Houston Journal of Mathematics. c 2012 University of Houston Volume 38, No. 1, Communicated by Kenneth R. Davidson

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1 Houston Journal of Mathematics c 2012 University of Houston Volume 38, No. 1, 2012 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS ZHANKUI XIAO AND FENG WEI Communicated by Kenneth R. Davidson Abstract. Motivated by the systemic work of Lu [21, 23] we mainly consider the question of whether any Jordan higher derivation on some operator algebras is a higher derivation. Let A be a torsion free algebra over a commutative ring R, D be the set of all Jordan higher derivations D = {d n} n=0 on A, and be the set of all sequences {δ n} n=0 of Jordan derivations on A with δ 0 = 0. Then there is a one to one correspondence between D and. It is shown via this correspondence that every Jordan higher derivation on some operator algebras is a higher derivation. The involved operator algebras include CSL algebras, reflexive algebras, nest algebras. At last, we describe local actions of Jordan higher derivations on nest algebras. 1. Introduction Let R be a commutative ring and A be an associative R-algebra. An R- linear mapping d : A A is called a derivation if it satisfies the Leibniz rule d(xy) = d(x)y + xd(y) for all x, y A. We say that an R-linear mapping d : A A is a Jordan derivation if it satisfies d(x 2 ) = d(x)x+xd(x) for all x A. Let N be the set of all non-negative integers. If we define a sequence {d n } n N of R-linear mappings on A by d 0 = id A (the identity mapping of A) and d n = 1 n! dn, 2000 Mathematics Subject Classification. 47B47, 47L35. Key words and phrases. Jordan higher derivation, CSL algebra, reflexive algebra, nest algebra. The work of the first author is supported by a Young s Research Foundation of Huaqiao University (Grant No. 10BS323) and the work of the second author is partially supported by the National Nature Science Foundation of China (Grant No ). 275

2 276 ZHANKUI XIAO AND FENG WEI then the Leibniz rule ensures us that d n s satisfy the condition n d n (xy) = d i (x)d n i (y) (1.1) for all x, y A and each non-negative integer n. This motivates us to consider the sequences {d n } n N of R-linear mappings on an algebra A satisfying (1.1). Such a sequence is called a higher derivation of A. Usually, we denote it by D = {d n } n N. This interesting notion of higher derivations was firstly introduced by Hasse and Schmidt in [7], and hence algebraists sometimes call them Hasse- Schmidt derivations. Although, if d : A A is a derivation, then the sequence D = {d n } n N consisting of d n = 1 n! dn is a higher derivation; this is not the only example of higher derivations. This kind of higher derivation is called an ordinary higher derivation. Let D = {d n } n N be a sequence of R-linear mappings of A. If there exist two sequences {a n } n N and {b n } n N in A satisfying the conditions a 0 = b 0 = 1 and n a ib n i = δ n0 = n b ia n i such that n d n (x) = a i xb n i (1.2) for all x A and each non-negative integer n, then it is easy to verify that D = {d n } n N is a higher derivation of A, where δ n0 is the Kronecker sign. Higher derivations of the form (1.2) are said to be inner higher derivations. Let K be a field. The divided powers operators d n = 1 n n! x on the polynomial algebra n i A = K[x 1,, x n ] are also basic examples of higher derivations. Let D = {d n } n N be a sequence of R-linear mappings of A such that d 0 = id A. D is called a Jordan higher derivation of A if for each n N, d n (x 2 ) = d i (x)d j (x) (1.3) i+j=n holds for all x A. D is called a Jordan triple higher derivation of A if for each n N, d n (xyx) = d i (x)d j (y)d k (x) (1.4) i+j+k=n holds for all x, y A. Obviously, any higher derivation on A is a Jordan higher derivation and is also a Jordan triple higher derivation. But, the converse statements are in general not true. Higher derivations are an active subject of research in noncommutative algebras and operator algebras. Ferrero and Haetinger gave the conditions under which Jordan higher derivations (or Jordan triple higher derivations) of a (semi-)prime

