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(1)DERIVATIONS ON BANACH–JORDAN PAIRS by A. FERNÁNDEZ LÓPEZ, H. MARHNINE (Departamento de Algebra, Geometrı́a y Topologı́a, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain) and C. ZARHOUTI (Département de Mathématiques, Faculté des Sciences, BP 2121, Tétouan, Morocco) (Dedicated to the memory of Eulalia Garcı́a Rus) [Received 4 May 2000. Revised 8 December 2000] 0. Introduction A basic continuity problem consists in determining algebraic conditions on a Banach algebra A which ensure that derivations on A are continuous. In 1968, Johnson established the continuity of derivations on semisimple commutative Banach algebras [15]. This result was extended by Johnson and Sinclair in [16] to arbitrary semisimple Banach algebras by making use of irreducible representations. The investigation into the problem of automatic continuity was developed later in different directions, specially for non-associative algebras, providing among other important results that derivations on A are continuous whenever A is a classical Banach–Lie algebra of operators on a Hilbert space [11], a semisimple Banach alternative algebra [25], a Mal’cev H∗ -algebra with zero annihilator [27] or, more generally, an H∗ -algebra with zero annihilator [24], and, as noticed by Youngson [28], a JB∗ -algebra. Of particular relevance to our work is the brilliant result, recently proved by Villena in [26], that derivations on semisimple Banach–Jordan algebras are continuous. In this paper we deal with derivations acting on Banach–Jordan pairs. On the one hand, the problem of automatic continuity arises in a natural way in order to generalize the main result in [26] (since Jordan pairs are a natural extension of Jordan algebras), and on the other hand, Banach–Jordan pairs are interesting in themselves by their connection with the geometry of bounded symmetric domains [19]. 1. Preliminaries Unless otherwise specified, Jordan pairs and Jordan algebras will be understood over an arbitrary ring of scalars . Nevertheless, we shall be mainly interested in the linear case ( 12 ∈ ), and very specially in the case when is the real or complex field. The definitions relating to Jordan pairs not explicitly stated in the paper can be found in [18]. We record in this section some notation and results. Given a Jordan pair V = (V + , V − ), we write Q : V ± → Hom (V ∓ , V ± ) to denote the quadratic maps of V . The multiplication Q x y is quadratic in x and linear in y and has linearizations {x, y, z} = Q(x, z)y = V (x, y)z = Q x+z y − Q x y − Q z y. Note that {x, y, x} = 2Q x y, so we only need to consider the triple product in the linear case. A typical example of a Jordan pair is given by taking V + = Hom (X, Y ), V − = Hom (Y, X ), Quart. J. Math. 52 (2001), 269–283. c Oxford University Press 2001.

(2) 270. A . FERN ÁNDEZ. et al.. linear maps between right vector spaces X and Y over a division associative -algebra , with Q a b = aba. Any associative, alternative or Jordan algebra A gives rise to a Jordan pair (A, A) with quadratic multiplications x yx or Ux y, where U denotes the usual U -operator of the Jordan algebra. In the opposite direction, given a Jordan pair V = (V + , V − ) and an element y ∈ V −σ , we can define a Jordan algebra on V σ by (y) Ua = Q a Q y and a (2,y) = Q a y. This Jordan algebra, denoted by (V σ )(y) , is called the y-homotope of V σ . If V is a linear Jordan pair, we just need to define the linear product: x · y z = 12 {x, y, z}. A Jordan pair V is non-degenerate if Q a = 0 implies that a = 0. Non-degenerate Jordan pairs are semiprime: Q I ± I ∓ = 0 implies that I = 0, for I an ideal of V ; but the converse is not true in general, even for Jordan algebras. A Jordan pair V is prime if Q B ± C ∓ = 0 implies that B = 0 or C = 0 for B and C ideals of V , and it is called strongly prime if it is prime and non-degenerate. Note that if I = (I + , I − ) is an ideal of a non-degenerate Jordan pair V and x ∈ I − is such that Q x I + = 0, then we have by [18, JP3] Q Q x V + V + = Q x Q V + Q x V + ⊆ Q x I + = 0, which implies that (1.1) If V is non-degenerate, then so is every ideal I = (I + , I − ). Moreover, an ideal I = (I + , I − ) is equal to zero if and only if I + = 0 or I − = 0. Local algebras of a Jordan pair. Let V = (V + , V − ) be a Jordan pair and y ∈ V −σ . By [18, 4.19], the set Ker(y) = {x ∈ V σ : Q y x = 0, Q y Q x y = 0} is an ideal of (V σ )(y) and the quotient (V σ )(y) / Ker(y) is a Jordan algebra called the local algebra of V at y, which we denote by Vy . As pointed out in [1, 1.2.4], the condition Q y Q x y = 0 is superfluous if V is linear or non-degenerate. Jacobson radical and primitive ideals. Following [18], the Jacobson radical of a Jordan pair V is defined as the ideal Rad(V ) = (Rad(V + ), Rad(V − )), where Rad(V σ ) is the set of properly quasi-invertible elements of V σ , that is, those elements of V σ which are quasi-invertible in every homotope (V σ )(y) . A Jordan pair V is called semiprimitive if Rad(V ) = 0. Following [13], an inner ideal K of a Jordan algebra J is called x-modular (x is called a modulus for K ) if the following conditions, which make sense even if J is not unital, hold: (i) U1−x J ⊆ K , (ii) {(1 − x), J, K } ⊆ K , (iii) (1 − x) ◦ K ⊆ K , and (iv) x − x 2 ∈ K , where x ◦ y := (x + y)2 − x 2 − y 2 is the linearization of the squaring operator. If K merely satisfies (i) and (ii), then it is said to be weakly modular. An x-modular inner ideal is said to be x-maximal if it is maximal among all proper x-modular inner ideals of J . A maximal modular inner ideal is an x-maximal inner ideal for some modulus x..

