EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 4, 2017, 749-762 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Hasse-Schmidt Derivations on Banach-Jordan Pairs Hassan Marhnine1, Chafika Zarhouti1,∗ 1 Av. My Abdelaziz, Souani, B.P. 3117 Tangier 90000, Morocco Abstract. The aim of this paper consists in establishing the automatic continuity of Hasse- Schmidt derivations on Banach-Jordan Pairs and Banach-Jordan Algebras satisfying some algebraic conditions. Namely, higher derivations on semiprimitive Banach-Jordan Pairs and semiprimitive Banach-Jordan Algebras are continuous. 2010 Mathematics Subject Classifications: 17C65, 46H40, 46H70 Key Words and Phrases: Derivation, Higher derivation, automatic continuity, Banach-Jordan pairs, Banach-Jordan algebras 1. Introduction Higher derivations were introduced first by Hasse and Schmidt [12], that’s why alge- braist sometimes call them Hasse-Schmidt derivations. For further algebraic properties about these operators, the reader is referred to [5,7,11,17,27,28] where they are studied in other context. Higher derivations are used in [30] to study generic solving of higher differential equations. Loy proved in [22] that if A is an (F )-algebra which is a subalge- bra of a Banach algebra B of power series, then every higher derivation {dn} : A −→ B (n = 0, 1, 2, ...) is automatically continuous. Jewell showed in [15] that any higher deriva- tion from a Banach algebra into a semisimple Banach algebra is continuous provided ker(d0) ⊆ ker(dn), for all n ≥ 1. S. Hejazian and T.L. Shatery show in [13] that every higher derivation {dn} from a JB∗−algebra A into a JB∗−algebra B is continuous pro- vided that d0 is a *-homomorphism. They also prove that every higher derivation from a commutative C∗−algebra or from a C∗−algebra which has minimal idempotents and is the closure of its socle is continuous. M. Mirzavaziri gives in [24] a characterization of higher derivations on algebras. In this paper, we deal with higher derivations on Banach-Jordan pairs. We intend to settle the automatic continuity of these operators provided that some algebraic conditions are satisfied. Our approach to this result consists in intensive use of local algebras theory frequently used by authors in Jordan structures. Let us note that Jordan pairs are a natural ∗Corresponding author. Email addresses: radimarhn@hotmail.com (H. Marhnine), chafikazar@hotmail.com (C. Zarhouti) http://www.ejpam.com 749 c© 2017 EJPAM All rights reserved. H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 750 extension of Jordan algebras and arise as well in a natural way in the geometry of bounded symmetric domains. Loos proved in [21] a strong dependence between homogeneous circled domains, in finite complex vector spaces, and Jordan pairs. 2. Preliminaries In this paper we shall deal with Jordan pairs and Jordan algebras over a commutative ring of scalars R of characteristic not two. The reader is referred to [18] for further details. However, we shall record in this section some notations and results. A Jordan pair over a commutative ring R of characteristic not two is a pair of R- modules P = (P+, P−) endowed with a couple (Q+, Q−) of quadratic operators Qσ : P σ −→ HomR(P−σ, P σ) such that the following identities hold for all (x, y) ∈ P σ × P−σ (σ = ±) V σ (x,y)Q σ x = QσxV −σ (y,x), V σ (Qσxy,x) = V σ (x,Q−σ y x), where V