3 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS 277 ring are higher derivations [4]. Furthermore, Ferrero and Haetinger established the connections between the algebraicity of derivations and the linear identities satisfied by higher derivations in a prime ring [5]. Jung in [15] proved that any generalized Jordan higher derivation on a 2-torsion free prime ring is a generalized higher derivation. Furthermore, it was shown that the same statement is still true for 2-torsion free semiprime rings [30]. The authors of the current article in [31, 32] studied Jordan higher derivations and higher derivations on triangular algebras. We observed that the notions of higher derivations, Jordan higher derivations and Jordan triple higher derivations on a triangular algebra are equivalent to each other. Moreover, we showed that any higher derivation on some classical triangular algebras, such as (block-)upper triangular matrix algebras, full matrix algebras, nest algebras and incidence algebras, are inner. Miller [26, 27] set out a formal general definition of higher derivations on a Banach algebra over a partially ordered topological abelian group and gave the basic representation theorem for a higher derivation as the exponential of a derivation-valued function. Qi and Hou [29] considered Jordan higher derivations on nest algebras and obtained the following result. Let N be a nest on a Banach space X and τ(n ) be the associated nest algebra. If there exists a nontrivial element in N which is complemented in X, then every Jordan (or Jordan triple) higher derivation on τ(n ) is a higher derivation. More recently, Lu and his cooperators [17, 22, 24] systemically investigated Jordan derivations on some operator algebras. In [22], he proved that every Jordan derivation from a CSL algebra into itself is a derivation. Then he studied Jordan derivations of a class of reflexive algebras [24]. Let L be a subspace lattice on a Banach space X, B(X) be the algebra of all bounded linear operator on X and AlgL be the reflexive algebra associated with L. It was shown that under mild assumptions any Jordan derivation from AlgL into B(X) is a derivation. It is natural to ask whether the corresponding higher versions of Lu s results are still true. The objective of this paper is to generalize Lu s results to the case of Jordan higher derivations and its framework is as follows. The second section is devoting to a new characterization of Jordan higher derivation on associative algebras, which enables one to transfer the problems of Jordan higher derivations into the same problems concerning Jordan derivations. We establish an one to one correspondence relation between the set of all Jordan higher derivations and the set of all Jordan derivations. In the third section we apply the corresponding relation to study Jordan higher derivations on some operator algebras and extend

4 278 ZHANKUI XIAO AND FENG WEI Lu s results to the case of Jordan higher derivation. The local actions of Jordan higher derivations on nest algebras are presented in the last section. 2. Jordan Higher Derivations on Algebras In this section we will give a new characterizations concerning higher derivations, Jordan higher derivations and Jordan triple higher derivations of associative algebras. These new properties easily enables us to transfer the problems of Jordan higher derivations (resp. higher derivations/jordan triple higher derivations) into the same problems related to Jordan derivations (resp. derivations/jordan triple derivations). We establish an one to one correspondence between the set of all Jordan higher derivations (resp. higher derivations/jordan triple higher derivations) and the set of all Jordan derivations (resp. derivations/jordan triple derivations). A higher derivation {f i } m (resp. {f i} ) on A is said to be order m (resp. infinite order) if n f n (xy) = f i (x)f n i (y) for all x, y A and for each n = 0, 1, 2,, m (resp. n = 0, 1, 2, ). Similarly, a Jordan higher derivation {f i } m (resp. {f i} ) on A is said to be order m (resp. infinite order) if n f n (x 2 ) = f i (x)f n i (x) for all x A and for each n = 0, 1, 2,, m (resp. n = 0, 1, 2, ). Throughout this paper, all involved higher derivations, Jordan higher derivations and Jordan triple higher derivations are of infinite order and all obtained results also hold for the case of finite order. We firstly need a straightforward technical lemma. Lemma 2.1. Let d be a Jordan derivation of algebra A and a 0, a 1,, a n A, then ( n ) n d a i a n i = [d(a i )a n i + a i d(a n i )]. Proof. Linearizing the relation d(x 2 ) = d(x)x + xd(x) leads to d(xy + yx) = d(x)y + xd(y) + d(y)x + yd(x)