(3) DERIVATIONS ON BANACH – JORDAN PAIRS. 271. A Jordan algebra J is primitive if it has a maximal modular inner ideal with zero core, where the core, C(K ), of an inner ideal K is the largest ideal of J contained in K . Such a maximal modular inner ideal is called a primitizer for J [13, p. 164]. As pointed out in [4, p. 534], since any proper x-modular inner ideal is contained in an x-maximal one [13, 3.2], to obtain a primitizer from a proper x-modular inner ideal K , it will be sufficient that K be comaximal to all non-zero ideals: K + I = J for any non-zero ideal I of J . Such an inner ideal K is then called a proto-primitizer for J . According to [2], a Jordan pair V = (V + , V − ) is said to be primitive at b ∈ V −σ if there exists a proper inner ideal K ⊂ V σ such that: (v) K is a c-modular inner ideal of the homotope (V σ )(b) for some c ∈ V σ , and (vi) K complements the (σ )-parts of non-zero ideals: I σ + K = V σ for any non-zero ideal I = (I + , I − ). By [1, 6.1], local algebras of primitive pairs are primitive. Anquela and Cortés proved in [2, 3.9], that the converse is true if the Jordan pair is strongly prime. Moreover, by [3, 2.2], primitivity is ubiquitous. (1.2) V is primitive at b ∈ V −σ if and only if Vb is a primitive Jordan algebra and V is strongly prime. (1.3) The primitivity of V at a non-zero element implies the primitivity at any other non-zero element. The above definition of primitivity for Jordan pairs is equivalent to the following, proposed by Hessenberger in [12]. An inner ideal K ⊆ V + of a Jordan pair V is called weakly (x, y)-modular, for some (x, y) ∈ V + × V − , if K is weakly x-modular in the homotope (V σ )(y) . An inner ideal L ⊆ V − is weakly (x, y)-modular if it is weakly (y, x)-modular in V op . By [3, 4.11], a Jordan pair V is primitive if and only if V σ (for σ = ±) is primitive in the following sense (see [12, A.2.1]): there exists a maximal weak modular inner ideal K ⊂ V σ with zero core. As in the case of associative algebras, an ideal P = (P + , P − ) of a Jordan pair V is called a primitive ideal if V /P is a primitive Jordan pair. By [12, A.3.5], an ideal P = (P + , P − ) is primitive if and only if it is the core, P σ = C(K σ ), of a pair (K + , K − ) of maximal weak modular inner ideals K σ ⊂ V σ . Moreover, it follows from [12, A.4.8], (or [29]). (1.4) The Jacobson radical of a Jordan pair V is the intersection of all primitive ideals of V . Further results related to primitivity in Jordan pairs and Jordan algebras can be found in [1–4, 12, 13, 29]. Socle and capacity. For a non-degenerate Jordan pair V , Soc(V ) := (Soc(V + ), Soc(V − )), where Soc(V ± ) denotes the sum of all minimal inner ideals of V ± , is an ideal of V called the socle of V . By [20, Theorem 1], for x ∈ V + the following conditions are equivalent: (i) x is in the socle, (ii) x is von Neumann regular and if x = e+ for some idempotent e then the Peirce space V2 (e) has finite capacity (the lengths of its chains of principal inner ideals are bounded), and (iii) the inner ideal x generated by x has dcc on principal inner ideals. Hence it is easy to obtain the following local characterization of the socle [21, 0.7(b)]..