σ (x,y)z = Qσ(x,z)y = {x, y, z}σ, Qσ(x,z) = Qσx+z −Qσx −Qσz and {x, y, x}σ = 2Qσxy. An example of Jordan pairs over a field K is given by taking P = A(M,R,ϕ)J , where M = (M+,M−) is a pair of R-vector spaces such that M+ is a left R-module and M− is a right R-module over an associative K-algebra R and ϕ : M+×M− −→ R is an R-bilinear form in the sense that ϕ(ax, yb) = aϕ(x, y)b. The product of P = A(M,R,ϕ)J is defined by: Qxy = ϕ(x, y)x and Qyx = yϕ(x, y) ∀(x, y) ∈M+ ×M−. A Jordan pair P = (P+, P−) is said to be normed (Banach) provided the vector spaces P+ and P− are endowed with norms (complete), both denoted by ‖.‖ , making continuous the triple products {x, y, z}σ of P, merely denoted {x, y, z}. A typical example of Banach-Jordan pairs is given by taking P+ = BL(X ,Y), P− = BL(Y,X ), the pair of linear bounded operators between real or complex Banach spaces X and Y with the multiplication Quv = uvu. Such pair is frequently denoted by B(X ,Y). A (linear) Jordan algebra is a vector space J endowed with a binary product (a, b) 7−→ ab satisfying the identities: ab = ba, and a2(ba) = (a2b)a. If a complete norm is defined on J and makes continuous its product ab, J is said to be a Banach-Jordan algebra. Jordan pairs are known by their intimate relationship with Jordan algebras. Indeed, Any associative, alternative or Jordan algebra A gives rise to a Jordan pair (A,A) with a quadratic multiplication xyx or Uxy, with U denoting the usual U -operator of a Jordan algebra defined by Uxy = 2x(xy)− x2y. In the opposite direction, given a Jordan pair V = (V +, V −) and an element u ∈ V −σ, the vector space V σ gives rise to a Jordan algebra by defining the U -operator Ua = U (u) a = QaQu, and the square a(2,u) = Qau. This Jordan algebra, denoted by V σ(u), is called the u-homotope of V at u. If V is a linear Jordan pair, we just need to define the linear product in V σ(u) as follows: a.b = 1 2 {a, u, b} . H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 751 Local algebras of a Jordan pair. Let V be a Jordan pair and 0 6= u ∈ V −σ. By [18, 4.19] the set ker(u) whose elements are those x ∈ V σ such that Qux = QuQxu = 0, turns out to be an ideal of V σ(u) and the quotient V σ(u)/ ker(u) is a Jordan algebra called the local algebra of V at u which we denote by Vu. As pointed out in [9, 1.2.4(ii)] the condition QuQxu = 0 is superfluous if V is linear or nondegenerate: Qx = 0 implies x = 0. If V is a normed Jordan pair, Then V σ(u) is a normed Jordan algebra for the norm |x| = ‖x‖σ ‖u‖−σ. Moreover, by [23, §II. Lemma 3.1] , the local algebra Vu is also normed for the quotient norm ‖x+ ker(u)‖ = inf z∈ker(u) |x+ z| which is complete if so are the norms of V. Socle and capacity. For a nondegenerate Jordan pair V , its socle, denoted by Soc(V ), is the ideal Soc(V ) =(Soc(V +), Soc(V −)), where Soc(V σ) denotes the sum of all minimal inner ideals of V ±. (2.1) Let V be a nondegenerate Jordan pair and u ∈ V σ. Then u ∈ Soc(V σ) if and only if Vu has finite capacity [25, 0.7(b)]. A nondegenerate Jordan pair V has a finite capacity if it contains an orthogonal system {e1, ..., en} of division idempotents (V2(ei) is a division Jordan pair) such that ∩ni=1V0(ei) = 0, equivalently the lengths of its chains of principal inner ideals are bounded Primitive Jordan pairs and Jacobson radical. A Jordan pair V = (V +, V −) is said to be primitive at b ∈ V −σ if there exists a proper inner ideal K of V σ such that: i) K is a c-modular inner ideal of the homotope V σ(b) for some c ∈ V σ, ii) K complements the (σ)-parts of nonzero ideals: Iσ +K = V σ for any nonzero ideal I = (I+, I−) of V. . Anquela and Cortés proved in [1] and [2] the following results: (2.2) V is primitive at b ∈ V −σ if and only if Vb is a primitive Jordan algebra and V is strongly prime. (2.3) If V is primitive at some 0 6= b0 ∈ V −σ then so is V at every element 0 6= b ∈ V ±. Further results on primitive Jordan pairs can be found in [1] , [2] and [3] . Following [18], the Jacobson radical of a Jordan pair V is defined as the 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 which are quasi-invertible in every homotope V σ(u). A Jordan pair is said to be semiprimitive is Rad(V ) = 0. As in the case of associative algebras, an ideal P of a Jordan system (algebra or pair) V is called primitive if the factor system (algebra or pair) V/P is primitive. Moreover, it follows from [14, A.4.8] , or either [31]. (2.4) The Jacobson radical of a Jordan pair is the intersection of all its primitive ideals. 3. Technical results Recall that we can measure the continuity of a linear operator acting between two normed spaces by considering its so called separating subspace. Indeed, if T is a linear H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 752 operator defined between two real or complex normed vector spaces X and Y , then its separating subspace S(T ) is defined by: S(T ) = {y ∈ Y : ∃{xn}n ⊂ X such that limxn = 0 and limT (xn) = y} . It is easily seen that the separating subspace of T is a closed subspace of Y. 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. Let V and W be two Jordan pairs. By a higher derivation of rank k (k may be infinite), we mean a family of linear mappings {ϕn = (ϕ+ n , ϕ − n )}kn=1 from V into W such that ϕσn {x, y, z} = ∑ i+j+h=n { ϕσi x, ϕ −σ j y, ϕσhz } , (x, z ∈ V σ, y ∈ V −σ, n = 0, 1, 2, ..., k), where ϕσ0 = IdV σ (σ = ±). Let D = (D+, D−) be a derivation from V into W, that is a pair of linear operators Dσ : V σ −→ V σ satisfying Dσ {x, y, z} = {Dσx, y, z}+ {x,D−σy, z}+ {x, y,Dσz} , for all (x, z ∈ V σ, y ∈ V −σ. Any derivation D = (D+, D−) from V into W gives rise to a standard example of higher derivations {ϕn = (ϕ+ n , ϕ − n )}n≥0 from V into W by setting ϕ+ n = 1 n! Dn +, and ϕ−n = 1 n! Dn −. Remark 1. i) It follows from the last definitions that ϕ1 = (ϕ+ 1 , ϕ − 1 ) is a derivation. ii) In order to simplify notations, the index σ = ± in expressions like D±i (x), ϕ±i (x), ... will be sometimes suppressed if there is no confusion. Lemma 1. Let V be a normed Jordan pair and let k ≥ 2 be a fixed positive integer. If ϕn = (ϕ+ n , ϕ − n )} is a higher derivation on V such that ϕσi is continuous for every i ∈ {0, 1, ..., k − 1} (σ = +,−), then the separating subspace S(ϕk) = (S(ϕ+ k ), S(ϕ−k )) of ϕk is a closed ideal of V. Proof. Since the characteristic of the ground field is zero, it suffices to prove that S(ϕn) is an outer ideal of V. That is QV −σS(ϕσk) ⊂ S(ϕσk) and {S(ϕσk), V −σ, V σ} ⊂ S(ϕσk). Let s be an element of S(ϕσk) and a be an arbitrary element of V −σ. Then there exist a sequence {xn}n ⊂ V σ such that limxn = 0 and limϕσk(xn) = s. Consider the sequence {Qaxn}. By continuity of the operator Qa, we get limQaxn = Qa limxn = 0. Moreover, using the continuity of the