M ) is weak ∗ -completely additive, i.e., (2.2) T weak ∗ - X i ∈ I p i ! = weak ∗ - X i ∈ I T ( p i ) for every family ( p i ) i ∈ I of mutually orthogonal projections in M. Proof. ( a ) Let ( p n ) n ∈ N be a sequence of mutually orthogonal projections in M. Let us consider the element z = P n ∈ I n p n . By Lemma 2.6 there existskew-hermitian elements a z , b z ∈ M such that T ( x ) = [ a z , x ] + b z ◦ x for all x ∈ W ∗ ( z ) . Since ∞ P n =1 p n , p m ∈ W ∗ ( z ) , for all m ∈ N , and the product of M is jointly strong ∗ -continuous, we obtain that T ∞ X n =1 p n ! = " a z , ∞ X n =1 p n + b z ◦ ∞ X n =1 p n ! = ∞ X n =1 [ a z , p n ] + ∞ X n =1 b z ◦ p n = ∞ X n =1 T ( p n ) , i.e. T | P ( M ) is a countably or sequentially strong ∗ -additive mapping.( b ) Let ϕ be a positive normal functional in M ∗ , and let k . k ϕ denote theprehilbertian seminorm given by k z k ϕ = ϕ ( zz ∗ + z ∗ z ) ( z ∈ M ). Let { p i } i ∈ I be an arbitrary family of mutually orthogonal projections in M. For every n ∈ N define I n = { i ∈ I : k T ( p i ) k ϕ ≥ /n } . We claim, that I n is a finite set for every natural n . Otherwise, passingto a subset if necessary, we can assume that there exists a natural k suchthat I k is infinite and countable. In this case the series P i ∈ I k T ( p i ) does notconverge with respect to the semi-norm k . k ϕ . On the other hand, since I k is a countable set, by ( a ), we have T X i ∈ I k p i = strong ∗ - X i ∈ I k T ( p i ) , which is impossible. This proves the claim.We have shown that the set I = n i ∈ I : k T ( p i ) k ϕ = 0 o = [ n ∈ N I n -LOCAL TRIPLE DERIVATIONS 9 is a countable set, and k T ( p i ) k ϕ = 0, for every i ∈ I \ I .Set p = P i ∈ I \ I p i ∈ M. We shall show that ϕ ( T ( p )) = 0 . Let q denote thesupport projection of ϕ in M . Having in mind that k T ( p i ) k ϕ = 0, for every i ∈ I \ I , we deduce that T ( p i ) ⊥ q for every i ∈ I \ I .Replacing T with b T = T − δ ( T ( ) , ) we can assume that T ( ) = 0(cf. Lemma 2.1) and T ( x ) = T ( x ) ∗ , for every x ∈ M sa (cf. Lemma 2.2).By Lemma 2.3, for every i ∈ I \ I there exists a skew-hermitian element a i = a p,p i ∈ M such that T ( p ) = a i p − pa i , and, T ( p i ) = a i p i − p i a i . Since T ( p i ) ⊥ q we get ( a i p i − p i a i ) q = q ( a i p i − p i a i ) = 0, for all i ∈ I \ I . Thus, since pa i p i q = p i a i q ,( T ( p ) p i ) q = ( a i p − pa i ) p i q = a i p i q − pa i p i q = a i p i q − p i a i q = ( a i p i − p i a i ) q = 0 , and similarly q ( p i T ( p )) = 0 , for every i ∈ I \ I . Consequently,(2.3) ( T ( p ) p ) q = T ( p ) X i ∈ I \ I p i q = 0 = q X i ∈ I \ I p i T ( p ) = q ( pT ( p )) . Therefore, T ( p ) = δ p, ( p ) = δ p, { p, p, p } = 2 { δ p, ( p ) , p, p } + { p, δ p, ( p ) , p } = 2 { T ( p ) , p, p } + { p, T ( p ) , p } = pT ( p ) + T ( p ) p + pT ( p ) ∗ p = pT ( p ) + T ( p ) p + pT ( p ) p, which implies that ϕ ( T ( p )) = ϕ ( pT ( p ) + T ( p ) p + pT ( p ) p )= ϕ ( qpT ( p ) q ) + ϕ ( qT ( p ) pq ) + ϕ ( qpT ( p ) pq ) = (by (2.3)) = 0 . Finally, by ( a ) we have T X i ∈ I p i = k . k ϕ - X i ∈ I T ( p i ) . Two more applications of ( a ) give: ϕ T X i ∈ I p i !! = ϕ T p + X i ∈ I