(4) 272. A . FERN ÁNDEZ. et al.. (1.5) Let V be a non-degenerate Jordan pair and x ∈ V σ . Then x ∈ Soc(V σ ) if and only if Vx has finite capacity. Moreover, u ∈ V σ generates a minimal inner ideal if and only if Vu is a division Jordan algebra. Further results on socle theory for Jordan pairs can be found in [20]. 2. Measuring the lack of continuity of certain operators in Banach–Jordan algebras By a normed Jordan algebra we mean a (real or complex) Jordan algebra J such that, as a vector space, it is endowed with a norm . making continuous the product of J . If the norm is complete, then J is said to be a Banach–Jordan algebra. Of course, J can be given an equivalent norm . so that x · y x y for x, y ∈ J . By using techniques developed in [26], we prove in this section that if D is a linear operator on a complex Banach–Jordan algebra J such that its commutator with every operator of the multiplication algebra of J is continuous, then its separating subspace, modulo any primitive ideal, has finite capacity. This result will be crucial in the next section to prove the continuity of derivations in semiprimitive Banach–Jordan pairs. We begin by stating the structure theorem for complex primitive Banach–Jordan algebras due to Cabrera, Moreno and Rodrı́guez in a slightly different way from how it is done in [8]. T HEOREM 2.1 A complex Banach–Jordan algebra J is primitive if and only if one of the following holds. (i) It is a simple Jordan algebra having finite capacity. (ii) There exist an infinite-dimensional complex Banach space X and an associative subalgebra A of B L(X ) acting irreducibly on X such that J can be seen as a Jordan subalgebra of B L(X ) containing A as an ideal, and the inclusion J → B L(X ) is continuous. (iii) There exist an infinite-dimensional complex Banach space X and an associative subalgebra A of B L(X ) acting irreducibly on X such that J can be seen as a Jordan subalgebra of B L(X ), the inclusion J → B L(X ) is continuous, the identity mapping of J extends to a linear algebra involution ∗ on the subalgebra B of B L(X ) generated by J , A is a ∗-invariant subset of B, H (A, ∗) is an ideal of J , and A is generated as an algebra by H (A, ∗). R EMARK . Actually, a simple complex Banach–Jordan algebra having finite capacity is either finitedimensional or it is defined by a continuous non-degenerate quadratic form on a complex Banach space; but the relevant information we shall use later is that such algebras have finite spectrum [6]. Let {an } be a sequence of linear operators from a vector space X into itself and let {xn } be a sequence in X . The couple ({an }, {xn }) is said to be a gliding hump sequence pair if the following conditions hold: an . . . a1 xn = 0,. an+1 an . . . a1 xn = 0,. and a1 . . . an = 0.. Basing his arguments on Theorem 2.1 and using Zelmanovian methods, Villena proved in [26, Corollaries 1 and 2] the following lemma. L EMMA 2.2 Let J be a complex primitive Banach–Jordan algebra. (i) If J is of type (ii) of Theorem 2.1, then there exists a gliding hump sequence pair in J ..

(5) DERIVATIONS ON BANACH – JORDAN PAIRS. 273. (ii) If J is of type (iii) of Theorem 2.1, then there exist sequences {an } either in A or in J and {xn } in X such that the couple ({an }, {xn }) is a gliding hump sequence pair. Recall that we can measure the continuity of a linear operator acting between two Banach spaces by considering its so-called separating subspace. In fact, the definition makes sense even for linear operators between normed spaces; although it is in the case of Banach spaces where it proves more interesting. Let T : X → Y be a linear operator between normed spaces. The separating subspace of T is defined by S(T ) = {y ∈ Y : there are xn → 0 in X with T xn → y in Y }. It is easy to verify that the separating subspace of T is in fact a closed subspace. Moreover, by the closed graph theorem, if both X and Y are Banach spaces, then T is continuous if and only if S(T ) = 0. Given a Banach space X , we denote by cl(E) the closure of a subset E of X . If T, S are linear operators on X , we write [S, T ] = ST − T S to denote their commutator. The proof of the following technical result can be found in [14, Lemma 1]. L EMMA 2.3 Let X be a Banach space and let {Tn } be a sequence of continuous linear operators on X . If D is a linear operator on X such that for every n ∈ N the commutator [D, Tn ] is continuous, then there is an integer N such that, for every n N , cl(T1 . . . Tn S(D)) = cl(T1 . . . TN S(D)). Given a (linear) Jordan algebra J , the multiplication algebra of J , M(J ), is the subalgebra of L(J ) (the algebra of all linear operators on J ) generated by all the right multiplications: Ra x = a · x(x ∈ J ) for all a ∈ J . Clearly, right multiplications on normed Jordan algebras are continuous. We are ready to prove the main result of this section. P ROPOSITION 2.4 Let J be a complex Banach–Jordan algebra, and let I be a primitive ideal of J . If D is a linear operator of J such that [D, T ] is continuous for every T ∈ M(J ), then the primitive Jordan algebra (S(D) + I )/I has finite capacity. Proof. First we remark that the separating subspace S(D) is an ideal under the condition that the commutator of D with every T ∈ M(J ) is continuous: Given a ∈ J , taking T = Ra , the right multiplication by a, we have that f (x) = [D, Ra ](x) is continuous. Now the formula D(ax) = a D(x) + f (x) is sufficient to verify that S(D) is an ideal. Since primitive ideals of Banach–Jordan algebras are closed [9, Lemma 6.5], J/I is a complex primitive Banach–Jordan algebra. Setting π to denote the canonical projection of J onto J/I , π(S(D)) = (S(D) + I )/I is an ideal of J/I , and its closure, cl(π(S(D))), is a complex primitive Banach–Jordan algebra, which we shall denote by J0 . Clearly, it is enough to show that J0 has finite capacity. Suppose on the contrary that J0 has no finite capacity. Then, by Theorem 2.1, J0 is a Jordan algebra of continuous linear operators on a complex Banach space X of one of the types (ii), (iii) described there. We shall show that in either of these two cases there exist sequences {Sn } in M(J0 ) and {xn } in X such that S1 . . . Sn+1 (J0 )xn = 0. but. S1 . . . Sn (J0 )xn = 0. (1). which will yield a contradiction. Suppose that such sequences exist. For each n ∈ N we can take Tn ∈ M(J ) such that π Tn = Sn π on cl(S(D)). Hence, by using the continuity of π and (1), we.