5 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS 279 for all x, y A. If n is even, then ( n ) n/2 1 d a i a n i = d (a i a n i + a n i a i ) + a 2 n/2 = n/2 1 [d(a i )a n i + a i d(a n i ) + d(a n i )a i + a n i d(a i )] + d(a n/2 )a n/2 + a n/2 d(a n/2 ) n = [d(a i )a n i + a i d(a n i )]. If n is odd, the proof is done by a similar way. Proposition 2.2. Let {d n } n=0 be a Jordan higher derivation on A. Then there is a sequence {δ n } n=0 of Jordan derivations on A such that (n + 1)d = for each non-negative integer n. n δ k+1 d n k ( ) Proof. Let us make induction for the index n. If n = 0, then d 1 (a 2 ) = d 0 (a)d 1 (a) + d 1 (a)d 0 (a) = ad 1 (a) + d 1 (a)a for all a A. If we set δ 1 = d 1, then δ 1 is a Jordan derivation on A. We now suppose that δ k is a well-established mapping and is a Jordan derivation on A for each k n. Let us define δ = (n + 1)d δ k+1 d n k. It is sufficient for us to show that δ is a Jordan derivation on A. For any a A, we have δ (a 2 ) = (n + 1)d (a 2 ) δ k+1 d n k (a 2 ) ( n k ) = (n + 1) d k (a)d k (a) δ k+1 d i (a)d n k i (a).

6 280 ZHANKUI XIAO AND FENG WEI It follows from Lemma 2.1 that ( n k ) δ (a 2 ) = (k + n + 1 k)d k (a)d k (a) δ k+1 d i (a)d n k i (a) = kd k (a)d k (a) + d k (a)(n + 1 k)d k (a) n k [δ k+1 (d i (a))d n k i (a) + d i (a)δ k+1 (d n k i (a))]. for all a A. For convenience, let us write n k A = kd k (a)d k (a) δ k+1 (d i (a))d n k i (a), n k B = d k (a)(n + 1 k)d k (a) d i (a)δ k+1 (d n k i (a)). Then δ (a 2 ) = A + B. In the expression of summation 0 k + i n and k n. Thus if we set r = k + i, then n A = kd k (a)d k (a) = kd k (a)d k (a) = r=0 r=0 0 k r,k n r r=0 δ k+1 (d r k (a))d n r(a) n k, we have δ k+1 (d r k (a))d n r(a) δ k+1 (d n k (a))a n r (r + 1)d r+1(a)d n r(a) δ k+1 (d r k (a))d n r(a) δ k+1 (d n k (a))a [ = (r + 1)d r+1(a) r=0 r=0 ] r δ k+1 (d r k (a)) d n r(a) + (n + 1)d (a)a δ k+1 (d n k (a))a. By the induction hypothesis, (r + 1)d r+1 (a) = r δ k+1(d r k (a)) for r = 0,, n 1. Therefore [ ] A = (n + 1)d (a) δ k+1 (d n k (a)) a = δ (a)a.

7 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS 281 On the other hand, a direct computation yields n k B = d k (a)(n + 1 k)d k (a) d i (a)δ k+1 (d n k i (a)) = d k (a)(n + 1 k)d k (a) = = n d i (a)(n + 1 i)d i (a) n n i d i (a)δ k+1 (d n k i (a)) + aδ (a) n n i d i (a)δ k+1 (d n k i (a)) + aδ (a) [ ] n n i d i (a) (n + 1 i)d i (a) δ k+1 (d n k i (a)) + aδ (a). By the induction hypothesis, (n + 1 i)d i (a) = n i δ k+1(d n k i (a)) for i = 1,, n. So we conclude that B = a [ Finally, we arrive at (n + 1)d (a) ] n δ k+1 (d n k (a)) + aδ (a) = aδ (a). δ (a 2 ) = A + B = δ (a)a + aδ (a). for all a A. This implies that δ is a Jordan derivation on A, which is the required result. We can extract a tedious but intuitive algorithm from the above proposition, which will be used in later. Algorithm 2.3. By the formula ( ) we can compute the first five terms of {d n }. d 0 = I, d 1 = δ 1, 2d 2 = δ 1 d 1 + δ 2 d 0 = δ δ 2, d 2 = 1 2 δ δ 2, 3d 3 = δ 1 d 2 + δ 2 d 1 + δ 3 d 0 = δ 1 ( 1 2 δ δ 2) + δ 2 δ 1 + δ 3,