p i = ϕ T ( p ) + T X i ∈ I p i = ϕ ( T ( p )) + ϕ T X i ∈ I p i = X i ∈ I ϕ ( T ( p i )) . By the Cauchy-Schwarz inequality, 0 ≤ | ϕT ( p i ) | ≤ k T ( p i ) k ϕ = 0, forevery i ∈ I \ I , and hence X i ∈ I ϕ ( T ( p i )) = X i ∈ I ϕ ( T ( p i )) . The arbitrarinessof ϕ shows that T (cid:18) weak ∗ - P i ∈ I p i (cid:19) = weak ∗ - P i ∈ I T ( p i ). (cid:3) Let φ be a normal functional in the predual of a von Neumann algebra M. Our previous Proposition 2.7 assures that for every (not necessarilylinear nor continuous) 2-local triple derivation T : M → M the mapping φ ◦ T | P ( M ) : P ( M ) → C is a completely additive mapping or a charge on M . Under the additional hypothesis of M being a continuous von Neumannalgebra or, more generally, a von Neumann algebra with no Type I n -factors(1 < n < ∞ ) direct summands (i.e. without direct summand isomorphicto a matrix algebra M n ( C ), 1 < n < ∞ ), the Dorofeev–Sherstnev theorem([32, Theorem 29.5] or [13, Theorem 2]) imply that φ ◦ T | P ( M ) is a boundedcharge, that is, the set {| φ ◦ T ( p ) | : p ∈ P ( M ) } is bounded. The uniformboundedness principle gives: Corollary 2.8.
Let M be a von Neumann algebra with no Type I n -factordirect summands (1 < n < ∞ ) and let T : M → M be a ( not necessarilylinear nor continuous ) T | P ( M ) is a bounded weak ∗ -completely additive mapping. (cid:3) Additivity of 2-local triple derivations on hermitian parts ofvon Neumann algebras.
Suppose now that M is a von Neumann algebra with no Type I n -factordirect summands (1 < n < ∞ ), and T : M → M is a (not necessarily linearnor continuous) 2-local triple derivation. By Corollary 2.8 combined with theBunce-Wright-Mackey-Gleason theorem [7, 8], there exits a bounded linearoperator G : M → M satisfying that G ( p ) = T ( p ), for every projection p ∈ M .Let z be a self-adjoint element in M . By Lemma 2.6, there exist skew-hermitian elements a z , b z ∈ M such that T ( x ) = [ a z , x ] + b z ◦ x, for every x ∈W ∗ ( z ) . Since G | W ∗ ( z ) , T | W ∗ ( z ) : W ∗ ( z ) → M are bounded linear operators,which coincide on the set of projections of W ∗ ( z ), and every self-adjointelement in W ∗ ( z ) can be approximated in norm by finite linear combinationsof mutually orthogonal projections in W ∗ ( z ), it follows that T ( x ) = G ( x )for every x ∈ W ∗ ( z ) , and hence T ( a ) = G ( a ) , for every a ∈ M sa , in particular, T is additive on M sa . The above arguments materialize in the following result. -LOCAL TRIPLE DERIVATIONS 11
Proposition 2.9.
Let T : M → M be a ( not necessarily linear nor con-tinuous ) n -factor direct summands (1 < n < ∞ ) . Then the restriction T | M sa isadditive. (cid:3) Corollary 2.10.
Let T : M → M be a ( not necessarily linear nor contin-uous ) T | M sa is additive. Next we shall show that the conclusion of the above corollary is also truefor a finite von Neumann algebra.First we show that every 2-local triple derivation on a von Neumannalgebra “intertwines” central projections.
Lemma 2.11. If T is a ( not necessarily linear nor continuous ) 2 -local triplederivation on a von Neumann algebra M, and p is a central projection in M, then T ( M p ) ⊂ M p.