(6) 274. A . FERN ÁNDEZ. et al.. have π(cl(T1 . . . Tn+1 (S(D))))xn ⊆ cl(π(T1 . . . Tn+1 (S(D))))xn = cl(S1 . . . Sn+1 (π(S(D))))xn = 0. On the other hand, since J0 = cl(π(S(D)), it follows from (1) that π(cl(T1 . . . Tn (S(D))))xn ⊇ π(T1 . . . Tn (S(D))) = S1 . . . Sn (π(S(D)))xn = 0. The announced contradiction arises by applying Lemma 2.3 to the sequence {Tn } in M(J ) to get a positive integer N such that, for all n N , cl(T1 . . . TN (S(D))) = cl(T1 . . . Tn (S(D))). Therefore, it remains to construct, in each one of the two possibilities for J0 : types (ii) and (iii), two sequences {Sn } in M(J0 ) and {xn } in X satisfying (1). The key tools to build such sequences are the gliding hump sequence pairs. Assume first that J0 is of type (ii), that is, there exist an infinite-dimensional complex Banach space X and an associative subalgebra A of B L(X ) acting irreducibly on X such that J0 can be seen as a Jordan subalgebra of B L(X ) containing A as an ideal, and the inclusion J0 → B L(X ) is continuous. Apply Lemma 2.2(i) to obtain sequences {an } in J and {xn } in X such that {π(an )} lies in J0 and the couple ({π(an )}, {xn }) is a gliding hump sequence pair. Taking Sn = Uπ(an ) for every n, we have S1 . . . Sn+1 (J0 )xn = Uπ(a1 ) . . . Uπ(an+1 ) J0 xn = π(a1 ) . . . π(an+1 )J0 π(an+1 ) . . . π(a1 )xn = 0, and S1 . . . Sn (J0 )xn = Uπ(a1 ) . . . Uπ(an ) J0 xn = 0. For, if π(a1 ) . . . π(an )J0 π(an ) . . . π(a1 )xn = 0 then π(a1 ) . . . π(an )Aπ(an ) . . . π(a1 )xn = 0; but yn = π(an ) . . . π(a1 )xn = 0, and hence, π(a1 ) . . . π(an ) = 0 since A acts irreducibly on X , which is a contradiction. We have thus proved that the pair of sequences {Sn } in M(J0 ) and {xn } in X satisfy the conditions (1). Thus, we may suppose that J0 is of type (iii), that is, there exist an infinite-dimensional complex Banach space X and an associative subalgebra A of B L(X ) acting irreducibly on X such that J0 can be seen as a Jordan subalgebra of B L(X ), the inclusion J0 → B L(X ) is continuous, the identity mapping of J0 extends to a linear algebra involution ∗ on the subalgebra B of B L(X ) generated by J0 , A is a ∗-invariant subset of B, H (A, ∗) is an ideal of J0 , and A is generated by H (A, ∗). Lemma 2.2(ii) is now applied to get sequences {an } in J and {xn } in X such that {π(an )} either lies in J0 or A, and the pair ({π(an )}, {xn }) is a gliding hump sequence. Assume first that {π(an )} lies in J0 . As in the previous case, take Sn = Uπ(an ) for every n and verify that S1 . . . Sn+1 (J0 )xn = 0. We now see that S1 . . . Sn (J0 )xn = 0. Otherwise, π(a1 ) . . . π(an )J0 π(an ) . . . π(a1 )xn = 0, and by [26, Lemma 1 and Remark 5], for every s ∈ J0 and a ∈ A, a ∗ sπ(an+1 ) + π(an+1 )sa ∈ J0 . Hence π(a1 ) . . . π(an )(a ∗ sπ(an+1 ) + π(an+1 )sa)π(an ) . . . π(a1 )xn = 0,.

(7) DERIVATIONS ON BANACH – JORDAN PAIRS. 275. which implies, since π(an+1 )π(an ) . . . π(a1 )xn = 0, that π(a1 ) . . . π(an )π(an+1 )J0 Aπ(an ) . . . π(a1 )xn = 0. Since π(an ) . . . π(a1 )xn = 0, the irreducible action of A on X yields π(a1 ) . . . π(an )π(an+1 )J0 = 0.. (2). By [26, Lemma 1 and Remark 5] again, for every s ∈ J0 and a ∈ A, as + sa ∗ lies in J0 . Thus, by (2), π(a1 ) . . . π(an )π(an+1 )as = π(a1 ) . . . π(an )π(an+1 )(as + sa ∗ ) = 0, and hence, since J0 = 0 and A acts irreducibly on X , π(a1 ) . . . π(an )π(an+1 ) = 0, which contradicts the conditions on the sequence {π(an )}. Suppose finally that J0 is of type (iii) with {π(an )} in A. By [26, Lemma 1 and Remark 5] for each n, the formula Sn (π(b)) = π(an )∗ π(b)π(an ), π(b) ∈ J0 defines a linear operator on J0 which lies in M(J0 ). We claim that the sequences {Sn } in M(J0 ) and {xn } in X satisfy the conditions (1). Indeed S1 . . . Sn+1 (J0 )xn = π(a1 )∗ . . . π(an+1 )∗ J0 π(an+1 ) . . . π(a1 )xn = 0. Thus, it remains to verify that S1 . . . Sn (J0 )xn = 0. Suppose otherwise that π(a1 )∗ . . . π(an )∗ J0 π(an ) . . . π(a1 )xn = 0.. (3). By [26, Lemma 1 and Remark 5] for s ∈ J0 and a in A, we have (a ∗ π(an+1 ))∗ s + s(a ∗ π(an+1 )) ∈ J0 . By substitution in (3), this gives π(a1 )∗ . . . π(an )∗ π(an+1 )∗ asπ(an ) . . . π(a1 )xn = 0 since π(an+1 ) . . . π(a1 )xn = 0. Thus, π(a1 )∗ . . . π(an+1 )∗ A J0 π(an ) . . . π(a1 )xn = 0.. (4). But π(a1 )∗ . . . π(an+1 )∗ = (π(an+1 ) . . . π(a1 ))∗ = 0, and A acts irreducibly on X . Hence it follows from (4) that J0 π(an ) . . . π(a1 )xn = 0. Again, by [26, Lemma 1 and Remark 5] for s ∈ J0 and a in A, a ∗ s + sa ∈ J0 . Hence saπ(an ) . . . π(a1 )xn = (a ∗ s + sa)π(an ) . . . π(a1 )xn = 0 and then J0 Aπ(an ) . . . π(a1 )xn = 0, which is a contradiction since π(an ) . . . π(a1 )xn = 0, A acts irreducibly on X , and J0 = 0..

