A linear preserver problem on maps which are triple derivable at orthogonal pairs
aa r X i v : . [ m a t h . OA ] S e p A LINEAR PRESERVER PROBLEM ON MAPS WHICHARE TRIPLE DERIVABLE AT ORTHOGONAL PAIRS
AHLEM BEN ALI ESSALEH AND ANTONIO M. PERALTA
Abstract.
A linear mapping T on a JB ∗ -triple is called triple derivableat orthogonal pairs if for every a, b, c ∈ E with a ⊥ b we have0 = { T ( a ) , b, c } + { a, T ( b ) , c } + { a, b, T ( c ) } . We prove that for each bounded linear mapping T on a JB ∗ -algebra A the following assertions are equivalent:( a ) T is triple derivable at zero;( b ) T is triple derivable at orthogonal elements;( c ) There exists a Jordan ∗ -derivation D : A → A ∗∗ , a central element ξ ∈ A ∗∗ sa , and an anti-symmetric element η in the multiplier algebraof A , such that T ( a ) = D ( a ) + ξ ◦ a + η ◦ a, for all a ∈ A ;( d ) There exist a triple derivation δ : A → A ∗∗ and a symmetricelement S in the centroid of A ∗∗ such that T = δ + S .The result is new even in the case of C ∗ -algebras. We next establish anew characterization of those linear maps on a JBW ∗ -triple which aretriple derivations in terms of a good local behavior on Peirce 2-subspaces.We also prove that assuming some extra conditions on a JBW ∗ -triple M ,the following statements are equivalent for each bounded linear mapping T on M :( a ) T is triple derivable at orthogonal pairs;( b ) There exists a triple derivation δ : M → M and an operator S inthe centroid of M such that T = δ + S . Introduction:
Suppose X is a Banach A -bimodule over a complex Banach algebra A . Aderivation from A onto X is a linear mapping D : A → X satisfying thefollowing algebraic identity(1) D ( ab ) = D ( a ) b + aD ( b ) , ∀ ( a, b ) ∈ A . Researchers working on preservers problems are recently exploring theidea of finding conditions, weaker than those in the original definition, underwhich a (continuous) linear mapping is a derivation, a (Jordan) homomor-phism, etc. For example,
Mathematics Subject Classification.
Key words and phrases. C ∗ -algebra; JB ∗ -algebra; JB ∗ -triple; derivations; triple deriva-tions; generalized derivations; maps triple derivable at zero; maps triple derivable atorthogonal elements; roblem 1.1. Suppose T : A → X is a linear map satisfying (1) only on aproper subset D ⊂ A . Is T a derivation? There is no need to comment that the role of the set D is the real core ofthe question. A linear map T : A → X is said to be a derivation or derivableat a point z ∈ A if the identity (1) holds for every ( a, b ) ∈ D z := { ( a, b ) ∈ A : ab = z } . In order to illustrate the interest on linear maps on Banachalgebras which are derivable at a concrete point, the reader can consult thereferences [7, 17, 36, 37, 47, 55], among other additional papers.H. Ghahramani, Z. Pan [30] and B. Fadaee and H. Ghahramani [24] haverecently considered certain variants of Problem 1.1 in their studies on con-tinuous linear operators from a C ∗ -algebra A into a Banach A -bimodule X behaving like derivations or anti-derivations at elements in a certain subsetof A determined by orthogonality conditions. Let us detail the problem. Problem 1.2.
Let T : A → X be a continuous linear operator which isanti-derivable at zero, i.e., (2) T ( ab ) = T ( b ) a + bT ( a ) for all ( a, b ) ∈ D .Is T an anti-derivation or a perturbation of an anti-derivation? Clearly, a mapping D : A → X is called an anti-derivation if the identity(2) holds for every ( a, b ) ∈ A . If A is a C ∗ -algebra, a ∗ -derivation (respec-tively, a ∗ -anti-derivation) from A into itself, or into A ∗∗ , is a derivation(respectively, an anti-derivation) d : A → A satisfying d ( a ∗ ) = d ( a ) ∗ for all a ∈ A .Concerning Problem 1.1, B. Fadaee and H. Ghahramani prove in [24,Theorem 3.1] that for a continuous linear map T : A → A ∗∗ , where A is a C ∗ -algebra, the following statement holds: T is derivable at zero ifand only if there is a continuous derivation d : A → A ∗∗ and an element η ∈ Z ( A ∗∗ ) (the centre of A ∗∗ ) such that T ( a ) = d ( a ) + ηa for all a ∈ A .They also obtain a similar conclusion when T is r- ∗ -derivable at zero (thatis, ab ∗ = 0 ⇒ aT ( b ) ∗ + T ( a ) b ∗ = 0). Similar results were established by H.Ghahramani and Z. Pan for linear maps on a unital ∗ -algebra which is zeroproduct determined (cf. [30, Theorem 3.1]).B. Fadaee and H. Ghahramani also study continuous linear maps froma C ∗ -algebra A into its bidual which are anti-derivable at zero (see [24,Theorem 3.3]). In [1], D.A. Abulhamail, F.B. Jamjoom and the secondauthor of this note prove that for each bounded linear operator T from a C ∗ -algebra A into an essential Banach A -bimodule X the following statementsare equivalent:( a ) T is anti-derivable at zero (i.e. ab = 0 in A implies T ( b ) a + bT ( a ) = 0);( b ) There exist an anti-derivation d : A → X ∗∗ and an element ξ ∈ X ∗∗ satisfying ξa = aξ, ξ [ a, b ] = 0 , T ( ab ) = bT ( a ) + T ( b ) a − bξa, and T ( a ) = d ( a ) + ξa, for all a, b ∈ A .he conclusion can be improved in the case in which A is unital, or if X isa dual Banach A -bimodule.A linear mapping δ on a JB ∗ -triple E with triple product { ., ., . } is calleda triple derivation if it satisfies the following triple Leibniz’s rule δ { a, b, c } = { δ ( a ) , b, c } + { a, δ ( b ) , c } + { a, b, δ ( c ) } , for all a, b, c ∈ E (see subsection 1.1 for the concrete definitions and relationsbetween the basic notions). The class of triple derivations have been inten-sively studied (see, for example, [4, 32, 33, 49]). Surjective linear isometriesbetween C ∗ -algebras need not be, in general, C ∗ -homomorphisms nor Jor-dan ∗ -homomorphisms (cf. the famous paper [39]). However, a remarkableresult by W. Kaup proves that a linear surjection between JB ∗ -triples is anisometry if and only if it is a triple isomorphism (cf. [41, Proposition 5.5]).Since early papers in the theory of JB ∗ -triple up to recently contributions,triple derivations can be applied to define one-parameter continuous semi-groups of surjective linear isometries, equivalently, triple isomorphisms onJB ∗ -triples (see, for example, [29]).A linear mapping T on a JB ∗ -triple E is said to be triple derivable at anelement z ∈ E if T ( z ) = T { a, b, c } = { T ( a ) , b, c } + { a, T ( b ) , c } + { a, b, T ( c ) } , for all a, b, c ∈ E with { a, b, c } = z . As we shall see in subsection 1.1,C ∗ -algebras are one of the first natural examples of JB ∗ -triples. In thisparticular setting, we recently proved in [23, Theorem 2.3 and Corollary2.5] that every continuous linear map on a unital C ∗ -algebra A which is atriple derivation at the unit of A is a triple derivation and a generalizedderivation, that is, T ( ab ) = T ( a ) b + aT ( b ) − aT (1) b, for all a, b ∈ A . If we additionally assume that T (1) = 0, then T is a ∗ -derivation and a triple derivation [23, Proposition 2.4]. We also knowfrom [23, Theorem 2.9, Corollary 2.10] that every bounded linear map ona unital C ∗ -algebra which is triple derivable at zero must be a generalizedderivation, if we further assume that T (1) = 0 then T is a ∗ -derivation anda triple derivation too. Corollary 2.11 in [23] proves that every boundedlinear map T on a unital C ∗ -algebra which is triple derivable at zero and T (1) ∗ = − T (1) must be a triple derivation. A more surprising conclusion isthat every linear map on a von Neumann algebra which is triple derivableat zero is automatically continuous (see [23, Corollary 2.14]).In this paper we shall extend the above results in two different direc-tions. Firstly, we shall consider linear maps whose domains will be in thewider (and more natural) classes of JB ∗ -algebras and JB ∗ -triples. Secondly,we shall consider a hypothesis which is, a priori, weaker than being tripleerivable at zero. The relation “being orthogonal” among elements in C ∗ -algebras, JB ∗ -algebras and JB ∗ -triples has turned out to an useful to under-stand and classify these structures through maps preserving zero productsand orthogonality (see, for example, [2, 9, 10, 11, 14, 29, 44, 46, 45, 53],among others). As we shall detail later, elements a, b in a JB ∗ -triple E areorthogonal if L ( a, b ) = { a, b, ·} = 0, equivalently, { a, b, x } = 0 for all x ∈ E .We shall say that a linear mapping T on a JB ∗ -triple E is triple derivableat orthogonal pairs if0 = T { a, b, c } = { T ( a ) , b, c } + { a, T ( b ) , c } + { a, b, T ( c ) } only for those a, b, c ∈ E with a ⊥ b . Clearly, every triple derivation on E is triple derivable at orthogonal pairs, and every linear map on E whichis triple derivable at zero is triple derivable at orthogonal pairs. In orderto present new examples, let us recall that the centroid, Z ( E ) , of a JB ∗ -triple E is the set of all continuous linear operators S : E → E satisfying S { a, b, c } = { S ( c ) , b, a } for all a, b, c ∈ E (see [20]). By [20, Lemma 2.6] foreach S ∈ Z ( E ) there exists a unique bounded linear operator R : E → E satisfying { a, S ( b ) , c } = R { a, b, c } for all a, b, c ∈ E . Let δ : E → E be atriple derivation and S ∈ Z ( E ). It is easy to see that the mapping T = δ + S satisfies that for each a, b, c ∈ E with a ⊥ b we have0 = { T ( a ) , b, c } + { a, T ( b ) , c } + { a, b, T ( c ) } . That is, T is triple derivable at orthogonal pairs. We are naturally led tothe following conjecture: Conjecture 1.3.