8 282 ZHANKUI XIAO AND FENG WEI d 3 = 1 6 δ δ 1δ δ 2δ δ 3, 4d 4 = δ 1 d 3 + δ 2 d 2 + δ 3 d 1 + δ 4 d 0 = δ 1 ( 1 6 δ δ 1δ δ 2δ δ 3) + δ 2 ( 1 2 δ δ 2) + δ 3 δ 1 + δ 4, d 4 = 1 24 δ δ2 1δ δ 1δ 2 δ δ 1δ δ 2δ δ δ 3δ δ 4. Recall that an algebra A is said to be torsion free if nx = 0 implies that x = 0 for all x A and for each positive integer n. Clearly, any associative algebra over a field of characteristic 0 is always torsion free. Now we are in a position to state the main theorem of this section. Theorem 2.4. Let A be a torsion free algebra, D be the set of all Jordan higher derivations {d n } n=0 on A, and be the set of all sequences {δ n } n=0 of Jordan derivations on A with δ 0 = 0. Then there is a one to one correspondence between D and. Proof. It follows from Proposition 2.2 that for any Jordan higher derivation {d n } n=0 D, there is a sequence {δ n } of Jordan derivations with δ 0 = 0 such that n (n + 1)d = δ k+1 d n k for each non-negative integer n. Hence the following mapping ϕ : D {d n } n=0 {δ n } n=0 is well-defined, where (n + 1)d = n δ k+1d n k. Note that the solution of the recursive relation of Proposition 2.2 is unique. Therefore ϕ is injective. We next prove that ϕ is also surjective. For a given sequence {δ n } n=0 of Jordan derivations with δ 0 = 0, one can define d 0 = I and (n + 1)d = n δ k+1 d n k for each no-negative integer n. It is sufficient for us to prove that {d n } n=0 is a Jordan higher derivation of A. Obviously, d 1 = δ 1 is a Jordan derivation of A. We assume that d k (a 2 ) = k d i(a)d k i (a) for all a A and for each k n.

9 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS 283 Note that (n + 1)d (a 2 ) = = n δ k+1 d n k (a 2 ) ( n n k ) δ k+1 d i (a)d n k i (a). Combining Lemma 2.1 with the induction hypothesis gives (n + 1)d (a 2 ) = = = n n k [δ k+1 (d i(a))d n k i (a) + d i(a)δ k+1 (d n k i (a))] ( n i n ) n δ k+1 d n i k (a) d i(a) + d i(a) ( n i ) δ k+1 d n i k (a) n n (n i + 1)d n i+1(a)d i(a) + d i(a)(n i + 1)d n i+1(a) i=1 n = id i(a)d i(a) + (n + 1 i)d i(a)d n i+1(a) = (n + 1) d k (a)d k (a) for all a A. Since A is torsion free, we get d (a 2 ) = d k (a)d k (a) for all a A. This implies that {d n } n=0 is a Jordan higher derivation of A. By using analogous arguments of Proposition 2.2 we have Proposition 2.5. Let {d n } n=0 be a higher derivation (resp. Jordan triple higher derivation) on A. Then there is a sequence {δ n } n=0 of derivations (resp. Jordan triple derivations) on A such that for each non-negative integer n. (n + 1)d = n δ k+1 d n k In view of Proposition 2.5, we can get another one to one corresponding relation, which is completely parallel to Theorem 2.4. ( )