(8) 276. A . FERN ÁNDEZ. et al.. 3. Derivations on Banach–Jordan pairs By a normed Jordan pair we mean a (real or complex) Jordan pair V = (V + , V − ) where the vector spaces V + and V − are endowed with norms making continuous the triple products of V . If these norms are complete, then we shall say that V is a Banach–Jordan pair. A typical example of Banach–Jordan pair is given by V = (BL(X, Y ), BL(Y, X )), bounded linear operators between Banach spaces X and Y with Q T S = T ST the usual mapping composition. It is clear that if V is a normed (Banach) Jordan pair then, for each y ∈ V −σ the homotope V σ (y) , with the same norm as V σ , is a normed (Banach) Jordan algebra. Moreover by the continuity of the operator Q y , Ker(y) = Ker Q y is a closed ideal of V σ (y) . Thus, for the quotient norm we have the following. (3.1) Local algebras of normed (Banach) Jordan pairs are normed (Banach) Jordan algebras. Let V = (V + , V − ) be a Jordan pair over an arbitrary ring of scalars . Following [18], a pair D = (D+ , D− ) ∈ End(V + ) × End(V − ) is called a derivation on V if the condition Dσ (Q x y) = {Dσ (x), y, x} + Q x D−σ (y) holds for every x ∈ V σ , y ∈ V −σ , and σ = ±. If 12 ∈ , a simple computation shows that this is the case if and only if Dσ ({x, y, z}) = {Dσ (x), y, z} + {x, D−σ (y), z} + {x, y, Dσ (z)}. As pointed out in [18, 3.11], if W is a Jordan pair then, for every (x, y) ∈ W , the pair (3.2) D = (V (x, y), −V (y, x)) is a derivation on W called the inner derivation of W defined by (x, y). Examples of derivations in Banach–Jordan pairs coming from Banach associative pairs can be found in [23]. A key fact in continuity of derivations on Banach (associative and Jordan) algebras is that the separating subspace of every derivation on a Banach algebra is a closed ideal. As could be expected, a similar result holds for derivations on Banach–Jordan pairs. L EMMA 3.3 Let V be a normed Jordan pair and let D = (D+ , D− ) be a derivation on V . Then the pair of subspaces S(D) = (S(D+ ), S(D− )) is a closed ideal of V . Proof. Since the characteristic of the base field is zero, it suffices to show that S(D) is an outer ideal of V , that is, invariant under all Q v and V (u, v) operators. Take y ∈ S(D− ) and v ∈ V + , and let xn → 0 be in V − such that D− (xn ) → y in V − . By the continuity of Q v , Q v xn → 0, and, by the continuity of the triple products, lim D+ (Q v xn ) = lim{D+ (v), xn , v} + lim Q v D− (xn ) = {D+ (v), lim xn , v} + Q v lim D− (xn ) = Q x y, which proves that Q V + S(D− ) ⊆ S(D+ ). Given s ∈ S(D+ ), let tn → 0 in V + be such that D+ (tn ) → s. Again by the continuity of the triple products, {u, v, tn } → 0.