triple product of V and that of ϕσj such that 0 ≤ j ≤ k−1, we see H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 753 that the terms { ϕσi a, ϕ −σ j xn, ϕ σ ha } converge to zero when n tends to ∞ and consequently we have limϕσkQaxn = 1 2 limϕσk {a, xn, a} = 1 2 lim ∑ i+j+h=k { ϕσi a, ϕ −σ j xn, ϕ σ ha } = 1 2 { a, limϕ−σk xn, a } = 1 2 {a, s, a} = Qas, which establishes QV −σS(ϕσk) ⊂ S(ϕσk). On the other hand, for arbitrary pair (u, v) of elements in V −σ × V σ, the sequence {xn, u, v} converges to 0. Using again the continuity of the triple product of V as well as that of ϕσi such that 0 ≤ i ≤ k− 1 and i+ j + h = k, we see that, for arbitrary pair (u, v) of elements in V −σ×V σ, the terms { ϕσi xn, ϕ σ j u, ϕ σ hv } converge to zero when n tends to ∞. Consequently, we do have limϕσk({xn, u, v}) = lim ∑ i+j+h=k {ϕσi (xn), ϕ−σj (u), ϕσh(v)} = ∑ i+j+h=k {limϕσi (xn), ϕ−σj (u), ϕσh(v)} = {limϕσk(xn), u, v} = {s, u, v}, which establishes {S(ϕσk), V −σ, V σ} ⊂ S(ϕσk) as required. Finally, S(ϕk) is an ideal of V which is closed since the separating subspace of any linear operator is closed as it is pointed out. Remark 2. Let {Dn = (D+ n , D − n )} be a higher derivation on a normed Jordan pair V = (V +, V −) and let b be a nonzero element in V −σ. Let us note that {Dσ n} is not a higher derivation on the Jordan algebra V σ(b) even if Dσ n vanishes at b for all positive integers n. However, the behavior of Dσ n towards V σ(b) conserves nice properties as it is clarified in the following. Lemma 2. Let V = (V +, V −) be a normed Jordan pair and let b be a nonzero element in V −σ. If {Dn = (D+ n , D − n )}n≥0 is a higher derivation on V such that Dσ i is continuous for every i ∈ {0, 1, ..., k − 1} where k is a fixed positive integer greater than 2. Then for every T in the multiplication algebra M(V σ(b)) of the Jordan algebra V σ(b), the linear operator [Dσ k , T ] is continuous. Proof. Consider the set B = { T ∈M(V σ(b)) : [Dσ k , T ] is continuous } . H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 754 It is clear that B is a subspace of M(V σ(b)). Moreover, a simple computation shows that the formula T [Dσ k , S] + [Dσ k , T ]S = [Dσ k , TS] holds for all T, S in M(V σ(b)). This proves that B is a subalgebra of M(V σ(b)). On the other hand, for all a ∈ V σ(b), the left multiplication La lies in B. Indeed, since La = 1 2V(a,b), for all x ∈ V σ we have [Dσ k , La]x = Dσ kLax− LaDσ kx = 1 2 (Dσ k {a, b, x} − {a, b,Dσ kx}) = 1 2 ( ∑ i+j+h=k {Dσ i a,D −σ j b,Dσ hx} − {a, b,Dσ kx}) = 1 2 ( ∑ i+j+h=k h≤k−1 {Dσ i a,D −σ j b,Dσ hx}). This shows that [Dσ k , La] = 1 2 ( ∑ i+j+h=k 1≤h≤k−1 V(Dσi a,D −σ j b)D σ h + ∑ i+j=k V(Dσi a,D −σ j b)), which shows that the operator [Dσ k , La] is continuous since so are V(Dσi a,D −σ j b) and Dσ h for all h ∈ {1, ..., k − 1}. Finally, since M(V σ(b)) is generated by all left multiplications La, we see that M(V σ(b)) = B. The first automatic continuity result concerns higher derivations on nondegenerate Banach-Jordan pairs with nonzero socle. Theorem 1. Let V = (V +, V −) be a nondegenerate Banach-Jordan pair with nonzero socle. If Dn = (D+ n , D − n )}n≥0 is a higher derivation on V , then Dσ k is continuous for every positive integer k. Proof. By the closed graph Theorem, it suffices to prove that S(Dσ k ) = 0. We proceed by induction on k. For k = 0, the identity operator Dσ 0 = IdV σ is