Every continuous linear mapping T : E → E which istriple derivable at orthogonal pairs must be of the form T = δ + S , where S is a triple derivation on E and S is an element in the centroid of E . In this paper we provide a complete positive proof of Conjecture 1.3 in thecase in which the JB ∗ -triple E is a JB ∗ -algebra (see Theorem 3.8). We alsostudy this conjecture for general JB ∗ -triples. In section 2 we shall presentsome new applications of the contractive projection principle to give a firstcharacterization of those bounded linear maps on a JB ∗ -triple which aretriple derivations. A more general characterization is established in Theorem4.1, where we prove that for any bounded linear map T on a JB ∗ -triple E in which tripotents are norm-total, the following statements are equivalent:( a ) T is a triple derivation;( b ) T is triple derivable at orthogonal elements and for each tripotent e in E ,the element P ( e ) T ( e ) is a skew symmetric element in the JB ∗ -algebra E ( e ) (i.e. ( P ( e ) T ( e )) ∗ e = − T ( e )).Section 5 is devoted to the study of Conjecture 1.3. Our main conclusion(see Theorem 5.4) affirms that if M is a JBW ∗ -triple such that linearityon it is determined by Peirce 2-subspaces, the the following statements areequivalent for any bounded linear map T on M :( a ) T is triple derivable at orthogonal pairs; b ) There exists a triple derivation δ : M → M and an operator S in thecentroid of M such that T = δ + S .The existence of JB ∗ -triples admitting no non-trivial pairs of orthogonalelements (i.e. JB ∗ -triples or rank one), makes Conjecture 1.3 nonviable.It should be noted that if a JBW ∗ -triple satisfies that linearity on it isdetermined by Peirce 2-subspaces, then it cannot have rank one. By applyingthat Cartan factors have a very trivial centroid, we also show that if M = L ∞ j C j is an atomic JBW ∗ -triple such that each C j is a Cartan factor withrank at least 2, then a bounded linear mapping T : M → M is triplederivable at orthogonal pairs if, and only if, there exists a triple derivation δ : M → M and an operator S in the centroid of M such that T = δ + S (see Theorem 5.5).1.1. Basic background. A Jordan algebra A is a (non-necessarily asso-ciative) algebra whose product is abelian and satisfies the so-called Jordanidentity ( a ◦ b ) ◦ a = a ◦ ( b ◦ a ) , ( a, b ∈ A ) . A Jordan Banach algebra is a Jordan algebra A equipped with a completenorm, k . k , satisfying k a ◦ b k ≤ k a k k b k ( a, b ∈ A ). Every real or complexassociative Banach algebra is a real Jordan Banach algebra with respect tothe natural Jordan product a ◦ b := ( ab + ba ). A JB ∗ -algebra is a complexJordan Banach algebra A equipped with an algebra involution ∗ satisfying k{ a, a, a }k = k a k , a ∈ A, where { a, a, a } = 2( a ◦ a ∗ ) ◦ a − a ◦ a ∗ . C ∗ -algebrasare among the examples of JB ∗ -algebras when they are equipped with thenatural Jordan product and the natural norm and involution. The references[31, 15, 54] can be consulted as reference guides for the basic notions andresults in the theory of JB ∗ -algebras.C ∗ - and JB ∗ -algebras are particular examples of Banach spaces in a widerclass of complex Banach spaces known under the name of JB ∗ -triples. A JB ∗ -triple is a complex Banach space E equipped with a continuous tripleproduct { ., ., . } : E × E × E → E, ( a, b, c )
7→ { a, b, c } , which is bilinearand symmetric in ( a, c ) and conjugate linear in b , and satisfies the followingaxioms for all a, b, x, y ∈ E :(a) Jordan identity: L ( a, b ) L ( x, y ) = L ( x, y ) L ( a, b ) + L ( L ( a, b ) x, y ) − L ( x, L ( b, a ) y ) , where L ( a, b ) : E → E is the operator defined by L ( a, b ) x = { a, b, x } ;(b) L ( a, a ) is a hermitian operator with non-negative spectrum;(c) k{ a, a, a }k = k a k .This definition is an analytic-algebraic approach, established by W. Kaupin [41], to study bounded symmetric domains in complex Banach spacesfrom the holomorphic point of view. Concretely, for each abstract boundedsymmetric domain D in an arbitrary complex Banach space there existsa unique (up to linear isometries) JB ∗ -triple E whose open unit ball isbiholomorphically equivalent to D .B ∗ -triples enjoy many geometric properties, for example, a linear bijec-tion T on a JB ∗ -triple E is an isometry if and only if it is a triple isomor-phism, that is, T { a, b, c } = { T ( a ) , T ( b ) , T ( c ) } for all a, b, c ∈ E (cf. [41,Proposition 5.5]).The triple products(3) { x, y, z } = 12 ( xy ∗ z + zy ∗ x ) , and(4) { x, y, z } = ( x ◦ y ∗ ) ◦ z + ( z ◦ y ∗ ) ◦ x − ( x ◦ z ) ◦ y ∗ , are employed to induce a structure of JB ∗ -triple on C ∗ - and JB ∗ -algebras,respectively. The first one is also valid to define an structure of JB ∗ -tripleon the space B ( H, K ) of all bounded linear operators between two complexHilbert spaces H and K . Along the paper we shall write B ( X ) for theBanach space of all bounded linear operators on a Banach space X .An element e in a real or complex JB ∗ -triple E is said to be a tripotent if { e, e, e } = e . Each tripotent e ∈ E , determines a decomposition of E,E = E ( e ) ⊕ E ( e ) ⊕ E ( e ) , known as the Peirce decomposition associated with e , where E j ( e ) = { x ∈ E : { e, e, x } = j x } for each j = 0 , , Peirce arithmetic : { E i ( e ) , E j ( e ) , E k ( e ) } ⊆ E i − j + k ( e ) if i − j + k ∈{ , , } , and { E i ( e ) , E j ( e ) , E k ( e ) } = { } otherwise, and { E ( e ) , E ( e ) , E } = { E ( e ) , E ( e ) , E } = 0 . Consequently, each Peirce subspace E j ( e ) is a JB ∗ -subtriple of E .The Peirce 2-subspace E ( e ) enjoys has an additional structure. Namely, E ( e ) is a unital JB ∗ -algebra with unit e , product x ◦ e y := { x, e, y } andinvolution x ∗ e := { e, x, e } , respectively.A tripotent e in a JB ∗ -triple E is called complete if E ( e ) = { } .Elements a, b in a JB ∗ -triple E are called orthogonal (denoted by a ⊥ b )if L ( a, b ) = 0. The reader is referred to [10, Lemma 1] for several reformula-tions of the relation “being orthogonal” which will be applied without anyexplicit mention. It should be also noted that elements a, b in a C ∗ -algebra A are orthogonal in the C ∗ - sense (i.e. ab ∗ = b ∗ a = 0) if and only if theyare orthogonal when A is regarded as a JB ∗ -triple.We shall consider the natural partial order ≤ on the set of tripotents ofa JB ∗ -triple E defined by e ≤ u if u − e is a tripotent with u − e ⊥ e ..2. Jordan derivations.
Let X be a Jordan-Banach module over a Jor-dan Banach algebra A . A Jordan derivation from A into X is a linear map D : A → X satisfying: D ( a ◦ b ) = D ( a ) ◦ b + a ◦ D ( b ) . If X is unital Jordan Banach module over a JB ∗ -algebra A , it is easy tocheck that D (1) = 0, for every Jordan derivation D : A → X . A Jordan ∗ -derivation on A is a Jordan derivation D satisfying D ( a ) ∗ = D ( a ∗ ) , for all a ∈ A . Following standard notation, given x ∈ X and a ∈ A , the symbols M a and M x will denote the mappings M a : X → X , x M a ( x ) = a ◦ x and M x : A → X , a M x ( a ) = a ◦ x . By a little abuse of notation, wealso denote by M a the operator on A defined by M a ( b ) = a ◦ b . Examples ofJordan derivations can be given as follows: if we fix a ∈ A and x ∈ X , themapping [ M x , M a ] = M x M a − M a M x : A → X, b [ M x , M a ]( b ) , is a Jordan derivation. A derivation D : A → X which can be written inthe form D = P mi =1 ( M x i M a i − M a i M x i ), ( x i ∈ X, a i ∈ A ) is called inner .Returning to the setting to triple derivations, by the Jordan identity,given a, b in a JB ∗ -triple E , the mapping δ ( a, b ) := L ( a, b ) − L ( b, a ) is a triplederivation on E and obviously continuous. Actually, every triple derivationon a JB ∗ -triple is automatically continuous (see [4, Corollary 2.2]).A JBW ∗ -algebra is a JB ∗ -algebra which is also a dual Banach space (cf.[31, Theorem 4.4.16]). The second dual, A ∗∗ , of a JB ∗ -algebra A is a JBW ∗ -algebra with a respect to a product extending the original Jordan product of A (see [31, Theorem 4.4.3]). The Jordan product of every JBW ∗ -algebra isseparately weak ∗ continuous (cf. [31, Theorem 4.4.16 and Corollary 4.1.6]).In the setting of JB ∗ -triples, a JBW ∗ -triple is a JB ∗ -triple which is also adual Banach space. The bidual of every JB ∗ -triple is a JBW ∗ -triple [19].The alter ego of Sakai’s theorem asserts that every JBW ∗ -triple admits aunique (isometric) predual and its product is separately weak ∗ continuous[5] (see also [16, Theorems 5.7.20 and 5.7.38]).Let E be a JB ∗ -triple. It is known that the JB ∗ -subtriple of E gener-ated by a single element a ∈ A is (isometrically) triple isomorphic to acommutative C ∗ -algebra (cf. [40, Corollary 4.8], [41, Corollary 1.15] and[26]). In particular, we can find an element y in this subtriple satisfyingthat { y, y, y } = a . The element y, denoted by a [ ] , is called the cubic root of a . The 3 n th roots of a are inductively defined by a [ n ] = (cid:16) a [ n − ] (cid:17) [ ] , n ∈ N . If a is an element in a JBW ∗ -triple M , the sequence ( a [ n ] ) con-verges in the weak ∗ -topology of M to a tripotent denoted by r ( a ), which iscalled the range tripotent of a . The tripotent r ( a ) is the smallest tripotent e ∈ M satisfying that a is positive in the JBW ∗ -algebra E ∗∗ ( e ) (compare[22, Lemma 3.3]).y the separate weak ∗ continuity of the triple product of every JBW ∗ -triple, we can conclude that for each triple derivation δ on a JB ∗ -triple E ,the bitranspose δ ∗∗ : E ∗∗ → E ∗∗ is a triple derivation too.It is known that every Jordan ∗ -derivation on a JB ∗ -algebra is a triplederivation [32, Lemma 2]. Reciprocally, for each skew symmetric element a ∈ A , the mapping M a ( x ) = a ◦ x is a triple derivation, and if δ : A → A is a triple derivation, the element δ (1) is skew symmetric and the mapping δ − M δ ( u ) is a Jordan ∗ -derivation on A (cf. [32, Lemma 1 and its proof]).1.3. The centre of a JB ∗ -algebra and the centroid of a JB ∗ -triple. Elements a, b in a JB ∗ -algebra A are said to operator commute if the cor-responding Jordan multiplication operators M a and M b commute in B ( A ),i.e. if ( a ◦ c ) ◦ b = a ◦ ( c ◦ b ) for all c ∈ A . The centre of A , Z ( A ), is the setof all elements of A which operator commute with every other element of A . By the Shirshov-Cohn theorem for JB algebras (cf. [31, Theorem 7.2.5])two self-adjoint elements a and b in A generate a JB ∗ -subalgebra which is aJC ∗ -subalgebra of some B ( H ) (see also [54]), and, under this realization, a and b commute in the usual sense whenever they operator commute in B ( H )(see [52, Proposition 1]). It is also known from the same sources that twoelements a and b of A sa operator commute if and only if a ◦ b = { a, b, a } (equivalently, a ◦ b = 2( a ◦ b ) ◦ a − a ◦ b ). If a C ∗ -algebra A is regarded as aJB ∗ -algebra with respect to its natural Jordan product, it follows from theabove that the centre of A in the Jordan sense is precisely the usual centre,i.e., the set of all a ∈ A such that ab = ba for all b ∈ A .Several authors have treated the notions of commutativity and opera-tor commutativity in C ∗ - and JB ∗ -algebras. Concerning orthogonality, B.Fadaee and H. Ghahramani showed in [24, Lemma 2.2] that given an ele-ment η in the bidual of a C ∗ -algebra A , the condition aηb = 0 for all a, b ∈ A with ab = 0 implies that η lies in the centre of A ∗∗ . An appropriate versionfor Banach bimodules over C ∗ -algebras is established in [1, Lemma 5]. Wepresent next a Jordan version of of this fact. Lemma 1.4.