(9) DERIVATIONS ON BANACH – JORDAN PAIRS. 277. for u ∈ V + , v ∈ V − , and lim D+ ({u, v, tn }) = lim({D+ (u), v, tn } + t{u, D− (v), tn } + {u, v, D+ (tn )}) = {D+ (u), v, lim tn } + {u, D− (v), lim tn } + {u, v, lim D+ (tn )} = {u, v, s}, which proves that {V + , V − , S(D+ )} ⊆ S(D+ ). By the symmetry of the arguments, we obtain that S(D) is an ideal of V , which is also closed since the separating subspace of a linear operator is always closed, as already pointed out. Let V = (V + , V − ) be a normed Jordan pair, let y be a fixed element in V −σ , and let D = (D+ , D− ) be a derivation on V . In general, Dσ is not a derivation on the y-homotope of V σ (although this is the case if D−σ (y) = 0). However, the action of Dσ on (V σ )(y) still retains good properties which will be sufficient for our purposes. P ROPOSITION 3.4 Let V = (V + , V − ) be a normed Jordan pair, let y be a fixed element in V −σ , and let D = (D+ , D− ) be a derivation on V . Then for every T ∈ M(V σ (y) ) the linear operator [Dσ , T ] is continuous. Proof. This is clear for T = Ra a right multiplication on V σ (y) , since Ra = 12 V (a, y) and [Dσ , Ra ] = 12 (V (Dσ (a), y) + V (a, D−σ (y))) which is not difficult to verify. Thus, the set A of all T ∈ M(V σ (y) ) whose commutator [Dσ , T ] with Dσ is continuous contains all right multiplications of V σ (y) ; but A is clearly a subalgebra of M(V σ (y) ), and therefore A = M(V σ (y) ), since M(V σ (y) ) is generated by all the right multiplications. Recall that an element 0 = c ∈ V σ , where V is a Jordan pair over , is called reduced if Q c V −σ = c. It is clear that if is a field, then every reduced element lies in the socle. In the opposite direction, and as a consequence of the Mazur–Gelfand theorem for Jordan algebras [17], reduced elements do exist in abundance in non-degenerate normed Jordan pairs with non-zero socle. L EMMA 3.5 Let V be a non-degenerate complex normed Jordan pair. If I is a non-zero ideal of V contained in Soc(V ), then I σ contains a reduced element of V σ . Proof. By [20, Proposition 3], the simple elements of I are just the simple elements of V contained in I , and Soc(I ) = Soc(V ) ∩ I , so we may assume that I = V . Let K be a minimal inner ideal of V σ and take 0 = u ∈ K . By (1.5) and (3.1), Vu is a normed division Jordan algebra. Thus, in virtue of the Mazur–Gelfand theorem for Jordan algebras, Vu is isomorphic to the complex field. Now, by [1, 1.7.1], Vu is isomorphic to Q u V −σ as complex vector spaces, and hence Q u V −σ = Cu, which proves that u is a reduced element of V σ . As a first step in the proof of the main result of this paper, namely, the continuity of the derivations on semiprimitive Banach–Jordan pairs, we show that the socle of a non-degenerate complex normed Jordan pair annihilates the separating ideal of any derivation, so non-continuous derivations cannot occur on complex Banach–Jordan pairs having a large socle. T HEOREM 3.6 Let V be a non-degenerate complex normed Jordan pair, and let D be a derivation of V . Then Soc(V ) ∩ S(D) = 0..

(10) 278. A . FERN ÁNDEZ. et al.. Proof. Since S(D) is an ideal of V (3.3), we only need to show that Soc(V + )∩ S(D+ ) = 0 by (1.1). Suppose on the contrary that Soc(V + ) ∩ S(D+ ) = 0. Then, by (3.5), S(D+ ) contains a reduced element u of V σ . By von Neumann regularity of the socle [20, Theorem 1], u = Q u x for some x ∈ V − . Replace x by y = Q x u to obtain u = Q u y, with y ∈ S(D− ) since S(D) is an ideal. Let {xn } be a sequence in V − such that lim xn = 0 and lim D− (xn ) = y. Then Q u V − = Cu implies that for each n there exists αn ∈ C such that Q u xn = αn u. Since Q u xn → 0 because of the continuity of the operator Q u , the sequence {αn } of complex numbers is forced to converge to zero. Hence D+ (Q u xn ) = D+ (αn u) = αn D+ (u) → 0, which implies, together with the continuity of the triple product, and that of the operator Q u , that u = Q u y = lim Q u D− (xn ) = lim D+ (Q u xn ) − lim {D+ (u), xn , u} = 0, which is a contradiction. The continuity of the derivations on primitive complex Banach–Jordan pair can now be achieved by combining the above theorem with Proposition 2.4. P ROPOSITION 3.7 Every derivation D on a primitive complex Banach–Jordan pair V consists of continuous linear operators D+ and D− . Proof. By the closed graph theorem, we just need to prove that S(D) = 0. As Soc(V ) ∩ S(D) = 0 by (3.6), it follows from primeness of V (1.2) that if V has non-zero socle, then S(D) = 0. Therefore, we may assume that Soc(V ) = 0. By (1.3) and (3.1), for every non-zero y ∈ S(D− ) the local algebra Vy of V at y is a complex primitive Banach–Jordan algebra. Moreover, by (3.4), for every T ∈ M(V +(y) ) the operator [D+ , T ] is continuous. Thus, the ideal Ker(y) of V +(y) and the operator D+ satisfy the requirements of (2.4) relative to the complex Banach–Jordan algebra V +(y) . Therefore, the Jordan algebra (S(D+ ) + Ker(y))/ Ker(y) has finite capacity. But Soc(V ) = 0 forces this algebra to be zero; equivalently, S(D+ ) ⊆ Ker(y), since otherwise the primitive Jordan algebra Vy has non-zero socle and hence so has V [1, 2.1.5], which is a contradiction. Therefore Q y S(D+ ) = 0 and hence y = 0 by non-degeneracy of S(D) inherited from that of V (1.1). Then S(D− ) = 0 and hence S(D) = 0, again by (1.1), as required. Having settled the continuity of derivations on any primitive complex Banach–Jordan pair, we now pay attention to the semiprimitive case; but unlike Villena’s procedure consisting in searching primitive ideals not invariant by a given derivation [26], we shall be directly interested in primitive ideals not containing the separating ideal of a derivation on V . L EMMA 3.8 Let D be a derivation on a complex Banach–Jordan pair V and let P1 , . . . , Pn be primitive ideals of V not containing S(D). If K is an ideal of V such that P1 ∩ · · · ∩ Pn ∩ K = 0, then K = 0. Proof. By [12, A.5.2], primitive ideals in Banach–Jordan pairs are closed. Note that a primitive ideal P of V not containing S(D) cannot be invariant under D. For, if D(P) ⊆ P, then D defines a derivation, say δ = (δ+ , δ− ), on the primitive Banach–Jordan pair V /P, such that π σ D σ = δσ πσ.