obviously continuous. Assume that Di is continuous for i = 1, 2, ..., k − 1 and prove that so is Dk. In virtue of Lemma 1, it is known that S(Dk) is an ideal of V . We claim that Soc(V +) ∩ S(D+ k ) = 0. Assume that this is not the case. We follow the pattern given in [10, Theorem 3.6] to look for a contradiction. By [10, Lemma 3.5], there exists a nonzero element r in S(D+ k ) ∩ Soc(V +) such that r is reduced : QrV − = C.r. By von Neumann regularity of Soc(V ), there exists a nonzero element v in V − such that r = Qrv. Replace v by u = Qvr to see that, using JP3 in [18] , (1) Qru = QrQvr = QrQvQrv = QQrvv = Qrv = r. H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 755 By idealness of S(Dk), u lies in S(D−k ) and u is nonzero because otherwise r = 0, which is a contradiction. Hence, there exists a sequence {xn} in V − such that limxn = 0 and limD−k xn = u. Since r is reduced, we have QrV − = C.r and consequently, for every non negative integer n, there exists a complex number λn such Qrxn = λnr. Now the boundedness of the operator Qr shows that limQrxn = Qr limxn = 0. This makes the sequence {λn} converging to zero in the complex field C. It follows that (2) limD+ k (Qrxn) = limD+ k (λnr) = limλnD + k (r) = 0. On the other hand, by making use of the triple product of V and that of D−j , such that 1 ≤ j ≤ k−1, we see that all terms like {D+ i r,D − j xn, D + h r} converge to zero when n tends to ∞. That is lim{D+ i r,D − j xn, D + h r} = {D+ i r, limD−j xn, D + h r} = {D+ i r,D − j limxn, D + h r} = 0. It follows that, taking into account (1), limD+ k (Qrxn) = 1 2 limD−k ({r, xn, r}) = 1 2 lim ∑ i+j+h=k {D+ i r,D − j xn, D + h r} = 1 2 {r, limD−k xn, r} = 1 2 {r, u, r} = Qru = r, which contradicts (2) since r is nonzero. Now, by idealness of S(Dk) and Soc(V ), we see that for all s ∈ Soc(V −) Qs(S(D+ k )) ⊂ Soc(V −) ∩ S(D−k ) = 0. This shows that S(D+ k ) ⊆ ker(Qs) for every s in Soc(V −), that is S(D+ k ) ⊆ ∩ s∈Soc(V −) ker(Qs). But in virtue of [18, Theorem 4.13] , we see ∩ s∈Soc(V −) ker(Qs) ⊆ rad(Soc(V +)) and rad(Soc(V +)) = Soc(V +) ∩ rad(V +). But, the McCrimmon radical rad(V ) is reduced to zero by nondegeneracy of V . This proves that S(D+ k ) = 0 and, by the closed graph Theorem, D+ k is continuous. By the symmetry of the argument we see that D−k is analogously continuous. As a fundamental example of Jordan pairs having nonzero socle, B(X ,Y) the Jordan pair of bounded linear operators between two Banach spaces X and Y . So we have the following. H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 756 Corollary 1. Any higher derivation Dn = (D+ n , D − n ) on the Banach-Jordan pair B(X,Y ) consists of continuous operators. Proof. It is known that the Banach-Jordan pair B(X ,Y) of bounded linear operators between two Banach spaces X and Y is nondegenerate and has Soc(B(X ,Y)) = (FL(X ,Y),FL(Y,X ), the Banach-Jordan pair consisting in bounded linear operators of finite rank. Now the continuity of {Dn = (D+ n , D − n )} follows immediately from Theorem 1. 