Let A be a JB ∗ -algebra. Let ξ be an element in A ∗∗ such that U a,b ( ξ ) = 0 for all a, b ∈ A sa with a ⊥ b . Then ξ lies in the centre of A ∗∗ .Proof. Let us fix a functional φ ∈ A ∗ . We shall consider the followingsymmetric bounded bilinear form V : A × A → C , V ( a, b ) = φU a,b ( ξ ). Itfollows from our assumptions that V ( a, b ) = 0 , for every a, b ∈ A sa with a ⊥ b . We have therefore shown that V is a symmetric orthogonal form (cf.[35, Corollary 3.14]). We deduce from [35, Theorem 3.6] that the mapping G V : A → A ∗ , G V ( a )( b ) = V ( a, b ) is a purely Jordan generalized derivation.Furthermore, by Remark 3.2 there exists ϕ ∈ A ∗ such that G V ( a ◦ b ) = G V ( a ) ◦ b + a ◦ G V ( b ) − U a,b ( ϕ ) , for all a, b ∈ A . Then G V ( a ◦ b )( c ) = ( G V ( a ) ◦ b + a ◦ G V ( b ) − U a,b ( ϕ ))( c ) , r equivalently, V ( a ◦ b, c ) = V ( a, c ◦ b ) + V ( b, a ◦ c ) − ϕU a,b ( c ) , or φU a ◦ b,c ( ξ ) = φU a,c ◦ b ( ξ ) + φU b,a ◦ c ( ξ ) − ϕU a,b ( c ) , for all a, b, c ∈ A . If in the later expression we replace c by the elements in abounded increasing approximate unit ( c λ ) λ in A , and apply that ( c λ ) λ → ∗ topology of A ∗∗ , by taking weak ∗ limits we have(5) φ (( a ◦ b ) ◦ ξ ) = φU a,b ( ξ ) + φU b,a ( ξ ) − ϕ ( a ◦ b ) , for all a, b ∈ A . Replacing, b with c λ and taking weak ∗ limits we get φ ( a ◦ ξ ) = φ (2( a ◦ ξ )) − ϕ ( a ) , for all a ∈ A , witnessing that ϕ ( a ) = φ ( a ◦ ξ ) for all a ∈ A . Combiningthis fact with (5) we prove that φU a,b ( ξ ) + φU b,a ( ξ ) = 2 φ (( a ◦ b ) ◦ ξ ) , for all a, b ∈ A . The arbitrariness of φ and the Hahn-Banach theorem prove that(6) 2 U a,b ( ξ ) = U a,b ( ξ ) + U b,a ( ξ ) = 2( a ◦ b ) ◦ ξ, for all a, b ∈ A . The weak ∗ -density of A in A ∗∗ (Goldstine’s Theorem) andthe separate weak ∗ -continuity of the product of A ∗∗ (cf. [31, Theorem 4.4.16and Corollary 4.1.6]) allow us to deduce that (6) holds for all a, b ∈ A ∗∗ .The same is true for the hermitian and the skew symmetric part of ξ in A ∗∗ , ξ = ξ + ξ ∗ and ξ = ξ − ξ ∗ .An element s in a unital JB ∗ -algebra is called a symmetry if s = 1. Ifwe replace a and b with a symmetry s ∈ A ∗∗ in the version of (6) for ξ , ξ in A ∗∗ , we see that U s ( ξ j ) = ( s ◦ s ) ◦ ξ j = ξ j for j = 1 ,
2. Lemma 4.3.2 in[31] asserts that ξ j lies in the centre of A ∗∗ for all j = 1 ,
2, which concludesthe proof. (cid:3)
The lacking of a binary product in a JB ∗ -triple disables a natural notionof centre. The closest concept is the notion of centroid. Let E be a JB ∗ -triple. The main motivation to introduce the centroid is the connectionwith the centralizer of a Banach space X . We recall that a multiplier on X is a bounded linear mapping T : X → X satisfying that each extremepoint φ of the closed unit ball of X ∗ is an eigenvector of the transposedmapping T ∗ : X ∗ → X ∗ , that is, T ∗ ( φ ) = λ T ( φ ) φ , where λ T ( φ ) is thecorresponding eigenvalue (see [6]). The centralizer , C ( X ), of X is the setof all multipliers T on X for which there exists another multiplier S on X satisfying λ S ( φ ) = λ T ( φ ). An attractive result by S. Dineen and R.Timoney proves that for each JB ∗ -triple E , the centroid of E coincides withits centralizer, and it is precisely the set of all bounded linear operators on E commuting with all Hermitian operators on E (cf. [20, Theorem 2.8 andCorollary 2.9]). By [20, Lemma 2.6] for each S ∈ Z ( E ) there exists a uniquebounded linear operator R : E → E satisfying { a, S ( b ) , c } = R { a, b, c } forall a, b, c ∈ E .he centroid of a JB ∗ -algebra A is the set of all bounded linear operators T on A satisfying T ( a ◦ b ) = T ( a ) ◦ b, for all a, b ∈ A. The centroid of A as JB ∗ -triple coincides with its centroid as JB ∗ -algebra,and if A is unital a bounded linear operator T lies in the centroid of A ifand only if T = M z for a unique element z in the centre of A (see [20,Propositions 3.4 and 3.5]).We shall gather next some other properties of the centroid. Let T bean element in the centroid of a JB ∗ -triple E and let e be a tripotent in E .Since T L ( e, e ) = L ( e, e ) T , we can clearly conclude that T ( E j ( e )) ⊆ E j ( e )for all j = 0 , ,
2. In particular, T | E ( e ) : E ( e ) → E ( e ) is a bounded linearoperator, and given a, b ∈ E ( e ) we can see that T ( a ◦ e b ) = T { a, e, b } = { T ( a ) , e, b } = T ( a ) ◦ e b, which proves that T | E ( e ) is an element in the centroid of the unital JB ∗ -algebra E ( e ), therefore there exists an element ξ e in the centre of E ( e )such that T ( a ) = ξ e ◦ e a for all a ∈ E ( e ).2. Contractive projections and derivations
Let us recall that linear maps on a JB ∗ -triple which are triple derivableat a point have not been described yet. So, linear maps which are triplederivable at orthogonal elements remain also unknown.A linear mapping P on a Banach space X is called a projection if P = P .A projection P on X is contractive if k P k = 1. The image of a contrac-tive projection on a C ∗ -algebra need not be, in general, a C ∗ -algebra; forexample, for a rank-one projection p in B ( H ), the mapping P ( x ) = px is a contractive projection whose image is isometrically isomorphic to theHilbert space H , which clearly is not a C ∗ -algebra whenever H is infinitedimensional. However, for each contractive projection P on a JB ∗ -triple E ,its image, P ( E ) , is always a JB ∗ -triple with respect to the inherited normand the triple product given by(7) { x, y, z } P := P { x, y, z } , ( x, y, z ∈ P ( E ))(cf. [50], [42, Theorem] and [27]). The triple product given in (7) willbe denoted by { ., ., . } P , to avoid confusion the JB ∗ -triple P ( E ) will be, ingeneral, denoted by ( P ( E ) , { ., ., . } P ). The image of a contractive projectionneed not be a JB ∗ -subtriple of E [42, Example 3], however, the JB ∗ -triple P ( E ) is isometrically isomorphic to a closed subtriple of E ∗∗ [28, Theorem2].Let P : E → E be a contractive projection on a JB ∗ -triple. The nextidentity was established by W. Kaup in [42, Identity (4) in page 97](8) P { P ( a ) , b, P ( c ) } = P { P ( a ) , P ( b ) , P ( c ) } , ( a, b, c ∈ E ) . . Friedman and B. Russo complemented Kaup’s result by showing that theidentity(9) P { P ( a ) , P ( b ) , c } = P { P ( a ) , P ( b ) , P ( c ) } , holds for all a, b, c ∈ E (see [28, Theorem 3]). It should be noted that when P ( E ) is already a JB ∗ -subtriple of E (for example, when P = P j ( e ) for sometripotent e ∈ E and j ∈ { , , } ), the right hand sides in (8) and (9) can bereplaced with { P ( a ) , P ( b ) , P ( c ) } . Lemma 2.1.
Let δ : E → E be a triple derivation on a JB ∗ -triple. Suppose P : E → E is a contractive projection and P ( E ) is a subtriple of E . Thenthe mapping D = P δ | P ( E ) : P ( E ) → P ( E ) is a triple derivation on theJB ∗ -triple ( P ( E ) , { ., ., . } P ) = ( P ( E ) , { ., ., . } ) .Proof. Given a, b, c ∈ P ( E ) we have D { a, b, c } P = P δ { a, b, c } = P ( { δ ( a ) , b, c } + { a, δ ( b ) , c } + { a, b, δ ( c ) } )= { P δ ( a ) , b, c } + { a, P δ ( b ) , c } + { a, b, P δ ( c ) } = { D ( a ) , b, c } + { a, D ( b ) , c } + { a, b, D ( c ) } = { D ( a ) , b, c } P + { a, D ( b ) , c } P + { a, b, D ( c ) } P , where we have applied (8), (9) and the fact that a, b, c ∈ P ( E ), the latterbeing a JB ∗ -subtriple of E . (cid:3) Suppose δ is a triple derivation on a JB ∗ -triple E . For each tripotent e in E , the Peirce projection P j ( e ) is a contractive projection for every j ∈ { , , } . It is further known that the projection P ( e ) + P ( e ) is con-tractive too (see [26, Corollary 1.2]). Moreover, in this case, the corre-sponding images of these contractive projections, i.e. the Peirce subspaces E j ( e ) = P j ( e )( E ) , j = 0 , , E ( e ) ⊕ E ( e ), are JB ∗ -subtriples of E .It follows from Lemma 2.1 that D ej = P j ( e ) δ | E j ( e ) : E j ( e ) → E j ( e ) and D e , = ( P ( e ) + P ( e )) δ | E ( e ) ⊕ E ( e ) : E ( e ) ⊕ E ( e ) → E ( e ) ⊕ E ( e ) aretriple derivations.What about the reciprocal implication? That is, suppose T : E → E is a(bounded) linear operator on a JB ∗ -triple satisfying that for each tripotent e ∈ E , the mapping D e , = ( P ( e ) + P ( e )) T | E ( e ) ⊕ E ( e ) : E ( e ) ⊕ E ( e ) → E ( e ) ⊕ E ( e ) is a triple derivation. Is T a triple derivation? Proposition 2.2.
Let T : E → E be a bounded linear mapping on a JB ∗ -triple. Suppose that the set of tripotents is norm-total in E , that is, everyelement in E can be approximated in norm by a finite linear combinationof mutually orthogonal tripotents. Then the following statements are equiv-alent: ( a ) T is a triple derivation; ( b ) For each tripotent e in E the mapping D e , = ( P ( e )+ P ( e )) δ | E ( e ) ⊕ E ( e ) : E ( e ) ⊕ E ( e ) → E ( e ) ⊕ E ( e ) is a triple derivation. c ) For each tripotent e in E the mapping D e = P ( e ) T | E ( e ) : E ( e ) → E ( e ) is a triple derivation and P ( e ) T ( e ) = 0 .Proof. The implication ( a ) ⇒ ( b ) follows from Lemma 2.1.( b ) ⇒ ( c ) Let e be a tripotent in E . By our hypotheses, D e , is a triplederivation on E ( e ) ⊕ E ( e ). It then follows that D e , ( e ) = D e , { e, e, e } = 2 { D e , ( e ) , e, e } + { e, D e , ( e ) , e } ∈ E ( e ) ⊕ E ( e ) , where in the final step we applied Peirce arithmetic. Consequently, P ( e ) T ( e ) = P ( e ) D e , ( e ) = 0 . Since D e , ( e ) is a triple derivation on E ( e ) ⊕ E ( e ), a new application ofLemma 2.1 implies that P ( e ) T | E ( e ) = P ( e ) D e , | E ( e ) is a triple derivationon E ( e ).( c ) ⇒ ( a ) By hypothesis,(10) P ( e ) T ( e ) = 0 , and the mapping D e = P ( e ) T | E ( e ) is a triple derivation. Therefore, P ( e ) T ( e ) = P ( e ) T { e, e, e } = 2 { P ( e ) T ( e ) , e, e } + { e, P ( e ) T ( e ) , e } , and thus2 { T ( e ) , e, e } + { e, T ( e ) , e } = 2 { P ( e ) T ( e ) , e, e } + 2 { P ( e ) T ( e ) , e, e } + { e, P ( e ) T ( e ) , e } = P ( e ) T ( e ) + 2 { P ( e ) T ( e ) , e, e } + { e, P ( e ) T ( e ) , e } = P ( e ) T ( e ) + P ( e ) T ( e ) = T ( e ) . We have therefore shown that(11) T ( e ) = T { e, e, e } = 2 { T ( e ) , e, e } + { e, T ( e ) , e } , for all tripotent e ∈ E . We can reproduce now an argument taken fromthe proof of [13, Theorem 2.4], to get the desired statement. If we considera finite linear combination of mutually orthogonal tripotents e , . . . , e n , wededuce from the above that(12) T n X j =1 λ j e j , n X j =1 λ j e j , n X j =1 λ j e j = n X j =1 | λ j | λ j T { e j , e j , e j } = n X j =1 | λ j | λ j (2 { T ( e j ) , e j , e j } + { e j , T ( e j ) , e j } ) . On the other hand, if we fix three index i, j, k ∈ { , . . . , n } with i, k = j ,it follows from the fact that e i , e k ∈ E ( e j ), (10) and Peirce arithmetic that(13) { e i , T ( e j ) , e k } = 0 . ince e i ± e j is a tripotent, (11) implies that T ( e i ) ± T ( e j ) = T ( e i ± e j ) = 2 { T ( e i ± e j ) , e i ± e j , e i ± e j } + { e i ± e j , T ( e i ± e j ) , e i ± e j } = 2 { T ( e i ) , e i , e i } + 2 { T ( e i ) , e j , e j } ± { T ( e j ) , e i , e i }± { T ( e j ) , e j , e j } + { e i , T ( e i ) , e i } ± { e i , T ( e i ) , e j } + { e i , T ( e j ) , e j } ± { e j , T ( e i ) , e i } + { e j , T ( e j ) , e i } ± { e j , T ( e j ) , e j } which combined with (11) gives0 = 2 { T ( e i ) , e j , e j } ± { T ( e j ) , e i , e i } ± { e i , T ( e i ) , e j } + { e i , T ( e j ) , e j } (14) ± { e j , T ( e i ) , e i } + { e j , T ( e j ) , e i } , and then 0 = { T ( e i ) , e j , e j } + { e i , T ( e j ) , e j } . (15)Now, we check the following summands(16)2 T n X j =1 λ j e j , n X j =1 λ j e j , n X j =1 λ j e j + n X j =1 λ j e j , T n X j =1 λ j e j , n X j =1 λ j e j = 2 n X j =1 ,k =1 | λ j | λ k { T ( e k ) , e j , e j } + n X i =1 ,j =1 ,k =1 λ i λ j λ k { e i , T ( e j ) , e k } = (by (13)) = 2 n X j =1 ,i =1 | λ j | λ i { T ( e i ) , e j , e j } + n X i =1 ,k =1 ,i = k | λ i | λ k { e i , T ( e i ) , e k } + n X i =1 ,j =1 ,i = j λ i | λ j | { e i , T ( e j ) , e j } + n X j =1 λ j | λ j | { e j , T ( e j ) , e j } = (by (15)) = 2 n X j =1 | λ j | λ j { T ( e j ) , e j , e j } + | λ j | λ j { T ( e j ) , e j , e j } . By combining (12) and (16) it ca be concluded that T { a, a, a } = 2 { T ( a ) , a, a } + { a, T ( a ) , a } , for every a = P nj =1 λ j e j , where e , . . . , e n are mutually orthogonal tripo-tents in E . Since, by hypotheses, tripotents in E are norm-total, we get T { a, a, a } = 2 { T ( a ) , a, a } + { a, T ( a ) , a } , for every a ∈ E . A standard polar-ization identity proves that T is a triple derivation. (cid:3) Remark 2.3.