10 284 ZHANKUI XIAO AND FENG WEI Theorem 2.6. Let A be a torsion free algebra, Ω be the set of all higher derivations (resp. Jordan triple higher derivations) {d n } n=0 on A, and Γ be the set of all sequences {γ n } n=0 of derivations (resp. Jordan triple derivations) on A with γ 0 = 0. Then there is a one to one correspondence between Ω and Γ. Remark 2.7. We must point out that the above one to one corresponding relation between higher derivations and derivations of fields was obtained by Heerema in [8]. However, we deal with the case of associative algebras, which is quite different from the case of fields. 3. Jordan Higher Derivations on Some Operator Algebras In this section, we will use Proposition 2.2 and Theorem 2.4 to study Jordan (triple) higher derivations on some operator algebras, such as C -algebras, CSL algebras, reflexive algebras and standard operator algebras. In view of the relations ( ) and ( ) we obtain Proposition 3.1. Let A be an associative algebra over a field of characteristic zero. If every Jordan derivation (resp. Jordan triple derivation) on A is a derivation, then every Jordan higher derivation (resp. Jordan triple higher derivation) on A is a higher derivation. Let JHD (resp. HD/JD/D) be the set consisting of all Jordan higher derivations (resp. higher derivation/jordan derivations/derivations) of A. We denote the one to one correspondence between JHD and JD by φ and denote the one to one correspondence between HD and D by ψ. Indeed, if every Jordan derivation on A is a derivation, then the set JD is equivalent to the set D. By Theorem 2.4 it follows that the set JHD is equivalent to the set HD. The mutual relation between four sets can be seen via the following diagram JHD HD φ ψ JD D. Likewise, we denote the one to one correspondence between JTHD and JTD by η. If every Jordan triple derivation on A is a derivation, then the set JTD is equivalent to the set D. By Theorem 2.6 it follows that the set JTHD is equivalent to the set HD. The mutual relation under this case can be observed through the

11 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS 285 following diagram JTHD HD η ψ JTD D. In fact, Proposition 3.1 can be obtained via the Algorithm 2.3. This algorithm shows that for any Jordan higher derivation {d n } n=0 on A, there exists a sequence {δ n } n=0 of Jordan derivations on A such that d 0 = I, d 1 = δ 1, d 2 = 1 2 δ δ 2, d 3 = 1 6 δ δ 1δ δ 2δ δ 3, d 4 = 1 24 δ δ2 1δ δ 1δ 2 δ δ 1δ δ 2δ δ δ 3δ δ 4, According to the assumption of Proposition 3.1, each δ i (i = 0, 1, 2, ) is a derivation on A. Then for any x, y A, we have d 0 = I, d 1 (xy) = δ 1 (x)y + xδ 1 (y) = d 1 (x)y + xd 1 (y) = d 2 (xy) = 1 2 δ2 1(xy) δ 2(xy) 1 d i (x)d 1 i (y). = 1 2 δ2 1(x)y + δ 1 (x)δ 1 (y) xδ2 1(y) δ 2(x)y xδ 2(y) = 1 2 d2 1(x)y + d 1 (x)d 1 (y) xd2 1(y) [(2d 2 δ 2 1)(x)]y x[(2d 2 δ 2 1)(y)] = d 1 (x)d 1 (y) + d 2 (x)y + xd 2 (y) = 2 d i (x)d 2 i (y).