4. Main result Before going on the proof the main Theorem in this paper, we recall the following technical results which seem to be useful in the sequel. Lemma 3. [29]. Let X be a Banach space, {Ti}i a sequence of continuous linear operators defined on X and let {Ri}i be a sequence of linear continuous operators whose domain is X but which may map into other Banach spaces. Let T be a possibly discontinuous map from X to itself. If the operator RnTT1...Tm is continuous for m greater than n then RnTT1...Tn is continuous when n is sufficiently large. Proposition 1. [10]. Let J be a Banach-Jordan algebra and I be a primitive ideal of J. If D is a linear operator defined on J such that [D,T ] is continuous for all T in M (J) , then the primitive Jordan algebra (S (D) + I) /I has finite capacity. Lemma 4. Let V a nondegenerate Jordan pair and let P1, ..., Pn be nonzero ideals of V. If H is an ideal of V such that H ∩ P1 ∩ ... ∩ Pn = 0 then. H = 0. Proof. We proceed by induction. For n = 1, by idealness of H and P1, we have, for all u ∈ P σ1 , QuH−σ ⊆ Hσ ∩ P σ1 = 0. Then, by [ 18, Proposition 4.19] together with [ 18, Theorem 4.13] H−σ ⊆ ∩u∈Pσ1 Ker(u) ⊂ rad(P σ1 ) = rad(V σ) ∩ P σ1 , and hence Hσ = 0 by nondegeneracy of V : rad(V ) = 0 . Suppose the statement is true for some natural integer n and let P1, ..., Pn, Pn+1 be nonzero ideals of V satisfying the condition stated in the lemma. Then the ideals P1 and K = P2∩ ...∩Pn+1∩H also satisfy the same condition. Therefore, by we have just proved in the case n = 1, K = 0 and hence H = 0 by induction. Given a Banach space X, we denote by Cl(E) the closure of a subset E of X. We can now state our main result in this paper. Theorem 2. Let {Dn = (D+ n , D − n )} be a higher derivation on a Banach-Jordan pair V = (V +, V −). If V is semiprimitive, then Dσ k is continuous for every non negative integer k. H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 757 Proof. We proceed by induction. For if n = 0, Dσ 0 = IdV σ is trivially continuous. Suppose that D1, ..., Dk are continuous and show that this is also the case for Dk+1, that is S(Dk+1) = 0. Suppose that Dk+1 is discontinuous. Then, there exists a primitive ideal P such that S(Dk+1) is not contained in P . As a first step we show that all primitive ideals contain S(Dk+1) except finitely primitive ideals P1, ..., Pn for which the quotient pairs V/Pi have finite capacity. In other words de set Γ = { P = (P+, P−) primitive ideal of V : S(Dk+1) * P } is finite and, for any P ∈ Γ, the quotient pair V/P has finite capacity. Take P = (P+, P−) in Γ and b ∈ V − such that b /∈ P−. Since P σ is closed in V σ (see [14, A.5.2]), V/P is a Banach-Jordan pair and hence by (2.2) (V/P )b is a primitive Banach- Jordan algebra where b = b+P− is the image of b under the canonical projection V − 7−→ V −/P−. The algebra (V/P )b is known to be isomorphic to V +(b)/I where I = Q−1 b (P−) is so a primitive ideal of the Banach-Jordan algebra V +(b). Moreover, by Lemma 2 the linear operator Dk+1 and the ideal I satisfy the conditions required in Proposition 1 with respect to the Banach-Jordan algebra V +(b). Therefore, (S(Dk+1)+I)/I has nonzero finite capacity. This implies that (V/P )b has itself nonzero finite capacity [26, Theorem 18]. Thus by (2.1), Soc(V/P ) = V/P and hence, by completeness, V/P has nonzero finite capacity. Suppose that the set Γ is infinite, then we can take an infinite sequence {Pn} of distinct primitive ideals in Γ. By we have just proved, V/Pn is simple with finite capacity and hence has finite spectrum (see [19, Theorem 1] and [20, Theorem 3.8]). By a similar process used in [6, Lemma 2.8], we show the existence of an element b in V − and a sequence {an} in V + such that b /∈ ∪nP−n , πm(an) is invertible