It should be noted here that every JBW ∗ -triple and everycompact JB ∗ -triple satisfies the hypotheses of the previous proposition (see[34, Lemma 3.11] and [8]).The novelty here is that, as we shall see in subsequent results, Lemmas2.1 and 2.4 give a hint to study linear maps which are triple derivable atorthogonal pairs in terms of generalized Jordan derivations. emma 2.4. Let T : E → E be a linear map on a JB ∗ -triple which is triplederivable at orthogonal pairs. Then for each contractive projection P on E with P ( E ) being a subtriple of E , the mapping P T | P ( E ) : P ( E ) → P ( E ) is triple derivable at orthogonal pairs. In particular, for each tripotent e ∈ E , the mappings P j ( e ) T | E j ( e ) : E j ( e ) → E j ( e ) ( j = 0 , , and ( P ( e ) + P ( e )) T | E ( e ) ⊕ E ( e ) : E ( e ) ⊕ E ( e ) → E ( e ) ⊕ E ( e ) are triple derivable atorthogonal pairs.Proof. The ideas are very similar to those given in the proof of Lemma 2.1.Given a, b, c ∈ P ( E ) with a ⊥ b we have0 = P T { a, b, c } P = P T { a, b, c } = P ( { T ( a ) , b, c } + { a, T ( b ) , c } + { a, b, T ( c ) } = { P T ( a ) , b, c } + { a, P T ( b ) , c } + { a, b, P T ( c ) } = { P T ( a ) , b, c } P + { a, P T ( b ) , c } P + { a, b, P T ( c ) } P . where we have applied (8), (9), the fact that a, b, c ∈ P ( E ), and the assump-tion that P ( E ) is a JB ∗ -subtriple of E . (cid:3) Generalized Jordan derivations via orthogonal forms
Lemma 2.4 justifies the importance of determining the structure of thosebounded linear maps on a JB ∗ -algebra which are triple derivable at orthog-onal pairs. The Jordan analogues of the results in [13, §
3] and [3, §
2] remainunexplored. The aim of this section is to complete our knowledge on Jor-dan and triple derivations by studying those linear maps on JB ∗ -algebraspreserving the natural relations with respect to orthogonality that triplederivations enjoy.As we have already mentioned, given a triple derivation δ on a JB ∗ -triple E and elements a, b, c ∈ E with a, c ⊥ b we have0 = δ { a, b, c } = { δ ( a ) , b, c } + { a, δ ( b ) , c } + { a, b, δ ( c ) } = { a, δ ( b ) , c } . Similarly, if T : E → E is a linear mapping which is triple derivable atorthogonal elements we deduce that T satisfies the following property(17) { a, T ( b ) , c } = 0 , for all a, b, c ∈ E with a, c ⊥ b .So, as in the case of bounded linear maps on C ∗ -algebras, it seems interestingto study those bounded linear maps on E satisfying the just commentedproperty (17). We note that property (17) is related to the property studiedin [23, Lemma 2.8] for linear maps triple derivable at zero.As in the associative case, for each mapping T on a JB ∗ -algebra A , wedefine a mapping T ♯ : A → A given by T ♯ ( x ) := T ( x ∗ ) ∗ . Clearly T is(bounded) linear if and only if the same property holds for T ♯ . The map-ping T will be called symmetric (respectively, anti-symmetric) if T ♯ = T (respectively, T ♯ = − T ). Every mapping T can be written as the sum of asymmetric and an anti-symmetric mapping and a linear combination of twosymmetric mappings, T = ( T ♯ + T ) + ( T − T ♯ ) = ( T ♯ + T ) + i i ( T − T ♯ ). emma 3.1. Let T be a linear mapping on a JB ∗ -algebra. The followingstatements hold: ( a ) T is triple derivable at orthogonal elements if and only if T ♯ satisfies thesame property; ( b ) The real linear combination of linear mappings which are triple derivableat orthogonal elements also satisfies this property; ( c ) T is triple derivable at orthogonal elements if and only if ( T ♯ + T ) and ( T − T ♯ ) are triple derivable at orthogonal elements.Proof. ( a ) Suppose T is triple derivable at orthogonal elements. Take a, b, c ∈ A with a ⊥ b . Since a ∗ ⊥ b ∗ , it follows from the hypothesis that0 = { T ( a ∗ ) , b ∗ , c ∗ } + { a ∗ , T ( b ∗ ) , c ∗ } + { a ∗ , b ∗ , T ( c ∗ ) } . By applying the involution on A we obtain0 = { T ( a ∗ ) ∗ , b, c } + { a, T ( b ∗ ) ∗ , c } + { a, b, T ( c ∗ ) ∗ } = { T ♯ ( a ) , b, c } + { a, T ♯ ( b ) , c } + { a, b, T ♯ ( c ) } . Since T ♯♯ = T , the “if” implication also follows from the “only if” one.( b ) can be straightforwardly checked, and ( c ) follows from ( a ) and ( b ). (cid:3) Let G : A → X be a linear mapping where A is a JB ∗ -algebra and X is aJordan-Banach A -module. Following [2, §
4] and [13, § G is a generalized Jordan derivation if there exists ξ ∈ X ∗∗ satisfying(18) G ( a ◦ b ) = G ( a ) ◦ b + a ◦ G ( b ) − U a,b ( ξ ) , for every a, b in A . If A is unital the element ξ = G (1) lies in X . Anexample can be given as follows, fix a non-zero a in A . The mapping M a is a generalized Jordan derivation on A which is not a Jordan derivationbecause M a (1) = a = 0. If a = − a ∗ , the mapping M a neither is a triplederivation because M a (1) = a = − a ∗ = − M a (1) ∗ (cf. [32, Lemma 1 and itsproof]). Remark 3.2.
In our case, we shall be mainly interested in generalizedJordan derivations G : A → A ∗ , where the Jordan module operation in A ∗ isgiven by ( ϕ ◦ a )( b ) = ϕ ( a ◦ b ) ( a, b ∈ A ). Suppose G : A → A ∗ is a generalizedderivation (which is automatically continuous by [35, Proposition 2.1]) forsome ξ ∈ A ∗∗∗ . Let ( c λ ) λ be a bounded increasing approximate unit in A (see [31, Proposition 3.5.4]). The net ( G ( a ◦ c λ )) λ → G ( a ) in norm, and( G ( a ) ◦ c λ ) λ → G ( a ) in the weak ∗ topology of A ∗ . The net ( G ( c λ )) λ isbounded in a dual Banach space, and hence we can find a subnet (denotedby the same symbol) converging in the weak ∗ topology of A ∗ to some φ ∈ A ∗ . Therefore, ( a ◦ G ( c λ )) λ → a ◦ φ in the weak ∗ topology. On the otherhand, for each d ∈ A , the net U a,c λ ( ξ )( d ) = ξ ( U a,c λ ( d )) → ξ ( a ◦ d ), as U a,c λ ( d ) − a ◦ d k →
0. Finally, by combining all these facts with (18) weget G ( a )( d ) = G ( a )( d ) + φ ( a ◦ d ) − ξ ( a ◦ d ) , for all a, d ∈ A , which clearly guarantees that we can take ξ = φ ∈ A ∗ .A purely Jordan generalized derivation from A into A ∗ is a generalizedJordan derivation G : A → A ∗ satisfying G ( a )( b ) = G ( b )( a ), for every a, b ∈ A (cf. [35, Definition 3.4]), while a Jordan derivation D from A into A ∗ is said to be a Lie Jordan derivation if D ( a )( b ) = − D ( b )( a ), for every a, b ∈ A (cf. [35, Definition 3.12]).The results in [35] reveal the strong connections between generalized Jor-dan derivations and orthogonal forms on JB ∗ -algebras. Let A be a JB ∗ -algebra. We recall that a continuous bilinear form V : A × A → C is called orthogonal if V ( a, b ∗ ) = 0 , for every a, b ∈ A with a ⊥ b . If V ( a, b ) = 0only for elements a, b ∈ A sa with a ⊥ b , we say that V is orthogonal on A sa (cf. [35]). Corollary 3.14 (see also Propositions 3.8 and 3.9) in [35] provesthat a bilinear form on a JB ∗ -algebra A is orthogonal if and only if it is or-thogonal on A sa . A bilinear form V on A is called symmetric (respectively,anti-symmetric) if V ( a, b ) = V ( b, a ) (respectively, V ( a, b ) = − V ( b, a )) forall a, b ∈ A .Let OF s ( A ) (respectively, OF as ( A )) denote the space of all symmetricorthogonal forms on A (respectively, of all anti-symmetric orthogonal formson A ), and let PJ GD er ( A, A ∗ ) and L ie J D er ( A, A ∗ ) stand for the spacesof all purely Jordan generalized derivations from A into A ∗ and of all LieJordan derivations from A into A ∗ , respectively.For each V ∈ OF s ( A ) define G V : A → A ∗ in PJ GD er ( A, A ∗ ) given by G V ( a )( b ) = V ( a, b ), and for each G ∈ PJ GD er ( A, A ∗ ) we set V G : A × A → C , V G ( a, b ) := G ( a )( b ) ( a, b ∈ A ). By [35, Theorem 3.6], the mappings OF s ( A ) → PJ GD er( A, A ∗ ) , PJ GD er(
A, A ∗ ) → OF s ( A ) ,V G V , G V G , define two (isometric) linear bijections which are inverses of each other.For each V ∈ OF as ( A ) we define D V : A → A ∗ in L ie J D er ( A, A ∗ )given by D V ( a )( b ) = V ( a, b ), and for each D ∈ L ie J D er ( A, A ∗ ) we set V D : A × A → C , V D ( a, b ) := D ( a )( b ) ( a, b ∈ A ). By [35, Theorem 3.13], themappings OF as ( A ) → L ie J D er(
A, A ∗ ) , L ie J D er(