12 286 ZHANKUI XIAO AND FENG WEI Furthermore, we inductively calculate that 3 d 3 (xy) = d i (x)d 3 i (y), for all x, y A. d 4 (xy) = 4 d i (x)d 4 i (y),, n d n (xy) = d i (x)d n i (y) 3.1. Algebras of bounded linear operators. Let H be a Hilbert space over the real or complex field F and B(H) be the algebra of all bounded linear operators on H. It is well-known that every Jordan derivation on B(H) is an inner derivation. Corollary 3.2. Any Jordan higher derivation on B(H) is an inner higher derivation C -algebras. Let us see Jordan higher derivations on C -algebras. Let A be a semiprime algebra over a field F with characteristic zero. Then any Jordan derivation (or any Jordan triple derivation) on A is a derivation [2]. Since every C -algebra is semiprime, we have Proposition 3.3. Every Jordan higher derivation (or Jordan triple higher derivation) on a C -algebra is a higher derivation. In particular, every von Neumann algebra is also semiprime. It is well-known that every derivation on a von Neumann algebra is inner. Thus we get Corollary 3.4. Every Jordan higher derivation (or Jordan triple higher derivation) on a von Neumann algebras is an inner higher derivation CSL algebras and reflexive algebras. Let us recall some basic facts related to CSL algebras and reflexive algebras. Throughout this subsection, all algebras and vector spaces are defined over the field F, where F is the real field R or the complex field C. Given a Hilbert space H, we denote by B(H) the algebra of all bounded linear operators on H. The terms operator and projection on H mean that bounded linear mapping of H into itself and self-adjoint idempotent operator on H, respectively. A subspace lattice of H is a family of projections on

13 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS 287 H which contains the zero operator 0 and the identity operator I, and is closed under the usual lattice operations and. A nest is a totally ordered subspace lattice; a commutative subspace lattice (in brief, CSL) is a subspace lattice in which all projections commute pairwise. Suppose that L is a subspace lattice on H. We define the associated subspace lattice algebra Alg L to be the set of operators in B(H) leaving every subspace in L invariant, that is Alg L = { A B(H) (I E)AE = 0 for all E L }. Dually, if S is a subalgebra of B(H), we can define the invariant subspace lattice Lat S to be the set of projections which are left invariant by each operator in S, that is Lat S = { E E is a projection such that (I E)AE = 0 for all A S }. An algebra S is said to be reflexive if Alg Lat S = S. Clearly, every reflexive algebra is of the form Alg L for some subspace lattice L and vice versa. Dually, a lattice is called reflexive if Lat Alg L = L. It is well-known that each CSL is reflexive. We refer the reader to [21, 22, 24] for some basic facts concerning CSL algebras. Lu in [22] proved the following theorem Theorem 3.5. [22, Theorem 3.2] Let L be a commutative subspace lattice on a Hilbert space H and Alg L be the associated CSL algebra. Then every Jordan derivation form Alg L into itself is a derivation. Combining Theorem 3.5 with Proposition 3.1 we conclude that Proposition 3.6. Let L be a commutative subspace lattice on a Hilbert space H and Alg L be the associated CSL algebra. Then every Jordan higher derivation from Alg L into itself is a higher derivation. Suppose that L is a subspace lattice on a Banach space X. For E L, we can define E = { F L F E } and E + = { F L F E }. Let us set J (L) = { K L K {0} and K X } Then L is said to be completely distributive if L = { E L E L} for all L L; L is called a J -subspace lattice if (a) { K K J (L) } = X; (b) { K K J (L) } = {0}; (c) K K = X for all K J (L);

14 288 ZHANKUI XIAO AND FENG WEI (d) K K = {0} for all K J (L). We refer the reader to [18, 20] about the just two notions. In [24], Lu studied Jordan derivations of a class of reflexive algebras, namely those reflexive algebras Alg L for which { L L J (L) } = X or { L L J (L) } = {0}. Such a class of reflexive algebras includes: (i) completely distributive subspace lattice algebras; (ii) J -subspace lattice algebras; (iii) reflexive algebras Alg L for which X X; (iv) reflexive algebras Alg L for which {0} + {0}. He proved Theorem 3.7. [24, Theorem 2.1] Let L be a subspace lattice on a Banach space X. Suppose that { F F J (L) } = X or { F F J (L) } = {0}. If δ : Alg L B(X) is a Jordan derivation, then δ is a derivation. In view of Proposition 3.1, we immediately get the corresponding higher version of Theorem 3.7. Proposition 3.8. Let L be a subspace lattice on a Banach space X. Suppose that { F F J (L) } = X or { F F J (L) } = {0}. Then every Jordan higher derivation from Alg L into B(X) is a higher derivation. Let us next see Jordan triple higher derivations on standard operator algebras. We denote by F(H) the subalgebra of all bounded finite-rank operators. A subalgebra A of B(H) is called standard operator algebra provided that A contains F(H). Proposition 3.9. [12, Corollary 3.10] Let X be Banach space with dimx 1 and A be a standard operator algebra on X. Then every Jordan triple derivation from A into itself is a derivation. Applying Proposition 3.1 yields the higher order case of Proposition 3.9. Proposition Let X be Banach space with dimx 1 and A be a standard operator algebra on X. Then every Jordan triple higher derivation from A into itself is a higher derivation. Yang [33] studied Jordan -derivations on standard operator algebras and described Jordan -derivation pairs of standard operator algebras. The anonymous referee pointed to us that if the approaches of this article can be applied to describe higher order case of Jordan -derivations on standard operator algebras, it would be very interesting. We will detailedly consider this question in a future separate article.