(11) DERIVATIONS ON BANACH – JORDAN PAIRS. 279. for σ = ±, where π : V → V /P is the canonical projection. Since δσ is continuous by (3.7), S(πσ Dσ ) = S(δσ πσ ) = 0, which implies that S(Dσ ) ⊆ Pσ (σ = ±), a contradiction. Now we proceed to prove the lemma by induction. For n = 1 a straightforward computation shows that D(P1 ) + P1 is an ideal of V satisfying Q K (D(P1 ) + P1 ) = 0. Hence, by primeness of the Jordan pair V /P1 (1.2), D(P1 ) not contained in P1 implies that K ⊆ K ∩ P1 = 0. Suppose now that the statement is true for some positive integer n, and let P1 , . . . , Pn+1 and K be ideals of V satisfying the conditions of the lemma. Then P1 and L = P2 ∩ · · · ∩ Pn+1 ∩ K also satisfy these conditions. Thus, we have just proved in the case n = 1, L = 0 and hence K = 0 by induction. We are now ready for the main result of this paper. T HEOREM 3.9 Every derivation D on a semiprimitive (real or complex) Banach–Jordan pair V consists of continuous linear operators D+ and D− . Proof. Assume first that V is complex. As already pointed out, we need just prove that S(D) = 0. Suppose on the contrary that S(D) is non-zero and let 0 = y ∈ S(D− ). By (1.4), there exists a primitive ideal P = (P + , P − ) of V such that y ∈ / P − . But V /P is a primitive Banach–Jordan pair, and hence, by (1.3) and (3.1), (V /P) y is a primitive Banach–Jordan algebra, where y = y + P − is the image of y under the canonical projection of V − onto V − /P − . Since (V /P) y is isomorphic − −1 − +(y) ; moreover, to V +(y) /Q −1 y (P ), Q y (P ) is a primitive ideal of the Banach–Jordan algebra V +(y) −1 by (3.4), [D+ , T ] is continuous for all T ∈ M(V ). Thus, the couple D+ , Q y (P − ) satisfies the requirement of (2.4) relative to the Banach–Jordan algebra V +(y) . Therefore, either (i) S(D+ ) ⊆ − −1 − −1 − Q −1 y (P ) or (ii) the Jordan algebra (S(D+ ) + Q y (P ))/Q y (P ) has non-zero finite capacity, equivalently, − −1 − Q y (S(D+ )) ⊆ P − or (S(D+ ) + Q −1 y (P ))/Q y (P ) has non-zero finite capacity. − −1 − But (S(D+ ) + Q −1 y (P ))/Q y (P ) can be regarded as an ideal of (V /P) y , via the canonical +(y) −1 isomorphism (V /P) y ∼ /Q y (P − ) just pointed out. Hence the second alternative implies = V that (V /P) y has non-zero finite capacity itself (see [22, Theorem 18]). Therefore, we have one of the two following possibilities:. Q y (S(D+ )) ⊆ P − or (V /P) y has non-zero finite capacity. We claim that the inclusion Q y (S(D+ )) ⊆ P − holds for all but finitely many primitive ideals P of V , that is, the set (y) = {P primitive ideal of V : Q y (S(D+ )) P − } = {P primitive ideal of V : (V /P) y has non-zero finite capacity}. (5). is finite. If (y) were infinite, we could take an infinite sequence {Pn } of distinct primitive ideals − in (y). For each n ∈ N, let In = Q −1 y (Pn ) be the corresponding primitive ideal of the Banach– − −1 − Jordan algebra V +(y) . We claim that In = Im for n = m. For, if Q −1 y (Pn ) = Q y (Pm ), then + −1 − + −1 − Pm ⊆ Q y (Pm ) implies that Pm ⊆ Q y (Pn ), equivalently, Q y (Pm+ ) ⊆ Pn− .. (6).

(12) 280. A . FERN ÁNDEZ. et al.. By (1.5), it follows from (1) that for each n, 0 = y + Pn− ∈ Soc(V − /Pn− ), and since V /Pn is strongly prime (1.2), Soc(V − /Pn− ) is simple. Hence, either Soc(V /Pn ) ∩ (Pn + Pm )/Pn = Soc(V /Pn ) or Soc(V /Pn ) ∩ (Pn + Pm )/Pn = 0. In the first case, Soc(V /Pn ) ⊆ (Pn + Pm )/Pn , and hence it follows from (6) that Q y+Pn− Soc(V + /Pn+ ) ⊆ Q y+Pn− ((Pm+ + Pn+ )/Pn+ ) = 0, which is a contradiction since Soc(V /Pn ) is non-degenerate. Thus Soc(V /Pn ) ∩ (Pn + Pm )/Pn = 0, and hence Pm ⊆ Pn by primeness of V /Pn . Similarly, Pn ⊆ Pm which yields a contradiction since Pn = Pm by our initial assumption. Therefore, {In } is an infinite sequence of distinct primitive ideals of the complex Banach–Jordan algebra V +(y) providing quotients V +(y) /In of finite capacity. Now [26, Theorem 3] can be applied since, as pointed out in the remark after Theorem 2.1, Banach– Jordan algebras with finite capacity have finite spectrum, to obtain a sequence {an } in V +(y) such that an ∈ Im for m < n and an + Im is invertible for m n. Hence, for all positive integer n, we have Ua1 +In . . . Uan+1 +In (S(D+ ) + In ) = Ua1 . . . Uan+1 (S(D+ )) + In = 0 and Ua1 +In . . . Uan +In (S(D+ ) + In ) = Ua1 . . . Uan (S(D+ )) + In = 0. This shows that the inclusion cl(Ua1 . . . Uan+1 (S(D+ )) cl(Ua1 . . . Uan S(D+ )) is strict, which is a contradiction with Lemma 2.3. This proves that the set (y) is actually finite, say (y) = {P1 , . . . , Pn }. Since Rad(V ) is the intersection of all primitive ideals of V , setting K = {P : P ∈ / (y)} we have that P1 , . . . , Pn and K satisfy the requirements of (3.8), and so K = P = 0. P ∈(y) /. Since every primitive ideal P = (P + , P − ) which does not lie in (y) satisfies Q y (S(D+ )) ⊆ P − , we have Q y (S(D+ )) ⊆ P − = 0, P ∈(y) /. which is a contradiction since S(D) is non-degenerate by (1.1). Thus S(D) = 0 and therefore D+ and D− are continuous..

(13) DERIVATIONS ON BANACH – JORDAN PAIRS. 281. If V is real, we can consider the complexification VC = (VC+ , VC− ) of V . This is a complex Banach–Jordan pair whose norm extends that of V and which is endowed with the conjugate linear involutive automorphism η = (η+ , η− ), where η± (x + i y) = x − i y for x, y ∈ V ± (see [7] for the complexification of a Banach algebra). As in [10, Lemma 2(i)], it can be shown that VC is also semiprimitive: Rad(VC ) ∩ V is a quasi-invertible ideal of V , and hence it is contained in Rad(V ) by [18, 4.2], so Rad(VC ) ∩ V = 0, which implies that Rad(VC ) = 0 since the Jacobson radical is invariant under automorphisms. Moreover, the pair of operators = (+ , − ) given by σ (x + i y) = Dσ (x) + i Dσ (y) for all x, y ∈ V ± defines a Jordan pair derivation on VC . Then the continuity of complex derivations just proved applies to obtain that of σ and therefore that of Dσ . A derivation on a Jordan algebra J (with squaring and quadratic operators x → x 2 and x → Ux ) is a linear operator D : J → J satisfying D(Ux y) = {D(x), y, x} + Ux D(y) and D(x 2 ) = x ◦ D(x) for all x, y ∈ J , where x ◦ y = (x + y)2 − x 2 − y 2 is the linearization of the squaring operator. Again, if 12 ∈ these conditions reduce to the usual one: D(x · y) = x · D(y) + D(x) · y, where x · y = 12 (x ◦ y) is the usual Jordan product. Since the Jacobson radical of the Jordan pair V associated with a Jordan algebra J is Rad(V ) = (Rad(J ), Rad(J )), where Rad(J ) is the Jacobson radical of J , the fundamental result by Villena on the continuity of derivations on semiprimitive complex Banach–Jordan algebras [26] can be derived from Theorem 3.9. C OROLLARY 3.10 Derivations on (real or complex) semiprimitive Banach–Jordan algebras, and pairs of derivations on Banach–Jordan triple systems are continuous. Following [5], by a derivation on a JB∗ -triple A we mean a linear operator δ : A → A satisfying δ({x, y, z}) = {δ(x), y, z} + {x, δ(y), z} + {x, y, δ(z)} for all x, y, z ∈ A. Since every JB∗ -triple A gives rise to a real and semiprimitive Banach–Jordan pair V = (A, A) and since every derivations δ on the B∗ -triple A defines a derivation D = (δ, δ) on the Jordan pair V , it follows from (3.9) that derivations on JB∗ -triples are continuous [5]. Acknowledgements We would like to thank Teresa Cortés for her relevant information about primitive ideals in Jordan pairs. We are also indebted to Angel Rodrı́guez Palacios and Armando Villena for their interesting comments and suggestions. This work is supported by the AECI under the project ‘Estructuras algebraicas asociativas y no asociativas’, and by DGI grant PB97-1069-C02-01..

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