A, A ∗ ) → OF as ( A ) ,V D V , D V D , define two linear bijections which are inverses of each other.We deal first with symmetric bounded linear maps which are triple deriv-able at orthogonal pairs. roposition 3.3. Let T : A → A be a symmetric bounded linear operator ona JB ∗ -algebra. Suppose that T is triple derivable at orthogonal pairs. Then T is a generalized Jordan derivation. Furthermore, there exist a centralelement ξ ∈ A ∗∗ sa and a Jordan ∗ -derivation D : A → A ∗∗ such that T ( a ) = D ( a ) + M ξ ( a ) = D ( a ) + ξ ◦ a, for all a ∈ A . Clearly D is a triple derivation.Proof. Pick an arbitrary functional φ ∈ A ∗ . We consider the following sym-metric bounded bilinear form V : A × A → C , V ( a, b ) := φ ( T ( a ) ◦ b + a ◦ T ( b )).Let ( c λ ) λ be a bounded increasing approximate unit in A (cf. [31, Proposi-tion 3.5.4]). Given a, b ∈ A sa with a ⊥ b , the hypotheses assure that0 = T { a, b, c λ } = { T ( a ) , b, c λ } + { a, T ( b ) , c λ } + { a, b, T ( c λ ) } = { T ( a ) , b, c λ } + { a, T ( b ) , c λ } , for all λ. Taking norm limits in λ we arrive at0 = { T ( a ) , b, } + { a, T ( b ) , } = T ( a ) ◦ b + a ◦ T ( b ) ∗ . By applying that T is symmetric and b ∈ A sa we get 0 = T ( a ) ◦ b + a ◦ T ( b ) , and thus V ( a, b ) = 0. We have therefore shown that V is orthogonal on A sa and hence on the whole A (see [35, Corollary 3.14]). Theorem 3.6 in[35] implies that the mapping G V : A → A ∗ , G V ( a )( b ) = V ( a, b ) is a purelyJordan generalized derivation, that is, by Remark 3.2 there exists ϕ ∈ A ∗ such that G V ( a ◦ b ) = G V ( a ) ◦ b + a ◦ G V ( b ) − U a,b ( ϕ ) , for all a, b ∈ A , or equivalently, G V ( a ◦ b )( c ) = ( G V ( a ) ◦ b + a ◦ G V ( b ) − U a,b ( ϕ ))( c ) ,V ( a ◦ b, c ) = V ( a, c ◦ b ) + V ( b, a ◦ c ) − ϕU a,b ( c ) , or φ ( T ( a ◦ b ) ◦ c + ( a ◦ b ) ◦ T ( c )) = φ ( T ( a ) ◦ ( c ◦ b ) + a ◦ T ( c ◦ b ))+ φ ( T ( b ) ◦ ( a ◦ c ) + b ◦ T ( a ◦ c )) − ϕU a,b ( c ) , for all a, b, c ∈ A. If we replace c with the elements in a bounded increasingapproximate unit ( c λ ) λ in A , and we take norm and weak ∗ limits in λ wearrive at(19) φ ( T ( a ◦ b ) + ( a ◦ b ) ◦ T ∗∗ (1)) = φ (2 T ( a ) ◦ b + 2 a ◦ T ( b )) − ϕ ( a ◦ b ) , for all a, b ∈ A, where we have applied the well known fact that ( c λ ) λ → ∗ topology of A ∗∗ . By replacing b with c λ in the above identityand taking norm and weak ∗ limits we are led to the identity φ ( T ( a ) + a ◦ T ∗∗ (1)) = φ (2 T ( a ) + 2 a ◦ T ∗∗ (1)) − ϕ ( a ) , or equivalently, ϕ ( a ) = φ ( T ( a ) + a ◦ T ∗∗ (1))for all a ∈ A. Now, by substituting this latter identity in (19) it follows that φ ( T ( a ◦ b )+( a ◦ b ) ◦ T ∗∗ (1)) = φ (2 T ( a ) ◦ b +2 a ◦ T ( b )) − φ ( T ( a ◦ b )+( a ◦ b ) ◦ T ∗∗ (1)) , hich simplified gives φT ( a ◦ b ) = φ ( T ( a ) ◦ b + a ◦ T ( b ) − ( a ◦ b ) ◦ T ∗∗ (1)) , for all a, b ∈ A . The arbitrariness of φ combined with the Hahn-Banachtheorem prove that(20) T ( a ◦ b ) = T ( a ) ◦ b + a ◦ T ( b ) − ( a ◦ b ) ◦ T ∗∗ (1) , for all a, b ∈ A . Since T is symmetric, T ∗∗ enjoys the same property, andhence T ∗∗ (1) ∈ A ∗∗ sa . We can actually deduce from the the weak ∗ -densityof A in A ∗∗ (Goldstine’s Theorem), the separate weak ∗ -continuity of theproduct of A ∗∗ (cf. [31, Theorem 4.4.16 and Corollary 4.1.6]) and the weak ∗ continuity of T ∗∗ that the identity in (20) holds for all a, b ∈ A ∗∗ by justreplacing T with T ∗∗ .We shall next prove that ξ = T ∗∗ (1) lies in the centre of A ∗∗ . By theconclusions above, for each projection p ∈ A ∗∗ we have(21) T ∗∗ ( p ) = 2 T ∗∗ ( p ) ◦ p − p ◦ ξ. Let T ∗∗ ( p ) = x + x + x and ξ = y + y + y denote the Peirce de-compositions of T ∗∗ ( p ) and ξ with respect to p , respectively (i.e., x j = P j ( p )( T ∗∗ ( p )) and y j = P j ( p )( ξ ) for j = 0 , , x + x = { p, p, T ∗∗ ( p ) } = p ◦ T ∗∗ ( p ) and y + y = { p, p, ξ } = p ◦ ξ. Thenthe previous identity (21) gives x + x + x = T ∗∗ ( p ) = 2 T ∗∗ ( p ) ◦ p − p ◦ ξ = 2 x + x − y − y , which guarantees that y = 0. Now, having in mind that ξ, p ∈ A ∗∗ sa wededuce that P ( p )( ξ ) = { p, { p, ξ, p } , p } = U p ( U p ( ξ ∗ ) ∗ ) = U p ( ξ ) = U p ( ξ ) (see[31, § U p ( ξ ) = P ( p )( ξ ) = y . On the other hand p ◦ ξ = p ◦ ξ = { p, p, ξ } = y + y = y . Therefore p ◦ ξ = U p ( ξ ), which guaranteesthat p and ξ operator commute (cf. comments in page 8). Since projectionsin A ∗∗ sa are norm-total (cf. [31, Proposition 4.2.3]), we conclude that ξ liesin the centre of A ∗∗ .Finally, by applying that ξ = T ∗∗ (1) is a central element in A ∗∗ , we deducefrom (20) that the identity T ( a ◦ b ) = T ( a ) ◦ b + a ◦ T ( b ) − U a,b ( ξ ) , holds for all a, b ∈ A , witnessing that T is a generalized Jordan derivation.It is easy to check that the mapping D : A → A ∗∗ , D ( a ) = T ( a ) − ξ ◦ a is aJordan ∗ -derivation, and T satisfies T ( a ) = D ( a ) + M ξ ( a ) for all a ∈ A . (cid:3) The next corollary seems to be a new advance too.
Corollary 3.4.
Let T : A → A be a symmetric bounded linear operator on aC ∗ -algebra. Suppose that T is triple derivable at orthogonal pairs. Then T isa generalized derivation. Furthermore, there exist a central element ξ ∈ A ∗∗ sa and a ∗ -derivation D : A → A ∗∗ such that T ( a ) = D ( a )+ M ξ ( a ) = D ( a )+ ξa, for all a ∈ A .roof. Proposition 3.3 proves the existence of a central element ξ ∈ A ∗∗ sa and a Jordan ∗ -derivation D : A → A ∗∗ such that T ( a ) = D ( a ) + M ξ ( a ) = D ( a ) + ξ ◦ a, for all a ∈ A . A celebrated result of B.E. Johnson asserts thatevery Jordan derivation on a C ∗ -algebra is a derivation (cf. [38, Theorem6.3]). Therefore, D is a ∗ -derivation. The rest is clear because ξ is a centralelement. (cid:3) Let A be a JB ∗ -algebra. Following [21], the (Jordan) multipliers algebra of A is the set M ( A ) := { x ∈ A ∗∗ : x ◦ A ⊆ A } . The space M ( A ) is a unital JB ∗ -subalgebra of A ∗∗ . Moreover, M ( A ) is the(Jordan) idealizer of A in A ∗∗ , that is, the largest JB ∗ -subalgebra of A ∗∗ containing A as a closed Jordan ideal (cf. [21, Theorem 2]).We deal next with the anti-symmetric operators which are triple derivableat orthogonal pairs. Proposition 3.5.
Let T : A → A be an anti-symmetric bounded linear oper-ator on a JB ∗ -algebra. Suppose that T is triple derivable at orthogonal pairs.Then T is a triple derivation, moreover, there exists an anti-symmetric ele-ment η in the multiplier algebra of A , such that T ( x ) = η ◦ x for all x ∈ A .Proof. Fix an arbitrary φ ∈ A ∗ . Let us consider the following continuousbilinear form V ( a, b ) = φ ( − a ◦ T ( b ) + T ( a ) ◦ b ) ( a, b ∈ A ). Clearly V is anti-symmetric. Let us take a, b ∈ A sa with a ⊥ b . By hypothesis,0 = T { a, b, } = { T ( a ) , b, } + { a, T ( b ) , } + { a, b, T (1) } = { T ( a ) , b, } + { a, T ( b ) , } , and thus0 = { T ( a ) , b, } + { a, T ( b ) , } = T ( a ) ◦ b + a ◦ T ( b ) ∗ = T ( a ) ◦ b − a ◦ T ( b ) , where in the last equality we applied that T ♯ = − T . If A is not unital, wechoose a bounded increasing approximate unit ( c λ ) λ in A (see [31, Proposi-tion 3.5.4]), and the hypotheses give0 = T { a, b, c λ } = { T ( a ) , b, c λ } + { a, T ( b ) , c λ } + { a, b, T ( c λ ) } = { T ( a ) , b, c λ } + { a, T ( b ) , c λ } for all λ. Taking norm limits in λ we obtain0 = T ( a ) ◦ b + a ◦ T ( b ) ∗ = T ( a ) ◦ b − a ◦ T ( b ) , where, as before, we applied that T ♯ = − T . We have therefore shown that V ( a, b ) = φ ( T ( a ) ◦ b − T ( b ) ◦ a ) = 0 , and thus V is orthogonal on A sa , andhence on A (cf. [35, Corollary 3.14]). It follows from [35, Theorem 3.13] thatthe mapping D V : A → A ∗ , D V ( a )( b ) = V ( a, b ) is a Lie Jordan derivation.Consequently, D V ( a ◦ c ) = D V ( a ) ◦ c + a ◦ D V ( c ) , for all a, c ∈ A, equivalently, V ( a ◦ c, b ) = D V ( a ◦ c )( b ) = ( D V ( a ) ◦ c + a ◦ D V ( c ))( b ) = V ( a, b ◦ c )+ V ( c, a ◦ b ) , nd φ ( T ( a ◦ c ) ◦ b − T ( b ) ◦ ( a ◦ c )) = φ ( T ( a ) ◦ ( b ◦ c ) − T ( b ◦ c ) ◦ a + T ( c ) ◦ ( a ◦ b ) − T ( a ◦ b ) ◦ c ) , for all a, b, c ∈ A . Now, having in mind the arbitrariness of φ , we deducefrom the Hahn-Banach theorem that T ( a ◦ c ) ◦ b − T ( b ) ◦ ( a ◦ c ) = T ( a ) ◦ ( b ◦ c ) − T ( b ◦ c ) ◦ a + T ( c ) ◦ ( a ◦ b ) − T ( a ◦ b ) ◦ c, (22)for all a, b, c ∈ A .Let us assume that A is unital. In this case, by replacing c = 1 in (22)we obtain − a ◦ T ( b ) + T ( a ) ◦ b = T ( a ) ◦ b − T ( b ) ◦ a + T (1) ◦ ( a ◦ b ) − T ( a ◦ b ) , for all a, b ∈ A , which implies that T ( a ◦ b ) = ( a ◦ b ) ◦ T (1) for all a, b ∈ A .In the non-unital case, we can consider a bounded increasing approximateunit ( c λ ) λ in A (see [31, Proposition 3.5.4]). Having in mind that ( c λ ) λ → ∗ topology of A ∗∗ , by replacing c with c λ in (22) and takingweak ∗ limits we get − a ◦ T ( b ) + T ( a ) ◦ b = T ( a ) ◦ b − T ( b ) ◦ a + T ∗∗ (1) ◦ ( a ◦ b ) − T ( a ◦ b ) , for all a, b ∈ A , and thus T ( a ) = T ∗∗ (1) ◦ a for all a ∈ A . Since A ∋ T ( a ) = T ∗∗ (1) ◦ a for all a ∈ A , we conclude that T ∗∗ (1) lies in the multiplier algebraof A . Finally, since T ∗∗ is anti-symmetric too, the element η = T ∗∗ (1) isanti-symmetric and satisfies the desired statement. (cid:3) Remark 3.6.