In particular, T ( px ) = pT ( x ) for every x ∈ M .Proof. Consider x ∈ M p , then x = pxp = { x, p, p } . T coincides with a triplederivation δ x,p on the set { x, p } , hence T ( x ) = δ x,p ( x ) = δ x,p { x, p, p } = { δ x,p ( x ) , p, p } + { x, δ x,p ( p ) , p } + { x, p, δ x,p ( p ) } lies in M p.
For the final statement, fix x ∈ M, and consider skew-hermitian elements a x,xp , b x,xp ∈ M satisfying T ( x ) = [ a x,xp , x ] + b x,xp ◦ x, and T ( xp ) = [ a x,xp , xp ] + b x,xp ◦ ( xp ) . The assumption p being central implies that pT ( x ) = T ( px ) . (cid:3) Proposition 2.12.
Let T : M → M be a ( not necessarily linear nor contin-uous ) T | M sa is additive.Proof. Since M is finite there exists a faithful normal semi-finite trace τ on M. We shall consider the following two cases.
Case 1.
Suppose τ is a finite trace. Replacing T with b T = T − δ ( T ( ) , )we can assume that T ( ) = 0 (cf. Lemma 2.1) and T ( x ) = T ( x ) ∗ , for every x ∈ M sa (cf. Lemma 2.2). By Lemma 2.3, for every x, y ∈ M sa thereexists a skew-hermitian element a x,y ∈ M such that T ( x ) = [ a x,y , x ] and T ( y ) = [ a x,y , y ] . Then T ( x ) y + xT ( y ) = [ a x,y , x ] y + x [ a x,y , y ] = [ a x,y , xy ] , that is, [ a x,y , xy ] = T ( x ) y + xT ( y ) . Further 0 = τ ([ a x,y , xy ]) = τ ( T ( x ) y + xT ( y )) , i.e. τ ( T ( x ) y ) = − τ ( xT ( y )), for every x, y ∈ M sa . For arbitrary u, v, w ∈ M sa , set x = u + v, and y = w. The above identity implies τ ( T ( u + v ) w ) = − τ (( u + v ) T ( w )) == − τ ( uT ( w )) − τ ( vT ( w )) = τ ( T ( u ) w ) + τ ( T ( v ) w ) = τ (( T ( u ) + T ( v )) w ) , and so τ (( T ( u + v ) − T ( u ) − T ( v )) w ) = 0for all u, v, w ∈ M sa . Take w = T ( u + v ) − T ( u ) − T ( v ) . Then τ ( ww ∗ ) = 0 . Since the trace τ is faithful it follows that ww ∗ = 0 , and hence w = 0 . Therefore T ( u + v ) = T ( u ) + T ( v ) . Case 2.
As in
Case 1 , we may assume T ( ) = 0. Suppose now that τ is a semi-finite trace. Since M is finite there exists a family of mutuallyorthogonal central projections { z i } in M such that z i has finite trace for all i and W z i = (cf. [30, § i , T maps z i M into itself. From Case 1, T | z i M : z i M → z i M is additive.Furthermore, z i T ( x + y ) = T | z i M ( z i x + z i y ) = T | z i M ( z i x ) + T | z i M ( z i y ) = z i T ( x ) + z i T ( y ) , for every x, y ∈ M and every i . Therefore T ( x + y ) = X i z i ! T ( x + y ) = X i z i T ( x + y ) = X i ( z i T ( x ) + z i T ( y ))= X i z i ! T ( x ) + X i z i ! T ( y ) = T ( x ) + T ( y ) , for every x, y ∈ M. The proof is complete. (cid:3)
Let T : M → M be a (not necessarily linear nor continuous) 2-local triplederivation on an arbitrary von Neumann algebra. In this case there existorthogonal central projections z , z ∈ M with z + z = such that:( − ) z M is a finite von Neumann algebra;( − ) z M is a properly infinite von Neumann algebra,(cf. [30, § k = 1 , , z k T maps z k M into itself. By Corol-lary 2.10 and Proposition 2.12 both z T and z T are additive on M sa . So T = z T + z T also is additive on M sa .We have thus proved the following result: Proposition 2.13.
Let T : M → M be a ( not necessarily linear nor contin-uous ) T | M sa is additive. (cid:3) -LOCAL TRIPLE DERIVATIONS 13 Main result.