in (V/Pm)b for n < m and πm(an) = 0 for m < n where πm : V + 7−→ V +/P+ m is the natural projection. Indeed, take b1 in V − such that b1 /∈ P−1 . By induction we can construct the sequences {bn} in V − and {λn} in the complex field such that λ1 = 1. Having defined b1, ..., bn−1 and λ1, ..., λn−1, we take bn in n−1 ∩ i=1 P−i with ‖bn‖ = 1, 1 < λn < 1 2n and n∑ i=1 λibi /∈ P−n . This last condition is satisfied since n−1 ∩ i=1 Pi is not contained in Pn. Since the series n∑ i=1 λibi, converges in V −, we write b = ∞∑ i=1 λibi. We see that b = n∑ i=1 λibi is nonzero in V −/P−n and hence b /∈ ∪nP−n . Now take u1 in V + such that u1 /∈ P+ 1 . We proceed by choosing {un} in V + and , for any natural number k, the scalars { λkn }∞ n=k such that λkk = 1. Having selected them up to n − 1, we take un and λkn such that un ∈ n−1 ∩ i=1 P+ i , πn(un) is the unit of the Banach-Jordan algebra (V/Pm)b, 0 < λkn < 1 2n‖un‖ and πn( n∑ i=k λki ui) is invertible. If we take an = ∞∑ i=n λni ui, then we will have πm(an) is invertible in (V/Pm)b for m > n and πm(an) = 0 for m < n as required. Now consider an arbitrary x in V + and positive integers m,n. We compute in (V/Pn)b H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 758 to have πnDk+1U (b) a1 U (b) a2 ...U (b) am(x) = 1 4 πnDk+1 { a1, { b, U (b) a2 ...U (b) am(x), b } , a1 } = 1 4 πn ∑ i+j+h=k+1 { Dia1, Dj { b, U (b) a2 ...U (b) am(x), b } , Dha1 } = 1 4 πn ∑ i+j+h=k+1 j≤k { Dia1, Dj { b, U (b) a2 ...U (b) amx, b } , Dha1 } + 1 4 πn { a1, Dk+1 { b, U (b) a2 ...U (b) amx, b } , a1 } = ϕ(x) + 1 2 πnQa1( ∑ i+j+h=k+1 { Dib,DjU (b) a2 ...U (b) amx,Dhb } ) = ϕ(x) + 1 2 πnQa1( ∑ i+j+h=k+1 j≤k { Dib,DjU (b) a2 ...U (b) amx,Dhb } ) +πnQa1QbDk+1U (b) a2 ...U (b) amx = ψ1(x) + πnU (b) a1 Dk+1U (b) a2 ...U (b) am(x), where ψ1(x) = ϕ(x) + 1 2 πnQa1( ∑ i+j+h=k+1 j≤k { Dib,DjU (b) a2 U (b) a3 ...U (b) amx,Dhb } ) ϕ(x) = 1 4 πn ∑ i+j+h=k+1 j≤k { Dia1, Dj { U (b) a1 U (b) a2 ...U (b) am(x) } , Dha1 } are clearly continuous operators. By iterating the same process, we show that there exits a continuous linear operator ψm such that πnDk+1U (b) a1 U (b) a2 ...U (b) am(x) = ψm(x) + πnU (b) a1 U (b) a2 ...U (b) amDk+1(x). But we have πnU (b) a1 U (b) a2 ...U (b) amDk+1 = 0 when n < m. It follows that the operator πnDk+1U (b) a1 U (b) a2 ...U (b) am is continuous. Now, Lemma 3 applies to the sequences {Ri} and {Ti} withRi = πi and Ti = U (b) ai to obtain the continuity of the operator πnDk+1U (b) a1 U (b) a2 ...U (b) an when the integer n is sufficiently large. That is S(πnDk+1U (b) a1 U (b) a2 ...U (b) an ) = 0. But since πn(ai) is invertible for i ≤ n, we have S(πnDk+1U (b) a1 U (b) a2 ...U (b) an ) = S(πnU (b) a1 U (b) a2 ...U (b) an Dk+1) H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 759 = Cl(U (b) π1a1U (b) π2a2 ...U (b) πnanS(πnDk+1)) = Cl(S(πnDk+1)), which is a contradiction because Cl(S(πnDk+1)) 6= 0 since otherwise we will have S(Dk+1) ⊆ Pn for any positive integer n. The set Γ is actually finite, say Γ = {P1, ..., Pn} . Set H = ∩ P /∈Γ P . The ideals P1, P2, ..., Pn and H satisfy the requirements of Lemma 4 since ( n ∩ i=1 Pi)∩H = Rad(V ) is the intersection of all primitive ideals of V (2.4) and Rad(V ) = 0. We conclude that H = 0. But S(Dk+1) ⊆ P for any primitive ideal P not contained in Γ, then S(Dk+1) ⊆ ∩ P /∈Γ P = H = 0, which is a contradiction. Dk+1 is finally continuous. As it is