Let us observe that in the case of unital JB ∗ -algebras, thehypothesis T being triple derivable at orthogonal elements in Propositions3.3 and 3.5 can be reduced to the property { T ( a ) , b, } + { a, T ( b ) , } = 0 , for all a, b ∈ A sa with a ⊥ b. It can be checked that the proofs given above remain valid under this weakerassumption.The C ∗ - version of Proposition 3.5 also is a new result worth to be stated. Corollary 3.7.
Let T : A → A be an anti-symmetric bounded linear opera-tor on a C ∗ -algebra. Suppose that T is triple derivable at orthogonal pairs.Then T is a triple derivation, moreover, there exists an anti-symmetric ele-ment η in the multiplier algebra of A , such that T ( x ) = η ◦ x for all x ∈ A . We are now in a position to address the characterization of those boundedlinear maps on a JB ∗ -algebra which are triple derivable at orthogonal ele-ments. Theorem 3.8.
Let T : A → A be a bounded linear maps on a JB ∗ -algebra.Suppose T is triple derivable at orthogonal elements. Then there exists aJordan ∗ -derivation D : A → A ∗∗ , a central element ξ ∈ A ∗∗ sa , and an anti-symmetric element η in the multiplier algebra of A , such that T ( a ) = D ( a ) + ξ ◦ a + η ◦ a, for all a ∈ A. oreover, the mapping δ : A → A ∗∗ , δ ( a ) = D ( a ) + η ◦ a ( a ∈ A ) is a triplederivation and T ( a ) = δ ( a ) + ξ ◦ a, for all a ∈ A . The mapping S : A → A ∗∗ , S ( a ) = ξ ◦ a is the restriction to A of an element in the centroid of A ∗∗ .Proof. Let us write T = T + T , where T = T + T ♯ is symmetric and T = T − T ♯ is anti-symmetric. Lemma 3.1 assures that T and T are triplederivable at orthogonal pairs. Propositions 3.3 and 3.5 applied to T and T prove the desired conclusion. (cid:3) Remark 3.9.
If in Theorem 3.8 the JB ∗ -algebra A is unital, the Jordan ∗ -derivation D and the triple derivation δ are A -valued, and ξ is a symmetricelement in the centre of A .Theorem 3.8 particularly holds when A is a C ∗ -algebra.We can now complete the conclusion on linear maps which are triplederivable at zero. Corollary 3.10.
Let T : A → A be a bounded linear maps on a JB ∗ -algebra ( and in particular on a C ∗ -algebra ) . The the following assertionsare equivalent: ( a ) T is triple derivable at zero; ( b ) T is triple derivable at orthogonal elements; ( c ) There exists a Jordan ∗ -derivation D : A → A ∗∗ , a central element ξ ∈ A ∗∗ sa , and an anti-symmetric element η in the multiplier algebra of A , such that T ( a ) = D ( a ) + ξ ◦ a + η ◦ a, for all a ∈ A ;( d ) There exist a triple derivation δ : A → A ∗∗ and a symmetric element S in the centroid of A ∗∗ such that T = δ + S .Proof. We have already commented in the introduction that ( a ) ⇒ ( b ). Theimplications ( b ) ⇒ ( c ) ⇒ ( d ) follow from Theorem 3.8. Finally, if ( d ) holdsand we take a, b, c ∈ A with { a, b, c } = 0. By the assumptions { T ( a ) , b, c } + { a, T ( b ) , c } + { a, b, T ( c ) } = δ { a, b, c } + { S ( a ) , b, c } + { a, S ( b ) , c } + { a, b, S ( c ) } = S { a, b, c } + R { a, b, c } + S { a, b, c } = 0 , where R is the corresponding element in the centroid of A ∗∗ satisfying { x, S ( y ) , z } = R { x, y, z } for all x, y, z ∈ A ∗∗ . (cid:3) A new characterization of triple derivations
This section is devoted to present a new characterization of those con-tinuous linear maps on a JBW ∗ -triple which are triple derivations in termsof a good local behavior on Peirce 2-subspaces. Let us begin with a prop-erty which was already implicit in [32] and in many other recent studies onriple derivations and local triple derivations (see for example [12, 13]). Let δ : E → E be a triple derivation. Given a tripotent e ∈ E , the identity δ ( e ) = δ { e, e, e } = 2 { δ ( e ) , e, e } + { e, δ ( e ) , e } , combined with Peirce arithmetic show that P ( e ) δ ( e ) = 0 and Q ( e ) δ ( e ) = Q ( e ) P ( e ) δ ( e ) = − P ( e ) δ ( e ). The later implies that P ( e ) δ ( e ) is a skew sym-metric element in the JB ∗ -algebra E ( e ) (i.e. ( P ( e ) δ ( e )) ∗ e = − P ( e ) δ ( e )).This is a necessary condition to be a triple derivation. Theorem 4.1.
Let T : E → E be a bounded linear map, where E is aJB ∗ -triple in which tripotents are norm-total. The following statements areequivalent: ( a ) T is a triple derivation; ( b ) T is triple derivable at orthogonal elements and for each tripotent e in E ,the element P ( e ) T ( e ) is a skew symmetric element in the JB ∗ -algebra E ( e ) ( i.e. ( P ( e ) T ( e )) ∗ e = − P ( e ) T ( e )) .Proof. The implication ( a ) ⇒ ( b ) has been already commented.( b ) ⇒ ( a ) Let us fix a tripotent e ∈ E . The mapping T is triple derivableat orthogonal elements, and hence T satisfies the property in (17). Since P ( e ) T ( e ) ⊥ P ( e )( e ) = e (cf. Peirce arithmetic in page 6) it follows that0 = { P ( e ) T ( e ) , T ( e ) , P ( e ) T ( e ) } = { P ( e ) T ( e ) , P ( e ) T ( e ) , P ( e ) T ( e ) } , and consequently P ( e ) T ( e ) = 0 by the axioms of JB ∗ -triples.Lemma 2.4 implies that the mapping P ( e ) T | E ( e ) : E ( e ) → E ( e ) istriple derivable at orthogonal elements. Since E is a (unital) JB ∗ -algebra,Theorem 3.8 (see also Remark 3.9) assures the existence of Jordan ∗ -derivation D e : E ( e ) → E ( e ), a central element ξ e ∈ E ( e ) sa , and an anti-symmetricelement η e in the multiplier algebra of E ( e ), such that P ( e ) T ( a ) = D e ( a ) + ξ e ◦ e a + η e ◦ e a, for all a ∈ E ( e ) . The remaining hypothesis on T assures that ( P ( e ) T ( e )) ∗ e = − P ( e ) T ( e ),equivalently − ξ e − η e = − ( D e ( e ) + ξ e + η e ) = − P ( e ) T ( e ) = ( P ( e ) T ( e )) ∗ e = ( D e ( e ) + ξ e + η e ) ∗ e = ( ξ e + η e ) ∗ e = ξ e − η e , witnessing that ξ e = 0 . Therefore, P ( e ) T ( a ) = D e ( a ) + η e ◦ e a, for all a ∈ E ( e ) , which shows that P ( e ) T | E ( e ) is a triple derivation (cf. thecomments in page 8 or [32, Lemmata 1 and 2]). Proposition 2.2( c ) ⇒ ( a )proves that T is a triple derivation. (cid:3) It should be remarked that JBW ∗ -triples and compact JB ∗ -triples satisfythe hypotheses of the above theorem (cf. [34, Lemma 3.11] and [8]).. Linear maps on a JB ∗ -triple which are triple derivable atorthogonal elements This section is devoted to the study of those continuous linear maps on aJB ∗ -triple which are triple derivable at orthogonal elements, that is, Conjec-ture 1.3 in full generality. Our aim is to establish a new result on preserversby providing sufficient conditions on a continuous linear map which is triplederivable at orthogonal pairs to be a triple derivation. As we shall see inthis section, the triple setting will provide several surprises.We recall some concepts. A subset S in a JB ∗ -triple E will be called orthogonal if 0 / ∈ S and a ⊥ b for all a = b in S . The rank of E , denotedby r = r ( E ), will be the minimal cardinal number satisfying card( S ) ≤ r, for every orthogonal subset S ⊂ E . As in the case of the contractiveprojection problem for real JB ∗ -triples (see [51]), and the study of linearmaps between JB ∗ -triples preserving orthogonality (cf. [10]), the existenceof rank one JB ∗ -triples makes Conjecture 1.3 invalid in this case, becauseevery bounded linear operator on a rank one JB ∗ -triple is trivially triplederivable at orthogonal elements. Let us see an example. A Cartan factorof type 1 is a JB ∗ -triple which coincides with the space B ( H, K ), of allbounded linear operators between two complex Hilbert spaces H and K ,equipped with the triple product given in (3). In the case K = C , theJB ∗ -triple B ( H, C ) identifies with H under the triple product(23) { a, b, c } = 12 ( h a | b i c + h c | b i a ) ( a, b, c ∈ H ) , where h . | . i denotes the inner product of H . The type 1 Cartan factor B ( H, C ) is an example of a rank one JB ∗ -triple.We continue with a more detailed interpretation of Theorem 3.8.Let z be a symmetric element in the centre of a unital JB ∗ -algebra A , andlet e be a tripotent in A . It is easy to check that { a, z ◦ b, c } = ( a ◦ ( b ◦ z ) ∗ ) ◦ c + ( c ◦ ( b ◦ z ) ∗ ) ◦ a − ( a ◦ c ) ◦ ( b ◦ z ) ∗ = z ◦ (( a ◦ b ∗ ) ◦ c + ( c ◦ b ∗ ) ◦ a − ( a ◦ c ) ◦ b ∗ ) = z ◦ { a, b, c } = { z ◦ a, b, c } , for all a, b, c ∈ A . Therefore, { e, z, e } = 2( e ◦ z ) ◦ e − e ◦ z = e ◦ z,P ( e )( z ◦ e ) = Q ( e )( z ◦ e ) = { e, { e, z ◦ e, e } , e } = z ◦ { e, { e, e, e } , e } = z ◦ e, and Q ( e )( z ◦ e ) = { e, z ◦ e, e } = z ◦ { e, e, e } = z ◦ e, witnessing that z ◦ e is a symmetric element in the JB ∗ -algebra A ( e ). Lemma 5.1.
Let T : A → A be a bounded linear map on a unital JB ∗ -algebra. Suppose T is triple derivable at orthogonal pairs. Let e be atripotent in A . Then the mapping T e = P ( e ) T | A ( e ) : A ( e ) → A ( e ) is triple derivable at orthogonal pairs ( see Lemma 2.4 ) . Let δ : A → A,δ e : A ( e ) → A ( e ) , ξ ∈ Z ( A ) sa and ξ e ∈ Z ( A ( e )) sa denote the triplederivations and the elements satisfying T ( a ) = δ ( a ) + ξ ◦ a ( a ∈ A ) and e ( a ) = δ e ( a ) + ξ e ◦ a ( a ∈ A ( e )) whose existence is guaranteed by Theorem3.8. Then ξ ◦ e = ξ e . Proof.
Since, by Theorem 3.8, T ( a ) = δ ( a ) + ξ ◦ a ( a ∈ A ), and thus T e ( a ) = P ( e ) T ( a ) = P ( e ) δ ( a ) + P ( e )( ξ ◦ a ) ( a ∈ E ( a )). On the other hand, byTheorem 3.8, T e ( a ) = δ e ( a ) + ξ e ◦ a ( a ∈ A ( e )). Lemma 2.1 proves that P ( e ) δ | E ( e ) is a triple derivation on E ( e ). Therefore(24) δ e ( e ) + ξ e = δ e ( e ) + ξ e ◦ e e = T e ( e ) = P ( e ) δ ( e ) + P ( e )( ξ ◦ e )= P ( e ) δ ( e ) + P ( e ) { ξ, , e } = P ( e ) δ ( e ) + P ( e )( ξ ◦ { , , e } )= P ( e ) δ ( e ) + P ( e )( ξ ◦ e ) = P ( e ) δ ( e ) + ξ ◦ e, where in the last equality we applied the comments before this lemma as-suring that ξ ◦ e is a symmetric element in the JB ∗ -algebra A ( e ).As we have commented before, since δ e and P ( e ) δ | A ( e ) are triple deriva-tions on the unital JB ∗ -algebra A ( e ), δ e ( e ) ∗ e = − δ e ( e ) and ( P ( e ) δ ( e )) ∗ e = − P ( e ) δ ( e ) (cf. [32, Lemma 1 and its proof]). By combining this observationwith the identity in (24) we arrive at ξ ◦ e = ξ e . (cid:3) Along the rest of this section T : M → M will stand for a bounded linearoperator on a JBW ∗ -triple and we shall assume that T is triple derivableat orthogonal pairs. Lemma 2.4 guarantees that for each tripotent e ∈ M the mapping T e := P ( e ) T | M ( e ) : M ( e ) → M ( e ) is triple derivable atorthogonal pairs. Since M ( e ) is a (unital) JBW ∗ -algebra, Theorem 3.8 (seealso Remark 3.9) there exist a triple derivation δ e : M ( e ) → M ( e ) and a(unique) central symmetric element ξ e ∈ Z ( M ( e )) sa such that T e ( a ) = P ( e ) T ( a ) = δ e ( a ) + ξ e ◦ e a, for all a ∈ M ( e ) . We shall try to define a mapping S on M defined by the elements ξ e . Letus observe that, by Lemma 2.4 and Theorem 3.8, ξ e = P ( e ) T ( e ) + ( P ( e ) T ( e )) ∗ e P ( e ) + Q ( e )) T ( e ) , for every tripotent e ∈ M .An element a in M will be called algebraic if it can be written as afinite (positive) combination of mutually orthogonal tripotents in M , thatis, a = P mj =1 λ j e j with λ j > e , . . . , e m mutually orthogonal tripotentsin M . Lemma 5.2.
Let e , . . . , e m and v , . . . , v m be two families of mutuallyorthogonal tripotents in M and let λ j , µ k be positive real numbers such that P m j =1 λ j e j = P m k =1 µ k v k . Then P m j =1 λ j ξ e j = P m k =1 µ k ξ v k . Proof.
Let b = P m j =1 λ j e j = P m k =1 µ k v k . The range tripotent of b coincideswith the element u = P m j =1 e j = P m k =1 v k . Clearly, u ≥ e j , v k for all j, k .By a new application of Theorem 3.8 we deduce the existence of a tripleerivation δ u : M ( u ) → M ( u ) and a (unique) central symmetric element ξ u ∈ Z ( M ( u )) sa such that T u ( a ) = P ( u ) T ( a ) = δ u ( a ) + ξ u ◦ u a, for all a ∈ M ( u ) . Evaluating at the element b ∈ M ( u ) we get m X j =1 λ j ξ e j = m X j =1 λ j ξ u ◦ u e j = ξ u ◦ u m X j =1 λ j e j = ξ u ◦ u m X k =1 µ k v k ! = m X k =1 µ k ξ u ◦ u v k = m X k =1 µ k ξ v k , where in the first and last equalities we applied Lemma 5.1. (cid:3) The last ingredient to define a mapping S : M → M satisfying S ( e ) = ξ e for every tripotent e ∈ M is the next lemma. Lemma 5.3.
Let e and v be two tripotents in M , and let a ∈ M be anelement such that a ∈ M ( e ) ∩ M ( v ) . Then ξ v ◦ v a = ξ e ◦ e a .Proof. Theorem 3.8 implies the existence of triple derivations δ e : M ( e ) → M ( e ) and δ v : M ( v ) → M ( v ) and (unique) central symmetric elements ξ e ∈ Z ( M ( e )) sa and ξ v ∈ Z ( M ( v )) sa such that T e ( b ) = P ( e ) T ( b ) = δ e ( b ) + ξ e ◦ e b, for all b ∈ M ( e ) , and T v ( c ) = P ( v ) T ( c ) = δ v ( c ) + ξ v ◦ v b, for all c ∈ M ( v ) . Let u = r ( a ) denote the range tripotent of a in M . It follows from thehypotheses that u ∈ M ( e ) ∩ M ( v ) , and by the Peirce arithmetic M ( u ) ⊆ M ( e ) ∩ M ( v ) . Since M ( u ) is a JBW ∗ -triple, its set of tripotents is norm-total. Therefore, for each ε > { e , . . . , e m } ⊂ M ( u ) and positive real numbers λ j such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ξ e ◦ e a − m X j =1 λ j ξ e ◦ e e j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε , and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ξ v ◦ v a − m X j =1 λ j ξ v ◦ v e j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . Now, we deduce from Lemma 5.1, applied to the JBW ∗ -algebras M ( e )and M ( v ), that ξ e ◦ e e j = ξ e j and ξ v ◦ v e j = ξ e j for all j ∈ { , . . . , m } . Itthen follows from the previous inequalities that k ξ e ◦ e a − ξ v ◦ v a k < ε . thearbitrariness of ε > ξ e ◦ e a = ξ v ◦ v a , which concludes theproof. (cid:3) For each a ∈ M , let us define S ( a ) := ξ e ◦ e a , where e is any tripotent in M such that a ∈ M ( e ). Lemma 5.3 proves that the assignment a S ( a )gives a well-defined mapping S : M → M . It follows from the definitionthat S ( e ) = ξ e for each tripotent e ∈ M . In particular, S | M ( e ) : M ( e ) → M ( e ) is a bounded linear mapping. Since ξ e is an element in the centre of ( e ), the mapping S | M ( e ) is in the centroid of M ( e ) (a conclusion whichis consistent with the observations we made in page 10). The linearity ofthe mapping S on the whole M is not an obvious property, actually we shallonly culminate our study of Conjeture 1.3 by assuming the following extraproperty:Let N be a JBW ∗ -triple. We shall say that that linearity on N is de-termined by Peirce 2-subspaces if every mapping S : N → N such that P ( e ) S | N ( e ) : N ( e ) → N ( e ) is a linear operator for every tripotent e ∈ N must be a linear mapping. Every JBW ∗ -triple N admiting a unitary tripo-tent u (i.e. N ( u ) = N ) clearly satisfies this property.Let us justify this property or relate it with the example we gave at thebeginning of this section. A non-zero tripotent e in a JB ∗ -triple E is calledminimal if E ( e ) = C e . Let H be a complex Hilbert spaces regarded as atype 1 Cartan factor with the product in (23). It is known, and easy tocheck, that tripotents in H are precisely the elements in the unit sphere of H (and they are all minimal tripotents). Let S : H → H be a non-linear1-homogeneous mapping on H , i.e., S ( λx ) = λS ( x ) for all x ∈ H , λ ∈ C .Since for each tripotent e ∈ S ( H ), H ( e ) = C e , and P ( e )( x ) = h x | e i e , it iseasy to see that P ( e ) S ( λe ) = λ h S ( e ) | e i e and hence P ( e ) S | H ( e ) is linear, aproperty which is not enjoyed by S . Theorem 5.4.
Let T : M → M be a bounded linear mapping on a JBW ∗ -triple M . Suppose that linearity on M is determined by Peirce 2-subspaces.Then the following statements are equivalent: ( a ) T is triple derivable at orthogonal pairs; ( b ) There exists a triple derivation δ : M → M and an operator S in thecentroid of M such that T = δ + S .Proof. The implication ( b ) ⇒ ( a ) holds for any JB ∗ -triple M without anyextra assumption.( a ) ⇒ ( b ) Let S : M → M be the mapping defined by Lemma 5.3and subsequent comments. We observe that, for each tripotent e ∈ M , S ( M ( e )) ⊆ M ( e ) by definition. Since, the mapping S | M ( e ) : M ( e ) → M ( e ) is linear for every tripotent e ∈ M , it follows from the hypothesis on M that S is a linear mapping.We shall next prove that S is continuous. Having in mind that, for eachtripotent e ∈ M , we have ξ e = S ( e ) = ( P ( e ) + Q ( e )) T ( e ) we deduce that k ξ e k ≤ k T k . Given a in M and any tripotent u ∈ M such that a ∈ M ( u )we know that S ( a ) = a ◦ u ξ u , and hence k S ( a ) k ≤ k T k k a k , which provesthe continuity of S .We shall next show that S lies in the centroid of M . For this purpose,let us fix a tripotent e in M and an arbitrary a ∈ M . Since Peirce sub-spaces M ( e ), M ( e ) and M ( e ) are JBW ∗ -subtriples, the elements P ( e )( a ), P ( e )( a ), and P ( e )( a ) can be approximated in norm by finite positive com-binations of mutually orthogonal tripotents in the corresponding Peirceubspace. Suppose P m j k =1 λ k e k is a positive combination of mutually or-thogonal tripotents in M j ( e ). We shall deal first with the case j = 0 , ξ e k ∈ M ( e k ) and e k ∈ M j ( e ), we deduce from Peirce arithmetic that ξ e k ∈ M ( e k ) ⊆ M j ( e ). It follows from the definition of S that(25) S m j X k =1 λ k e k ! = m j X k =1 λ k ξ e k ∈ M j ( e ) , for j = 0 , ≤ k ≤ m j . The continuity of S implies that(26) S ( P j ( e )( a )) ∈ M j ( e ) , for every tripotent e ∈ M, and j = 0 , . The case j = 1 is more delicate. We assume first that e is a completetripotent in M . Let us take a tripotent v in M ( e ). We write y k = P k ( e )( ξ v )for k = 0 , ,
2. Since y = 0 and ξ v ∈ ( M ( v )) sa , we deduce that y + y = ξ v = { v, v, ξ } = { v, ξ v , v } . Now, by Peirce arithmetic with respect to e we get y = { v, v, y } = { v, y , v } = 0 , and y = { v, v, y } = { v, y , v } . It follows that ξ v = y = P ( e )( ξ v ) ∈ M ( v ). Since tripotents in the JBW ∗ -triple M ( e ) are norm total, by employing the definition of S and an identitylike in (25), a similar argument to that given above implies that S ( M ( e )) ⊆ M ( e ) and therefore(27) S ( M j ( e )) ∈ M j ( e ) , for all complete tripotent e ∈ M and j = 0 , , . We shall next show that (27) holds for all tripotent v ∈ M . Let us fixa tripotent v ∈ M . By [34, Lemma 3.12] there exists a complete tripotent e ∈ M such that v ≤ e . We can assume that v is non-complete, and hence e − v = 0. Let us make some observations. Since v ∈ M ( e ) the Peirceprojections P j ( e ) and P k ( v ) commute for all j, k ∈ { , , } . The Peirce1-subspace M ( v ) decomposes in the form(28) M ( v ) = ( M ( e ) ∩ M ( v )) ⊕ ( M ( e ) ∩ M ( v ))= ( M ( e ) ∩ M ( v ) ∩ M ( e − v )) ⊕ ( M ( e − v ) ∩ M ( e )) , the first equality being clear because e is complete and the correspondingPeirce projections commute. For the second one we observe that if x ∈ M ( e ) ∩ M ( v ), by orthogonality we have x = { e, e, x } = { v, v, x } + { e − v, e − v, x } = 12 x + { e − v, e − v, x } , which shows that x = { e − v, e − v, x } , and thus x ∈ M ( e − v ). Assumenext that x ∈ M ( e ) ∩ M ( v ). That is,12 x = { e, e, x } = { v, v, x } + { e − v, e − v, x } = 12 x + { e − v, e − v, x } , itnessing that x ∈ M ( e − v ). Finally, if x ∈ M ( e − v ) ∩ M ( e ), again byorthogonality we have12 x = { e, e, x } = { v, v, x } + { e − v, e − v, x } = { v, v, x } , which implies that x ∈ M ( v ).The summands W = M ( e ) ∩ M ( v ) ∩ M ( e − v ) and W = M ( e − v ) ∩ M ( e ) are JBW ∗ -subtriples of M and M ( v ) = W ⊕ W . By (27) S ( M ( e )) ⊆ M ( e ) because e is complete; and by (25) S ( M ( e − v )) ⊆ M ( e − v ). Therefore(29) S ( W ) ⊆ M ( e − v ) ∩ M ( e ) = W . Take now a tripotent w in the JBW ∗ -triple W = M ( e ) ∩ M ( v ) ∩ M ( e − v ). In this case, by Lemma 5.1, applied to the JBW ∗ -algebra M ( e ) and thetripotents v, e − v and e , we get ξ v = ξ e ◦ e v , ξ e − v = ξ e ◦ e ( e − v ), ξ e = ξ e ◦ e e = ξ e ◦ e v + ξ e ◦ e ( e − v ) = ξ v + ξ e − v , with v, ξ v ⊥ ξ e − v , e − v, because v and e − v are two orthogonal projectionsin M ( e ) and ξ e is a symmetric central element in the latter JBW ∗ -algebra.Lemma 5.1 also implies that ξ w = ξ e ◦ e w = { ξ e , e, w } ∈ M ( e ) , and by the above properties { ξ e , e, w } = { ξ e , v, w } + { ξ e , e − v, w } = { ξ v , v, w } + { ξ e − v , e − v, w } . By Peirce arithmetic { ξ v , v, w } ∈ M − ( v ) ∩ M − ( e − v ) = M ( v ) ∩ M ( e − v ) , and { ξ e − v , e − v, w } ∈ M − ( e − v ) ∩ M − ( v ) = M ( v ) ∩ M ( e − v ) . Therefore, ξ w ∈ M ( e ) ∩ M ( v ) ∩ M ( e − v ) = W , for every tripotent w ∈ W . Having in mind that the set of tripotents in theJBW ∗ -triple W is norm-total, a linearization like the one in (25) and thenorm continuity and linearity of S assert that S ( W ) ⊆ W . Combining thelatter conclusion with (29), (28) and the linearity of S we get S ( M ( v )) ⊆ W ⊕ W = M ( v ). It then follows that(30) S ( M j ( v )) ∈ M j ( v ) , for every tripotent v ∈ M, and j = 0 , , , (cf. (26) for j = 0 , v ∈ M , SL ( v, v )( a ) = S ( P ( v )( a ) + 12 P ( v )( a ))= S ( P ( v )( a )) + 12 S ( P ( v )( a )) = L ( v, v ) S ( a ) . ince the set of tripotents is norm-total in M and S is continuous, theidentity SL ( a, a ) = L ( a, a ) S holds for every a ∈ M , witnessing that S is anelement in the centroid of M (cf. [20, pages 330, 331]).Finally, the linear mapping δ = T − S : M → M is linear and continuous.Furthermore, δ is triple derivable at orthogonal pairs because T and S are(cf. Lemma 3.1( b )). It follows from the definition of S (see also Theorem3.8) that for each tripotent e ∈ M , we have12 ( P ( e ) + Q ( e )) δ ( e ) = 12 ( P ( e ) + Q ( e ))( T − S )( e ) = ξ e − ξ e = 0 . Theorem 4.1 assures that δ is a triple derivation. Clearly T = δ + S . (cid:3) A subspace I of a JB ∗ ’triple E is called a triple ideal if { E, E, I } + { I, E, E } ⊆ I . A JBW ∗ -triple which cannot be decomposed as a directorthogonal sum of two non-trivial ideals is called a JBW ∗ -triple factor (cf.[34, (4 . ∗ -factor reduces to the complex multiplesof the identity mapping (see [20, Corollary 2.11]). In particular the centroidof each Cartan factor is one-dimensional.The simplicity of the centroid in the case of Cartan factors and atomicJBW ∗ -triples allows us to relax some of the hypothesis in the previous The-orem 5.4. An atomic JBW ∗ -triple is a JBW ∗ -triple which coincides withthe ℓ ∞ -sum of a family of Cartan factors. We have already presented theCartan factors of type 1 at the beginning of this section, the definition ofthe remaining Cartan factors reads as follows: Suppose j is a conjugation(i.e. a conjugate linear isometry of period 2) on a complex Hilbert space H and define a linear involution on B ( H ) given by x x t := jx ∗ j . TheJB ∗ -subtriple of B ( H ) of all t -skew-symmetric (respectively, t -symmetric)operators in B ( H ) is called a Cartan factor of type 2 (respectively, of type3). A Cartan factor of type 4, is a a complex Hilbert space provided with aconjugation x x, where triple product and the norm are given by { x, y, z } = ( x | y ) z + ( z | y ) x − ( x | z ) y, and k x k = ( x | x ) + p ( x | x ) − | ( x | x ) | , respectively. The Cartan factors oftypes 5 and 6 (also called exceptional
Cartan factors) are spaces of matricesover the eight dimensional complex algebra of Cayley numbers; the type 6consists of all 3 by 3 self-adjoint matrices and has a natural Jordan algebrastructure, and the type 5 is the subtriple consisting of all 1 by 2 matrices(see [43] for a detailed presentation of Cartan factors).We recall that two tripotents u, v in a JB ∗ -triple E are called collinear (written u ⊤ v ) if u ∈ E ( v ) and v ∈ E ( u ). We say that u governs v ( u ⊢ v in short) whenever v ∈ U ( u ) and u ∈ U ( v ). According to [18, 48], anordered quadruple ( u , u , u , u ) of tripotents in a JB ∗ -triple E is calleda quadrangle if u ⊥ u , u ⊥ u , u ⊤ u ⊤ u ⊤ u ⊤ u and u = 2 { u , u , u } (the latter equality also holds if the indices are permutated cyclically, e.g. u = 2 { u , u , u } ). An ordered triplet ( v, u, ˜ v ) of tripotents in E , is calleda trangle if v ⊥ ˜ v , u ⊢ v , u ⊢ ˜ v and v = Q ( u )˜ v .e say that a tripotent u ∈ E has rank two if it can be written as thesum of two orthogonal minimal tripotents. Theorem 5.5.
Let T : M → M be a bounded linear mapping on an atomicJBW ∗ -triple M . Suppose that M = L ∞ j C j , where each C j is a Cartanfactor with rank at least . Then the following statements are equivalent: ( a ) T is triple derivable at orthogonal pairs; ( b ) There exists a triple derivation δ : M → M and an operator S in thecentroid of M such that T = δ + S .Proof. We have already commented that ( b ) ⇒ ( a ) holds for any JB ∗ -triple M without any extra assumption.( a ) ⇒ ( b ) We consider again the mapping S : M → M defined by Lemma5.3 and subsequent comments. We observe that S ( M ( e )) ⊆ M ( e ) bydefinition. We have seen above that for each tripotent e ∈ M the mapping S | M ( e ) : M ( e ) → M ( e ) is a bounded linear operator in the centroid of M ( e ) (see the comments after Lemma 5.3).By [32, Proposition 3] Cartan factors of type 1 with dim( H ) =dim( K ),Cartan factors of type 2 with dim( H ) even or infinite and all Cartan factorsof type 3 admit a unitary element, and hence they are unital JBW ∗ -algebras.Clearly, the Cartan factor of type 6 also admits a unitary element and enjoysthe same structure. Suppose we can find a unitary element u j ∈ C j . In thiscase, P ( u j ) T | ( C j ) ( u j )= C j : ( C j ) ( u j ) = C j → C j is linear, continuous andtriple derivable at orthogonal pairs with k P ( u j ) T | C j k ≤ k T k . It followsfrom Theorem 3.8 that there exists a triple derivation δ j : C j → C j anda symmetric bounded linear mapping S j in the centroid of C j such that P ( u j ) T | C j = δ j + S j . Since C j is a factor S j = α j Id C j for a real number α j with | α j | ≤ k T k , and(31) P ( u j ) T ( a ) = δ j ( a ) + α j a, for all a ∈ C j . We suppose now that C j does not contain any unitary element. Considertwo minimal tripotents e and v in C j . By [25, Lemma 3.10] one of thefollowing statements holds:( a ) There exist minimal tripotents v , v , v in C j , and complex numbers α , β , γ , δ such that ( e, v , v , v ) is a quadrangle, | α | + | β | + | γ | + | δ | = 1, αδ = βγ , and v = αe + βv + γv + δv ;( b ) There exist a minimal tripotent ˜ e ∈ C j , a rank two tripotent u ∈ C j ,and complex numbers α, β, δ such that ( e, u, ˜ e ) is a trangle, | α | +2 | β | + | δ | = 1, αδ = β , and v = αe + βu + δ ˜ e .This result, in particular, implies that for each rank two tripotent u in a Car-tan factor C of rank at least two the Peirce subspace C ( u ) is again a factor.It follows from the above that we can find a rank two tripotent u ∈ C j suchthat e, v ∈ ( C j ) ( u ). By Lemma 2.4 the mapping P ( u ) T | M ( u )=( C j ) ( u ) :( C j ) ( u ) → ( C j ) ( u ) is a bounded linear mapping which is triple derivablet orthogonal pairs. Theorem 3.8 proves the existence of a triple deriva-tion δ u on ( C j ) ( u ) and a symmetric element S u in the centroid of ( C j ) ( u )such that P ( u ) T | M ( u ) = δ u + S u . Since ( C j ) ( u ) is factor, there existsa real number α u such that S u ( x ) = α u x for all x ∈ ( C j ) ( u ), that is, ξ u = α u u . Consequently, by Lemma 5.1, ξ e = ξ u ◦ u e = S u ( e ) = α u e and ξ v = ξ u ◦ u v = S u ( v ) = α u v . We have therefore concluded that for anycouple of minimal tripotents e, v ∈ C j ∃ α e,v ∈ R such that ξ e = S u ( e ) = α e,v e, and ξ v = S u ( v ) = α e,v v, and thus(32) ∃ α j ∈ R such that ξ e = α j e, for all minimal tripotent e ∈ C j . Let a be any element in C j and let u be a tripotent such that a ∈ ( C j ) ( u ).Actually, ( C j ) ( u ) is another Cartan factor (cf. [48, Lemma 3.9, StructureTheorem 3.14 and Classification Theorem 3.20]). By Lemma 2.4 and The-orem 3.8, there exists a triple derivation δ u on ( C j ) ( u ) and a symmetricelement in the centroid of ( C j ) ( u ), that is, a real number α u such that P ( u ) T ( a ) = δ u ( a ) + α u a, for all a ∈ ( C j ) ( u ) . When the previous identity is evaluated at a minimal tripotent e ∈ ( C j ) ( u )we deduce that α u = α j , equivalently, ξ u = α j u (cf. (32)), and thus S ( a ) = α j a for all a ∈ C j . We observe that | α j | ≤ k T k for all j .Let P j denote the projection of M onto C j . Since P j is a contractiveprojection, Lemma 2.4 assures that P j T | C j : C j → C j is triple derivable atorthogonal pairs. The operator δ j = P j T | C j − S | C j also is triple derivableat orthogonal pairs, and by the arguments above, ( P ( u ) + Q ( u )) δ j ( u ) = 0for all tripotent u ∈ C j . Theorem 4.1 implies that δ j is a triple derivationon C j and(33) P j T ( a ) = δ j ( a ) + α j a, for all a ∈ C j . For each two different indices j = j , let us take two tripotents e j ∈ C j and e j ∈ C j . Since e j ⊥ e j it follows from the hypotheses that0 = { T ( e j ) , e j , e j } + { e j , T ( e j ) , e j } . Let us consider the second summand. Since e j ∈ M ( e j ), e j ∈ M ( e j ), byPeirce arithmetic, { e j , T ( e j ) , e j } = { e j , P ( e j ) T ( e j ) , e j } = 0, because P ( e j )( M ) ⊆ C j ⊂ M ( e j ). This implies that 0 = { T ( e j ) , e j , e j } , whichproves that T ( e j ) ⊥ e j for every tripotent e j ∈ C j . The arbitrariness of j = j and the fact that tripotents are norm-total in each C j assure that T ( C j ) ⊆ C j for every j .Finally, since for each j , T ( C j ) ⊆ C j and T | C j : C j → C j can be writtenin the form T | C j = δ j + α j Id C j for a triple derivation δ j on C j and a realnumber α j with | α j | ≤ k T k (cf. (31) and (33)), we deduce that T (( a j ) j ) = δ (( a j ) j ) + S (( a j ) j ) , here δ : M → M is the triple derivation given by δ (( a j ) j ) = ( δ j ( a j )) j and S is the bounded linear mapping given by S (( a j ) j ) = ( α j a j ) j , which is anelement in the centroid of M . (cid:3) Acknowledgements
Second author partially supported by the SpanishMinistry of Science, Innovation and Universities (MICINN) and EuropeanRegional Development Fund project no. PGC2018-093332-B-I00, ProgramaOperativo FEDER 2014-2020 and Consejer´ıa de Econom´ıa y Conocimientode la Junta de Andaluc´ıa grant number A-FQM-242-UGR18, and Junta deAndaluc´ıa grant FQM375.
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