15 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS Jordan Higher Derivable Points Recently, there are a number of papers which are contributed to the study of local behavior of derivations and its generalizations on operator algebras. The main purpose of these works is to consider under what conditions derivations and its generalizations of operator algebras can be completely determined by the action on some points or sets of operators (see [1, 10, 11, 14, 13, 16, 23, 34, 36, 37, 38] and references therein). Let R be a commutative ring, A be an associative R-algebra and τ be an R-linear mapping from A into itself. We say that τ is derivable at point z A if τ(xy) = τ(x)y+xτ(y) for all x, y A with xy = z. τ is said to be Jordan derivable at point z A if τ(x y) = τ(x) y + x τ(y) for all x, y A with x y = z, where x y = xy + yx. An element z A is called an all-derivable point (resp. a Jordan all-derivable point) in A if every derivable mapping (resp. Jordan derivable mapping) at z is a derivation (resp. Jordan derivation). Obviously, the condition of mappings derivable (resp. Jordan derivable) at some point is weaker than the condition of being a derivation (resp. Jordan derivation). The following question has a number of interests for researchers: will the linear mappings derivable (resp. Jordan derivable) at some points imply the mappings are in fact derivations (resp. Jordan derivation)? Let B(H) be the algebra of all bounded linear operator on the Hilbert space H. Jing in [11] showed that the identity operator I is a Jordan allderivable point in B(H). Zhu and Xiong [37, 39] proved that: (1) every invertible operator in nest algebra algn is an all-derivable point for the strong operator topology; and (2) a matrix G is an all-derivable point in the full matrix algebra M n (R) of n n matrices over the real field R if and only if G 0. Hou and Qi [10] showed that every unit operator in the J-subspace lattice algebra AlgL of a complex Banach space X is an all-derivable point. In this section we will to generalize the above results to the case of higher derivations and Jordan higher derivations. Let A be an R-algebra. A family of R-linear mappings D = {d n } n=0 of A is called a Jordan higher derivable family at x if d n (a b) = i+j=n d i(a) d j (b) for each n and for all a, b A with a b = x. An element x A is called a Jordan higher all-derivable point if every Jordan higher derivable family at x is a Jordan higher derivation. We refer the reader to [34, 36, 38, 39] and the references therein for more details of this field. Lemma 4.1. Let A be a torsion free algebra and x a Jordan all-derivable point. Then x is a Jordan higher all-derivable point.