We can state now the main result of this paper.
Theorem 2.14.
Let M be an arbitrary von Neumann algebra and let T : M → M be a ( not necessarily linear nor continuous ) T is a triple derivation ( hence linear and continuous ) . Equivalently,the set Der t ( M ) , of all triple derivations on M, is algebraically 2-reflexivein the set M ( M ) = M M of all mappings from M into M . We need the following two Lemmata.
Lemma 2.15.
Let T : M → M be a ( not necessarily linear nor continuous ) T ( ) = 0 . Thenthere exists a skew-hermitian element a ∈ M such that T ( x ) = [ a, x ] , for all x ∈ M sa . Proof.
Let x ∈ M sa . By Lemma 2.3 there exist a skew-hermitian element a x,x ∈ M such that T ( x ) = [ a x,x , x ] , T ( x ) = [ a x,x , x ] . Thus T ( x ) = [ a x,x , x ] = [ a x,x , x ] x + x [ a x,x , x ] = T ( x ) x + xT ( x ) , i.e.(2.4) T ( x ) = T ( x ) x + xT ( x ) , for every x ∈ M sa .By Proposition 2.13 and Lemma 2.2, T | M sa : M sa → M sa is a real linearmapping. Now, we consider the linear extension ˆ T of T | M sa to M definedby ˆ T ( x + ix ) = T ( x ) + iT ( x ) , x , x ∈ M sa . Taking into account the homogeneity of T, Proposition 2.13 and the iden-tity (2.4) we obtain that ˆ T is a Jordan derivation on M. By [5, Theorem1] any Jordan derivation on a semi-prime algebra is a derivation. Since M is von Neumann algebra, ˆ T is a derivation on M (see also [33] and [16]).Therefore there exists an element a ∈ M such that ˆ T ( x ) = [ a, x ] for all x ∈ M. In particular, T ( x ) = [ a, x ] for all x ∈ M sa . Since T ( M sa ) ⊆ M sa ,we can assume that a ∗ = − a , which completes the proof. (cid:3) Lemma 2.16.
Let T : M → M be a ( not necessarily linear nor continuous ) T | M sa ≡ , then T ≡ . Proof.
Let x ∈ M be an arbitrary element and let x = x + ix , where x , x ∈ M sa . Since T is homogeneous, if necessary, passing to the element(1 + k x k ) − x, we can suppose that k x k < . In this case the element y = + x is positive and invertible. Take skew-hermitian elements a x,y , b x,y ∈ M such that T ( x ) = [ a x,y , x ] + b x,y ◦ x,T ( y ) = [ a x,y , y ] + b x,y ◦ y. Since T ( y ) = 0 , we get [ a x,y , y ] + b x,y ◦ y = 0 . By Lemma 2.4 we obtain that[ a x,y , y ] = 0 and ib x,y ◦ y = 0 . Taking into account that ib x,y is hermitian, y is positive and invertible, Lemma 2.5 implies that b x,y = 0 . We further note that0 = [ a x,y , y ] = [ a x,y , + x ] = [ a x,y , x ] , i.e. [ a x,y , x ] = 0 . Now, T ( x ) = [ a x,y , x ] + b x,y ◦ x = [ a x,y , x + ix ] = [ a x,y , x ] , i.e. T ( x ) = [ a x,y , x ] . Therefore, T ( x ) ∗ = [ a x,y , x ] ∗ = [ x , a ∗ x,y ] = [ x , − a x,y ] = [ a x,y , x ] = T ( x ) . So(2.5) T ( x ) ∗ = T ( x ) . Now replacing x by ix on (2.5) we obtain from the homogeneity of T that(2.6) T ( x ) ∗ = − T ( x ) . Combining (2.5) and (2.6) we obtain that T ( x ) = 0 , which finishes theproof. (cid:3) Proof of Theorem 2.14.
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Proc. Amer. Math. Soc. (3) , 209-214 (1970). Ch. Abdirov 1, Department of Mathematics, Karakalpak State University,Nukus 230113, Uzbekistan
E-mail address : [email protected] Department of Mathematics, University of Illinois, Urbana IL 61801, USA
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