pointed out, any Jordan algebra gives rise to a Jordan pair (J, J) with the quadratic map Qa = Ua defined by Uab = 2a(ab)− a2b. A family {dn} ( n = 0, 1, 2, ..., k, k may be ∞) of linear operators defined on J is said to be a higher derivation if, for all a, b in J, we have dn(ab) = k=n∑ k=1 dk(a)dn−k(b). A tedious computation enables to prove that any higher derivation {dn}n≥0 on a Jordan algebra gives rise to a higher derivation {(dn, dn)}n≥0 on the Jordan pair (J, J) with respect to the triple product {x, y, z} = (xy)z + (yz)x− (zx)y. The Jordan pair is semiprimitive if so is J. Hence, according to Theorem 2, we have the following. Corollary 2. . Any higher derivation {dn}n≥0 on a semiprimitive Banach-Jordan algebra consists of continuous operators. 5. Higher derivations on Banach alternative pairs and JB∗-triples The reader is referred to [18] for definitions and basic results on alternative pairs. Given an alternative pair A = (A+, A−), we write (x, y, z) 7−→ 〈xyz〉 to denote the triple product of (x, y, z) in Aσ ×A−σ ×Aσ (σ = ±). By a normed alternative pair we mean a complex alternative pair A = (A+, A−), where the vector spaces A+ and A− are equipped with norms ‖.‖σ making continuous the triple product 〈xyz〉 . A = (A+, A−) is said to b Banach alternative pair provided the norms ‖.‖σ are complete. The Banach spaces Mp,q(C), Mq,p(C) of rectangular matrices with entries in the complex field C define a Banach alternative pair A = (Mp,q(C),Mq,p(C)), with respect to the triple product 〈RST 〉 = RST, the usual matrices product. H. Marhnine, C. Zarhouti / Eur. J. Pure Appl. Math, 10 (4) (2017), 749-762 760 A higher derivation on an alternative pairA = (A+, A−) is a sequence {Dn = (D+ n , D − n )}n≥0 of linear operators Dσ n : Aσ 7−→ Aσ satisfying the formula Dσ n(< xyz >) = ∑ i+j+h=n < Dσ i (x)D−σj (y)Dσ h(z) >, (x, z ∈ Aσ, y ∈ A−σ, n = 0, 1, 2, ... with Dσ 0 = IdAσ . Corollary 3. Let {Dn = (D+ n , D − n )}n≥0 be a higher derivation on a Banach alternative pair A = (A+, A−). If A is semiprimitive, then Dσ k is continuous for every positive integer k. Proof. It is known that any alternative pair A = (A+, A−) gives rise to a Jordan pair frequently denoted by AJ (see [18, Theorem 7.1]) by considering the quadratic operators Qxy = 〈xyx〉 for all (x, y) in Aσ ×A−σ. ClearlyAJ is a Banach-Jordan pair wheneverA is a Banach alternative pair. By [18, 7.9(1)] , AJ is semiprimitive if and only if so is A. Moreover, a simple computation enables to verify that every higher derivation {Dn = (D+ n , D − n )} on A induces a higher derivation on AJ with respect to its triple product defined by {x, y, z} = Q(x,z)y = 〈xyz〉+ 〈zyx〉 for all (x, y, z) in Aσ ×A−σ ×Aσ. Actually, Theorem 2 applies to deduce that Dσ n is continuous for every natural number n. Following [4] , we mean by a higher derivation on a JB∗−triple E, a sequence {δn}n≥0 of linear operators δk : E −→ E satisfying δn({x, y, z}) = ∑ i+j+k=n {δix, δjy, δkz} , for all x, y, z in E, where δ0 = IdE . Corollary 4. Any higher derivation {δn}n≥0 on a JB∗-triple E is continuous. Proof. Since every JB∗−triple E gives rise to a complex semiprimitive Banach-Jordan pair V = (V +, V −), where V + = E as vector space and V − is the conjugate complex vector space of E that is the vector space with the new scalar multiplication λ.x = λx for x ∈ E and λ ∈ C. 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