16 290 ZHANKUI XIAO AND FENG WEI Proof. Let D = {d n } n=0 be a Jordan higher derivable mapping at x A. Note that d 1 is a Jordan derivable mapping at x A and hence d 1 is a Jordan derivation. Simulating the proof of Proposition 2.2 step by step, we conclude that there exists a sequence {δ n } of Jordan derivations on A such that n (n + 1)d (a b) = δ k+1 d n k (a b) for all a, b A satisfying a b = x and each nonnegative integer n. Since the algebra A is torsion free, a direct induction shows d (a b) = d i (a) d j (b) i+j= for all a, b A. Therefore D = (d n ) n=0 is a Jordan higher derivation and hence x is a Jordan higher all-derivable point of A. In view of [11, Theorem 2.6] we immediately get Proposition 4.2. Let B(H) be algebra of all bounded linear operators on the Hilbert space H. Then I is a Jordan higher all-derivable point of B(H). Let s recall the definition of nest algebras. Let H be a complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. Let I be a index set. A nest is a set N of closed subspaces of H satisfying the following conditions: (1) 0, H N ; (2) If N 1, N 2 N, then either N 1 N 2 or N 2 N 1 ; (3) If {N i } i I N, then i I N i N ; (4) If {N i } i I N, then the norm closure of the linear span of i I N i also lies in N. If N = {0, H}, then N is called a trivial nest, otherwise it is called a non-trivial nest. The nest algebra associated with N is the set T (N ) = { T B(H) T (N) N for all N N }. A nontrivial nest algebra is a triangular algebra. Indeed, if N N \{0, H} and E is the orthogonal projection onto N, then N 1 = E(N ) and N 2 = (1 E)(N ) are nests of N and N, respectively. Moreover, T (N 1 ) = ET (N )E, T (N 2 ) = (1 E)T (N )(1 E) are nest algebras and T (N ) = [ T (N1 ) ET (N )(1 E) O T (N 2 ) ].

17 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS 291 Proposition 4.3. Let T (N ) be a nest algebra associated with N over a complex Hilbert space H. Then every invertible operator of T (N ) is a higher all-derivable point. Proof. If N is a trivial nest, then T (N ) = B(H) is a von Neumann algebra. At this case, the claim follows from Lemma 4.1 and [6, Corollary 2.5]. If N is a nontrivial nest, at this case it is known that every invertible operator in the nest algebra τ(n ) is an all-derivable point of the nest algebra for the strongly operator topology [37]. Then the claim follows from Lemma 4.1 and [32, Theorem 2.5]. Acknowledgement. We are sincerely grateful to Professor Kenneth R. Davidson for his kind considerations and warm help. We would like to thank the anonymous referee for a very thorough reading of the manuscript and for many valuable comments. In particular, the referee suggest that we should use the approaches of this paper to study Jordan higher -derivations on standard operator algebras. References [1] J. Alaminos, J. Extremera and A. R. Villena, Characterizing homomorphisms and derivations on C -algebras, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007) 1-7. [2] M. Bre sar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc., 104 (1988), [3] K. R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Series, 191, Longman, London/New York, [4] M. Ferrero and C. Haetinger, Higher derivations and a theorem by Herstein, Quaest. Math., 25 (2002), [5] M. Ferrero and C. Haetinger, Higher derivations of semiprime rings, Comm. Algebra, 30 (2002), [6] M. E. Gordji, A characterization of (σ, τ)-derivations on von Neumann algebras, arxiv:[math. OA] [7] H. Hasse and F. K. Schmidt, Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (German), J. Reine Angew. Math., 177 (1937), [8] N. Heerema, Higher derivations and automorphisms of complete local rings, Bull. Amer. Math. Soc., 76 (1970), [9] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), [10] J. -C. Hou and X. -F. Qi, Additive maps derivable at some points on J-subspace lattice algebras, Linear Algebra Appl., 429 (2008), [11] W. Jing, On all Jordan all-derivable points of B(H ), Linear Algebra Appl., 430 (2009), [12] W. Jing and F. Lu, Additivity of Jordan (triple) derivations on rings, Comm. Algebra, to appear.

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19 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS 293 [38] J. Zhu and C. -P. Xiong, All-derivable points in continuous nest algebras, J. Math. Anal. Appl., 340 (2008), [39] J. Zhu, C. -P. Xiong and L. Zhang, All-derivable points in matrix algebra, Linear Algebra Appl., 430 (2009), Received August 6, 2011 (Xiao) School of Mathematical Sciences, Huaqiao University, Quanzhou, , P. R. China address: (Wei) School of Mathematics, Beijing Institute of Technology, Beijing, , P. R. China address: address: